Can the mass of longitudinal and transverse W bosons be measured separately?

Some higgsless unified models of particle physics predict that the mass of longitudinally polarized W bosons and the mass of transversely polarized W bosons are different. In those models, a difference in masses is also predicted to appear for the Z boson. Is there a way that the existing data from the LHC, the LEP or the Tevatron can be used to check this prediction? Or maybe it has already been checked?


Why do Nuclei lose mass?

When it comes to things like gravity and the electromagnetic force, masses aren't reduced-but with nuclei the mass difference is noticeable. What about nuclear forces makes them capable of putting mass into binding energy?


Infinite universe - Jumping to pointless conclusions

I watched an episode of thee BBC Horizon series titled 'To infinity and beyond'. In this program a number of respected physicists and mathematicians were talking about the nature of infinity and an infinite universe. They argued that if the universe is infact infinite then every possibility must play out somewhere, even going so far as to get a physicist to calculate how far you would have to travel to find an exact replica of the earth. They also used the classic example of infinite monkeys producing the works of Shakespeare. Now to me the monkey example makes sense considering that the works of Shakespeare are finite and so are the number of possibile outcomes of a monkey mashing away at a keyboard for a given period of time (although very large still finite). But the logic of applying this to the universe seems flawed to me, surely in the context of the universe the possibilities are also infinite as it is with a sequence of numbers. the numbers 1,2,3,4... go on infinitely without repeating themselves so to say because something is infinite every possibility must play out seems to me incorrect. Now perhaps you could argue that a finite number of atoms can only assemble themselves in a finite number of ways - again I can see this although I am not entirely sure that two lots of atoms arranged in the same way are necessarily the same since we currently rely on probablity calculations to predict the outcome of subatomic processes (this could arguably be due to slight variations at a more fundamental level...) but even if this were true. They went further to suggest that in an infinite universe there is an exact copy of you or I doing exactly the same thing and indeed every possible thing. Now it seems to me at this point things have become ridiculous, even by the time you have calculated the possibility of an exact atomic copy, the number of possible interactions I could have at any time with the universe, their affects on my atomic makeup and indeed future interactions, surely at this point we are dealing with infinite possibilites. Any feedback on any of these points welcome but my central assumption here is this: Infinite possibilities will not play out even over an infinite period of time. Is this correct?


Equivalent temperature: laser and cell containing Rb

What's the meaning of "equivalent temperature" related to a cell containing rubidium and crossed by laser?


Do gases have a general upper limit of density?

Is there some limit for the density of gases, at which no change in condition could make it more dense without making it fluid, or solid - or something 'in between'?


Eigenenergies and eigenkets given the Hamiltonian

For a two level system the Hamiltonian is: $$ H=a(|1\rangle \langle1|-|2\rangle\langle2|+|1\rangle\langle2|+|2\rangle\langle1|) $$ where $a$ is a number with the dimension of an energy. I need to find the energy eigenvalues and the corresponding eigenkets (as a combination of $|1\rangle$ and $|2\rangle$). I used: $$ H |ψ\rangle=E|ψ\rangle $$ And using the fact that : $|a\rangle = \sum_i c_i |a_i\rangle$ I wrote $|ψ\rangle$ as a combination of the two system kets $|ψ\rangle=c_1|1\rangle + c_2|2\rangle$ ($c_1$,$c_2$ are complex numbers). so $$\begin{aligned} H|ψ\rangle =& a(|1\rangle\langle1|-|2\rangle\langle2|+|1\rangle\langle2|+|2\rangle\langle1|)\cdot(c_1|1\rangle+c_2|2\rangle) \\\ =& a(c1|1\rangle-c_2|2\rangle+c_2|1\rangle c_1|2\rangle)=a ((c1+c2)|1\rangle+(c1-c2)|2\rangle) \\\ =& E|ψ\rangle. \end{aligned}$$ How do I continue?


Electron Relaxation/Polarization for and n-type Semiconductor

Please help me understand the following (general) statement, referring to electrons in a full valence band of an n-type semiconductor: "An electron filling up the last empty state in the valence band will in doing so prevent the other valence electrons from reducing their energy through polarization and relaxation around the impurity center" I don't understand what is meant by the last part of this. I know that in a full band, electrons have no empty state to move into, but I don't know what is meant by polarization and relaxation "around the impurity." The impurity referred to is a donor (n-type SC). Another statement (which I think is basically the same) says: "In the case of a completely filled valence band, no (impurity, I assume) scattering can take place, because all states are occupied. Now, if a hole is created at, for example, the top of the valence band, the remaining valence electrons will relax around the impurity ions... this new flexibility of the system makes the system lower in its energy." What does this mean "relax around impurity ions"? To sum it up, what I'm not understanding here is: a) what is meant by "relax around the impurity" (I have no idea what they mean by "relax"), and b) reducing energy by polarization or relaxation (What/how/why? What do they mean by polarization?) Edit: I spoke with a professor about this, and I _think_ that this all has to do with screening. In other words, if there are empty states in the valence band, the charge density is modified to a new equilibrium to screen the impurity, "relaxing" around the impurity. I think the point on polarization follows the same logic (screening). Any input on my thoughts would be appreciated.


Properties of an object whose collision with Earth would completely disintegrate Earth

This picture got me thinking, whether such collision would be possible, and would it really look like that from the moon. What would be the properties (mass, size, velocity etc.) of an object, whose collision with Earth would completely destroy it in the manner of that picture? That is, after the collision, a large portion (if not most) of Earth would continue straight along the same path as that of the colliding object, but from the opposite side of the planet. If such an event were to happen, would the sight in that picture be an accurate depiction when observed from the surface of the moon?


Black hole (classical or quantum?)

Theoretically we can convert a body into black hole by compressing its mass below some radius (known as Schwarzschild Radius). Suppose such object after becoming black hole has a radius which is comparable to atom and nucleus radius. Now my question is: what happens if we run this object into double slit experiment setup (single black hole at a time)? Shall we get interference pattern on the detector wall because its radius is in quantum domain (say we somehow know when on detector screen this black hole will hit) or simple-particle pattern (i.e. classical way) because of the mass associated with that black hole? If you have a really tiny black hole, will it behave as a classical particle, or will it exhibit quantum behavior?


Is 4-volume element a scalar or a pseudoscalar in special relativity?

In general relativity 4-volume element $\mathrm{d}^4 x = \mathrm{d} x^0\mathrm{d} x^1 \mathrm{d} x^2\mathrm{d} x^3$ is clearly a pseudoscalar (or scalar density) of weight 1 since it transforms as $\mathrm{d}^4 x \to \mathrm{d}^4 x' = J \,\mathrm{d}^4 x$, with $J$ Jacobian determinant of the coordinates transformation. In special relativity we consider only Lorentz transformations, which have $J = \pm 1$, so shouldn't $\mathrm{d}^4 x$ be a pseudoscalar also in special relativity? Landau states ("The Classical Theory of Fields", § 6) that «the element is a scalar: it is obvious that the volume of a portion of four-space is unchanged by a rotation of the coordinate system», but this doesn't proof that $\mathrm{d}^4 x$ is a scalar rather than a pseudoscalar since a pseudoscalar looks like a scalar under a proper rotation of the coordinate system ($J = 1$), differences arise only for an axis inversion ($J = -1$).


What are the differences in using an eyepiece with a barlow instead of a shorter eyepiece?

I intend to start making visual observations, and I want to know more about the difference between using barlow lenses and one or two eyepieces and using a complete kit of eyepieces, both for planetary and deep space objects.


Physical intepretation of nodes in quantum mechanics

I am taking my second course in QM, and my head is starting to spin as it probably should. But I would very much like to clear up my head about a few details regarding the wave function. As I know it is impossible to predict where particles are and one can only give a probability of where it _should_ be. The simplest case is a "frictionless" particle "bouncing" back and forth inside a infinite square well. Eg a particle in the following potential $$ V(x) = \left( \begin{array}{cc} 0 \ , & \text{for} \ 0 \leq x \leq a \\\ \infty \ , & \text{elsewhere} \end{array} \right) $$ Which gives rise to the following normalized solution $$ \psi_n(x) = \sqrt{\frac{2}{a}} \sin\left( \frac{\pi n}{a}x \right) $$ My problem is what the **nodes** in the square function represents. If I draw $|\Psi_2(x,0)|^2=|\psi_2(x)|^2$ I obtain a graph similar to the one below. ![enter image description here](http://i.stack.imgur.com/5Z2US.png) What is the physical explenation that finding the particle around a small region around $a/2$ is close to zero? Or why is it so much less likely to find it near $a/2$ than $a/4$? Eg why is $$ P(a/2 -\varepsilon \leq X \leq a/2+\varepsilon) = \int_{a/2-\varepsilon}^{a/2+\varepsilon} \left| \psi_2(x) \right|^2 \,\mathrm{d}x \sim 0 $$ for small $\varepsilon$


What would happen if a supermassive blackhole at the center of a galaxy evaporates?

Well the title says it all but I will just explain it in detail a bit. We know that a blackhole eventually has to evaporate due to the hawking radiation. It is also said that the blackholes at the center of galaxies are the galaxies' source of energy. So what would happen if that blackhole evaporates?


Does gravity cause spacetime to bend?

Does gravity cause spacetime to bend even though space has no mass?


What are the demographics of stars visible to the naked eye?

Of the stars that can be seen with the naked eye, what are the distributions of each type? For example, how many are main sequence? how many are Super Red giants/white dwarfs/neutron stars...etc. What's the distribution by mass and distance?


Coulomb interaction as virtual particles exchange?

I've been reading about virtual particle exchanges in physics books and in Physics SA posts, where a particle interpretation of gravity and Coulomb interaction is established. The Feynman Diagram picture (combined with the snowball picture that a fellow contributor provided in a related post) serves to account for the repulsion part: _grosso modo_ two particles in motion, a virtual photon/graviton, momentum change induces opposite motions ($\Delta \vec p \ $ and $ \ -\Delta \vec p$). Still, how can we account for the attraction (as a layman's analogy if not in formalism)?


Physical vapor deposition of SiO2

Could you suggest any if it's possible to deposit a layer of SiO2 via PVD? Why doesn't it widely used? Is that correct that I need ~1600C in vacuum? Could you suggest any books/papers about that? Unfortunately cannot tag this properly :-)


Lean angle of a turning bicycle

I'm asked to derive a relationship for the leaning angle of a bicycle with the following specs:   * Center of gravity for bike and rider is a distance $L$ above the ground when vertical, and the total mass of bike and rider is $M$.   * The wheel radius is $r$ and the wheel mass is $m$   * The radius of curvature for the turn is $R$, the speed is $v$ I am to show that $$\tan \theta = \frac{v^2}{R} \frac{2mr + ML}{MLg}$$ but I am unsure. If I take $M$ to be the total mass, including the wheels, then I would guess that the condition is that centrifugal force and gravity act in such a way that the total torque is zero, i.e. $$M g L \sin \theta = M \frac{v^2}{R} \cos \theta L$. But then the mass of the wheels and their radius doesn't enter. If, on the other hand, I exclude the wheels from $M$, wouldn't I still have to add them to both sides, i.e. $$(MgL + 2mgr) \sin \theta = \frac{v^2}{R} \cos \theta (ML + 2mr)$$ or where does the extra term 2mr come from?


does current product push or pull effect

Since i was a child i was told that current has pushing or pulling effect. However, i never got a dc shock and i don't remember ac shock (i got it very long before). The day before yesterday, an experienced electrician told me that dc has pulling effect ( most of the batteries supply only 12V it's not so much to worry but if supplied with very high voltage it is very deadly comparison to ac and) and ac has pushing effect. Logically i can't find any explanations. electricity is just flow of charge, and in our body, we have more electrons than protons (from C-14 decay) but still very less than enough to give shock or pushing or pulling effect.


Is there a finite frequency of physical interactions?

If particles use gauge bosons to convey information about various forces then is there something like a finite "refresh rate" at which they update information about themselves?


How to add magnitude data in an ENU coordiante system

I have water velocity data taken in an ENU (East, North, UP) or XYZ coordinate system. The data is contained in 3 columns like this:               E   N   U     1   3   .1     2   5   .2     2   6   .1     -1  4   .1     -2  4   .1      So the first column represents velocity (in m/s) in the East (or X) direction (negative numbers represent velocity in the West (or -X) direction. The second column indicates velocity in the North (or Y) direction. And the 3rd column represents velocity in the Up (or Z direction). The data is scalar, but I'm assuming I want to end up with speed and directional data. For ease of understanding, if I were to just consider 2 dimensional data (ignore the Z or up data), is it acceptable to add the E and N data to get useful magnitude and directional data? For example if in the last row East is -2 and North is 4. So I've got 2 m/s West and 4 m/s North. But does the number 2 (-2 + 4) mean anything? Or do I need to use $a^2 +b^2= c^2 = 4 +16= 20 \rightarrow \sqrt{20}= 4.47$ But If I were to do this, then I loose the - sign (West direction) information, and it would not have mattered if the data was originally West (-2) or East (2). In the end, I'm looking to get useable data on current speed. But I'm not even sure if the end result of this kind of data should be (A) magnitude + direction (ie 2m/s at 20 degrees) or just (B) a single number that is only a combined magnitude with no reference to direction(ie 2m/s). I know for my calculations I do need an absolute magnitude that would represent current speed (option B) but am not sure how to calculate this.


Does the moon affect the Earth's climate?

So, this morning I was talking to a friend about astronomical observations, and he told me that lately there has only been good weather when there was a full moon in the sky, which was a shame. I jokingly said: 'maybe there's a correlation!', but then I started thinking: wait, if the moon can affect the oceans, why shouldn't it also make an impact on the atmosphere, which is just another fluid. So... are there atmospheric tides? Does the moon affect the weather or the climate in a significant way?


Charging using a CRT TV set

I was doing an experiment with my CRT TV where I put aluminum foil on the screen wire it to a can and open the TV and a ring between 2 can will swing if I ground the other can. http://scitoys.com/scitoys/scitoys/electro/electro4.html#franklin Anyway I tried something out, I used the same way to charge a metal sphere while the tv was on, the sphere was charged and everything seemed to be alright only when removed the wire (connecting the sphere and the foil) from the foil on the TV the sphere lost its charge, why is that? If the charge just decayed, is that normal to decay so fast (in less than a minute) while it is not grounded?


Applicability of Baxter's method for IRF models

In a interaction-round-a-face model of $n^2$ particles in a lattice, a weight $W(a,b,c,d)$ is assigned to each face in the lattice based on the spins $a,b,c,d$ (listed say from the bottom-left corner in counter-clockwise fashion) of the particles at the corners of the face. Based on this, a partition function is formed $Z=\sum\prod W(a,b,c,d)$, where the the sum is over all possible spins of all particles and the product is over all the faces in the lattice. The well-known text _Exactly Solved Models in Statistical Mechanics_ by Baxter shows how to obtain exact solutions for $Z$ in a couple cases, like the six-vertex model and the eight-vertex model, using "commuting transfer matrices." In reading Baxter's book, I am uncertain of the applicability of this method. For instance, I am working on a particular application where particles in a square lattice can have either up spin (+1) or down spin (-1) and the weight of any face $W(a,b,c,d)=1$ unless all four spins around the face alternate, in which case $W(1,-1,1,-1)=W(-1,1,-1,1)=4$. (The boundary condition here is that the all the spins on the boundary must be up.) Does anyone know if the commuting-transfer-matrix method (or any other method) should yield an exact solution for this particular weighting of the spins? More generally, should the commuting-transfer-matrix method yield exact solutions for _any_ weighting of the spins (even in the two-spin case)?


Giant string in space

I saw an interview with Ed Witten, where he said one way to confirm string- theory is to observe a giant string floating in space, left over from the Big- Bang. How does one observe such a string, does it have thickness? Is this string any different then the strings that (hypothetically) make up elementary particles? What is required to produce such a large string, why is this string so much bigger then others? What happends if one tries to interact with this string, can one stretch/cut it etc? Can we have a stable mega-string on earth, and can it be used for anything?


Has the use of the holographic principle of string theory in condensed matter physics silenced the skeptics?

It seems to me that the use of string theory in calculations of strongly- interacting matter in condensed matter physics is an example of the theory being on the right track. And then there's the application of string theory to black holes. Have these quieted the skeptics? If not, what's the deal? Could a theory be used in making accurate calculations and still be wrong? Aren't the odds of that happening rather small?


What keeps a gas giant from falling in on itself?

There is not enough gravity at the center to start nuclear fusion, but it seems that there would be plenty enough to collapse the planet.


Electric Field in Conductor Zero?

My textbook claims that the electric field in a conductor is zero in a static condition, as otherwise, a current would flow. But what if I go infinitely close to a proton; there will be an electric field right? Is my idea wrong or is the textbook flawed?


The Z-Torque: how can it be shown intuitively that it does not work?

There is a new kickstarter project that claims to increase torque and power compared to a normal crank on a bicycle (Z-Torque on kickstarter). If this patented (US Patent Number 5899119) approach works it would be a revolution for personal transport. ![enter image description here](http://i.stack.imgur.com/TWcjx.jpg) The idea behind this is that a Z-shaped crank should allow somehow better leverage compared to a straight crank. There are quite a few comments that this should not work as intended but as in a lot of cases it sounds plausible at first but it is incompatible with physics. As the inventor and some of the supporters are convinced that this actually works I am looking for a clear and simple argument why this is not possible. Using vectors and cross products ($\tau = \vec{r}\times\vec{F}$) this is easy to show but can it be shown with as few abstractions as possible?


On the double slit experiment with 4 slits

I'm not a physicist nor a science savvy person, but I was wondering if this experiment was ever performed in a simultaneous fashion on screens with fixed references(marks) and firing different wavelengths particles. So, to recap, 4 slits, 2 emitters firing at different times (where the $\Delta T$ is known and certain) on two different screens. I hope this question is not too stupid. P.S. I'm sorry, I've been too vague. I ask this because I've recently learned that this interference pattern emerge even firing large atom clusters instead of single particles. This is astonishing and it's clearly stating that there's a limit in mass aggregation where it stops behaving in duality. Cause, you know, firing bricks will hardly produce interference. And, in the end, I was wondering if it's not possible that it's spacetime itself vibrating at a certain $f$ and if there's a limit when mass stop to be affected (relatively) by this vibration. Like pinning a vibrating rubber sheet. I'm so confused.


pattern of moving electrons in wire under AC

when we use a DC battery into a circuit the flow of electrons in wire creates a clear picture as they flow from negative terminal to positive terminal. under AC(Alternating current) we know how energy flows and what does it mean by $I_{rms}$ or average current etc. But what about the pattern in which electrons move?


Can an electric motor force angular momentum not to be conserved in an isolated system?

An ice skater is in a spin, she pulls her arms in and she spins faster, she lets her arms extend outward and then she starts to slow down. She will probably weigh on a weigh scale about the same weight during this spin cycle. Now let us connect an electric motor to her to make her spin faster as she extends her arms outward. Would angular momentum still be conserved? It seems to me that it would not. Also would she still have the same weight? I think she will weigh less as she is being accelerated by the electric motor.


Why the breath sometimes warm and sometimes cold? (2 different explanations!)

If you blow air against your hand with your mouth open, you feel warm breath. If you do with with your lips closed except for a small opening, you feel cold breath. One explanation from here says "With your mouth open most of the air which reaches your hand has come out of your mouth. With pursed lips lots of cold air is entrained so the air reaching your hand is cold and moving quickly." Another explanation from Paul Hewitt video,(go to min 24:00), is that air comes out cold it because it expands coming out of small opening. So which explanation is right?


difference between battery and capacitor in urdu languege

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Why should a holomorphic function be expanded in Laurent series rather than Taylor series?

In 2d free conformal field theory, there is an operator equation: $$ \partial\bar\partial\hat{X}^\mu\left(z,\bar z\right)=0 $$ Why can it have Laurent expansion like this below rather than Taylor $$ \partial X^\mu\left(z\right)=-i\left(\frac{\alpha'}{2}\right)^{1/2}\sum_{m=-\infty}^{\infty}\frac{\alpha_m^\mu}{z^{m+1}}, \\\ \bar\partial X^\mu\left(\bar z\right)=-i\left(\frac{\alpha'}{2}\right)^{1/2}\sum_{m=-\infty}^{\infty}\frac{\tilde{\alpha}_m^\mu}{\bar z^{m+1}}. $$


Inside a conductor?

My textbook says the field inside a conductor must be zero in order for the system to be equilibrium and therefore there must be no excess charge inside. Their proof: 1) Place a gaussian surface inside the conductor. Since the system is at equilibrium, all points on the surface must have an electric field of zero. 2) Therefore the net flux is zero, implying the charge inside is zero. 3) If there is no charge inside, all excess charge must lie on the surface. My problem: "Since the system is at equilibrium, all points on the surface must have an electric field of zero." Why? Consider an conductor with no excess charge. To simplify things, say the conductor consists of a 100 free electrons and a 100 ions of charge 1e+. The conductor is neutral and at equilibrium. Now consider the field on one specific electron. There a total of 99 other electrons and 100 ions. So there _would_ be a net field acting on that specific electron. So even though the system is at equilibrium, the net field on the individual particles is non-zero. So a system can be at equilibrium, even if the net field inside is non-zero. What am I misunderstanding? **Edit:** Another example: Consider a system consisting of one electron and one proton. If nothing is holding them back they will attract each other until they touch. After they touch, the force acting on them is still non-zero, yet they wouldn't accelerate.


A bird problem and trains

So that the problem: > _There are 2 trains at speed of 30km/h in the same trail and opposite > directions and moving for a collision at 60km of distance of each other. One > bird in front of one of the trains fly at 60km/h toward of other train. When > he reaches he flies back to the other train until the trains collide._ I got a question: as the bird travels in all the way to the collision?


Value of of gravity 9.8m/sec^2 and centrifugal force

I had a doubt regarding centrifugal force on earth is not accounted in any equations as object falling towards earth.eg f=ma.i read questions asked by few about the same and got to a conclusion that value of g(9.8m/s^2) is a relative number derieved.but I still cant find it in gravitational force equation.why is that.or is my interpretation wrong.this might be a repetitive question. But please clarify.


free energy and entropy

my understanding of free energy and entropy is that as entropy of a system increases its free energy decreases. As sun has free energy and this energy is being converted to useful work, and its entropy is increasing. Its entropy will continue to increase till its material is exhausted to cease free energy production. Are permanent magnets source of free energy? Because of permanent dipoles they are magnetic and do they lose their orientation very slowly to become disordered and increase its entropy?


How damaging is light?

On _Surely You're Joking Mr Feynman_ , when talking about the Trinity test, the author states: > the only thing that could really hurt your eyes (bright light can never hurt > your eyes) is ultraviolet light. I got behind a truck windshield, because > the ultraviolet can't go through glass, so that would be safe, and so I > could see the damn thing. This is also clear in this video, where the glasses are seen as black by the UV camera. If that's the case, why are sunglasses said to `prevent bright sunlight and high-energy visible light from damaging or discomforting the eyes`? _(from Wikipedia)_ Is it for marketing purposes or was Feynman wrong?


Does a perfect mirror behave the same as a blackbody radiator?

If I put a perfect mirror(i.e. reflects with no attenuation) next to a blackbody radiator its spectra should be the same as the blackbody radiator. Looking only at the spectra - is there any difference between a blackbody radiator and a perfect mirror? For instance, suppose the mirror is accelerating as in the dynamic Casimir effect - does the spectra change in the same way as the blackbody radiator?


How can the product of two real linear operators be not real?

I'm puzzled about a statement from Diracs book "The principles of quantum mechanics" (§8, p.28): > As a simple examples of this result, it should be noted that, if $\xi$ and > $\eta$ are real, in general $\xi\eta$ is not real. This is an important > difference from classical mechanics. However, $\xi\eta + \eta\xi$ is real, > and so is $i(\xi\eta - \eta\xi)$. Only when $\xi$ and $\eta$ commute is > $\xi\eta$ itself also real. Here $\xi$ and $\eta$ are linear operators, so (I think) these could be represented as matrices. However how can a product of two real matrices be "not real"? What does Dirac mean when he says "is not real"? Is Dirac maybe talking about eigenvalues? So does Dirac mean, that the product of two matrices with real eigenvalues could have imaginary eigenvalues? And does he want to say, that a real symmetric matrix like $\xi\eta + \eta\xi$ and a purely imaginary antisymmetric matrix like $i(\xi\eta - \eta\xi)$ always have real eigenvalues?


Mie theory "upper limit"?

I have read most often that objects experimented with in Mie theory are on a scale not much larger than the incident wavelength (usually a fiber of diameter $5$ microns with an incident wavelength of $630$ nm or so). Can Mie theory be used to describe the scattering pattern of an object any size, as long as a sufficient number of terms in the Mie solution can be calculated?


Is air resistance affected by both mass and surface area?

If so, which has the biggest impact?


Impedance Matching

When viewing fast pulses on an oscilloscope, why is impedance matching required? I'm not totally clear on why. I'm guessing it is because I don't want 'ringing' and because I want signal integrity.


Determining Water Level

Given a cylinder of a constant shape and size, how can the water level be calculated using the inside surface of the container? For instance, could the inside lining of the container be a pressure sensitive foil of some sort, and using that pressure could the water level be calculated?


Coriolis effect misunderstanding

My wife asked me what the Coriolis Effect was, so I did some research. I think I get it now but I'm surprised at the number of "explanations" people have uploaded that actually seem wrong. This one is an example. http://www.classzone.com/books/earth_science/terc/content/visualizations/es1904/es1904page01.cfm This is wrong, right?


What are the facts that allow accepting the Oort cloud theory?

I admit without any fact that the Oort cloud (comet reservoir) should exist, and it seems to be accepted by far by the large majority of astronomers. But it is still a theory without any direct observation. Except the fact that we guess that comets come from this cloud (a cloud of comets). So what are the facts or indirect observations that may validate this comet cloud theory? Collateral to this question, is it easy to observe these indirect facts, and is there any special way to observe these Oort comets to validate the theory? Is it acceptable to say there is no cloud and just a few solar and most inter- solar objects?


About muons, taus and the number of generations

there! Why cannot consider the muon or the tau just like excited states of the electron (and similarly, with the (u,d) (c,s) (t,b) pairs? I guess that it is related to the absence of decays like muon-> electron+photon and similar stuff. But, is there any other reason?


Why do we feel a sudden chill?

When we are in a room, especially in some old houses, we feel this sudden rush of cold air. How does a strong air current arise and pass away so quickly?(lasts for about a second) I know air-currents are caused by pressure differences. But can pressure difference of the magnitude required to generate an air current be created within a second? If so , please answer with equation that relates the pressure and Temperature with time interval. Thanks..i hope I am now much clearer with my question. (I am inclined to think in the direction of something similar to a weather situation)


Orbital plot to explain bonding and anti-bonding molecular orbitals

Related is this question: https://chemistry.stackexchange.com/questions/2586/molecular-and-atomic- orbitals-and-antibonding. In my textbook on organic chemistry they represent an orbital as a 2d wave function to explain the formation of bonding and anti-bonding molecular orbitals (I displayed it here as a simple $\sin x$ function, just for illustration. I understand real wave function may be quite different). ![Example wave function, sin x](http://i.stack.imgur.com/ktOWh.png) They say the bonding atomic orbitals can either combine in phase, resulting in a bonding molecular orbital (the electron density becomes larges between the molecules, creating an electrostatic astringent force exerted by the electron in the center on the nuclei of the combining atoms), here represented as $2\sin x$ ![Example bonding molecular orbital wave function, 2sin x](http://i.stack.imgur.com/ctM61.png) or combine out of phase resulting in a nodal plane between the nuclei: an anti-bonding molecular orbital. Here represented as $\sin x + \sin (x+\pi)$ ![Example anti-bonding molecular orbital wave function, sin x + sin \(x + pi\)](http://i.stack.imgur.com/DMLtU.png) Now the question: how can a single orbital, that is either in phase or out of phase with the orbital of the bonding atom (i.e. the other atom) form both a bonding molecular orbital and an anti-bonding molecular orbital, while it is clear that an orbital is either in one phase or an other and not both? Related is, what this 2d diagram actually represents (related question, but not quite the same: https://chemistry.stackexchange.com/questions/12397/meaning-of-depiction-of- atomic-orbitals). Please ask if anything is unclear. Your help is appreciated!


How is the direction of Magnetic/Electric Lines of Force Known?

It is shown that the direction of magnetic line is from north to the south and that of the electric line is from positive to negative. How do we/scientists know that the imaginary lines of force or the field goes from this to that, what is the meaning of this direction? Do not give me mathematical definitions which I already know; I want the fundamental meaning of this.


How do particle scattering cross sections scale with energy in colliders? Particularly photons, electrons, protons, and gold or lead nucleii? (If necessary, break this into four separate questions.) It is stated that due to the Heisenberg uncertainty principle, it takes more energy to measure a smaller distance and the inverse wavelength energy- wavelength relationship for photons is well known.   Therefore, I would conclude that the (total scattering) cross section for photons at least would vary as the inverse square of the energy. On the other hand, general relativity says the the longitudinal dimension shrinks with increasing velocity, but the transverse dimensions do not.   Therefore, I would conclude that the cross section of say a gold atom in RHIC or a lead atom in the LHC would basically remain constant as the energy increased. But how about the electron, supposedly a point particle?   Due to PEP and LEP, we should have good data on this. Looking at PDG figure 41.6,   http://pdg.lbl.gov/2011/reviews/rpp2011-rev-cross-section-plots.pdf   I see lots of big spikes for resonant particles, but the underlying bacground is clearly a falling powerlaw that I think looks like it could be an inverse square law. The proton cross sections in the next few figures, e.g. 41.7 and 41.11, show a falling elastic scattering cross section, but they also show an approximately constant total cross section. What little i've found for RHIC supports the approximately constant cross section. Therefore I conclude that the cross sections of photons and electrons (neglecting resonances) scale as the inverse square of the energy, but the cross sections of protons and nucleii are roughly constant with energy. Comments or corrections? So is it true that the photons and electrons are different from the protons and nucleii? And why don't the protons scatter as three or six point particles (quarks and gluons) instead of one big blob?


What is the nature of electric field? is it quantized? is it a wave?

What I seek here is to understand whether the electric field in its pure form as in between the electron and the proton is uniform or does it have some kind of wave/particle nature or both, does it have frequency or wavelength, or is it quantized?. I do not want any explanations of the effects of electric field or thereof, I want to understand the field itself. Please tell me something I all ready do not know from my engineering education.


How would I calculate the work function of a metal?

In the photoelectric effect, the work function is the minimum amount of energy (per photon) needed to eject an electron from the surface of a metal. Is it possible to calculate this energy from the atomic properties of the metal (atomic number, atomic mass, electron configuration, etc.), and the properties of the lattice structure, in a way that doesn't require a complex computational model? (If so, how?) Or, if not, is there a simple formula in terms of the metal's atomic/lattice properties that can approximate it to within a few percent?


Could there be a planet just as the Earth, but bigger?

Could there be a planet twice or thrice the size of Earth but while being the same? and by the same I mean, hosting the same life as Earth, having a day last 24 hours and a year 365 days. Basically, could there be a bigger Earth? What would be needed to replicate Earth but in a bigger planet?


Which is more efficient Water heating method : Electric heating coil OR Induction stove?

Which is more efficient Water heating method : Electric heating coil Vs Induction stove


Why isn't pressure a measure of energy?

Hey guys, I'm having a problem in understanding the 1st Law of Thermodynamics. If i would increase the pressure of a closed system by compressing a gas in a cylinder isothermal, the 1.Law states, that all the work i put in to the system, i get out by heat. So my 2.state would be the gas at the same temperature (same internal energy), but compressed (at a higher pressure). But isn't this compressed gas able to do more work, than the not compressed?   If i drilled a hole in the cylinder my high pressure gas could do some work. Where does this energy come from, since all my input energy i put in by work, came out in form of heat?


How can one make sure that one had understood the material studied?

I do not fully understand the process of understanding of a material one reads. Suppose someone reads a chapter from a physics book. How does one make sure that one had really understood what she/he had read? I must confess that at times I read from text and I even feel that I had understood the content but later (may be at the time of exams or solving assignments or when somebody raises a question from that topic), I realize that the feeling was illusory and I feel so stupid. At times I feel that my understanding is incomplete even at basic levels. I am eager to know from experienced people in this forum if there is any method to test whether one had really understood the content of what one had read. I am talking of advanced level physics courses. May be one can also suggest useful skills one can develop for better grasp of items one studies. Thanks. I have some other related thoughts. Perhaps this is not just a problem with me but with many others, including even researchers. I have seen during one seminar given by an well-known faculty member of an institute, another well- known researcher from audience started asking questions and making comments which revealed lack of understanding of the speaker and he was almost in tears! Not that the speaker was dishonest but this shows that even experts like him can have "illusory" sense of understanding of a topic (not only what he/she had read but done some research work). So may be there is nothing like full understanding of a topic but deeper and deeper understanding of topics. So the question now becomes, how can one attempt to understand a topic deeply and keep going deeper? And for students, how can they be sure their understanding of a topic is good enough?


Is there the smallest particle that can be guaranteed to be unable to be broken down into smaller particles?


Information or Matter or Energy

**Which one is the most fundamental of the universe? What it is made of?** Physicists are searching for unique and fundamental form. The question is , information is more fundamental than energy ? According to the thermodynamic Physics **information is any kind of event that affects the state of a dynamic system that can interpret the information.** But as far as I know its a bit of energy as I heard from the Physicist Lineard Susskind's Lectures. Will you please clarify according to the article ? Actually I was confused reading the head line article.http://www.pbs.org/wgbh/nova/blogs/physics/2014/04/is-information- fundamental/ I want to know that why are they referring the information as fundamental unity? Isn't it that Information is a mathematical concept or its a bit of energy i mean quantized energy?


Introductory derivations of Heisenberg uncertainty principle

I'm not an expert when it comes to quantum mechanics, so correct me wherever I'm wrong, but: I've always been a little bit bothered by introductory derivations of the Heisenberg uncertainty relations (for, for example, position and momentum) that involve carrying out an imaginary "observation" of a particle by bonking it with photons. The argument goes (at least the way I've heard it) that the more precision you want in the position measurement, the higher frequency (and so energy) the photon must have. Higher energy bonking leads to more uncertainty in the momentum afterward, and so there's a tradeoff between position and momentum precision. The derivations then manage to work out that the precisions satisfy the Heisenberg uncertainty relations. The problem I have with those kinds of derivations is that they start out assuming that a particle actually _has_ a position and momentum to speak of, and if you were lucky enough to guess them, you'd be able to predict the particle's motion exactly - it's entirely classical reasoning. The way I understand it, the position and momentum representations of the state are fourier transforms of each other, and so it's just a simple mathematical fact that as one becomes more localized, the other one becomes less localized. Another way of saying it (and please feel free to comment on the validity/value of this way of thinking about it), that I think makes it qualitatively more obvious, is that each observable corresponds to an eigenbasis. It's pretty intuitive that, in general, if one aligns the state vector closely with any one basis vector in one eigenbasis (corresponding to a nearly definite value of that observable), then the state vector _won't_ be aligned any one basis vector of the other observable, but will instead be a mixture of several basis vectors, and so will have an uncertain value of the other observable. If this sort of reasoning is correct, then it'd seem like the classical derivations of the Heisenberg uncertainty principle are complete gibberish - rather than having to do with a particular way the measurement is made, the limits on measuring simultaneous, definite values of position and momentum are a result of the fact (which the vector space representation makes plain) that a state cannot even have simultaneous definite values. If it really is true that the classical derivations of the Heisenberg uncertainty relations is whack, is there any value in them at all? Thanks!


Is everything moving at c in a c unit circle

I was trying to explain special relativity to a few friends in a simple way and wound up with an analogy using a c unit circle. I was using y as travelling in time, x moving in space; move in space and you are borrowing from your clock-speed. E.g. the twin paradox: Your travelling twin has borrowed clock-speed. Twins clock was slower -> twin is younger. What is the problem with representing relativity using a (c) unit circle from an observers point of view? Anything Lorentz or Minkowski would be out of the question of course, I'm looking for a layman friendly description that is close to true. The circle discussed, was in my mind a half circle $y>=0$. Where $y=0$ would be a photon.


What type of magnet to use?

I am asking this question to find out practical sizes of magnets, but this will be in a fictional setting. Say I have a person dressed in iron armor, medieval knight-style. Are there types of magnets that weigh on the order of 500 kg or so that would be strong enough to pull the armor to it? I'm not thinking of super-strength magnets like neodymium. Maybe a kg of force per kg of the magnet? Sorry, I'm not really familiar enough with magnets to know how to express their strength. What material would such a magnet be made of? Would the costs of making such a magnet be overly inhibitive, or is a magnet of that size even feasible?


If there turns out to be matter with negative mass, would there also be antimatter with negative mass?

I just heard of the concept of **negative mass**. Googling led me to see that the concept of negative mass is often contrasted with **antimatter**. This made me to wonder whether there would then also be a fourth category of **exotic matter** being **antimatter with negative mass**. Of this I can find no mention. If it turns out that we have **1)** normal matter **2)** antimatter and **3)** normal matter with negative mass, would that imply that we would also have **4)** antimatter with negative mass?


How many atoms does one need to get reliable crystal information with X ray or electron diffraction?

I know that, in theoretical derivations of X ray diffraction (XRD) or electron diffraction (ED), we always assume an infinite lattice, and, as you make the crystal smaller, edge effects start to matter more. As you make it even smaller, you start to lose information about actual orientation of the crystal and such (obviously, the diffraction from two atoms really tells you literally nothing). I know it's a continuum, so "reliable" is a little subjective, but at what point can you not get any useful information from XRD or ED? I've seen ED images inset in TEM pictures of ~10nm structures, which comes out to about 100 atoms.


Wick rotation and the arrow of time

It is well known that we can switch from a statistical system to a quantum mechanical system by a Wick rotation. Has this rotation some implication on the way the time flow? namely, this is an accident or has a deep meaning?


A flyby of orbiting supermassive black holes

Consider two supermassive black holes of equal mass orbiting about their common centre of mass. Is it the case that a free-fall trajectory along the axis of rotation would be outside of either event horizon at all black hole separation distances > 0 (based on the symmetry of the situation)? To rephrase this, would you be able to navigate a rocket along a path at right angles to the orbital plane and bisecting the line between the two black holes with no ill effects whatsoever even when the black holes were very close to each other? Supplementary question: What is the shape of each of the event horizons prior to coalescence? _Note: I’m assuming that tidal forces would be small until the two singularities were extremely close. Note also that the point midway between the two BHs is the L1 Lagrangian point._


Practice AP Physics B Exam Question regarding Momentum

I am trying to review momentum for the AP exam coming up. I will be taking the AP Physics C exam for Mechanics, but I was just practicing on any free response questions I could find and I came across this one. It is from 1983, free response number 2. It says: > _A block of mass $M$ is resting on a horizontal, frictionless surface, and > it is attached as shown above to a relaxed spring of spring constant $k$. A > second block of mass $2M$ and initial speed $v_0$ collides with and sticks > to the first block. Develop expressions for the following quantities in > terms of $M$, $k$, and $v_o$._ > > _a.) $v$, the speed of the blocks immediately after impact._ > > _b.) x, the maximum distance that the spring is compressed._ ![The diagram that came with the problem](http://i.stack.imgur.com/tP5l8.png) So I got (a) pretty easily, just by using conservation of momentum. I found the result of $$v=\frac23 v_0$$ For part b, I began by finding the change in Kinetic Energy of the 2 blocks from immediately after impact to the point where their velocity is zero as a result of the negative work that the Spring does on the blocks. I found that $$K_i=\frac12(3M)\left[\frac23v_0\right]^2 = \frac23Mv_0^2$$ and that $$K_f=0$$ I then equated this change in $K$ with the Work done by the spring and got $$-\frac23Mv_0^2 = \frac12kx_i^2 - \frac12kx_f^2 = \frac12kx_0^2 - \frac12k(x_0-x)^2$$ I then tried to solve this equation for x, which is the distance that the spring is compressed, and found that $$x=\frac12*(x_0+sqrt({x_0^2+\frac43\left(\frac{Mv_0^2}k\right)})$$ from the quadratic formula. Now, in theory, I don't really see anything wrong with this, but according to solutions I found online, people have been equating the change in K with the change in momentum, and having this equation: $$\frac23Mv_0^2 = \frac12kx^2$$ Can anyone explain why this equation would be true, or failing that, where I may have gone wrong in my own calculation? I appreciate the help.


Asymptotic symmetry algebra

So after a lot of research, and tons and tons of papers that I've went through, I finally have some idea how to solve the equations that will give me candidates for the asymptotic symmetry group for Kerr/CFT correspondence. One has boundary conditions $h_{\mu\nu}$, and metric that behaves like $g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu}$, where $\bar{g}_{\mu\nu}$ is the background metric (in this case it's taken as near horizon extreme Kerr metric), and $h_{\mu\nu}$ are perturbations (given). So the task is finding diffeomorphisms $\xi$ for which this metric will transform according to $$\bar{g}_{\mu\nu}+\mathcal{L}_\xi \bar{g}_{\mu\nu}=g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu}$$ And one solves that by solving equations $$\mathcal{L}_\xi g_{\mu\nu}=\mathcal(r^m)$$ where $\mathcal{O}(r^m)$ are boundary conditions, given in terms of radial coordinates. From this, and the fact that $h_{\mu\nu}=h_{\nu\mu}$ I get ten equations. Now my issue is this: What to take as ansatz for $\xi$? In some papers, they take this: $$\xi^\mu=\xi^\mu_0(t,\theta,\phi)+r\xi^\mu_1(t,\theta,\phi)+\mathcal{O}(r^2)$$ But in certain papers I've found that authors take this form (see after eq. 4.4 here, or here eq 5.1) $$\xi^\mu=\sum\limits_{n=-1}^\infty \xi^\mu_n r^{-n}$$ such that we have subleading contributions (a falloff in radial coordinate). Now, I'm inclined to use the second one, since the diffeomorphism given in Guica et. al is given with subleading terms. I should solve this by putting a certain few terms in, and see what the boundary condition says. For instance we have $$\mathcal{L}_\xi g_{tt}=\mathcal{O}(r^2)$$ That means that all $\mathcal{O}(r^2)$ contributions will cancel, and only equations with $r$, or smaller $r$ contributions will survive. But the answer is drastically different if I take first or second ansatz. So what to take? Am I on the right track? If I take second one, what n should I start from? $n=-1$ or $n=-2$? Oh and I finally got how to solve this by looking at this article and following how they got it in appendix A **EDIT:** I've been thinking. Does the fact that in their article, the metric, and the ansatz goes with powers of $v$, means that since their equations are limited by $\mathcal{O}(v)$ onward all higher power will cancel each other? In that case, since I'm taking only subleading terms in metric, and boundary conditions (different for each of 10 terms), and if I take that my ansatz falls off as $r^{-n}$, that means that all lower order correction cancel out? The question is still: what n to take? Since it's not the same if my $\xi$ starts with $r^2$, or $r$ :\ **EDIT2:** I'll put one component to show what I'm doing. Although I think something is wrong, since I'm getting complicated equations (when things get too complicated, it's kind of a show that things might not be going in the right direction, at least that's what experience showed me) So, my metric is $g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu}$, as explained above. The first question that comes to my mind: do I put manually $h_{\mu\nu}$ in this expression, or do I use the given boundary conditions? I could say that $h_{\mu\nu}$ is arbitrary, and my metric has to have a power fall off perturbations, so that my $tt$ component would be $$g_{tt}=-\Omega^2(\theta)(1+r^2(1-\Lambda^2(\theta)))+r^{-1}h_{tt}(t,\theta,\phi)+\mathcal{O}(r^{-2})\quad (\star)$$ I removed $2GJ$ since it's just a factor in front of the metric, not depending on any of the variables ($t,r,\theta,\phi$), and it doesn't contribute to diffeomorphisms. I will try this after I finish what I did. However, I just put the given boundary conditions. For $tt$ component I have $$\mathcal{L}_\xi g_{tt}=\mathcal{O}(r^2)=\xi^\sigma\partial_\sigma g_{tt}+g_{\sigma t}\partial_t\xi^\sigma+g_{t\sigma}\partial_t\xi^\sigma$$ Since $g_{tt}$ depends on $r$ and $\theta$, I have two components in the first argument. In the second and third arguments are the same since $g_{t\phi}=g_{\phi t}$. So I have $$\xi^\theta\partial_\theta g_{tt}+\xi^r\partial_r g_{tt}+2(g_{tt}\partial_t\xi^t+g_{t\phi}\partial_t \xi^\phi)=\mathcal{O}(r^2)$$ $$(\xi^\theta_{-1}r+\xi^\theta_0+\xi^\theta_1r^{-1}+\mathcal{O}(r^{-2}))(-2\Omega\Omega'+2\Omega(\Lambda\Lambda'\Omega-\Omega'(1-\Lambda^2))r^2+\mathcal{O}(r^2))+(\xi^r_{-1}r+\xi^r_0+\xi^r_1r^{-1}+\mathcal{O}(r^{-2}))(-2\Omega^2(1-\Lambda^2)r+\mathcal{O}(r))+2((-\Omega^2-\Omega^2(1-\Lambda^2)r^2+\mathcal{O}(r^2))(\partial_t\xi^t_{-1}r+\partial_t\xi^t_0+\mathcal{O}(r^{-1}))+(\Lambda^2\Omega^2r+\mathcal{O}(1))(\partial_t\xi^\phi_{-1}r+\partial_t\xi^\phi_0+\partial_t\xi^\phi_1 r+\mathcal{O}(r^{-2})))=\mathcal{O}(r^2)$$ After sorting all out, and saying that all the corrections of $\mathcal{O}(r^2)$ vanish, I end up with $$(\Lambda\Lambda'\Omega-\Omega'(1-\Lambda^2))\xi^\theta_{-1}-\Omega(1-\Lambda^2)\partial_t\xi^t_{-1}=0$$ But, this one is relatively ok, the other ones sometimes have six or more components ($\xi^\mu$). I'll work all of them out, and then try with $(\star)$ as an assumption and see if things simplify... **EDIT 3:** So things aren't simplifying. But one thing I've learned: I need to solve asymptotic Killing equation. My next try, the one I think is a fast and relatively good try, is to just see what the leading orders of components should be, such that they satisfy the asymptotic Killing equations, given by boundary conditions. Then I'll try to do explicit calculation. For instance (and I hope I'm on a right track) for $tt$ component I would have $$\mathcal{L}_\xi g_{tt}=\mathcal{O}(r^2)$$ $$\xi^\theta\partial_\theta g_{tt}+\xi^r\partial_r g_{tt}+2(g_{tt}\partial_t\xi^t+g_{t\phi}\partial_t \xi^\phi)=\mathcal{O}(r^2)$$ $$\xi^\theta\mathcal{O}(r^2)+\xi^r\mathcal{O}(r)+\mathcal{O}(r^2)\partial_t\xi^t+\mathcal{O}(r)\partial_t\xi^\phi=\mathcal{O}(r^2)$$ So for this to be true (LHS=RHS) I should have $$\xi^r\to\mathcal{O}(r)$$ $$\xi^\theta\to\mathcal{O}(1)$$ $$\partial_t\xi^t\to\mathcal{O}(1)$$ $$\partial_t\xi^\phi\to\mathcal{O}(r)$$ I hope I'm on a right track. I got a hint how Guica et. al did it. First we need to guess what the symmetries are and then generate boundary conditions by acting with those diffeomorphisms, and then check that the charges are finite and integrable. So it's more a guessing game than exact solving :S **EDIT 4:** So far (by solving asymptotic Killing equations) I only get the $\xi^\theta$ component right. That is I have $$\xi^\theta_0=\xi^\theta_{-1}r$$ which, when I put in the ansatz I get $$\xi^\theta=\xi^\theta_{-1}r+\xi^\theta_0+\xi^\theta_1r^{-1}+\mathcal{O}(r^{-2})$$ $$\xi^\theta=\xi^\theta_1 r^{-1}+\mathcal{O}(r^{-2})$$ which is exactly the one in the article by Guica et. al. But no luck with other components. If only there was some info on what I need to put in asymptotic Killing equation after the equal sign... **EDIT 5:** I'll write the equations I got by putting $\mathcal{L}_\xi g_{\mu\nu}=\mathcal{O}(r^n)$ where $\mathcal{O}(r^n)$ are given by boundary conditions: $$tt:\quad (\Lambda \Lambda' \Omega+(\Lambda^2-1)\Omega')\xi^\theta_{-1}+(\Lambda^2-1)\Omega \partial_t\xi^t_{-1}=0$$ $$t r:\quad (\Lambda^2-1)r(\xi^t_{-1}+\xi^t_0r^{-1})+(\Lambda^2-1)r^3(\xi^t_{-1}+\xi^t_0r^{-1}+\xi^t_1r^{-2}+\xi^t_2r^{-3})-$$ $$-r(\xi^t_{-1}+\xi^t_0r^{-1})+\Lambda^2\xi^\phi_{-1}+\Lambda^2r^2(\xi^\phi_{-1}+\xi^\phi_0r^{-1}+\xi^\phi_1r^{-2})+\partial_t\xi^r_{-1}=0$$ $$t\theta:\quad (\Lambda^2-1)r^2(\partial_\theta\xi^t_{-1}r+\partial_\theta\xi^t_0+\partial_\theta\xi^t_1r^{-1}+\partial_\theta\xi^t_2r^{-2})-(\partial_\theta\xi^t_{-1}r+\partial_\theta\xi^t_0)+$$ $$+\Lambda^2r(\partial_\theta\xi^\phi_{-1}r+\partial_\theta\xi^\phi_0+\partial_\theta\xi^\phi_2r^{-1})+(\partial_t\xi^\theta_{-1}r+\partial_t\xi^\theta_0)=0$$ $$t\phi:\quad (2\Lambda(\Lambda'\Omega+\Lambda\Omega'))(\xi^\theta_{-1}r+\xi^\theta_0)+\Lambda^2\Omega\xi^r_{-1}+$$ $$+(\Omega(\Lambda^2-1))(\partial_\phi\xi^t_{-1}r+\partial_\phi\xi^t_0+\partial_\phi\xi^t_1r^{-1})-\Omega\partial_\phi\xi^t_{-1}+$$ $$+\Lambda^2\Omega(\partial_\phi\xi^\phi_{-1}r+\partial_\phi\xi^\phi_0)+\Lambda^2\Omega(\partial_t\xi^\phi_{-1}r+\partial_t\xi^t_0)+\Lambda^2\Omega\partial_t\xi^\phi_{-1}=0$$ $$rr:\quad \xi^\theta_{-1}r+\xi^\theta_0=0$$ $$r\theta:\quad \partial_\theta\xi^r_{-1}+(\xi^\theta_{-1}r+\xi^\theta_0)=0$$ $$r\phi:\quad (\xi^t_{-1}r+\xi^t_0)+\xi^\phi_{-1}=0$$ $$\theta\theta:\quad \Omega'(\xi^\theta_{-1}r+\xi^\theta_0)+\Omega(\partial_\theta\xi^\theta_{-1}r+\partial_\theta\xi^\theta_0)=0$$ $$\theta\phi:\quad (\partial_\phi\xi^\theta_{-1}r+\partial_\phi\xi^\theta_0)+\Lambda^2r(\partial_\theta\xi^t_{-1}r+\partial_\theta\xi^t_0+\partial_\theta\xi^t_1r^{-1})+$$ $$+\Lambda^2(\partial_\theta\xi^\phi_{-1}r+\partial_\theta\xi^\phi_0)=0$$ $$\phi\phi:\quad (\Omega\Lambda'+\Lambda\Omega')\xi^\theta_{-1}r+\Lambda\Omega r(\partial_\phi\xi^t_{-1}r+\partial_\phi\xi^t_0)+$$ $$+\Lambda\Omega\partial_\phi\xi^\phi_{-1}r=0$$ The problem is, the only equation I can get some information, about how my vector should behave in components is $rr$, from which I get the general form of $\xi^\theta$ component of diffeomorphism. From $r\theta$ I get that the $\xi^r_{-1}$ term is a constant that doesn't depend on $\theta$, and $\theta\theta$ equation will just confirm that $rr$ one is correct. I guess from the fact that $\xi^r_{-1}$ is not zero I could say that the $r$ component will start at that term (like in the article). **EDIT 6:** I have added the bounty to raise awareness to this, if anyone knows if what I did so far is correct, and what to do next, please do tell :)


Equations describing the liquifaction of gases

I would like to find out how much energy is required to liquify gases (oxygen or nitrogen, for example). **What are some relevant equations?** I'm curious whether creating small quantities of dry ice or cryogenic fluids is feasible at home. If James Dewar was able to liquify hydrogen in 1898 surely it must not be that difficult!


Where does the reverse current come from in a organic semiconducting diode?

Unlike traditional inorganic diodes, organic semiconducting materials hold few thermally exited free carriers under room temperature, so the reverse current should be small if pinholes of the films is well avoided. However the fact is that the reverse current varies a lot between devices using different types of organic materials. Is it because there are other sources of reverse current? Or is it mostly dominated by charge transport layers which have relatively higher conductivity compare to active layer?


Use microwave cavity in atomic clock

In most of the papers regarding atomic clocks, the author talks about a microwave cavity. In this box, all the unwanted frequencies of the electromagnetic radiation are absorbed and the other frequencies are maintained. But how does this work with an atomic clock? In this case, one uses atoms rather than radiation. What is the difference between these applications of the cavity? I thought this might have to do something with the duality between atomic particles and waves, although I couldn't find any sources that could verify this.


What is a Free Body Diagram?

I have this problem: > A 10 kg box is resting on a frictionless table and someone drags it by a > string at a 30 degree angle to the horizontal with a force of 40 Newtons. > Find the acceleration of the box. Draw a free body diagram of the system. What is a free body diagram?


If electrons behave as standing waves when they are bound to an atom then how do they carry charge?

Today in my physics lesson we learnt that the best way of describing the behaviour of an electron that is bound to an atom is to treat it as a standing wave. I understand that this is the wave-particle duality however I was taught that in the wave-particle duality something either acts as a wave or a particle, not both at the same time. With this in mind I'm curious as to how electrons can still carry charge when they are behaving as a wave. I asked my teacher this question and he wasn't able to answer it so hopefully someone on here can explain it to me :)


Are there any astronomical objects at night that are dangerous to look at (through a telescope)?

Are there any astronomical bodies that would be dangerous to my vision to view through a telescope? Obviously the sun is dangerous, but are there other bodies at night I should avoid?


2nd order variation of Hilbert-Einstein action + Gibbons-Hawking-York boundary term

While the first order metric variation of Hilbert-Einstein action plus Gibbons-Hawking-York boundary term is well-known and takes the form: $\delta S_{HE}+\delta S_{GHY}=-\frac{1}{16\pi G}\int d^3x \sqrt{-\gamma}~(K^{\mu\nu}-K\gamma^{\mu\nu})\delta \gamma_{\mu\nu}$ ($K_{\mu\nu}$ and $\gamma_{\mu\nu}$ are the extrinsic curvature and the projection metric), does anybody know the second order variation of this combined action? Thanks to anyone who can help me!


Integral over a product of two Green's functions

Need some help here on a frequently encountered integral in Green's function formalism. Forgive me since I am a junior student. I have an integral/summation as a product of a retarded and advanced Green's functions, looks simply like $$\sum_{p^{\prime}}\frac{1}{p^{2}-{p^{\prime}}^{2}-i\epsilon}\frac{1}{p^{2}-{p^{\prime}}^{2}+i\epsilon}$$ where I have omitted the mass $m$ to make the notation simple; $\epsilon$ is a positive infinitesimal. I can convert the summation to an integral over momentum $p$ or energy, this is standard. Then I follow the complex analysis and identify 4 poles (if the integration is over momentum) and further apply the residue theorem. But, what I get is an expression proportional to $\frac{1}{\epsilon}$. This means it is divergent since $\epsilon\rightarrow 0$. Could any one please point out to me what have I missed in this calculation? What mistakes did I make? Many many thanks.


How do we perceive hotness or coldness of an object?

Some objects, especially metallic ones, feel cold on touching and others like wood, etc. feel warm on touching. Both are exposed to same environment and are in their stable state, so some kind of equilibrium must be being reached. What is this equilibrium? And how do we perceive hotness or coldness of an object? Does skin have some kind of heat sensors, etc., which transmit signals to brain? Like, how do the eye transmit/convey an image formed on the retina to brain?


How long does it take to weigh an uncooperative mass?

(This is curiosity, not a practical question. It was inspired by standing still for a digital scale, and considering whether it would be possible to make a scale that could handle being jumped on, juggled on, etc.) Setup: An impermeable rigid box with arbitrary contents on top of (fixed to) an ideal scale, in a known uniform gravitational field and no electromagnetic field. Assume classical mechanics (i.e. no relativistic effects or uncertainty). Question: How long must data (i.e. the force on the scale) be collected in order to know the mass of the box and contents, given that the contents of the box are _doing their best to make it difficult_ (which I think can only consist of accelerating masses vertically)? I think that it is not possible to get an exact answer after a finite time; for example, consider if the box contains a mass being accelerated downward arbitrarily slowly — thus reducing the total weight by a small fraction, until it touches bottom. Is there an upper bound on the error, given a time interval and some property of the collected data (e.g. the maximum or mean force)?


Holonomy twisting

There is Witten's topological twist of standard SUSY QFTs with enough SUSY into Witten-type TQFTs. What is a holonomy twist?


Wigner characteristic function

I came across the "representation of a Gaussian state by its Wigner characteristic function". I don't know what Wigner characteristic function is and google results are not precise enough. Neither have I been able to find this in some quantum mechanics book. It will be great if you could explain what it means, but even better would be citing a source from where I could read it up. I came across this term in this paper: "Inseparability criterion for continuous variable systems". Duan, Giedke, Chirac and Zoller. Physical Review Letters (year 2000)


Polar vs non-polar fluid

In the book "Vectors, Tensors, and the Basic Equations of Fluid Mechanics" by Rutherford Aris I read the following: > If the fluid is such that the torques within it arise only as the moments of > direct forces we shall call it nonpolar. A polar fluid is one that is > capable of transmitting stress couples and being subjected to body torques, > as in polyatomic and certain non-Newtonian fluids. Can someone help me understand this? Say by giving me another definition of polar and nonpolar fluids. Thanks.


How do we know that nonperturbative canonical quantum gravity is wrong?

In these forums and elsewhere it is routinely agreed that "we do not have a theory of quantum gravity." My question is, how do we know that canonical quantum gravity is "wrong"? I understand that the theory is perturbatively nonrenormalizable, but doesn't that just mean that we can't apply perturbation theory to it? It seems the theory is nonperturbatively renormalizable. A theory being nonperturbative doesn't make it "wrong". My understanding is that in modern QFT nonrenormalizability is not anymore considered such a big deal, and that even the Standard Model is expected to be an effective theory modified by nonrenormalizable terms appearing at higher energies. So barring practical considerations, why is canonical quantum gravity considered "not a theory of quantum gravity?" EDIT: I am not sure that the term "Canonical Quantum Gravity" is correct. I am referring to what is called "Quantum Einstein Gravity" in this paper. If someone knows better please help me correct my terminology. I was not meaning to imply anything about LQG because I thought that LQG "brought more to the table" than just a nonperturbative approach to the same QFT, but I could be wrong about that.


Which areas in physics overlap with those of social network theory for the analysis of the graphs?

I am studying social networks in terms of graph theory and linear algebra. I know that physicists have published and worked a lot in this field. This causes me to assume that there are sub-fields in physics which overlap in the essence of their problems with those of small world networks. Which natural phenomena exhibit these kind of features similar to small world networks? I would like to know that, so maybe I can look at those problems to get inspiration that can be taken to social network theory.


What reasons are there to suspect string theory may be an incorrect theory?

It's a truism that in science it is at least as important to state the reasons why a theory or idea might be wrong, as to state the reasons why it is might be correct. For example, early renormalization methods made experimental predictions that were very accurate, but the inventor's famously worried for decades about the conceptual problems of divergences; and many of their papers/articles/books on the subject include a long discussion about how acutely worried they were that their method for handling divergences was deeply flawed somehow. My intention with this post is not to engage any user in a debate about whether string theory is true or not. I think anyone who takes string theory seriously should be happy to acknowledge as clearly as possible all the potential theoretical issues/problems with the theory. Therefore I would like to ask, what are the major theoretical problems which would lead a reasonable person to be worried that string theory is not an accurate theory? I am excluding the fact that string theory has made no (major) predictions that have been confirmed, I am only interested in conceptual and theoretical ideas which might discredit the theory. Please keep all answers to one issue per post.


Current division in ideal inductors?

The Question is:- > Find the current distribution in the inductors at steady state (i.e. > assuming $t\rightarrow \infty$ after the switch is closed and circuit > established and consider the inductors without any resistance and ideal). ![enter image description here](http://i.stack.imgur.com/noa0t.jpg) **My Efforts**   Since both of the are in parallel, the equations for current at any time $t$ after establishing the circuit is given by ($i_1$=corrent in $5mH$ inductor and $i_2$ in the other, and the total current $i$):- $$i(t)=i_1(t)+i_2(t)=4(1-e^{-\frac{5t}{L_{net}}})$$ where $L_{net}=\frac{10}{3}mH$. But I do not know how this current divides into two in the inductances at any time $t$ or in the steady state. Had the inductors not been ideal, i.e. if they had different resistances, the current could divide in the ration of their resistances in the final steady state, but since they are ideal, this method won't work. How can I solve this? the answers given are $\frac83A$ in the $50mH$ inductor and $\frac43A$ in the other.


The matrix elements of the momentum operator in position representation are: $$\langle x | \hat{p} | x' \rangle = -i \hbar \frac{\partial \delta(x-x')}{\partial x}$$ Does this imply that $\langle x | \hat{p} | x' \rangle = 0 $ whenever $x \neq x'$? Is the momentum operator diagonal in position representation? I know that the momentum operator shouldn't be diagonal in position representation (otherwise solving for the eigenergies and eigenfunctions of most Hamiltonians would be trivial). I am obviously confused here. I need more than just yes/no answers to these questions. I need some explanation, or some intuition.


Relativistic transformation of the wave packet length

Let us suppose we have an excited atom at rest. It has a certain mean lifetime $\tau_0$. If we wait sufficiently long time $t>>\tau_0$, we will find a deactivated atom and a (spherical) electromagnetic wave function of photon with about $\tau_0\cdot c$ long layer with non zero probability to find a photon within. Something like a fast expanding probability "ring" with a $\tau_0\cdot c$ width of the ring. Now, let us consider this system in a moving reference frame. It seems to me that this width $\tau_0\cdot c$ is relativistic invariant: it is a difference between two "fronts" of electromagnetic wave rather than a length of a material body subjected to the Lorentz contraction. Is it correct? In other words, whether this picture relativistic invariant? ![Wave function evolution](http://i.stack.imgur.com/vtenD.jpg)


Is the Copenhagen interpretation merely an approximation to quantum mechanics?

So, I'm reading Max Tegmark's _Our Mathematical Universe_ (Knopf edition, p. 229). He's discussing Everett/MWI for a bit and I'm not really paying attention and then I wake up to this: > [I]t's time to update the quantum textbooks to mention decoherence (many > still don't) and to make clear that the Copenhagen interpretation is better > thought of as the _Copenhagen approximation:_ even though the wave function > probably doesn't collapse, it's a very useful approximation to do the > calculations as if it does collapse when you make an observation. Now, I believe I have read (elsewhere, e.g., on this site) that these interpretations do not make any difference _whatsoever_ * when it comes to predictions. But here, apparently (at least that is how I read the above), it is claimed that MWI and Copenhagen actually _differ_ in predictions (and that MWI has the _right_ predictions). **Is that correct?** (This would make it slightly more difficult to be agnostic about choosing your favourite interpretation for the time of day, right?) NB: I'm not specifically asking whether the wavefunction collapses or not. *As in: zip, nada, zilch, _exactly_ $0$.


Problem from Sakurai about a delta-function potential

Can you help me with this problem from Sakurai: A particle of mass m in one dimension is bound to a fixed center by an attractive delta-function potential: $$V(x) ~= ~-a\delta(x) , \qquad a>0.$$ At $t = 0$, the potential is suddenly switched off (that is, $V = 0$ for $t > 0$). Find the wave function for $t > 0$.


Who first provided a string realization of dual resonance models?

After the $N$-particle generalization of the Veneziano amplitude was written down and studied, who was the first (or who were among the first) to realize that the amplitudes could be understood in terms of a relativistic string in the early 1970s?


Does shadow have mass?

I know it sounds like a foolish question but I have a reason for asking and I'm hoping someone here, can give a convincing response. Here is why I pose the question...it seems to me that all this discussion of photons, and other sub-atomic particles possessing mass or negative mass is really a discussion about the mass of shadows. Although we can't really weigh shadow we can see it, we can measure its area, observe its outline. a photon- which is massless - may seem heavier than shadow. It is really driving me crazy when I think about it.


Gravity on the International Space Station - General Relativity perspective

My question is an extension to this one: Gravity on the International Space Station. If all the outside views of the ISS was sealed, then the crew inside would not be able to tell whether they were in orbit around the earth in orbital speed or free floating in space beyond the orbit of Neptune, right? How would time dilation due to gravitation fields be affected? Supposing you have three atomic clocks: 1 - One on surface of the Earth, at Sea Level, 2 - One in the ISS, 3 - One in deep space beyond the orbit of Neptune. At what speed would each clock run compared to the other two?


Calculating Forces via Feynman diagrams?

How would one go about calculating forces that test objects feel using Feynman diagram methods? For example, say we have a massive object in GR so that the metric takes on the standard Schwarzschild form, $g^S_{\mu\nu}$. Now add some test matter in which is described by some lagrangian $\mathcal L'$ and has corresponding stress tensor $T'^{\mu\nu}$. In the action, the first order perturbation will then be $S\supset \int d^4x\, \sqrt{-g^S}\, h_{\mu\nu}T'^{\mu\nu}$ where $h_{\mu\nu}$ is the perturbation of the metric away from the Schwarzschild metric. I know how to calculate the force on the test particle in GR, but how would one do so by using field theory techniques like Feynman diagrams? Relatedly, in some modified gravity theories you get some action like $\mathcal L\sim \frac{1}{2}(\partial\phi)^2+\mathcal{L}_{\rm int}(\phi)+\phi T/M_{\rm pl}$ where $\phi$ is related to the modification of gravity. If $\phi$ takes on a non-zero profile, say $\phi\sim1/r$, then this results in a fifth force on test particles, according to the literature. How do I see this by using Feynman diagrams? I know that usually to figure out the potential generated by an interaction you can used the Born approximation, but I dont' quite see how that will work out in this case.


Static discharge out of 3 volts button battery

I am starting to get interested in static electricity I have found this which is very cool but I do not know how I can control the output voltage. http://www.instructables.com/id/How-to-build-the-Worlds-Smallest-Electronic- Shock/#intro What I need to do is to use a 3 Volts Button Battery and get a 50-500 volt static charge from it. And I need the device to be very small. What can I change in order to control the voltage? and is a 3 volts button battery good to keep this thing alive for months? and lastly, can I flip the battery to control the out put charge (negative or positive)?


Is it possible to increase the electromagnetic repulsion between two particles?

Given the fact that the force stopping me falling through the ground is the electromagnetic repulsion between the electrons in my feet and the ground, would it be possible to increase this repulsive force? Also is friction generated due to this repulsion of angular faces at a very small scale ? if so could a material be made so smooth it had zero friction because the electromagnetic force simply kept the two apart instead of causing friction? Sorry if its a stupid question but im very interested to find out and why not if i am wrong.


Renormalizing composite operators

Consider the QED Lagrangian, \begin{equation} {\cal L} = \bar{\psi} ^{(0)} ( i \partial_\mu \gamma^\mu - m ) \psi ^{(0)} - e A _\mu ^{(0)} \bar{\psi} ^{(0)} \gamma ^\mu \psi ^{(0)} - \frac{1}{4} F ^2 _{ (0) } \end{equation} If we apply the typical renormalization procedure, \begin{align} & A _\mu = \frac{1}{ \sqrt{ Z _A } } A _\mu ^{ ( 0 ) } \\\ & \psi = \frac{1}{ \sqrt{ Z _\psi } } \psi ^{ (0) } \\\ & e = \frac{1}{ Z _e } e ^{(0)} \\\ & m = \frac{1}{ Z _m } m ^{(0)} \end{align} Then we get a mass counterterm, \begin{equation} m ( 1 - Z _m Z _\psi ) \bar{\psi} \psi \end{equation} Up to here, I have no issues. Now suppose we want to consider the renormalized of some operator, $ {\cal O} ( x ) $. To be explicit, lets take the operator to be $ \bar{\psi} \psi (x) $. The way I would naively think about it is that we should insert this operator into the Lagrangian in terms of the bare fields and then allow its coupling to get renormalized in the same way that we did above. So we would have, \begin{equation} \Delta {\cal L} = g ^{(0)} {\cal O} ^{(0)} ( x ) = g ^{(0)} \bar{\psi} ^{(0)} \psi ^{(0)} \end{equation} then we would say \begin{align} & g = \frac{1}{ Z _g } g ^{(0)} \end{align} and we get a counterterm, \begin{equation} \Delta {\cal L} = g ( 1 - Z _g Z _\psi ) \end{equation} However Wise and Manohar in their book, _Heavy Quark Physics_ , get a slightly different result. They get a counterterm of, \begin{equation} \Delta {\cal L} = g \left( 1 - \frac{ Z _\psi }{ Z _g } \right) \end{equation} What is wrong about my understanding of composite operators?


Non-covariance of the higher rank propagator (from Weinberg's QFT textbook)

In chapter 6.2 of Weinberg's  QFT Vol1 , he gave the general form of Wick contractions of all possible fields(scalar, spinor, vector, etc.), he showed $\Delta_{lm}(x,y)=\theta(x-y)P^{(L)}_{lm}(-i\frac{\partial}{\partial x})\Delta_+(x-y)+\theta(y-x)P^{(L)}_{lm}(-i\frac{\partial}{\partial x})\Delta_+(y-x)\cdots(6.2.8)$ where $\Delta_+(x)=\int d^3p(2p^0)^{-1}e^{ip\cdot x}$, and $P^{(L)}_{lm}$ is a covariant polynomial if its argument is a on-shell 4-momentum(i.e. $P^{(L)}_{lm}(\sqrt{\mathbf{p^2}+m^2},\mathbf{p})$), but may not be covariant for off-shell 4-momentum. Then by a series of mathematical identities, he was able to show $\Delta_{mn}(x,y)$ can be written in a 4-momentum integral: $\Delta_{lm}(x,y)=\int d^4q\frac{P^{(L)}_{lm}(q)\exp(iq\cdot (x-y))}{q^2+m^2-i\epsilon}\cdots(6.2.18)$ From here he argued since the 4-momentum $q$ in $(6.2.18)$ is not always on- shell, $P^{(L)}_{lm}$ may not be covariant, and in turn the $\Delta_{mn}(x,y)$ may not be covariant. It's indeed not for a vector field as what he showed immediately after, and that's why we need to add a non-covariant term in the Hamiltonian to make the propagtor covariant ect.etc. I did follow the steps he showed, but something strange occured to me when I looked back at the expression $(6.2.8)$: $(6.2.8)$ looks completely covariant, since $\theta$ is invariant, $\Delta_+$ is invariant and is written in 3-momentum integral so that $P^{(L)}_{lm}(-i\frac{\partial}{\partial x})\Delta_+(x-y)$ must be covariant, yet $(6.2.18)$-which is supposed to be equivalent to $(6.2.8)$- is not covariant. I am wondering if there's any subtlety I missed. Also there's another mystery to me: in chapter 3.5 (page 144), he gave a perturbative proof of Lorentz invariance based on Dyson series, the invariance of Hamiltonian density and micro-causality condition, and he did mention Lorentz invariance could be disturbed by the reasonings given in chapter 6.2, but the proof in chap 3.5 is completely formal and I can't see exactly how the reasonings in 6.2 can jeopardize it. (PS: I wish I could have formulated my question in a more self-contained way, but I couldn't unless I copied some whole pages of Weinberg, so I apologize for the potential vagueness in advance)


Height of a piston in a heated cylinder containing H2O

I'm a first time user and I hope I won't be too enigmatic asking the following question: I have a cylinder (radius= $6$ $\text{cm}$) with a frictionless piston on top of it and inside $30$ $g$ of $H_2 O$ (molar weight is $M=30$). External pressure is $1 \text{ atm}$ and initial temperature is $T_1=273$ $K$. $Q=24$ $KJ$ of heat are put into the system. What height does the piston reach? I'm pretty sure I didn't get this right, but here is what I did: $$Q = Q_\text{water} - Q_\text{Latent evaporation heat} + Q_\text{steam}= mc \times (373-273) - m L + (m/M)Cv \times (T_2-373)$$ from this equation I found $T_2$ and knowing that pressure is constant, I found the volume of the steam and thus the height of the piston which resulted something crazy like $27 \text{m}$. What have I done wrong?


Dividing two star spectra

I am doing some work that involves dividing two stellar spectra from the same star. Those stellar spectra are constructed by summing random samples of multiple spectra from the same star to improve the signal to noise ratio (SNR). I know that for a sum, the $SNR$ will vary with $\sqrt N$ where $N$ is the number of spectra I am adding to construct each one of my spectra. My problem is: Assuming $SNR_1$ and $SNR_2$ are the signal to noise ratios of the spectras I am dividing by each other, how can I obtain the $SNR$ of the resulting spectrum? Do you recommend any documentation where thse operations are explained in detail? Thanks!


Polarization photon and Stokes parameters

I have the following situation: About the polarization of the photon, I introduce the basis: Horizontal polarization $|\leftrightarrow>=\binom{1}{0}$ Vertical polarization $|\updownarrow>=\binom{0}{1}$ The density matrix in this problem is: $$\rho =\frac{1}{2}\begin{pmatrix} 1+\xi _{1} & \xi_{2}-i\xi _{3}\\\ \xi_{2}+i\xi _{3} & 1-\xi _{1} \end{pmatrix}$$ The Stokes parameters are: $\xi _{1}, \xi _{2}, \xi _{3}$ The probability that if the photon has got lineal polarization whose axis forms an angle $\theta$ with de horizontal is: $$|w>=cos\theta |\leftrightarrow>+sin\theta|\updownarrow>$$ $$ P_{\theta}=<w|\rho|w>=\frac{1}{2}\left ( 1+\xi_{1}cos(2\theta)+\xi_{2}sin(2\theta) \right )$$ Is there any value of the Stokes parameters for which this probability is zero?


A Type of Pendulum

![http://i.stack.imgur.com/vAnpP.png](http://i.stack.imgur.com/Z9V2R.png) Is there any chance that $$rtl(\ddot\omega+\ddot\phi)\cdot\sin{(\phi+\omega t)}- gl\dot\phi \cdot \sin{\phi} + ltr\dot\omega(\dot\phi^2-\dot\omega)\cdot \cos{(\phi+\omega t)}-gtr\cos{(\omega t)}+l^2\ddot\phi-t^2r^2\ddot\omega=0$$ is equation of motion for the ball in the picture? * * * To describe x and y coordinates I wrote $x = l\cdot \sin{\phi} + r\cdot \cos{\omega t} $ and $y = l\cdot \cos{\phi} + r\cdot \sin{\omega t}$ Than express kinetic energy as $E_k = \frac{1}{2} m \cdot (\dot x^2 + \dot y^2)$ and potential energy as $E_p = mgy$. Next step was Langrangian $\mathcal{L} = E_k - E_p$ I put $\frac{d}{dt}\left( \frac{\partial{\mathcal{L}}}{\partial \dot \phi} \right) - \frac{\partial \mathcal{L}}{\partial \phi } = \frac{d}{dt}\left( \frac{\partial{\mathcal{L}}}{\partial \dot \omega} \right) - \frac{\partial \mathcal{L}}{\partial \omega}$ and that lead to the equation above, but I might do some mistakes. I hope it's ok to put the two equations together like this. Is it? * * * So if $q$ is only $\phi$ than $$\dot x = \dot \phi l \cos{(\phi)}-r \omega \sin{(\omega t)}$$ $$\dot y = - \dot \phi l \sin{(\phi)} + r\omega \cos{(\omega t)}$$ $$E_k = \frac{1}{2}m(\dot \phi^2 l^2 \cos^2{(\phi)} - 2r \omega l \dot \phi \cos{(\phi)} \sin{(\omega t)}+r^2 \omega^2 \sin^2{(\omega t)} + r^2 \omega^2 \cos^2{(\omega t)} - 2r \omega l \dot \phi \sin{(\phi)} \cos{(\omega t)}+l^2 \dot \phi^2 \sin^2{(\phi)})$$ $$Ek = \frac12 m(\dot \phi^2 l^2 (\cos^2{(\phi)}+\sin^2{(\phi)}) + r^2 \omega^2 (\sin^2{(\omega t)}+\cos^2{(\omega t)})-2rl \omega \dot \phi (\cos{(\phi)}\sin{(\omega t)}+\sin{(\phi)}\cos{(\omega t)}))$$ $$E_k = \frac12 m \dot\phi^2 l^2 + \frac12 m r^2 \omega^2 - mrl \omega \dot \phi \sin{(\phi + \omega t)}$$ So it would be kinetic energy of the ball. Than I need potential energy. $$E_p = mgl\cos{(\phi)} + mgr\sin{(\omega t)}$$ Therefor Lagrangian should be: $$\mathcal{L} = \frac12 m \dot\phi^2 l^2 + \frac12 m r^2 \omega^2 - mrl \omega \dot \phi \sin{(\phi + \omega t)} - mgl\cos{(\phi)} - mgr\sin{(\omega t)}$$ It seems like right time for Lagrange equation. I'll compute every part of equation on the new line. $$\frac{\partial \mathcal{L}}{\partial \phi} = -mrl\omega \dot \phi^2 \cos{(\phi+ \omega t)} + mgl \dot \phi \sin{(\phi)}$$ $$\frac{\partial \mathcal{L}}{\partial \dot \phi} = ml^2 \dot \phi - mrl \omega \sin{(\phi + \omega t)}$$ $$\frac{d}{dt} \left( \frac{\partial \mathcal{L}}{\partial \dot \phi} \right) = ml^2\ddot \phi - mrl \omega^2 \dot \phi \cos{(\phi + \omega t)}$$ If this was right, the equation would look like: $$ml^2\ddot \phi - mrl \omega^2 \dot \phi \cos{(\phi + \omega t)} + mrl\omega \dot \phi^2 \cos{(\phi+ \omega t)} - mgl \dot \phi \sin{(\phi)} = 0$$ If I put cos together and multiply whole equation by $\frac{1}{ml^2}$ I get $$\ddot \phi - \frac rl \omega \dot \phi \cos{(\phi + \omega t)} (\omega + \dot \phi) - \frac gl \dot \phi \sin{(\phi)}$$ which is not right.


How many how many planck areas would capture of a resting electron add to the surface of a black hole?

In other words, what is the 'information content' of an electron (or of other supposedly elementary particles for that matter) as revealed by their effects on black hole horizon area?


Raising and lowering operators of orbital angular momentum

For the orbital angular momentum, the raising and lowering operators are given by, $$ L_+ = e^{i\phi} \bigg(\frac{\partial}{\partial\theta} + i\: cot\theta\frac{\partial}{\partial\phi}\bigg) $$ $$ L_- = -e^{-i\phi} \bigg(\frac{\partial}{\partial\theta} - i\: cot\theta\frac{\partial}{\partial\phi}\bigg) $$ With this I obtain $$ L_+^\dagger = - L_- $$ But with the actual definition in terms of $ L_x $ and $ L_y $ with $$ L_+ = L_x + i L_y $$ $$ L_- = L_x - i L_y $$ $$ L_+^\dagger = L_- $$ How do I reconcile between these two results ? Or is there any mistake I have committed ? PS : My professor hinted saying it had something to do with compactness of angular momentum, but I didn't understand !! (My problem is not in obtaining the result of $L_+$ and $L_-$, but in reconciling these two facts).


Can a wave possess spin?

Since a matter wave is associated with a particle in quantum mechanics, does the wave spins? I mean, can we visualize the spinning of wave or is it possible that the wave spins?


Snooker/billiards cueing

I understand the 2 body dynamics very well, but I can't see how the professionals apply their **chalked** cue and hit the way they do. As a few for instances:   1. How much does the chalk come into play in terms of friction?   2. How off center is it possible to play?   3. How do they aim the off center? Also in a situation like this http://www.youtube.com/watch?v=McThDshEgU0 Ronnie O'Sullivan's fantastic play. What useful measurements are there to grade his skill?


What is the relation of sound propagation to air pressure?

Hi there i am wondering, air is making sound propagate. So in vacuum there is no sound. but what is the relation of pressure to sound volume? is it linear? if i have a source of sound and lets say i am in a room with air pressure 0.5 of the pressure outside would the sound be 0.5 lighter? or does air pressure have no impact on the sound volume? Is there a comprehensible formula for that? thanks a lot!


Why the trap is needed in cold atom experiment?

In ultracold atomic gas experiments, the optical lattice provides a periodic optical potential to trap the atoms, why an extra trap, usually a harmonic trap is needed to trap the atoms?


Set theory, category theory, realism and the recent "reality of the wavefunction" papers

I will add a better phrased question here. Do we need to consider quantum foundations to form a quantum theory of gravity? The kind of foundational question I am thinking of is expressed in the recent "reality of the wavefunction" paper. I have become convinced that if we move physics into a category theoretic foundation (one that eschews the presentation of the theory of categories IN SET) then we might avoid problems of ontic and epistemic states for quantum theory. Does anyone strongly disagree? I have been working on a presentation of the theory of categories that is more like type theory and would avoid things like "The category of groups consists of the set of all groups". The reason we avoid problems of ontic/epistemic is because the mathematics is grounded in the concept that "morphism" equals "cuasal relation" rather than "my set of states is ...well...a set of elements which exists by virtue of the fact that the elements exist". We might trade "causal relation" for a more generic relation that reflects "causally neutral probability theories". Does anyone strongly disagree?...or agree :)


Which solution to the electromagnetic wave equation is the most accurate model of monochromatic light?

When a photon is modeled as a monochromatic electromagnetic wave its electric and magnetic components are usually taken to be sine waves (for example here http://hyperphysics.phy-astr.gsu.edu/hbase/waves/emwv.html). I believe the practical reason for this is that any solution of the electromagnetic wave equation can be expressed as a sum of sine waves. But physically, when a photon can be interpreted as a wave, how is it best modeled? Do we have empirical evidence to think that it is best modeled by a single sine wave for the E and B fields, or if not which solution of the electromagnetic wave equation would best model it? Are there experiments that could show that light waves resemble more say square waves than sine waves? **Update** To rephrase the core of my question more properly: If a photon / electromagnetic wave travels through a point in space, in vacuum, and we measure the electric and magnetic fields at this point with a very high temporal resolution, would we measure the electric and magnetic fields to fluctuate exactly as sine waves, or as something else? Has such an experiment ever been made?


Vector Equilibrium

I'm trying to understand more about this 'vector equilibrium' apparently coined by R.Buckminster Fuller. In this site (http://cosmometry.net/), it claims: > "The Vector Equilibrium, as its name describes, is the only geometric form > wherein all of the vectors are of equal length and angular relationship (60° > angles throughout). This includes both from its center point out to its > circumferential vertices, and the edges (vectors) connecting all of those > vertices." I'm wondering, how can this be the case, when even in the image on that site, it is clear that some sides of the shape are squares => 90 angles between the vectors connecting the vertices. Is this site incorrect then?


Interaction energy between dipole and potential

It is known that interaction energy = $-\vec{p}.\vec{E}$ where $\vec{p}$ is dipole moment and $\vec{E}$ is the electric field. I have to calculate the interaction energy of a system whose dipole moment and potential are the only available data. My assumption is since the negative gradient of potential is electric field, can we calculate the dipolar component of the potential as an alternative to electric field, i.e. $$\begin{align} \phi_{dx}&=\phi_x(x-x_c) \\\ \phi_{dy}&=\phi_y(y-y_c) \\\ \phi_{dz}&=\phi_z(z-z_c) \end{align}$$ where $x$, $y$, $z$ are the positions of atoms and $x_c$, $y_c$, $z_c$ is the geometrical centre, and then calculate the dot product with the dipole moment $(p_x,p_y,p_z)$. Does it sounds meaningful?? Kindly provide some suggestions in solving my problem.


How can contracting dimensions lead to cosmological inflation?

Using the Kasner metric, given by $$ ds^2 = -dt^2 + \sum_{j=1}^D t^{2p_j}(dx^j) $$ it is possible to not only describe the cosmological expansion of some space directions (the ones with positive Kasner exponents $p_j$, but this metric allows for some dimensions to contract too, those have negative $p_j$. The two Kasner conditions $$ \sum_{j=1}^{D-1} p_j = 1 $$ and $$ \sum_{j=1}^{D-1} (p_j)^2 = 1 $$ say that there have to be contracting and expanding dimensions at the same time, as the $p_j$ can not all have the same sign. In a comment I have read, that in models with for example 3 expanding and $n>1$ contracting dimennsions, the contracting dimensions _drive_ the inflation in the other directions by leading their expansion to accelerate without a cosmological constant. This is interesting and about this I'd like to learn some more. So can somebody a bit more explicitely explain how such inflation models work? For example what exactly would the vacuum energy from a physics point of view be in this case? Up to now I only heard about inflation models where the vacuum energy density is the potential energy of some inflaton field(s) in a little bit more detail.


What is fidelity in experimental QM?

I often comer across the term high fidelity in QM papers. Does fidelity imply ratio of entangled photons / total photons? Is there some other metric to measure how good the source is?


What would happen to the Moon if Earth is turned into a black hole?

Assume that all of sudden the Earth is turned into a black hole. And the moon revolves around the Earth (before turning into a black hole). What would happen to the Moon after earth changes to black hole will it be sucked to the black hole or continue to revolve the black hole?


How to find replacement function of a mass?

![enter image description here](http://i.stack.imgur.com/8OAWc.png) I wonder how I can find the replacement function of the center of the blue mass? The center of mass of the blue mass is $(0,0)$ and the blue mass is homogeneous. The masses do not move at t=0 in the system (Their initial speeds are zero). There is no friction in the system and the red mass is a point mass. $u$ is the angle of the red mass with $y$ axis. I need help in learning the theory behind solving such systems. Thank you. EDITED: I would like to show what I did till now to solve the problem. I followed the hints that Manishearth offered $u=\alpha _0$ The definations of the blue mass center replacement: $X_b(t)$ The replacement function that I try to fnd $Y_b(t)$ The definations of the red mass Center replacement: $X_r(t)$ $Y_r(t)$ 1- Mass of center of entire system is in the beginning: $$r_{0x}=\frac{m. r. sin(\alpha _0)}{m+M}$$ $$r_{0y}=-\frac{m. r. cos(\alpha _0)}{m+M}$$ Mass center will not depend on time. If So, $$mX_r(t)+MX_b(t)=(m+M)r_{0x}=m. r. sin(\alpha _0)$$ (1) $$mY_r(t)+MY_b(t)=(m+M)r_{0y}=-m. r. cos(\alpha _0)$$ (2) 2-Energy is conserved $$\frac{m (\dot X_r(t))^2 } {2}+ \frac{m (\dot Y_r(t))^2 } {2}+\frac{M (\dot X_b(t))^2 } {2}+\frac{M (\dot Y_b(t))^2 } {2} +mg(a+Y_r(t))+Mg(a+Y_b(t))=mg(a-r.cos(\alpha _0))+Mga$$ (3) I dont know how to apply Hint 3 and Hint 4. Till now I have 4 unknown and 3 equations. Am I in correct way to solve $X_b(t)$ ? Thanks in advice


Evidence that "space exists rather than just particles"

What is (theoretical) evidence that you need to define all of space with properties rather than just stating where all particles are? I mean, does every single coordinate x, y, z have a meaning or would it be sufficient to keep track of particle positions and type. In other words, to specify reality, do you need all of $\Psi(x,y,z,a)$ where $a$ is another dimension (charge, spin, etc.) or is there any framework where you just specify a _countable_ number or particles with coordinates $(x_{P_i}, y_{P_i}, z_{P_i})$ and you do not care about coordinates inbetween particles (obviously there would be distant interaction laws). Can you disprove sufficiency of the second representation?


M-theory and many-worlds interpretation

I am getting some confusion on whether M-theory accepts many-worlds interpretation. Can anyone show me the reasons or rebuttals for the possibility of the many-worlds interpretation in M-theory? Thanks.


Where the extra power comes from?

Suppose we have two radio waveforms each has amplitude of 1, then the total power is 2. Suppose these two waveforms add up some where constructively, then the amplitude become 2, and the total power is 4, so where is the extra power comes from?


How many watts of light do instruments in modern telescopes need?

I just calculated that if the European extremely large telescope(978 m^2 area) would be pointed at the Andromeda galaxy once it is built, it will only get mere 1.2*10^-6 Watts of power from Andromeda. And I didn't account for atmospheric factors or the fact that the telescope is only going to read visible and infra red light. That is to me a really small number that we achieve even after using the highest aperture that would be available to us. Are my calculations correct(I have double checked but I still really doubt it.)? I used the relation between intensity, luminosity and distance. Please, I am just an amateur. If you guys find an embarrassing arithmetic mistake, let me know so. If my calculations are correct, do the scientific instruments on say, Hubble work with such a small amount of light? How many Watts of light does the Camera on these telescopes need to work properly?


Mathematical question on Collisions

> A `2.5`kg ball travelling with a speed of `7.5`m/s makes an elastic > collision with another ball of `1.5`kg and travelling at the speed of > `3.0`m/s in the same direction. > > \-- What are the velocities of the balls immediately after collision?