Does the Pauli exclusion principle instantaneously affect distant electrons?

According to Brian Cox in his _A night with the Stars_ lecture$^1$, the Pauli exclusion principle means that no electron in the universe can have the same energy state as any other electron in the universe, and that if he does something to change the energy state of one group of electrons (rubbing a diamond to heat it up in his demo) then that must cause other electrons somewhere in the universe to change their energy states as the states of the electrons in the diamond change. But when does this change occur? Surely if the electrons are separated by a significant gap then the change cannot be instant because information can only travel at the speed of light. Wouldn't that mean that if you changed the energy state of one electron to be the same as another electron that was some distance away, then surely the two electrons would be in the same state until the information that one other electron is in the same state reaches the other electron. Or can information be transferred instantly from one place to another? If it can, then doesn't that mean it's not bound by the same laws as the rest of the universe? \-- $^1$: The Youtube link keeps breaking, so here is a search on Youtube for Brian Cox' _A Night with the Stars_ lecture.


The shape of the earth$\ldots$

....is an oblate spheroid because centrifugal force stretches the tropical regions to a point farther from the center than they would be if the planet did not rotate. So we all learned in childhood, and it seems perfectly obvious. However... I am at $45^\circ$ north latitude. Does that mean   * An angle with vertex at the center of the earth and one ray pointing toward the equator at the same longitude as mine, and one ray pointing toward me, is $45^\circ$ (that would mean I'm closer to the north pole than to the equator, measured along the surface, as becomes obvious if you think about really extreme oblateness); or   * The normal to the ground where I stand makes a $45^\circ$ angle with the normal to the ground at the equator at the same latitude (this puts me closer to the equator than to the north pole); or   * something else? If for the sake of simplicity we assume the earth is a fluid of uniform density, it seems one's potential energy relative to the center of the earth would be the same at all points on the surface.   * Would the force of gravity at my location, assuming no rotation, be directly toward the center? Would it be just as strong as if the whole mass of the earth were at the center and my location is just as far from the center as it is now?   * Would the sum of the force of gravity (toward the center or in whichever direction it is) and the centrifugal force (away from the axis) be normal to the surface at my location?   * Given all this, how does one find the exact shape?   * How well does that shape in this idealized problem match that of the actual earth?


how does dynamic casimir effect generate correlated photons

There is a recent paper on arxiv receiving lot of acclaim http://arxiv.org/abs/1105.4714 The authors experimentally show that moving a mirror of a cavity at high speeds produces light from high vacuum. The usual doubts about the experimental techniques seem to be very clearly addressed and reviewed (as per Fred Capasso's comments) http://www.nature.com/news/2011/110603/full/news.2011.346.html) My question is : Can someone explain how correlated/squeezed photons are generated in this process ? I can get a feel for how a moving mirror can generate real photons by imparting energy to the vacuum (correct me if this is not consistent with the detailed theory). But, I don't see how photons are generated in pairs. Could someone describe the parametric process happening here ?


Landau's ambiguous statement about the existence of inertial frames

Landau writes "It is found, however, that a frame of reference can always be chosen in which space is homogeneous and isotropic and time is homogeneous." Does he mean that we can prove the existence of an inertial frame or does he want to say that it is assumed by doing enough number of experiments? Can we start with some axioms and definitions of properties of space and time and then deduce the existence of such a frame in which space is homogeneous and isotropic and time is homogeneous?


Why is quantum mechanics based on probability theory?

What makes us formulate quantum mechanics based on probability theory? Isn't the real quantum world based on unknown laws to us? Is it possible that results of an experiment will be measurable in another way but not expected value?


Resource about Supercontinuum Generation in Fibers

I would like to consult a nice reference that explains the theoretical background of SC generation in optical fibers in detail but more or less self- contained. I would also like to have your opinions on nonlinear optic books that present material at introductory level. Thanks in advance


What is the difference between spin glass and spin liquid? Do they both originate from frustration?


Harmonic oscillator and Lorentz symmetry

There is a analog between harmonic oscillator $x=\frac{1}{\sqrt{2\omega}}(a+a^\dagger)$ and quantum field $\phi=\int dp^3\frac{1}{(2\pi)^3}\frac{1}{\sqrt{2\omega_p}}(a_p e^{ipx}+a^\dagger e^{-ipx})$, which is used to quantize the field operator. However, one thing confuse me is about the coefficient $\frac{1}{\sqrt{2\omega}}$. For field operator, this comes from Lorentz invariance, just because we have integrated time t. However, for harmonic oscillator, there seems no apparent Lorentz symmetry give me this. Is there any hidden symmetry behind the harmonic oscillator?


Does the attached "poster" work as a hook into the arXiv paper cited, "Nonlinear Wightman fields"?

"Nonlinear Wightman fields" are my current response to a wish to do interacting quantum field theory differently, no matter how successful what we currently do may be. The following image of a single page "poster" attempts to get across why I think the mathematics is interesting, with the intention that an ordinary physicist would find the hook curious enough to look at the 22 page arXiv paper. The mathematics of section IV of that paper is really as simple as "use the Hadamard product of matrices" to construct something nontrivial that might have more relevance to Physics than the Wightman axioms have had since they were created more than 50 years ago. This is at the same time familiar stuff and far up a different mountain, so there is some hope at the same time as there is concern that physicists generally will not understand the motivation or that I've made a mistake in the mathematics. [Note that the image itself is legible even if what is displayed inline is marginal.] ![Nonlinear Wightman fields -- a poster](http://i.stack.imgur.com/hzGN8.jpg) The paper this cites is something of an outcome from the thinking that led to my question "Does renormalization make quantum fields into (slightly) nonlinear functionals of test functions?", 18 months ago, when I was using PhysicsSE quite actively as a way to help my research. Some of the content of the arXiv paper may also be found in a much rawer form as an entry in this year's FQXi essay competition (so raw that I can't recommend that you chase the link). People who've been on PhysicsSE long enough may remember my attempts, for over a year, to use PhysicsSE as a tool for research, which I eventually felt to be counterproductive because it led to too little long-term thinking, and also because I felt unable to do much to help people whose research directions were different from mine. I will submit the arXiv paper to a journal, probably JMathPhys, fairly soon, where it will hopefully generate a worthwhile critique either of the whole or of parts, but I would prefer to iron out kinks in the presentation or in the mathematics before doing so (or, if someone is kind enough to make a killing remark about the mathematics, you could save the editors or referees of JMathPhys or of some other journal some trouble). As an aside, anyone who takes the time to read sections II and III of the arXiv paper will find calculations and an unconventional perspective on interacting quantum field theory that I would expect anyone who has thought for a while about foundational issues in QFT to find curious and perhaps useful.


Keeping air in a giant gravitationally-bound space balloon

Let's say a space-faring society wants to make a space station that has a large volume filled with air (or other gas), but no gravity. Using normal pressure tanks will require gathering an amount of material proportional to the volume, which is a ratio set by the tensile strength and desired pressure. It's likely that some other method for containing gas would be more economic for anything over a certain volume. Could you confine a large volume of air in space using self-gravitation to hold the container together? Well obviously you can. Consider: a spherical volume of air surrounded by a solid sphere of matter through which the air cannot leak or diffuse and the pressure balances the gravity of the walls. Think of a big balloon ball in space, or alternatively, injecting the middle of the moon with air until it starts expanding. A hollow planet, if you will. There will be little gravity within the air because the walls don't contribute to the gravity inside and air has a low density. So here is what I'd like to ask:   * For a given pressure, what surface mass density ($kg/m^2$) would you need? What would the wall thickness be?   * Going by the amount of material required, at what size would this approach be more economic?   * How stable would this thing be?   * Hypothetically, could you use the same principles to drape a airtight tarp over Mars and keep atmosphere from leaking out? It would be the surface mass density that determines the altitude at which it rests, right? Disclosure type statements: I can do a lot of the calcs for this myself, but I don't want to because I have doubts about certain parts and I want to avoid influencing other people with my potentially incorrect thought process. If you look at my Physics SE activity, you might notice that I'm fascinated by self-gravitation problems. I came up with this question reading Keeping air in a well.


What happens to angular momentum when matter is converted to energy?

Let's say a spinning star radiates mass-energy only from it's pole regions. How does the loss of mass-energy effect the angular momentum of the star?


Meaning of negative frequency of sound wave

Suppose that Alice and Bob are both holding speakers emitting sound at a frequency $f$. Alice is stationary while Bob is moving towards Alice at twice the speed of sound. In the case of Alice, if I forget for a minute that she is holding a speaker herself, and just think about the Doppler effect from Bob moving towards her, I get $$f_{\text{effective}}=f\frac{1}{1-\frac{2v_a}{v_a}}=-f$$ I do not know what "negative" frequencies mean in the context of sound waves, and so I do not really know how to describe what Alice will hear as Bob becomes very close to her. I also do not know how the fact that she is emitting sound at a frequency $f$ will affect, if at all, what she hears. And, from Bob's perspective, I get $f_{\text{effective}}=3f$, which is a bit more meaningful, but does not factor in the fact that Bob is emitting sound himself. Any clarification is appreciated.


Does String Theory explain wave-particle duality?

Does string theory explain the weird things that happens at the quantum level, especially wave-particle duality?


Alternate layman's metaphors for illustrating curved space-time

The metaphor of a surface (typically a pool table or a trampoline) distorted by a massive object is commonly used as a metaphor for illustrating gravitationally induced space-time curvature. But as has been pointed out here and elsewhere, this explanation seems (to a layman like me, at least), to be "hopelessly circular", and in the end contributes little to an understanding of how modern theories of gravitation work. Are there other (or additional) metaphors that might be helpful in illustrating to lay readers (a) what motivates modern gravitational theory and (b) why it has greater explanatory power than Newtonian gravitation?


Why closed bottle change its weight when I put effervescence tablet?

I use several shape, several type of material (glass, metal, plastic), I use two different balances with 0.01g of accuracy. I put oil on gasket, and put upside down (like that I can see if water escape). But always it's the same result, when I put one, two or more effervescence tablets, the weight decrease: one tablet => -0.03g two tablets => -0.06g three tablets => -0.09g Sure the difference is not big. But if I put an object without effervescence tablet, the weight is always the same (move sometimes +0.01 or -0.01g but never more). The time for decreasing is about 2 minutes so it's not enough for change something from temperature I think especially with glass container. So, maybe someone has done this experimentation before and know where is my error ? Or maybe someone can test and try this experimentation ? @Luboš Motl: "Recall that the air density is about 1.3 grams per cubic centimeters", you're sure ? it's not 0.0013 ? Yes, I have the same values for glass (2 mm of thickness and one of 3 mm of glass) with a metal cover of 0.8 mm of metal. I done about more than 100 measures. After 2 minutes, the tablet is full dissolved in water and weight move very few after (-0.01g next 2 minutes) but never more. Maybe I found: when bubbles move up in water, they have gas in it (CO2), this gas move up with a speed, so there is a quantity of movement, the weight losses is the sum of mass of bubbles multiply by speed. Like quantity of movement is conserved, the weight don't change. It's possible to use the formula of Newton: force=2mv if top speed is 0.25 m/s the weight losses is 2*0.3/1000*0.25/10 = 0.015 g with 0.3 g of CO2, image show speed that I found on Internet. If it's that, the weight must change with a ball full of air (ping-pong ball) in water. But the CO2 in a tablet is very powerfull, 0.3g give 0.187 liter of CO2, even the ball move faster the volume is not great in a bottle. The ball must be down before close the container. And the gas must relax when the ball reached surface. Is it possible it is the rotation of Earth that change the pressure in water due to the centripetal forces ? This give -0.03N for each kg of water. Good day


Desperately Need Help with Grade 9 Static Electricity

I am preparing to teach Grade 9 Static Electricity next week and am going crazy trying to figure out what is happening in one of my experiments. I have a short piece of PVC pipe, 4 inches diameter, and I rub it with wool to charge it negatively. I can observe excellent repulsion when I touch it with my foil bit (dangling from a thread). Here is the problem: I am holding down the PVC pipe down on a wooden base with two brass-plated wood screws, and these screws somehow collect an INSANE amount of positive charge, even when I am very careful not to touch them with the wool. The foil bit is strongly attracted to the screws, and when it touches them it bounces off more violently than anything I've seen in any of my other static electricity experiments. Can anybody explain what I am seeing?


Correct way to calculate torque produced by axle

For my electrical engineering course, we had to build a simple DC motor that can lift a coin. I have tested the motor, and here are the results:   * rotational speed (no load): 3630 RPM (380 rad/sec)   * Current with load (9g): 3A (limited)   * Current with no load: 0.7A   * Maximum weight lifted: 0.1422 N   * Max voltage 12V I now want to plot a torque-speed curve and calculate its efficiency. My problem is that the weight it lifted is a force quantity. To convert it to a torque quantity, I multiply by the axle radius (6mm) which gives 0.00085 Nm. This seems wrong to me becausekjg when I calculate the efficiency $\epsilon = \frac{\omega \tau}{VI}$ I get $\epsilon = 0.9 \% $ which seems extremely low. Is this the correct way to calculate the torque produced by an axle when lifting a load?


What is the fundamental reason of the fermion doubling?

Recall that the fermion doubling is the problem in taking the $a \to 0$ limit of a naively discretized fermionic theory (defined on a lattice with lattice spacing $a$). After such a limit one finds themselves with an additional amount (precisely $2^d$) of fermionic fields. One can fix this by considering different discretizations of the action that make unwanted fields decouple in the continuum limit. The downside is that the additional terms have to spoil some nice features of the theory (chiral symmetry, locality, lattice symmetry, ...). Now, I wonder what is the true reason for the appearance of new fields. Is it the fermionic nature of the theory? (In other words, is a similar problem ruled out for bosonic fields?) And do all (naive?) fermionic theories (that is, independent of the continuum form of the action) suffer from this problem? More generally, how can one tell _a priori_ what will the field content of a lattice theory in the continuum limit be? Or is the field content fundamentally a continuum limit property that has to be calculated?


Nature of Microscopic space-time

I am going through the introductory chapter's of Schwinger's Source theory. He writes, > It [Source Theory] is a phenomenological theory, designed to describe the > observed particles. No speculations about the inner structure of the > particles are introduced. No abstract definition of particle is devised. The > theory is thereby firmly grounded in space-time, where the experimenter > manipulates his tools, **but the question of ultimate limitation to > microscopic space-time description is left open, with the decision reserved > to experiments. Correspondingly, no Operator-fields are used.** Now in this regard, I would want to know how operator fields answer the question of ultimate limitations to microscopic space-time (If they are related to each other)? **EDIT 1 :** It just struck me that the limitation could be due to canonical commutation between field operators and their conjugates. However, I don't see how to formalize a restriction using this commutation.


does the current's direction on a loop with a magnetic field affect the torque?

I have a question in my book ![http://i.imgur.com/P5R55sR.png](http://i.stack.imgur.com/7yZhT.png) The question is very easy, but then my solutions manual gave me unexpected answers ![http://i.imgur.com/tSMmJEY.png](http://i.stack.imgur.com/cam6h.png) I don't get how in d) they conclude the vectors go in the opposite direction, but in b) they go in the same direction. The only thing I could guess is that there's some assumed current direction but I can't find anything in my book about that. Thanks for any help


If the temperature of 2 materials are the same does that mean the molecules are vibrating at the same speed?

Pretty much what the title says. My base question is this. Assuming I take a piece of steel, and a piece of PVC plastic and I measure both their temperatures and find they are the same. I then take a look at the vibration speeds of the individual molecules would they be the same as well? Here's a rough example: I measure both the steel and PVC and find them both at 100F, and then I measure the vibrations of a molecule in the steel and find it to be moving at 10 miles an hour. Would the PVC molecules also be moving at 10 miles an hour? I'm sure I'm not using the correct units of measure to measure the vibration, but I didn't know what the correct unit of measure is for something like that. Hopefully it gets the point across.


Why is there a phase factor when the two composite angular momentum is exchanged in Clebsch–Gordan coefficients

An identity exists for CG coefficients: $$\langle j_1 m_1 j_2 m_2 |J M \rangle = (-1)^{j_1+j_2-J} \langle j_2 m_2 j_1 m_1|J M\rangle,$$ But why is there a phase factor $(-1)^{j_1+j_2-J}$? It seems to me that $$|JM\rangle =\sum_{m_{1},m_{2}}|j_{1}m_{1}\rangle\otimes|j_{2}m_{2}\rangle\langle j_{1}m_{1}j_{2}m_{2}|JM\rangle =\sum_{m_{1},m_{2}}|j_{2}m_{1}\rangle\otimes|j_{1}m_{2}\rangle\langle j_{2}m_{2}j_{1}m_{1}|JM\rangle $$ And since $|j_{1}m_{1}\rangle\otimes|j_{2}m_{2}\rangle$ and $|j_{2}m_{1}\rangle\otimes|j_{1}m_{2}\rangle$ are the same physical state, there should be no difference between $\langle j_1 m_1 j_2 m_2 |J M \rangle$ and $\langle j_2 m_2 j_1 m_1|J M\rangle$. What do I get wrong?


Physics for mathematicians

How and from where does a mathematician learn physics from a mathematical stand point? I am reading the book by Spivak Elementary Mechanics from a mathematicians view point. The first couple of pages of Lecture 1 of the book summarizes what I intend by physics from a mathematical stand point. I wanted to find out what are the other good sources for other branches of physics. Thanks


Prereqs for The Geometry of Physics by Frankel

I'm interested in giving The Geometry of Physics a read, and I was wondering what the mathematical and (more importantly) physical prerequisites are. My background is a bit stronger on the mathematical side, and I'm worried that if heavy (or even moderate) exposure to physics is required that I'll miss out on a lot. If someone has read it and has a good sense of this sort of thing, I'd very much appreciate the input! A link to the book.


What is a phonon?

I am trying to understand intuitively what a phonon is, but for the moment I find it quite difficult (having a limited background in quantum mechanics, an undergraduate course in non-relativistic QM). In fact, I find it hard to formulate good questions, so I hope my questions below make some sense. I read that phonons are (the quantum mechanical analog of) normal modes of vibration in a crystalline system of atoms or molecules, so I guess a superposition, i.e. a general vibration should also be a phonon. Is that so? Why would they then be described as normal modes? Could we say that a phonon is a particle whose position wave function extends over the whole crystal? Are the quantum mechanical frequency and wave vector the same as the frequency and wave vector of the corresponding classical oscillation (vibration in the crystal)? In what sense is it (like) a particle? In that it is always observed or it always interacts at a specific location?


What is "charge discreteness"?

I assume it is some kind of quantity. Google only made things more confusing. I get that it has something to do with circuits. I also get what a discrete charge is. In fact, I thought charges were, by definition, discrete - because each individual proton/electron/hole contributed one unit of charge.


Speed astronauts measure moving at the speed of light

Lets suppose a spaceship travels with v = 0.9c relative to the Earth. The time inside the spaceship would pass slower than on Earth. Would the astronauts measure a different speed (that means, a different one that the observer on Earth does) in relation to the same reference frame (Earth)?


Is the speed of light related to the mass of the universe?

If the mass of the universe were cut in half, would it affect the speed of light? Would it be twice as fast? Would it stay the same? Do we have instruments that are sensitive enough to measure the speed of light at different positions relative to high-mass objects to empirically answer this question? The speed of light is (something of) a universal constant, but is it really dependent on the universe or on something intrinsic to photons? **EDIT:** Related question: Since gravity is a relationship between one atom and every other atom in the entire universe, and it takes all the energy in the universe to travel at the speed of light, is there something about the energy/gravity/mass of the universe that "slows" light from going a faster speed?


Examples of systems with energy as an intensive variable

I need to consider a couple of examples of systems which have energies that are intensive variables - not extensive. I'be been thinking about this and I am not coming up with anything. My understanding is that extensive variables (at least wrt usual energies) scales with mass or length (system size). It also seems that some 'energies' depend upon the model used, such as how strong the interactions are in neighbors of atoms or dipoles, etc., or whether one is considering chemical potential or not, etc. Any good suggestions?


Electrical properties of molten gallium arsenide

I'm looking for the resistivity and magnetic permeability of molten gallium arsenide, but can only seem to find the values for the solid material at room temperature (e.g., Wikpedia). Not even temperature-dependency of these parameters seems to be available. Is this because the value just doesn't change with temperature? If it does, what's a good resource for such parameters?


Is quantum uncertainty principle related to thermodynamics?

Would like to ask a question, but first i would like to say **Hello Everybody** in a way that plays the system, since some geniouses decided that one should not be able to say hello in a question. The uncertainty principle in quantum mechanics is well known and considered one of most basic properties of natural reality. The 2nd Law of thermodynamics is also well known and also considered one of the most basic processes of natural reality. The uncertainty principle uses and is related to Planck's constant. Planck's constant has the dimensions of _action_ and in a statistical mechanics approach, also relates nicely with the partitioning of the phase-space providing the basic measure for the entropy functional (this answer provides a nice outline of this). Apart from that, there are relatively recent papers which relate the Heisenberg Uncertainty Principle in quantum mechanics directly and intuitively to the 2nd Law of Thermodynamics. Is this relation correct? And if so can we derive one from the other? Thank you PS. One can also check this question, which although not the same, is related in an interesting way. UPDATE: anna's answer is accepted since by mentioning the derivation of (part of) the 2nd law from unitary dynamics, answers the question at least in one way. Please consider this as still open so you can add another answer. There are more alternatives (and one of which is my stance, ie thermodynamics -> uncertainty)


What kind of light is needed to light up Venus?

Let's say I would want to light up Venus, such that we can see Venus all day long and not have to wait for a Venus Transit. What kind of light would I need for it? How powerful would it need to be?


What is a fluid flux?

I need a concise definition of a fluid flux and an accompanying example. I've never taken a single physics course before, but I'm required to understand this concept so I can do the calculations for a Complex Analysis class.


Neutrino Oscillation and their gravitational implications

As I understand neutrinos, there are three different flavors, all with different masses. Although the masses of these neutrinos have not been directly measured, their mass differences have been. Current experiments, KATRIN and Project8 are going to measure neutrino masses and we shall know soon enough. Regardless, their mass states change as they travel through space. This leads to my question... Since an object's gravitational field is related to its mass and neutrinos have different mass states while they are traveling, it must mean that every point in space must be constantly altering in gravity intensity! Although every object alters the gravitational field intensity as it travels and passes a given point, neutrinos would do it differently because they keep changing mass states! Let's assume a constant stream of neutrinos pass by a point in space versus neutrinos that don't oscillate (This is hypothetical) doing the same thing. Wouldn't these extremely weak gravitational waves be different given oscillations than not?


G(2) lattice and the M-theory landscape

In a previous question (Calabi-Yau manifolds and compactification of extra dimensions in M-theory), I was told that the $G(2)$ lattice can be used to compactify the extra 7 dimensions of M-theory and preserve exactly $\mathcal N=1$ supersymmetry. However, since there is only 1 $G(2)$ lattice, there should be only 1 4-dimensional M-theory. Then, why is there such a huge fuss about the M-theory landscape? Thanks!


How does one show using QED that same/opposite electric charges repel/attract each other, respectively?

Why do same charges repel each other and opposite charges attract each other (please explain the phenomenon using real laws of nature (QED) not with the approximation model)?


Are all elementary particles of the same type exactly the same?

Are all elementary particles of the same type EXACTLY the same? Is there some variation in what an electron is, for example, or are they all the same?


Can low temperature plasma exist?

Plasma is ionized gas which as far as I know only occurs at high temperatures. When plasma cools down it tends to recombine with the electrons present and turn back into gas. But what if the disassociated electrons in the plasma were removed and the plasma were allowed to cool down in a vacuum, while being held in place by a strong magnetic field. Would this substance still be plasma? Is this possible?


Help calculating field of view for ball lens I just bought

![Diagram](http://imgur.com/KwER0d8) I have a 5 mm (diameter) solid ball lens that I intend to place in front of a 36mm X 24mm camera sensor (I have already made preparations for the image to be focused and for the sensor to not be flooded with light). I want to calculate the resulting horizontal and vertical field of view from the ball lens when I take a picture. Please see the link for a diagram of the setup if the image does not work. ![http://imgur.com/KwER0d8](http://i.imgur.com/KwER0d8.jpg) I have already calculated the back focal length of the ball lens to be 1.479 mm.


Which bonds are the cross-links and which are secondary bonds (in elastomers)?

Elastomeres are "defined" as: "linear-chain polymers with widely spaced cross-links attaching each molecule to its neighbours" Now I found sentences (talking about glass transition): "This means that at room temperature the secondary bonds are melted and the molecules can slide relative to each other with ease. Were it not for the cross-links, the material would be a viscous liquid, but the cross-links give it a degree of mechanical stability." So, what exactly are these bonds that melt? and which ones do not? An example would be nice. Chemistry is not my strongest field so be rigorous. The quotations are from cellular solids by gibson and ashby.


Has the Double-Slit Quantum Eraser Experiment ever been tried on a large scale?

I was just reading about quantum entanglement and the example was the Double- Slit Quantum Eraser Experiment. Then this was used as a basis for saying that particles might be half a universe apart and still be just as connected. So I was just wondering if entanglement experiments have been done to show that this is really the case or if it is considered "academic" to conclude that there's no difference when the distances are huge? As a side query, quantum mechanical interference seems to be based on the existence of probability - does this mean probability should be thought of as something as tangible as matter/energy?


Frequency of earthquakes

Are earthquakes getting less frequent across the centuries? I know that more seismic stations have register more earthquakes in the last century, but that doesn't imply there were more. I am interested on the geophysical side of it. The logic is that things settle down, so there is less stuff to shuttle. Add to it that the earth (at its core) get colder with the passage of time.


Work done by isothermal expansion from two different viewpoints

Consider an adiabatic system as follows. It consists of a gas in a container and a piston. Initially, the system is at equilibrium and the gas inside it occupies a volume $V_i$ at a pressure $p_i$ which is equal to the outside pressure. Suddenly, the outside pressure changes and reduces to $p_{atm}$. The piston moves to equalize the pressure and the gas expands isothermally to obtain equilibrium. The gas now occupies a volume $V_f$ at a pressure of $p_f$ which is equal to $p_{atm}$ Now, two textbooks I have define the work done from two different viewpoints. **1: From the viewpoints of the surroundings** : The work done on the system by the surroundings equals $-p_{atm}\Delta V$. Since $p_{atm}$ is pretty much constant for the whole of the whole of the process, we can say that the work done equals: $$W_1 = -p_{atm}(V_f - V_i)\tag1$$ **2: From the viewpoint of the system:** We can write the internal pressure of the system as a function of its volume: $p_{in}(V)$. As during the expansion, the internal pressure changes, the work done by the system equals $$W_2 = \int_{V_i}^{V_f}p_{in}(V)dV$$ Now, I don't know which definition to use. The work is done by the system (from definition 2) is done on the surroundings. But what about the negative work that the surroundings did on the system? Where did that energy go? Maybe, the two definitions express the same thing: the work done by the system. The negative sign in definition 1 signifies that the work is done by the system. But that would mean that $$W_1 = W_2$$ We can simplify $W_2$ as follows. Clearly, $$p_{in}(V) = \frac{p_iV_i}{V}$$ $$W_2 = \int_{V_i}^{V_f}\frac{p_iV_i}{V}dV = p_iV_i \ln{\frac{V_f}{V_i}} \tag2$$ The internal pressure when the volume is equal to $V_f$ is $p_{atm}$ $$\implies p_{atm} = p_{in}(V_f) = \frac{p_iV_i}{V_f}$$ $$\implies V_f = \frac{p_iV_i}{p_{atm}}$$ Putting this into $(1), (2)$ gives us, $$W_2 = p_iV_i\ln{\frac{p_{i}}{p_{atm}}}$$ $$W_1 = -p_{atm}\left(\frac{p_iV_i}{p_{atm}} - V_i\right) = -V_i(p_{atm} - p_i)$$ Consider some values. Let $p_i = 5 \ \rm{Pa}, V_i = 1 \ \rm{m^3}, p_{atm} = 1 \ \rm{Pa}$. $$W_1 = -1\cdot(1-5) = 4$$ $$W_2 = 1\cdot1\cdot\ln{\frac{5}{1}} = 1.609$$ Where am I making a mistake?


Does there exist a single plate capacitor(conductor)? _if yes_ How will you define the **capacitance** and potential(difference) of such conductor?


Probability and the propagator

Due to the Wiki article, "...In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum...". Let's have the expression for the propagator of some field $\hat {\Psi}_{l}$ in the free theory: $$ \tag 1 D_{lm}(x - y) = \langle |\hat{T}\left( \hat {\varphi}_{l}(x)\hat {\varphi}_{m}^{\dagger}(y)\right) |\rangle = -\frac{i}{(2 \pi )^{4}}\int \frac{F_{lm}(p)e^{-ip(x - y)}d^{4}p}{p^{2} - m^{2} - i\varepsilon}. $$ Let's use two examples: spinor and scalar fields: for them $(1)$ takes the form $$ \tag 2 D_{lm}(x - y) = D(x - y) = -\frac{i}{(2 \pi )^{4}}\int \frac{e^{-ip(x - y)}d^{4}p}{p^{2} - m^{2} - i\varepsilon}, $$ $$ \tag 3 D_{lm}(x - y) = -\frac{i}{(2 \pi )^{4}}\int \frac{(\gamma^{\mu}p_{\mu} + m)_{lm}e^{-ip(x - y)}d^{4}p}{p^{2} - m^{2} - i\varepsilon} $$ respectively. How to convert $(2), (3)$ into the probability (which lies at interval $[0, 1]$)? Particularly i don't understand what to do with spinor indices in $(3)$ (the probability must be Lorentz scalar, so I need to sum over the indices (?)). And also, why doesn't probability depends on momentum (the propagator doesn't contain info about momentum)?


Why is the potential energy of a particle $mgh$ regardless of force?

Consider a particle raised to height $h$ by applying a constant force of $F$ vertically opposite to the direction of force of gravity on it. The potential energy of the particle is $$W = mgh$$ But, according to the definition of work, the work done on the particle is $$W' = Fh$$ According to conservation of energy, $W = W'$. But what if $F>mg$?


Dead time in data acquisition

I am creating a data acquisition software, based on Sparrow's Kmax. There I would like to add a feature that will show the system's dead time. I have already a text field that shows the real time and live. My question has to do, with the dead time. Is it right to say that the dead time is $\text{dead time}=\dfrac{\text{total time of recording events}}{\text{total time of acquisition}}=\dfrac{\text{live time}}{\text{real time}}$


Quantum Anomalies for Bosons

We know that there is Adler and Bell-Jackiw(ABJ) type anomalies for fermions. In some case, the ABJ anomaly affecs particle physics pheonomelogy, such as pion decays or kaon decays(in the case of pion, we still have a calculation on left/right chiral fermions running on the 1-loop triangle diagram). In some other case, there is 1+1D QED Schiwinger or axial anomaly for chiral fermions. The commutation of fermionic anomaly is usually done by, either a 1-loop Feynman diagram, or a Fujikawa path integral method. The above may be some examples of anomalies for fermions. **Is there any example of quantum anomalies for bosons (pure bosonic systems)?**


Electric Field Change Freezing Temperature of Water?

I assume that the reason water freezes is because as you decrease the temperature, the kinetic energy of the water molecules decreases and the dipole bonding potential eventually over comes the escape velocity of the molecules and they form a crystal structure. What would happen if you attempted to freeze water in the presence of a very powerful electric field? Will the water molecules align to the field and make it harder to form a solid or easier to form a solid?


Could quarks and leptons mix if they carried flavor charges?

If quarks and leptons carried flavor charges that differed across generations (as they do in some theories), then could mixing take place?


Calculating Average Velocity

I understand that the concept of an average of a data list means finding a certain value 'x', which ensures that the sum of the deviations of the numbers on the left of 'x' and on the right of 'x' yields the value 0. Assume we are given the values 2,6,9 and 12. I label these values from left to right in the sequence I have provided as: a,b,c and d respectively. The definition of finding these values' average (let's call this average value 'x') in mathematical form is:      $${(x - a) + (x - b) = -(x - c) - (x - d)}$$      So this equation can be reduced to:      $$\Large x = \frac{(a + b + c + d)}{4}$$      This will provide you with the value for the average of any number of values. Similarly there is a formula in physics that calculates the average velocity from a graph by dividing the sum of the initial and final velocities by 2.      $$\Large\frac{v_f + v_i}{2} = \bar{v}$$      There is also another formula which calculates the average velocity from a graph by calculating the slope of the line through two points on the graph. This is done by dividing the change in position by the change in time:      $$\Large \frac{x_f - x_i}{t_f - t_i} = \bar{v}$$      I can understand the third equation from the top of the page, as I can understand this calculation of average velocity in terms of what I explained in the first two equations from the top of the page. I am unable to reconcile the calculation of the average of velocities as seen in the last equation with the method as listed above in the first two equations. Can someone help me solve this dilemma?


Time evolution of Gaussian wave packet

I'm slightly confused as to answer this question, someone please help: Consider a free particle in one dimension, described by the initial wave function $$\psi(x,0) = e^{ip_{0}x/\hbar}e^{-x^{2}/2\Delta^{2}}(\pi\Delta^2)^{-1/4}.$$ Find the time-evolved wavefunctions $\psi(x,t)$. Now I know that since it is a free particle we have the hamiltonian operator as $$H = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2},$$ which yields the energy eigenfunctions to be of the form $$\psi_E(x,t) = C_1e^{ikx}+C_2e^{-ikx},$$ where $k=\frac{\sqrt{2mE}}{\hbar}$, and the time evolution of the Schrödinger equation gives $$\psi(x,t)=e^{-\frac{i}{\hbar}Ht}\psi(x,0)$$ but the issue I face is what is the correct method to find the solution so that I can then calculate things such as the probability density $P(x,t)$ and the mean and the uncertainty (all which is straight forward once I know $\psi(x,t)$. In short - how do I find the initial state in terms of the energy eigenfunctions $\psi_E(x,t)$ so that I can find the time evolved state wavefunction.


Does the Lorentz transformation not apply to light?

Since you would know that light always travels at the constant velocity with respect to all frame of reference ....according to relativity whenever we are traveling at speed of light our time with respect to relative rest observer would become stopped ..if. it means light travels with respect to all frame of reference at light speed, so it implies that light from the sun would never reach us ..but sadly it would reach us within 8 minutes ..How is that possible?


Here is a question about the canonical momentum that I had asked some days ago, but I still have one point that I am not understand. Considering a particle moves in a magnetic field with charge $q$ and mass $m$, its hamiltonian is $$H=\frac{\vec{P}^2}{2m}=\frac{(\vec{p}+q\vec{A})^2}{2m}$$ where $\vec{p}$ is the momentum of the particle, $\vec{A}$ is the vector potential of the magnetic field and $\vec{P}$ is the canonical momentum of the particle. I think, because of the expression of the hamiltonian, the canonical momentum $\vec{P}$ is a conserved quantity. But by the answer in the previous link, it seems that the canonical momentum is not conserved even in a simple example that a particle moves in a homogeneous magnetic field. I am confused about this question. Is the canonical momentum conserved when a particle moves in magnetic field?


Are O-planes not dynamical?

In Polchinski's book "String theory", he is saying "Unlike the case of D-branes, there are no string modes tied to the orientifold plane to represent fluctuations in its shape. Our heuristic argument that a gravitational wave forces a D-brane to oscillate does not apply to the orientifold plane. Essentially, the identifications (8.8.3) become boundary conditions at the fixed plane, such that the incident and reflected waves cancel. For the D-brane, the reflected wave is higher order in the string coupling." I want know how to check "For O-plane,the incident and reflected waves cancel. For the D-brane, the reflected wave is higher order in the string coupling." Is there any papers or books or Web sites calculating it?


How does a Fresnel rhomb work (half and quarter wave plate)? I am aware of birefringence, which creates a phase shift of $\Delta\phi=\dfrac{2\pi\Delta nL}{\lambda_0}$. But this doesn't explain how a plane polarised light shifts it's polarization angle after a half-waveplate or how a linearly polarised light becomes elliptical after a quarter-waveplate. For the half one let's assume that wikipedia has an answer http://en.wikipedia.org/wiki/Waveplate but why this electric field $E(\hat{f}+i\hat{s})e^{i(kz-\omega t})$ describes an elliptical polarised light? What I need is a mathematical description of how the retarders work...


How do you clean a dusty lens?

The lenses of my telescope and binoculars are dusty. What is the best way to clean them without damaging the optic coating?


Flat space limit of the Schwarzschild metric and Hawking temperature

The Schwarzschild metric reduces to the Minkowski metric in the limit of vanishing $M$, but the Hawking temperature which is proportional to $1/M$ diverges in the same limit. This would imply that flat spacetime has infinite rather than zero temperature. What am I missing? EDIT: This question is back up on the front page because I've posted an answer of my own \- let me know if I'm thinking about this right.


Interpreting the results

I have preformed the muon lifetime experiment at my uni's lab, and got the data. It's text file with 8190 numbers. My TDC unit was set so that the time gates were at 10 $\mu s$, and it has 8192 channels (first and last contained some kind of noise so they were not included in data analysis). Now, I made a program in python that will sum every n data, and make a new list that I can then plot in bar plots and from that calculate the mean muon lifetime, because it decays by exponential law. I also found that I can estimate the muon lifetime by using this formula: $$<t>=\frac{\sum\limits_{i=n_l}^{n_u}N_i t_i}{\sum\limits_{i=n_l}^{n_u}N_i},$$ where $N_i$ is the number of counts in the bin, $t_i$ is the 'time bin', and the bins go from $n_l$ (lower) to the $n_u$ (upper) bin. Using that as a guide, I found that for my data I get the best estimate if I set the bin count at 7 (7 bins), and if I put 10, I get higher result. Now if I set the bin number to 20, I get even higher number. But that sounds wrong. I mean, there must be some kind of catch. If I divide the time I get for 20 bins by two I get the correct result (~2.5 $\mu s$). The pictures are like this: ![enter image description here](http://i.stack.imgur.com/hZ1Z4.png) ![enter image description here](http://i.stack.imgur.com/QgNQJ.png) ![enter image description here](http://i.stack.imgur.com/eS8a0.png) So what am I doing wrong with interpreting this?


Is building your own reflector telescope worth it?

I have always fancied building my own reflector telescope. I am wondering - For a certain budget, can you get better results with a home made rig? Or is this a folly, and really it would be better to buy a ready made unit?


What is the fundamental reason for noise?

I have read that noise is a result of there being "no such thing as a perfect one-way valve". That energy transfer is never perfectly one-way; there will always (at some level) a finite flow of energy from output to input. I understand this much, but what is the fundamental reason for this? Does it have its roots in thermodynamics? Edit: The idea of a "one-way valve" is merely a construction in the mind, it doesn't refer to an actual valve, like in a car engine or something. It's just a general way of describing 'what connects two systems'. Additionally, although I am looking for the _fundamental_ reason for noise, it may be easier to let people know that I am referring specifically to _measurement noise_.


Newton's cradle

Why, when one releases 2 balls in Newton's cradle, two balls on the opposite side bounce out at approximately the same speed as the 1st pair, rather than one ball at higher speed, or 3 balls at lower speed?


Information of things inside a black hole

**Can we get the information of things that are gone in black hole?**


Boundary conditions in AdS/CFT

This question is in reference to this very famous paper of Witten.   * In general through the whole paper why is the author able to just focus on the scalar field propagating in the bulk and not need to take into account all the other fields and the complicated Lagrangian in the bulk (a Type IIB superstrings?)    * To construct the example in equation 4.1 (middle of page 6) why did the author choose half-BPS operators and is there a simple way to see that an example of ${\cal O}$ written down is a half-BPS operator? (..what are other such?..is there a classification?..)    * How generic is the argument in equation 4.8 (top of page 8) to get the RG flow equation? Or is this a special case which works here for some special reason?  With a change in the mass/renormalization scale/cut-off one usually asks for the connected n-point functions or the effective potential to be invariant - but here the author seems to want to have the scalar field's boundary asymptotics to be invariant - I found this renormalization condition very new and mysterious.   * I guess the most exciting analysis in this paper is the argument in the first paragraph on the top of page 9. Can someone help understand that?      * To start off how one know that the operators ${\cal O}_1$ and ${\cal O}_2'$ related to the boundary values of the two scalar fields are actually (super?)conformal primaries of the boundary (S?)CFT?      * I did not understand how one sees that the deformation as stated in equation 4.12 (and the line before it) preserves quantum conformal invariance.      * and the main point about the structure of equation 4.12 and the conformal invariance of the boundary being maintainable for $f \neq 0$..


Bicycle Wheel Drag in Slipstream

I was recently driving behind a car that had a bicycle mounted on a carrier over the rear bumper. The bicycle wheels were not bound so they were rotating in the slip-stream of the car. I wonder, the fact that the wheels are turning; does this theoretically increase or decrease the drag on the car?


Motivation behind studying the asymptotic structures

I am trying to explain to myself the motivation behind studying the asymptotic structures at null, time-like and space-like infinities (For the purposes of this post, I will stick to four dimensional Minkowski spacetime). I think I have the motivation behind the first two of them down, but not the third.   1. Null Infinity is interesting since radiation and fluxes of massless particles can reach it. One must understand these in order to answer questions such as energy loss due to gravitational radiation, or charge fluxes. Further, information can travel from the bulk to null infinity only, and is therefore an important thing to study in that context as well. Geroch's paper notes that > At null infinity, one's description involves "what the system is doing > through time", i.e. the dynamics of the system. I agree with this.   1. Time-like infinity corresponds to the starting and ending points of massive particles travelling along geodesics. I would presume that these would be of importance when describing "In" and "Out" states of a quantum theory.   2. Space-like infinity is a little harder for me to motivate. In Geroch's paper, he notes > At spatial infinity, "what the system does" is never even recorded. Rather, > one sees only the state of the system", once and for all. Here are my questions: >   1. Are the motivations I provided above for null and timelike infinity > correct? Is there anything more to add? > >   2. What does Geroch's comment mean? Surely, in general, a system will > evolve in time. What then describes the "state of a system"?? Is it the > initial conditions? What, precisely is the information about the system that > is stored at space-like infinity? > > I have often seen null and spacelike infinities being discussed thoroughly without much literature on time-like infinity. While the extensive study of null infinities does not surprise me, the study of spacelike infinity does. I therefore feel like I am missing something quite important here. EDIT: I just had a thought. It seems to me that both null and spatial infinity can be reached in a finite value of coordinate time (appropriately defined). On the other hand, time-like infinity can only be reached asymptotically. This implies that study of time-like infinity requires taking two limits $r \to \infty$ and $t \to \pm \infty$, whereas, study of null and spacelike infinity involve single limits like $r \to \infty,~v$ fixed and $r \to \infty,~t$ fixed respectively. Maybe this has something to do with my answer, but I'm not quite sure how.


Understanding boundary conditions on slices of AdS5

This is a thing Iïve seen on many papers dealing with Warped Extra Dimensions, specifically on slices of AdS5. But the one where it appears more clearly is a lecture by Tony Gherghetta: http://arxiv.org/abs/hep-ph/0601213 Essentially what is done is that one builds a 5-dimensional theory in a slice of a Anti-De-Sitter space, the slice meaning that the fifth dimension has a small size and ends with a 4D brane at each extreme. The principle of least action applied to the 5D action leads us to two terms (eq. 7 in the paper above), one on the bulk (the vanishing of this one leads us to the bulk equation of motion) and one on the branes. The vanishing of this second term can be accomplished in two ways: 1) the variation of the field on the branes is zero 2) the term multiplying the variation is zero on the branes (let?s call it B) Now, all authors affirm that, in absence of extra terms on the brane, this two conditions lead to Neumann or Dirichlet conditions for the field on the branes (in the paper above this is said just after equation 10). My question is: why is that? For scalar fields it can be shown that B equals the derivative of the field, so I can see the Neumann condition there (am I wrong?). But the 1st condition says the variation of the field is zero not the field itself. Clearly I am misunderstanding something here...


Why is charge not taken as a fundamental unit?

According to the definition of electric current, it appears to be a derived quantity. Charge on the other hand seems more fundamental than electric current. Then why is current taken as fundamental quantity instead of charge? Is it arbitrary choice? Is it because we can measure current more efficiently than charge or some other reason?


Is electron volt an alternate unit for electric potential?

My question is: Can an electron volt be considered an alternate unit for electric potential?


Function of air conditioner

What i always thought that air conditioners blow the cool air without knowing that they actually take the warm air from indoor and from outdoor. But whats the point of taking air from indoor and outdoor, Shouldn't they just be blowing the cool air inside and out air outside? How does it really make the difference? Also, Does all Air conditioners function the same way as in shown in this figure, even split A.Cs? ![enter link description here](http://i.stack.imgur.com/iXrvX.png) I've run out of **tags** for questions :\


Fluid Mechanics from a variational principle

It is posible to define a good variational principle to describe Fluid Mechanics? if so, wath is the correct tratement of the issue. I guess something like: $I=\int d^4x (\frac{1}{2}\rho v^2-P-\rho g x)$


How and why will the Milky way collide with the Andromeda?

Hubble's law says that the universe is expanding.How come the milky way and the andromeda are on a collision course?How will they end up colliding with each other?


How does the Milkovic Two-Stage Mechanical Oscillator Pendulum-Lever System work?

See http://peswiki.com/index.php/Directory:Milkovic_Two- Stage_Mechanical_Oscillator The Two-Stage Mechanical Oscillator Pendulum-Lever System is very simple, yet very puzzling because it appears that more energy is going out (since it is able to pump water) than is put in (only lift one lift of the pendulum and an occasional push). Where is the apparent excess energy coming from? The reason I am asking this question is because all of the explanations I have seen claim that somehow Newton's Laws are violated, which I cannot believe.


Microphones, Loudspeaker and their analogies to spring mass system

I have just started studying Microphones and Loudspeakers. I need a good text to refer which can explain their mechanical analogies with simplicity and basics too.


Why do we think of light as a wave?

I've read that light travels in a straight line and has a wavelength of 400nm to 700nm. But I don't understand why does it have a wavelength and what creates its wavelength? I agree with the concept of sound which also has wavelength, thus called sound waves which are created by the vibrational movement in air. I'm not aware of calling light a wave. Does the light vibrate too? If so, then how?


Nuclear fission difference in energy calculation?

If you have a uranium atom of radius $10^{-14}m$ that undergoes fission into two fragments each with 46 protons and radius $8\times10^{-15}m$. There are two ways you can calculate the electrostatic energy realised (according to my textbook):   1. Find the electric potential energy of the two fragments as if they where point charges at a distance of $2\times8\times10^{-15}m$ apart (i.e. find the potential energy one of these charges would have in the field of the other). This gives an answer of around 180 MeV.   2. Find the electrostatic energy of the original nucleus by imagining the nucleus was built layer by layer each of thickness $dr$ and then integrating over it to find the energy of the whole nucleus. If this is then repeated for the two fragments and the sum of the electrostatic energies of the fragments is taken from that of the original nucleus that is the electrostatic energy released. But this gives an energy of around 270MeV. Here is my question, why is there a difference between these two values (the 180MeV and the 270MeV)?


Basics of osmosis. What about excluded volume?

I may not understand osmosis very well. Let us suppose two compartments filled with water, separated by a semi-permeable membrane. At equilibrium, both levels are equals. Let us introduce now a given volume of solute in one of the compartments (say right). At first, the level of the right compartment will increase, to accommodate the extra volume of solute. Because of this, the concentration of water in the right compartment has decreased, and is no more at equilibrium with the left compartment. Thus a net flux of water from left to right occurs until equilibrium is reached. That's what I understand from what I read so far, but I have a problem with this. In this particular example, it is true to say that the water concentration on the right side decreased when the solute is introduced, but because of the excluded volume (from the solute), the water pressure should remain the same, and thus I would not expect a net flux. Unless there is other effect I don't consider? Thank you for your explanation.


Does the speed of medium affect the path of light?

Let's say I shine a laser from a stationary medium into a moving medium (suppose the water is moving very quickly) perpendicular to the interface and back to a stationary medium like this: ![Scenarios](http://i.stack.imgur.com/SiL6P.png) (Note: left and right sides of the image are stationary mediums, center is a medium moving in the direction indicated by the arrow) Which of the above scenarios (A, B, C, or "I'm way off") correctly reflects the path the light will take (even if the translation is incredibly small)? **Edit:** To answer some good questions (and things I left out of the original question):   * The center (moving) medium is water   * The left and right medium (stationary) is air   * The first angle of incidence (on the left) between the air and the water is perpendicular


When one monkey accelerates up a rope, what happens to the other?

The question is: Two equal weight monkeys each hang from the ends of a rope passing over a weightless, frictionless pulley. If one accelerates up the rope, what happens to the other? I'm going to assume here that if one monkey climbs, the other monkey goes up (I tried to imagine how it would look in real life). But in physical terms, why would this be? Because of the additional force exerted by the climbing monkey?


Good book on deriving approximate solutions from first principles?

I have always been excited by examples in which a few simple assumptions and first principles are used to characterize a system. For example, I did an exercise in which Crawford estimates a lake to be rectangular and then predicts the frequency of its waves through the harmonic motion of its center of mass. I'd love to come up with questions and simple answers about natural phenomena, but I am not very good at coming up with either the questions or the answers, and I'd like to see lots of examples. What are some of the books you like on the subject?


Will a precessing spinning wheel fall down if there is no friction at all?

If there where no friction at all, would a spinning wheel held up by one end of the axis spin precess forever without falling down? ![Spinning wheel](http://i.stack.imgur.com/S3f2n.gif) I just asked another question about the same problem: Direction of torque precession of a spinning wheel Since it seems to be a good practice on stackexchange not to ask several questions in one post, I splitted them up into two questions. However if I am wrong, feel free to merge this questions.


Time of a ball going up and down with air resistance

A ball is shot directly upward, and then it comes back to the place where it was shot. Suppose we have air resistance. Suppose $t_1$ is the time period from the moment that the ball was shot to the moment that it reached its highest altitude, and $t_2$ is the time period from the moment it reached its highest altitude to the moment it reached its original position. Is $t_1=t_2$? Why or why not?


Why is there no oxidizer in space?

Just as a thought experiment. One factor in the economies of space exploration is that of fuel. This may be split as   * MSL - Earth Orbit    * Earth Orbit to Inner Solar System    * Beyond. In each of these cases, the rocket demands an oxidizer to be loaded at the point of launch as neither oxygen, nor oxidizers exist in free space. Why does space not have free oxygen/oxidizers? p.s. According to wikipedia Oxygen is the third most abundant element in the known Universe ...


What is a resonating valence bond (RVB) state?

There's something known as a "resonating valence bond" (RVB) state, which plays a role in at least some attempts to understand physics of high-$T_c$ superconductors. This, roughly, involves a state that's in a superposition (hence the "resonating" part of the name, if I understand correctly) of different ways to pair electrons into strongly-bonded spin singlets. My question is: what is a more precise definition of this sort of state? What's the underlying physics, when does it arise, and why is it interesting? Points an answer might address: is there a simple toy model for which this is the ground state, that sheds light on what sort of system it could arise in? Is there an interesting continuum limit, in which we can characterize this state in a more field-theoretic language? Are there particular kinds of instabilities such a state tends to be subject to? I think I know where I would start digging if I wanted to really understand this for myself, but mostly I'm asking it to probe the community and see what kind of expertise might be lurking here, since there haven't been so many condensed matter questions.


A question to clarify the use of divergent series in calculating the casimir effect

Some time ago I posted a question here on this forum. I would like to ask some questions regarding the way the energy per unit area between metallic plates is calculated. The full calculation is on wikipedia. At some point in the calculation on the relevant wikipedia page (see the link above), we have the equation: $$\frac{ \langle E \rangle }{ A} = - \frac{ \hbar c \pi^2 }{6a^3}\cdot\zeta(-3) . $$ In the next step, it is written rather casually that $\zeta(-3) = - \frac{1}{120} \qquad (*) $. This is true when considering the analytic continuation of the riemann zeta function or the Ramanujan Summation method. Therefore, it is concluded, that $$\frac{ \langle E \rangle }{A} = - \frac{ \hbar c \pi^2}{720 a^3} . $$ I am wondering under which circumstances people decided to assume the $(*)$-marked equation is 'true'. I can think of a couple of scenarios:   1. The formula for $\frac{ \langle E \rangle }{A} $ was already derived by means of another method which did not require the use of (regularised) divergent sums. Therefore, physicists could infer that $\zeta(-3)$ _had_ to be equal to $ - \frac{1}{120} $, making the derivation of the formula by means of this method, which does use divergent series, correct.    2. The exact formula for $\frac{ \langle E \rangle }{A} $ was not already known. Physicist did have some data points that roughly showed them how the formula should look. Therefore, they tried some different constants for $\zeta(-3)$. At some point they guessed $\zeta(-3) = - \frac{1}{120} $, which yielded a formula that coincided with the known data points. They might have already known that $\zeta(-3) = - \frac{1}{120} $ by means of zeta function regularisation, making it easier to use this equation as a "guess" to find a suitable formula for $\frac{ \langle E \rangle }{A} $ .    3. Some other scenario.  Which scenario roughly describes how the formula for $\frac{ \langle E \rangle }{A} $ came into existence? If it was scenario 1, which other method did physicists formerly employ to derive the formula? If it was scenario 3, how _did_ this whole process unfold? Thanks a lot, Max


Work done or not in this case?

I have a very simple question. A motorboat directed upstream is seen to be at rest from the bank of a river. Is the engine doing any work? Is it right to say that since it is not causing any displacement, so it is not doing any work.


The Ultimate Hand Dryer

I have come across many hand dryers that attempt to dry your hands really fast after you wash them. Here are two of them: XLERATOR http://www.exceldryer.com/ Dyson Airblade http://www.dysonairblade.com/homepage.asp So I guess I have a ridiculously high standard cause I think even these are too slow. Would it be possible to create a large static electric field to attract the water molecules off of your hand? Can someone offer some ideas that would remove the water off of your hand using some type of electric field generated by charged plates or something?


Dropping an anchor from a boat

> A yacht on a lake drops its anchor overboard. What happens to the water > level in the lake? > >   1. It rises very slightly. >   2. It falls very slightly. >   3. It stays exactly the same. >   4. It's impossible to say. > My understanding is that due to Archimedes principle, when the anchor is in the boat, it contributes to the mass of the boat, and thus the mass of water displaced. When it is thrown overboard, it is now the volume of the anchor which contributes to the amount of water displaced. Therefore, without knowing the density, mass and volume of the anchor, it is impossible to determine the effect on the water level, therefore the correct answer is option 4. Can anyone tell me if my reasoning is correct as there is no mark scheme for the test where this question came from. Thanks in advance!


What really is the smallest "mass" or "object" in the universe?

Look at this here. With respect to the sciences, the atom is obviously not the smallest piece of mass. Apparently, if people have already broken down the atom in to particles smaller than so, why haven't particles been understood yet? Old scholars reasoned that everything has smaller parts, so what's smaller than subatomic particles? Or is there a limit in the size of mass, only being able to be small to an extent? Because mass always seems to keep growing, but when scaling opposite in size we reach a limit. Therefore, big always gets bigger, but why does small have limits?


explicit matrix elements for a representation decomposed into subgroup by branching rules

I'm looking for a way to construct a representation for a simple Lie group such that one particular subgroup is manifest. I learned the branching rules from Cahn, Georgi and Slansky, but I'm still not sure how to derive the explicit representation from the weight system deformed by extended Dynkin diagram? To decompose a irreducible representation under a maximal regular subalgebra, we look at the extended Dynkin diagram, which has one more node from the minus highest root added on the original Dynkin diagram, and we then eliminate another node to deform the extended Dynkin diagram into the Dynkin diagram for the maximal regular subalgebra. The key point is to replace the Dynkin coefficient corresponding to the eliminated node by the one corresponding to the extra node of the minus highest root. Thus, deforming the weight system of the given irrep with above replacement gives us the branching rules and the Cartan matrices for the subalgebra in the given irrep. For example, under the subgroup $SU(2)\times SU(2)\times SU(2) \Subset SO(7)$, we have the 8 dimensional irrep of SO(7) deformed as $8 \rightarrow (1,2,2) + (2,1,2)$, see P33-34 LieART. To construct the explicit matrix representation, we define the coordinates for each simple roots in an orthonormal basis (Cartan-Weyl basis) and therefore the coordinates for the fundamental weights as well. Thus, the generators corresponding to Cartan matrices have its matrix entries as the coordinate of every weights. Now, my question is what are the fundamental weights for the deformed Dynkin diagram? Should I keep the old fundamental weights except the replaced one or the new fundamental weights are resolved from the deformed Dynkin diagram? Most importantly, what's the new fundamental weight corresponding to the extended root? I would also need help to reveal how generators of the subalgebra are in the subset of generators of the mother algebra. When I replace the Dynkin coefficient with the extended root, how could the corresponding Cartan matrix belong to linear combination of Cartan matrices of the mother algebra? I'm a Physics student with poor knowledge in Lie algebra. Please reinterpret my description with proper language whenever you needed to.


How we can manipulate the momentum of a particle?

Is there any way to affect a particle's momentum value?


Eigenvalue of the adiabatic Hamiltonian of Farhi's three qubit 2-SAT problem

I was trying to reproduce example 3.3 of Quantum Computation by Adiabatic Evolution by Edward Farhi et. al. This is an adiabatic algorithm to solve an instance of three qubits 2-SAT problem. I think I have created the initial Hamiltonian, $H_B$ correctly. $$H_B = \left(\frac{1}{2}\left(I_2 - \sigma_x \right)\right)\otimes I_2\otimes I_2+I_2\otimes \left(\frac{1}{2}\left(I_2 - \sigma_x \right)\right)\otimes I_2+\left(\frac{1}{2}\left(I_2 - \sigma_x \right)\right)\otimes I_2\otimes I_2+I_2\otimes I_2\otimes \left(\frac{1}{2}\left(I_2 - \sigma_x \right)\right)+I_2\otimes \left(\frac{1}{2}\left(I_2 - \sigma_x \right)\right)\otimes I_2+I_2\otimes I_2\otimes \left(\frac{1}{2}\left(I_2 - \sigma_x \right)\right)$$ When I evaluate it, the matrx is: $$H_B=\left( \begin{array}{cccccccc} 3 & -1 & -1 & 0 & -1 & 0 & 0 & 0 \\\ -1 & 3 & 0 & -1 & 0 & -1 & 0 & 0 \\\ -1 & 0 & 3 & -1 & 0 & 0 & -1 & 0 \\\ 0 & -1 & -1 & 3 & 0 & 0 & 0 & -1 \\\ -1 & 0 & 0 & 0 & 3 & -1 & -1 & 0 \\\ 0 & -1 & 0 & 0 & -1 & 3 & 0 & -1 \\\ 0 & 0 & -1 & 0 & -1 & 0 & 3 & -1 \\\ 0 & 0 & 0 & -1 & 0 & -1 & -1 & 3 \\\ \end{array} \right)$$ According to the example, the unique satisfying assignment is $011$. The problem Hamiltonian is a sum of three sub-Hamiltonians each correspond to a clause. $$H_P = H^{12}_{imply} + H^{13}_{disagree}+H^{23}_{agree}$$ Here are my results for sub-Hamiltonians. $H^{12}_{imply}$ clause can be satisfied with any of $00$, $01$ or $11$. So, $$H^{12}_{imply} = I_8 - \left(\left(\frac{1}{\sqrt{6}}\right)\left(|000\rangle+|001\rangle+|010\rangle+|011\rangle+|110\rangle+|111\rangle\right)\right) \left(\left(\left(\frac{1}{\sqrt{6}}\right)\left(|000\rangle+|001\rangle+|010\rangle+|011\rangle+|110\rangle+|111\rangle\right)\right)\right)^{\dagger} $$ The ground state of this Hamiltonian satisfies the assignment constraint. $H^{13}_{disagree}$ clause can be satisfied with any of $01$ or $10$. So, $$H^{13}_{disagree} = I_8 -\left(\left(\frac{1}{2}\right)\left(|001\rangle+|011\rangle+|100\rangle+|110\rangle\right)\right) \left(\left(\left(\frac{1}{2}\right)\left(|001\rangle+|011\rangle+|100\rangle+|110\rangle\right)\right)\right)^{\dagger}$$ The ground state of this Hamiltonian satisfies the assignment constraint. $H^{23}_{agree}$ clause can be satisfied with any of $00$ or $11$. So, $$H^{23}_{agree} = I_8 - \left(\left(\frac{1}{2}\right)\left(|001\rangle+|011\rangle+|100\rangle+|110\rangle\right)\right) \left(\left(\left(\frac{1}{2}\right)\left(|001\rangle+|011\rangle+|100\rangle+|110\rangle\right)\right)\right)^{\dagger}$$ The ground state of this Hamiltonian satisfies the assignment constraint. So, $$H_P = \left( \begin{array}{cccccccc} \frac{17}{6} & -\frac{1}{6} & -\frac{1}{6} & -\frac{1}{6} & 0 & 0 & -\frac{1}{6} & -\frac{1}{6} \\\ -\frac{1}{6} & \frac{7}{3} & -\frac{1}{6} & -\frac{2}{3} & -\frac{1}{2} & 0 & -\frac{2}{3} & -\frac{1}{6} \\\ -\frac{1}{6} & -\frac{1}{6} & \frac{17}{6} & -\frac{1}{6} & 0 & 0 & -\frac{1}{6} & -\frac{1}{6} \\\ -\frac{1}{6} & -\frac{2}{3} & -\frac{1}{6} & \frac{7}{3} & -\frac{1}{2} & 0 & -\frac{2}{3} & -\frac{1}{6} \\\ 0 & -\frac{1}{2} & 0 & -\frac{1}{2} & \frac{5}{2} & 0 & -\frac{1}{2} & 0 \\\ 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0 \\\ -\frac{1}{6} & -\frac{2}{3} & -\frac{1}{6} & -\frac{2}{3} & -\frac{1}{2} & 0 & \frac{7}{3} & -\frac{1}{6} \\\ -\frac{1}{6} & -\frac{1}{6} & -\frac{1}{6} & -\frac{1}{6} & 0 & 0 & -\frac{1}{6} & \frac{17}{6} \\\ \end{array} \right)$$ This Hamiltonian, which represents the instance of the problem is expected to be satisfied only with the assignment 011. So, the ground state should be $|011\rangle$. But in reality, the ground eigenvalue is greater than $0$ and the ground state is $\left(1,4,1,4,3,0,4,1\right)^{\dagger}$. Here, $I_n$ is the $n \times n$ identity matrix. **What was I doing wrong?** My Mathematica code is available here.


Energy in an electromagnetic wave

A radio antenna creates EM waves through switching the polarization in the antenna at a certain frequency. I assume the the energy of the photons produced in this process amount to E=hf for each photon. So good so far, but classically, i read that the EM field is an oscillation of the electric field between positive and negative values, as would seem reasonable from what the antenna is doing. If individual photons simply have a single positive energy relating to the frequency of the wave, how is this positive-negative oscillation represented in that stream of photons? I guess to put it simply, how is a changing electric field represented in a stream of photons in general, and from a radio antenna specifically?


doppler shift through two mediums

When considering the Doppler shift, the 'canonical equation' is $$f=\frac{c+vr}{c+vs}f_0$$ However, this equation seems to run into trouble in the following situation: A light source inside water is moving at a speed $v$ towards a receiver outside the water. Can we modify the above equation to deal with a transition between two media? Or is there a completely different formula that applies here?


A Different Lasing Medium

Powerful lasers are highly intense, diverge negligibly and are also coherent. These radiations are emitted through partially reflecting mirrors after simultaneous reflections within the lasing medium. Don't these EM radiations affect the lasing medium or reflecting mirrors? Also, How is plasma used as a lasing medium in certain lasers?


Maximum Possible Information in the universe?

I remember hearing about this in one of the programs in discovery science. The physicist claimed that the maximum possible information in the universe is (10)^(10^123) whereas the maximum possible information that can be known by man is (10)^(10^90). Can anyone explain to me how can we arrive at such a specific number, and also how can information be represented by only numbers?


What is the advantage of segmented particle traps?

I'm currently trying to familiarize myself with the physics of ion and particle traps, especially with linear Paul traps. Many scientific experiments I've come across use segmented electrodes (like in this image from a Nature article). My question is _what do you gain by segmenting the electrodes?_ I could imagine that you can move the trapped particles along the axis to some extent, or determine their position in the trap. But in both cases I believe there are more precise methods.


Compactification and off-diagonal terms of the metric tensor

In standard 3+1 dimensional spacetime, the metric tensor is of order 4 and had ten independent coefficients, hence there are 6 terms off the diagonal in the corresponding $4\times 4$ real symmetric matrix. On the other hand, superstring theory postulates that there are 6 additional spatial dimensions that have to be compactified in a Calabi-yau manifold. My question is thus: are these two facts somehow related or is it just mere coincidence? For example, would the observable dimensions correspond to eigenvalues of the metric tensor?


Picture of supports

This questions stems from Axiomatic Quantum Field Theory and is mathematical in nature. However, I feel that an answer from physicists is more in line with what I will be asking. Let $\phi$ be a real quantum field, namely $\phi$ is an operator-valued distribution. One of the requirements of $\phi$ is that it is _local_. In case $f\in C{^\infty _0 }_{real}$, then the assumption of locality requires that $$\phi(f)\phi(g)=\phi(g)\phi(f)$$ when the supports of f and g cannot be connected by a light ray. One says that such supports are _space-like separated_. **Reference** : Page 7 http://www.arthurjaffe.com/Assets/pdf/Quantum- Theory_Relativity.pdf I'd like to gather further insight into this statement. Namely, how does one picture the supports? Should I have light cones in mind?


Does a magnet contain (and potentially produce) energy?

Very quick question, does a magnet contain energy? The general consensus seems to be, it does not. And this is generally confirmed by the fact that it would break the first law of thermodynamics. Whatever the hell that is (joke:) The reason I ask is because a) I'm no genius and b) because I'm perplexed. So maybe some of you smart people could help me out please. Here's the scenario; Now, if I took, oh I dunno, say a metal ball and lifted it say six inches. I have converted some of my man boob calories into energy that is now stored in the ball. When I release it and it drops to original level, the energy is released. Makes sense. Now if I took the same metal ball and rolled it along the ground, it would continue to roll until the kinetic energy was depleted, through friction and stuff like that. Now if I take the same ball and roll it along the ground, but this time with a magnet suspended 6 inches from the ground and directly in the line of movement. The magnet is strong enough to attract the ball and is therefore lifted 6 inches and sticks to the magnet. Where has that energy come from? It can't have come from me putting the magnet there, as once I put the magnet back on the ground, I have released that energy. As I said, I'm not smart, nor educated, just been pondering this question for a couple of days. Would be great if you could allow my brain to get back to menial tasks. Thanks


Help with the understanding of boundary conditions on $AdS_3$

So I am trying to reproduce results in this article, precisely the 3rd chapter 'Virasoro algebra for AdS$_3$'. I have the metric in this form: $$ds^2=-\left(1+\frac{r^2}{l^2}\right)dt^2+\left(1+\frac{r^2}{l^2}\right)^{-1}dr^2+r^2d\phi^2$$ And I have the boundary conditions. So if I'm correct, I should find the most general diffeomorphism, by solving $\mathcal{L}_\xi g_{\mu\nu}=\mathcal{O}(h_{\mu\nu})$, where $h_{\mu\nu}$ are the boundary conditions (subleading terms). So, if I'm doing things right, I get 5 equations. Because the $t\phi$ term of Lie derivative vanishes. Now, I should use the power expansion of $\xi$, as given in the paper, and solve these 5 differential equations or? I'm not certain if I'm on a right path, so any advice is welcome...


Entropy inequality

Assume that you have two bipartite systems $\rho_1^{AB},\rho_2^{AB}$ then I would like to prove the following: $$S(\frac{1}{2}( \rho_1^{AB}+I^A\otimes\rho_2^B))+S(\frac{1}{2}(\rho_2^{AB}+I^A\otimes\rho_1^B)) \geq S(\frac{1}{2}(\rho_1^{AB}+I^A\otimes\rho_1^B))+S(\frac{1}{2}(\rho_2^{AB}+I^A\otimes\rho_2^B))$$ where $S$ is the von Neumann entropy, $\rho_1^B=tr_A(\rho_1^{AB}),\rho_2^B=tr_A(\rho_2^{AB})$ and $I^A$ is the maximally mixed state on $A$. It looks like it should pass with some monotony property, any hints or counterexample are welcome.


What is Quantization?

In classical mechanics you construct an action (involving a Lagrangian in arbitrary generalized coordinates, a Hamiltonian in canonical coordinates [to make your EOM more "convenient & symmetric"]), then extremizing it gives the equations of motion. Alternatively one can find a first order PDE for the action as a function of it's endpoints to obtain the Hamilton-Jacobi equation, & the Poisson bracket formulation is merely a means of changing variables in your PDE so as to ensure your new variables are still characteristics of the H-J PDE (i.e. solutions of the EOM - see No. 37). All that makes sense to me, we're extremizing a functional to get the EOM or solving a PDE which implicitly assumes we've already got the solution (path of the particle) inside of the action that leads to the PDE. However in quantum mechanics, at least in the canonical quantization I think, you apparently just take the Hamiltonian (the Lagrangian in canonical coordinates) & mish-mash this with ideas from changing variables in the Hamilton-Jacobi equation representation of your problem so that you ensure the coordinates are characteristics of your Hamilton-Jacobi equation (i.e. the solutions of the EOM), then you put these ideas in some new space for some reason (Hilbert space) & have a theory of QM. Based on what I've written you are literally doing the exact same thing you do in classical mechanics in the beginning, you're sneaking in classical ideas & for some reason you make things into an algebra - I don't see why this is necessary, or why you can't do exactly what you do in classical mechanics??? Furthermore I think my questions have some merit when you note that Schrodinger's original derivation involved an action functional using the Hamilton-Jacobi equation. Again we see Schrodinger doing a similar thing to the modern idea's, here he's mish-mashing the Hamilton-Jacobi equation with extremizing an action functional instead of just extremizing the original Lagrangian or Hamiltonian, analogous to modern QM mish-mashing the Hamiltonian with changes of variables in the H-J PDE (via Poisson brackets). What's going on in this big Jigsaw? Why do we need to start mixing up all our pieces, why can't we just copy classical mechanics exactly - we are on some level anyway, as far as I can see... I can understand doing these things if they are just convenient tricks, the way you could say that invoking the H-J PDE is just a trick for dealing with Lagrangians & Hamiltonians, but I'm pretty sure the claim is that the process of quantization simply must be done, one step is just absolutely necessary, you simply cannot follow the classical ideas, even though from what I've said we basically are just doing the classical thing - in a roundabout way. It probably has something to do with complex numbers, at least partially, as mentioned in the note on page 276 here, but I have no idea as to how to see that & Schrodinger's original derivation didn't assume them so I'm confused about this. To make my questions about quantization explicit if they aren't apparent from what I've written above: **a)** Why does one need to make an algebra out of mixing the Hamiltonian with Poisson brackets? (Where this question stresses the interpretation of Hamiltonian's as Lagrangian's just with different coordinates, & Poisson brackets as conditions on changing variables in the Hamilton-Jacobi equation, so that we make the relationship to CM explicit) **b)** Why can't quantum mechanics just be modelled by extremizing a Lagrangian, or solving a H-J PDE? (From my explanation above it seems quantization smuggles these idea's into it's formalism anyway, just mish-mashing them together in some vector space) **c)** How do complex numbers relate to this process? (Are they the reason quantum mechanics radically differs from classical mechanics. If so, how does this fall out of the procedure as inevitable?) Apologies if these weren't clear from what I've written, but I feel what I've written is absolutely essential to my question. **Edit:** Parts **b)** & **c)** have been nicely answered, thus part **a)** is all that remains, & it's solution seems to lie in this article, which derives the time dependent Schrodinger equation (TDSE) from the TISE. In other words, the TISE is apparently derived from classical mechanical principles, as Schrodinger did it, then at some point in the complicated derivation from page 12 on the authors reach a point at which quantum mechanical assumptions become absolutely necessary, & apparently this is the reason one assumes tons of axioms & feels comfortable constructing Hilbert spaces etc... Thus elucidating how this derivation incontravertibly results in quantum mechanical assumptions should justify why quantization is necessary, but I cannot figure this out from my poorly-understood reading of the derivation. Understanding this is the key to QM apparently, unless I'm mistaken (highly probable) thus if anyone can provide an answer in light of this articles contents that would be fantastic, thank you!


Under which representation of U(1) transform electron and photon gauge field?

I know that under $SU(2) \times SU(2)$, the left-handed electron transforms under $ ( \frac{1}{2},0 ) $ representation and the vector gauge field $A_\mu$ under $ ( \frac{1}{2},\frac{1}{2}) $. Since the electron transforms under $U(1)$, there must be a represenation under which it transforms. What is this representation? Does it have a name? Apparenly $A_\mu$ does not transform under the same representation, which would mean $e^{\alpha(x) Q} A_\mu$, but instead as $A_\mu + i \partial_\mu \alpha(x)$ ? What representation is this? Of course I realize that the transformation of $A_\mu$ can't be different for the Lagrangian to be invariant, but that shouldn't be used to define the it.


Any suggestions for units conversion?

> **Possible Duplicate:**   >  Photometer: measured Irradiance L converted to photon rate I am conducting a experiment where stimulus output of $470\ nm$ is measured by a radiometer at $30\ \mu W\ cm^{-2}$. The stimulus is $1$ inch from the detector. Any suggestion on how I might go about converting $\mu W\ cm^{-2}$ to log photon $cm^{-2} s^{-1}$?


What does the * mean in spherical harmonics?

In Jackson's book about classical electrodynamics, this formula comes up: $$q_{lm} = \int \mathrm d^3 x' \, Y^*_{lm}\left(\theta', \phi'\right) r'^l \rho\left(\vec x'\right)$$ What does that $^*$ mean?


Symmetry and overlapping of ground states

In a quantum mechanics, there is the following formula to derive the zero energy $E_0$ of a perturbed Hamiltonian $$H = H_0 + V$$ knowing the zero energy $W_0$ of the free Hamiltonian $H_0$: $$E_0 = W_0 + i\frac{d}{dt}\text{ln}R(t)|_{t\rightarrow\infty(1-i\eta)}$$ The exponential killing the excited states faster than the lowest energy one. However, one needs to suppose a non-vanishing overlap of the two ground states $|\phi_0\rangle$ for $H_0$ and $|\psi_0\rangle$ for $H$. I read that if two have different symmetry they must be orthogonal but I didn't manage to derive why. Let's suppose that $G$ is a symmetry of $|\phi_0\rangle$ then $G|\phi_0\rangle=0$ and $\langle\psi_0|G|\phi_0\rangle=0$ and if $|\psi_0\rangle$ is not invariant under $G$ I still need to have $$G|\psi_0\rangle \propto |\psi_0\rangle$$ to derive $\langle\psi_0|\phi_0\rangle=0$. ($|\psi_0\rangle$ needs to be an eigenvector of $G$ with non-vanishing eigenvalue)


Electrical flow in a simple parallel circuit

I'm having trouble understanding something in one of my text books: > Let’s have a look at the implications of each circuit configuration. Figure > 3.13 shows the Conventional representation of a parallel circuit. If you > assume that the resistance of the wires can be neglected, then the voltage > drop across each bulb is equal to the e.m.f. of the source, the dynamo. The > Current flowing from the source is divided between each bulb depending on > its resistance (remember I = E/R), Removing one of the bulbs would not > affect the voltage drop across the other bulb and would therefore not affect > the Current in it, although the overall Current from the dynamo would drop > as the demand has been reduced. Specifically, the line about removing one of the bulbs not affecting the voltage drop. My understanding is that the sum of voltage drops in a circuit must be equal to the output of the emf, but the way I'm reading the text suggests that if you have three bulbs in a parallel circuit and you remove one of them, the voltage drop remains the same - this is what I don't understand, though. If you a lamp is removed, then that's one less lamp consuming emf. Does it not get redistributed to the other two lamps? Likewise, the text mentions the overall current from the dynamo dropping because demand has been reduced. My initial understanding was that current from an emf is only affected by the emf - not the demands of components further along the circuit, as in this case. What have I misunderstood?


Quantum Field Theory Variants

I am a math guy, so sorry for the naivety. When I peruse the wikipedia I see many "variants" of quantum field theory...conformal quantum field theory, topological quantum field theory, axiomatic/constructive quantum field theory, algebraic quantum field theory, etc. Whether or not these are actually variants of something is unclear to me. I don't really have a specific question, but I was wondering if you guys could help me understand what these different things are and/or point me to somewhere to get a clearer picture.


Has anyone theorized a connection between entropy and quantum uncertainty?

I apologize if this kind of idle theorizing is frowned upon here, but I was wondering if it is possible that the Second Law of Thermodynamics is a consequence of quantum uncertainty. I've heard entropy of a system defined as the number of micro-states that it can have to correspond to the macro-states it has. So that definition makes it sound like entropy is simply losing information. As we know, entropy increases as time goes on. Now this seems contradictory to me; we know more as time goes on, not less. Is it possible that, because you can gain more and more information about a system as time goes on as you can interact with it more that, some information needs to be "hidden" from you. And that this process of losing information is entropy? P.S. I know that what I "know" about any system does not approach the limits set out by the uncertainty principle. But as System A interacts with System B, over time System A's state is more influenced by System B and in that sense System A has gained knowledge of System B.


The quantum state can be interpreted statistically, again

Now there are two papers **The quantum state cannot be interpreted statistically** http://arxiv.org/abs/1111.3328 (It was discussed here the consecuences of this "no-go theorem") And this one (two of the authors are the same as the previous paper): **The quantum state can be interpreted statistically** http://arxiv.org/abs/1201.6554 I would like to note this: **titles give only poor information about the content** , and they seem even maliciously chosen, but the mere existence of the two papers is funny anyway.. The question is : **Which is more general!?** From the paper: _"Recently, a no-go theorem was proven [21] showing that a $\psi$-epistemic interpretation is impossible. A key assumption of the argument in [21] is preparation independence situations where quantum theory assigns independent product states are presumed to be completely describable by independently combining the two purportedly deeper descriptions for each system. Here, we will show via explicit constructions that without this assumption, $\psi$-epistemic models can be constructed with all quantum predictions retained"_ **About being general** As I understand the second one just _seems_ more general (because of "less assumptions"), but by no means Newton's dynamics is more general than Einsein's relativity because "it lacks of c=constant assumption". It's weird anyway, because it would mean that if " wavefunction is a real physical object" (a funny phrase from www.nature.com article) would depend on assumptions!, then what kind of realism depend on assumptions? Perhaps a point to discuss (assuming the theorem is well proven) is whether those assumptions have sense, if they come from experiments, or if they are just limiting the scope (toy model), or if those are random assumtions that have no source.


Do all black holes have a singularity?

If a large star goes supernova, but not enough mass collapses to form a black hole, it often forms a neutron star. My understanding is that this is the densest object that can exist because of the Pauli exclusion principle: It's made entirely of degenerate matter, each particle of which cannot occupy the same quantum state of any other. So these objects are so massive that they gravitationally lens light. If you make them more massive, they bend the light more. Keep going and going until they bend the light _so much_ that light passing near the surface can barely escape. It's still a neutron star. Add a bit more mass, just enough that light passing just over the surface cannot escape. Now it's a black hole with an event horizon (I think?). Does this mean the neutron star has become a singularity? Isn't it still just a neutron star just beneath the event horizon? Why are black holes treated as having a singularity instead of just an incredibly massive neutron star at its center? Does something happen when an event horizon is "created?"


E&M and geometry - a historical perspective

Recently, I was contemplating the beautiful formulation of electromagnetism (specifically Maxwell's equations) in terms of differential forms: $$F=\mathrm{d} A\implies \mathrm{d}F=0 \hspace{1cm}\text{and}\hspace{1cm} \mathrm{d}\star\mathrm{d}F=\mu_0 J $$ I started thinking about the history of this way of looking at things, and realized that I don't know much about it at all. My first question was therefore: Was it known already at the time of Maxwell (or soon after) that electromagnetism could be cast in this geometric form? How was this first introduced and who did it? After consulting Maxwell's treatise, it became clear that at least Maxwell himself was not aware of this formulation. But maybe someone else immediately recognized the geometric formulation once Maxwell published his results... In modern times, one is - at least as a physicist - usually first introduced to the field strength tensor $F$ through the covariant formulation of Maxwell's equation using tensor calculus, where it is defined as $F_{\mu\nu}=\partial_\mu A_\nu -\partial_\nu A_\mu$. When one then learns about differential forms etc. it is then obvious that $F=\mathrm{d}A$ and the geometric formulation follows quite naturally. However, was this also the case historically? Did 'they' come up with the tensor calculus formulation of $F$ first, and did they only then recognize the geometric description? Or was the geometric description discovered first? Another possibility is that it took the introduction of Einstein's general relativity for anyone to realize that fields can be interpreted in terms of geometry. In conclusion, I am interested in a chronological description of the development of the different formulations of electromagnetism, with emphasis on the following points:   1. Who first came up with the geometric formulation in terms of differential forms?   2. Is it known at all how this person arrived at this?    3. Was the geometric interpretation discovered _before_ tensor calculus became popular, or only after it was know that $F_{\mu\nu}=\partial_\mu A_\nu -\partial_\nu A_\mu$? Was this after the introduction of GR, and was it at all influenced by Einstein's work?


Suns emission spectrum

I learnt that in astrophysical spectroscopy, the emission spectrum of distant stars is used to determine what they're made of. So why is it that our own Sun is emitting the whole spectrum ? (or is that information incorrect)


Why do my ice cubes stick together?

When I put ice cubes in a glass of water, I find that sometimes they will stick together and form a sort of "bridge" between them as they melt. There is usually a visible line where one ends and the other begins, and they break apart if pushed (but for the most part they stick together if they aren't interfered with). I'm wondering how they form this bridge in the first place if they're melting, and why it stays together.


Follow up question on "Wilson Loops as Raising Operators"

This is a follow-up question on the topic that I opened a few days ago, Wilson Loops as raising operators. The paper > Topological Degeneracy of Quantum Hall Fluids. X.G. Wen, A. Zee. _Phys. Rev. > B_ **58** no. 23 (1998), pp. 15717-15728. arXiv:cond-mat/9711223. gives a nice derivation of the explicit ground states of the $U(1)$ Chern- Simons Theory on a torus in Section 2 on Abelian Quantum Hall States. In particular Eq. (12) gives the generic form of a ground state $\psi(y) = \sum_{n=-\infty}^{\infty} c_{n} \ e^{i\ 2\pi ny}$. Due to the fact that the theory lives on a torus the ground state manifold is found to be $k$-fold degenerate. My question: Is it possible (by direct calculation) to obtain the relations \begin{align} W(b)|n \rangle &= |n + 1 \text{ mod } |k| \rangle, \nonumber \\\ W(a) |n \rangle &= e^{2\pi i n /k} |n \rangle. \end{align} from the previous question? I don't have a particularly strong background in field theory so I am feeling somewhat uneasy when it comes to the explicit evaluation of the Wilson Loop (with its exponentiated gauge field and the path ordering) acting on the constructed state. I am looking forward to your responses.


Gravity of Light

I'm reading Quantum field theory in a Nutshell and I find a very interesting calculation that leads to the gravitational interaction between 2 light beam. Is this kind of interaction permitted in general relativity? (not necessarily a quantum gravity theory or the like) Is there a energy tensor $T_{\mu\nu}$ asscociated with a photon so that it creates $R_{\mu\nu}\ne 0$? It's awesome if someone can give a simple derivation in the framework of general relativity.


principle in a water Faucet

If you try to block with your finger the flow of the water out of the faucet or a water hose, the water's speed increases. But while you're blocking the water flow out of the faucet by turning the handle, the speed doesn't increase. So what's the difference between the two ?


Differences in the behaviour of pinching a garden hose and closing a tap

Let's say you have a garden hose connected to an ordinary water tap which is opened fully. If you pinch the end of the hose, water leaves the hose at a higher speed (and this can be useful while watering plants, to reach pots which are further away). However when a tap (with no hose connected) is opened only slightly, water flows out at a low speed, possibly even in drops. The actions of pinching the end of a hose and of almost-closing an open tap seem similar, so why the difference in behaviour?