title: What are the generators of spherical symmetry?

text: The title says it all. I think this should be a pretty simple question but I just couldn't find the answer. Ok -- I'll give a bit more context to my question. I'm encountering this in the context of general relativity, when we're trying to solve for the Schwarzschild solution. We write down a metric that respects spherical symmetry. According to the textbook (Carroll), "we argue that a spherically symmetric space-time can be foliated by two-spheres - in other words, that (almost) every point lies on a unique sphere that is left invariant by the generators of spherical symmetry." Some time later he mentions the Killing vectors of $S^2$, which are $$R=\partial_\phi$$ $$S=\cos \phi \partial_\theta - \cot \theta \sin \phi \partial_\phi$$ $$T=-\sin\phi\partial_\theta - \cot\theta \cos\phi\partial_\phi$$ and he observes that these vectors obey the following algebra: $$[R,S]=T\quad [S,T]=R\quad [T,R]=S.$$ I'm not sure how the Killing vectors are exactly linked to spherical symmetry, and why such an algebra between the Killing vectors is important (specifically with regard to spherical symmetry). And I also don't know what he means when he says "generators of spherical symmetry", hence the original question.


title: Amount of 2 amperage

text: I have 2 equation describing the alternating amperage $I_1$ and $I_2$. I need to get amount of these amperages. My equations: $$I_1=10\sin(\omega t+30)$$ $$I_2=20\sin(\omega t-50)$$ How can i make it? I forgot. Thank you.


title: Going left or going right of the free particle wave function?

text: In this excerpt from Griffith's quantum mechanics book: > # THE FREE PARTICLE > > We turn next to what should have been the simplest case of all: the free > particle [$V(x) = 0$ everywhere]. As you'll see in a momentum, the free > particle is in fact a surprisingly subtle and tricky example. The time- > independent Schrödinger equation reads > > $$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} = E\psi\tag{2.74}$$ > > or > > $$\frac{d^2\psi}{dx^2} = -k^2\psi,\quad\text{where } k \equiv > \frac{\sqrt{2mE}}{\hbar}.\tag{2.75}$$ > > So far, it's the same as inside the infinite square well (Equation 2.17), > where the potential is also zero; this time, however, I prefer to write the > general solution in exponential form (instead of sines and cosines) for > reasons that will appear in due course: > > $$\psi(x) = Ae^{ikx} + Be^{-ikx}\tag{2.76}$$ Equation 2.76 is the solution for equation 2.74. I don't understand why Griffiths says the first term in equation 2.76 represents a wave traveling to the right and the second to the left. Some website says when we apply momentum operator to first term then we get positive value in the positive x direction, but why do we separate two terms? If I apply momentum operator on equation 2.76, I can't get the wavefunction back. Since nobody know the physical meaning of wavefunction, then why do we simply separate it?


title: does the beam of a laser have 'throw'?

text: I was thinking about Einstein's train and platform experiment and was wondering if a beam of light experiences throw? Let me explain, if I take a water hose and point it straight out and then swing and stop my arm, the water will for a moment go past the end of the hose in the direction I was swinging my arm... Does light exhibit the same behavior? If I took a laser and started from the origin of a Cartesian system and accelerated it to let's say 75% of the speed of light along the x axis with it pointing in the y direction. When said laser pointer hits a brick wall at x0 will the laser beam actual cross to x0+delta?


title: Are valence electrons in nonpolar molecules localized/delocalized?

text: Are valence electrons in non-polar molecules localized or delocalized? I'm quite confused about the whole electron localization and delocalization business. I'd love a simplified explanation that a high-schooler can understand.


title: Could there be a body of water the size of a planet?

text: My friend and I were reading the news of the discovery of a black hole spewing huge amounts of water vapor into space, and it got us thinking: could there be a blob of liquid water in space the size of, say, a planet? Just in the interest of curiosity, I want to ask about several scenarios:   * Could the water being ejected by that black hole ever condense into something of the size I'm describing?   * Would it boil over immediately?   * If it would boil over, supposing the water "planet" could somehow have an atmosphere or maybe even a glass enclosure that keeps the liquid water away from the vacuum, what might happen to the planet as a result of other forces, namely gravity? Would the elements near the middle heat up and maybe fuse?   * Roughly how large a body of water could you have that's stable given that it could be protected from the vacuum?   * And just for fun, supposing the body of water was "bootstrapped" with primitive life or the ingredients for it, could it potentially support life?


title: What is the largest sphere of liquid water that could exist in space held together under it's own gravity?

text: What is the largest theoretical sphere of liquid water that could exist in space held together under it's own gravity? I've always wondered if a planet the size of earth could exist as a single volume of water? If in orbit of a star (which I guess it would need to be to remain a liquid), would the star's UV radiation evaporate such a watery world? Do the laws of physics permit enormous volumes of liquid water to float around in space?


title: Exact summation of a sub-class of diagram: do we know the exact solved problem?

text: In quantum field theories (to be relativistic, (non-)relativistic statistical or whatever), we have the powerful diagrammatic approach at our disposal. Most of the time we can not sum up all the diagrams, only a sub-class of them. I was thinking of the Hartree-Fock ones to give an example, but they are several classes we can re-sum exactly (skeleton, ladder, ...). Somehow this re- summation of diagrams is _exact_. I will keep italics to refer to this _exactness_. Clearly the solution we find that way is not the full solution of the problem we considered at the beginning. By "problem" I obviously mean the eigenstates/eigenvalues problem of a secular equation given by a Hamiltonian/Lagrangian. My question is the following: **do we know the problem we _exactly_ solved summing only partial class(es) of diagram ?** I mean: can we write out an effective Hamiltonian/Lagrangian that our _exact_ summation is the _exact_ solution of ? Can we at least know the _exact_ sub-space (eigenstates and/or eigenvalues), the solutions we found _exactly_ belong to ?


title: From the Poincare group to Minkowski space

text: In special relativity one assume that spacetime can be locally described by 4 coordinates, so it makes sense to model it as a 4-dimensional manifold. I had the impression that it is assumed that there is an atlas of special charts called inertial reference frames. Their transition maps are restrictions of affine maps on $\mathbb{R}^4 \rightarrow \mathbb{R}^4$. The subgroup of invertable affine maps which are allowed as transition maps is called the kinematical group. Einstein shows in his paper, I guess, that one can choose the kinematical group to be the Poincare group and for example [Bacry Levy-Leblond] showed that there would only be a few possible choices for these kinematical groups. In the books, I am aware of, comes now a huge step and one immediately assume one is in Minkowski space. So my questions are the following: Given a manifold $M^4$ with an atlas such that the transition maps are all restrictions of the affine maps of the Poincare group.   1. How can one construct a Lorentz metric (a (3,1)-semi-Riemannian metric) on $M$.   2. Does anyone know a reference for the fact that a manifold $M^4$ with affine transition maps is covered by an affine space (i.e. A simple connected one is topologically $\mathbb{R}^4$)? One might be tempted to consider the above as something belonging to general relativity, as one uses charts etc. But note affine transition maps are a huge restriction and at the end (maybe after taking a covering map) one talking about Minkowski space anyways.


title: Is it possible to have a geostationary satellite over the poles?

text: My understanding of orbital mechanics is very limited, but as I understand geostationary satellite, they stay in place by having an orbital speed corresponding to the spot they're orbiting over. So my basic intuition tells me that it's not possible to have a geostationary satellite over the poles, since they're effectively standing still, and to my knowledge, a satellite must keep a certain velocity to keep from getting pulled down by gravity. Am I missing something?


title: Proof of rotational and translational invariance?

text: If $ϕ^† ϕ$ is invariant under $SU(2) \times U(1)$, a Phys.SE question I recently posted, then does that mean $ϕ^† ϕ$ is invariant under rotations and translations?


title: Is the Fock state a superposition of product states?

text: I know that in principle, the Fock state is not the same as a product state because if it were, the photons would be uncorrelated. What I don't understand is, can it be expanded in product states? For example, if two photons share the same mode, is the state $|2,0\rangle$ the same as the product $|1,0\rangle|1,0\rangle$? If they are in different modes, is the Fock state $|1,1\rangle$ the same as superposition of two Fock states, $\frac{1}{\sqrt{2}}\left(|1_a,1_b\rangle+|1_b,1_a\rangle\right)$ (with $a$ and $b$ labeling the photons)? If yes, then is this superposition, in turn, the same as $\frac{1}{\sqrt{2}}\left(|1_a,1_b\rangle+|1_b,1_a\rangle\right)$ ?


title: Polarizable molecule in E-field

text: If we have a linear molecule with a dipole moment $\mu$ in a static electric field $E$, the potential is given by $V = - \langle \mu,E \rangle$. What is the appropriate equation for the potential if we have a polarizability $\epsilon_1$ along the direction of the dipole moment? If anything is unclear, please let me know.


title: Understanding mathematically the free expansion process of an ideal gas

text: I'm trying to understand mathematically that for the free expansion of an ideal gas the internal energy $E$ just depends on temperature $T$ and not volume $V$. In the free expansion process the change in internal energy $\Delta Q = \Delta W = 0$, therefore by the first law $\Delta E=0$. Energy is a function of state so $E=E(V,T)=E(p,T)=E(p,V)$, using $V$ and $T$ as the state variables we have $E=E(V,T)$. I have the following written mathematically in my notes but do not understand how it is derived. If $E=E(V,T)$ then $dE=\left(\frac{\delta E}{\delta V}\right)_T dV+\left(\frac{\delta E}{\delta T}\right)_V dT$ since we know experimentally that $dT=0$ and $dE=0$ from the first law we can write: $\left(\frac{\delta E}{\delta V}\right)_T=0$, as required. I have two main problems with this, firstly I do not understand where $dE=\left(\frac{\delta E}{\delta V}\right)_T dV+\left(\frac{\delta E}{\delta T}\right)_V dT$ comes from. My guess is that it is the differential of $E(V,T)$ but I do not fully understand the notation of what $E(V,T)$ represents that is, I understand that $E$ is a function of temperature and volume but I do not know how this can be written mathematically other than $E(V,T)$ so I can't differentiate it myself to check this. Again if I were to guess I would get an expression for $V$ and $T$ using the ideal gas equation and try to differentiate that? Secondly even with the results I do not understand why the end result is $\left(\frac{\delta E}{\delta V}\right)_T=0$ and not $\left(\frac{\delta E}{\delta V}\right)_T dV=0$. Has $dV$ simply been omitted because multiplying it by 0 in turn makes it 0? This question has arisen from me looking back over my lecture notes so it is possible I have written something down incorrectly which is now confusing me. If you could help to clarify any of the points above that would be great. Please let me know if more information is needed. Edit: I should also add I am not clear on the significance of the subscripted values $T$ and $V$ on either of the fractions represent. Update: I found the suffixes $T$ and $V$ mean that the temperature and volume is constant in the process.


title: How Can force exist without acceleration?

text: From what I understand acceleration does not cause a force, but rather forces cause acceleration, so if I have a ball moving with CONSTANT velocity, that hits a wall, then the wall must apply a force to deflect the ball (i'am assuming no force by the ball since it is moving with constant velocity), but where does the force by the wall come from? Also if the wall does not accelerate to cause the force, how would you calculate it?


title: Energy conversion efficiency of Q-switched laser?

text: This is kind of an elementary question, but since i cannot find a reliable source on this i wanted to ask what is the formula for energy conversion efficiency? I know efficiency is calculated as useful energy out/energy in, but i've been reading some papers on Q-switching, and some people have used Energy(pump)/Energy(Q-switch) which is the same as Energy(in)/Energy(out). Could anyone please clarify this? If it isn't as i said, can the energy conversion efficiency be more than 1? Also, i was wondering what is the max efficiency that Q-switched (Nd:YAG) lasers get? I ended up getting around 58%


title: Is there any optical experiment that can not be explained by photon model defined in QED?

text: Some author(in the paper of Evolution of the modern photon) mentioned that photon cannot account for the zero intensity through two crossed polarizators by 90 degree. Why so?


title: Can all canonical transformation be obtained through generation function approaches?

text: The question can be formulated as following: Suppose $$\delta \int_{t_1}^{t_2}{[p\cdot \dot{q} - H(p,q,t) ]dt} = 0$$ $$\delta \int_{t_1}^{t_2}{[P\cdot \dot{Q} - K(P,Q,t) ]dt} = 0$$ in which $$P = P(p,q,t), Q = Q(p,q,t)$$ is an invertible transformation. Can we prove that there must exist a $\lambda$ and function $G(p,q,t)$ (or $G(p,Q,t)$, $G(P,Q,t)$, $G(P,q,t)$), such that $$\lambda[p\cdot \dot{q} - H(p,q,t) ] = [P\cdot \dot{Q} - K(P,Q,t) ] + \frac{dG}{dt}~?$$


title: LS Coupling - weird image in the book

text: In the book by Arthur Beiser, _Concepts of modern physics_, in the chapter LS coupling there is this image: ![enter image description here](http://i.stack.imgur.com/BTWyz.png) **QUESTION:** How do we get total orbital angular momentum $L=3$ ( _image (a)_ ) out of quantum numbers $\ell_1=1 \Rightarrow m_{\ell 1}=-1,0,1$ and $\ell_2=2 \Longrightarrow m_{\ell 2} = -2,-1,0,1,2$? What about $L=2$ and $L=1$? **My attempt:** I tried to combine the $z$ components using equations $L_z=m_\ell \hbar$ and got these results for $\boxed{L\equiv L_z}$: \begin{align} L_{1z}=& \left\\{ \begin{aligned} -1&\hbar\\\ 0&\\\ 1&\hbar \end{aligned} \right.\\\ L_{2z}=& \left\\{ \begin{aligned} -2&\hbar\\\ -1&\hbar\\\ 0&\\\ 1&\hbar\\\ 2&\hbar \end{aligned} \right. \end{align} If I sum these up one of the results is $-3\hbar$, which is weird and isn't listed as one of the possible values in the picture. Also the picture states that one of the possible total orbital angular momenta is $L=3$ and not $L=3\hbar$. What is this a mistake or what?


title: Is it correct that tires quickly get bald when used at high speed?

text: In my investigation, tires will quickly get bald if we ride a motorcycle at high speed. Is it correct? How to explain it in physics? Does the friction increase at high speed rotation? Update: The word "quickly" means that the tire gets bald with less distance traveled.


title: Braess's Paradox in Electrical Circuits

text: According to this and other similar papers, http://www.nature.com/nature/journal/v352/n6337/abs/352699a0.html, adding a current-carrying path can increase the voltage drop across a circuit. What is the simplest example of a circuit (preferably consisting solely of two- terminal devices) that exhibits such behavior?


title: What are the strongest sources of collimated neutrons and protons?

text: I am imagining an unusual experiment which will require intense beams of either protons or neutrons. The experiment would work better with neutrons, but neutron sources are much ~~weaker~~ messier so I am considering both. I am interested in beams of almost any non-relativistic energy. I am not sure yet about how focused I will need the beam to be. I know this is a beginner question, but what are the basic options here for the strongest source (as measured by neucleons per second)? * * * Neutron apparently come from two sources: reactors and accelerators. (As Sir Patrick Stewert can attest.) In the latter case ("spallation"), a proton beam is just slammed into a heavy-metal target and the neutrons that fly out are caught. It appears that the highest intensity protons beams are often used to produce the highest intensity neutron beams in this manner. Saclay High Intensity Proton Injector Project (IPHI): $6 \times 10^{17}$ protons/second. Low-Energy Demonstration Accelerator (LEDA): $6 \times 10^{17}$ protons/second. (These agree with Andre's chart below showing 100 mA. Note also that the LHC current is about 500 mA, although the protons are grouped into bunches. And as Andre pointed out, the LHC circulates the same protons and so could not provide a continuous 500 mA beam into a target/experiment that consumed them.) Neutron sources, being the result of either fission or spallation, are messy. I still don't have a good handle on what kind of neutron intensities are possible for what momentum ranges. The Lujan Neutron Scattering Center at LANCE is a neutron source for many experiments. * * * In spallation, the number of neutrons produced per incident proton is usually on the order of 40, but the phase space distribution is much wider than the incident protons. (Protons beams can be directly focused using magnets.) According to a CERN report by Lengeler, reactors typically can produce thermal neutrons (~0.025 eV) fluxes of order $10^{14}$ neutrons/cm${}^2$s, whereas spallation sources can exceed $10^{17}$/cm${}^2$s. * * * It is probably best, then, to think of continuous neutron sources as phase- space-density limited, with the above numbers giving a rough estimate of the currently achievable densities. Apparently, spallation sources are limited by the need to dissipate the heat from the incident proton beam. If high fluxes are needed only briefly, laser pulses can produce $4 \times 10^{9}$/cm${}^2$ over a nanosecond, i.e. an instantaneous flux of $4 \times 10^{18}$/cm${}^2$s. As for protons, it's worth noting that the high-intensity 50 mA beam at the European Spallation Source is limited by "space charge effects at low energy, by the power that can be delivered to the beam in each cavity at medium and high energies, and by beam losses."


title: Confusion on entropy in the universe

text: I was researching about the heat-death of the universe, and I was wondering if my understanding of it so far is factual to the theory? However, the 3rd Law of Thermodynamics prevents the universe’s entropy from reversing because that law states that the temperature cannot equal 0 Kelvin. The heat of the universe does not equal 0 Kelvin and the only way to get it to 0 Kelvin would be to have an infinite amount of objects that are at 0 Kelvin to bring the universe’s temperature down to near 0 K.


title: Expectation value of energy from the position state of hydrogen atom

text: I was given with the position state of hydrogen atom: $$ R_{21} =\left(\sqrt{\frac{1}{3}}Y^0_1 + \sqrt{\frac{2}{3}}Y^1_1\right) $$ I am getting confused about getting the expectation value of energy, I know from the form that: $$ R_{21} \Rightarrow n=2, l=1$$ according to this website. I should just: $$ \langle E_n\rangle = - \frac{1}{2} \alpha^2\mu c^2 \frac{\frac{1}{3} \cdot \frac{1}{2^2} + \frac{2}{3} \cdot \frac{1}{2^2}}{2^2} \\\= - \frac{1}{2} \alpha^2\mu c^2 \frac{\frac{1}{12} + \frac{2}{12}}{4} \\\= - \frac{1}{2} \alpha^2\mu c^2 \frac{\frac{3}{12}}{4} \\\= - \frac{1}{2} \alpha^2\mu c^2\frac{1}{16}$$ Am I right?


title: Calculate electric potential and electric field of uniformly charged wire

text: > Say you have a uniformly charged straight wire of length $2L$ going from > $-L$ to $+L$ along the $z$ axis with its midpoint located at $z=0$. What is > the electrostatic potential, in cylindrical coordinates, for points in the > midplane of the wire ($x,y$ plane)? From this electric potential, calculate > the electric field in the $x,y$ plane. My intuition is telling me that, because of the symmetry, the only direction the electric can be directed is in the xy-plane. So we have $$dq=\lambda dz$$ Thus, since $$dV=\frac{kdq}{r}$$ We have $$V=2\int_0^L \frac{kdq}{r}$$ Also $$r=\sqrt{x^2+y^2+z^2}$$ The integral is then $$2k\int_0^L \frac{\lambda dz}{\sqrt{x^2+y^2+z^2}}$$ Evaluating this I get $$V = 2 k \lambda \ln ( \sqrt{(x^2 + y^2 + z^2)} + z)$$ I know it's not in spherical coordinates, but can someone tell me if this is right so far? If so, the electric field is then given by the negative gradient, correct?


title: Size of a photon

text: When detecting radio waves in space, we use very large telescopes or arrays of telescopes. But according to QM, aren't photons point particles when measured? Does a photon with a large wavelength have a wider probability distribution with regard to where it can be expected to be detected? Is that why photons with long wavelengths require large telescopes to be detected relative to shorter wavelength photons?


title: Base states with hamiltonian matrix

text: It is better for you to have studied "Feynman lectures on Physics Vol.3", because I cannot distinguish whether the words or expressions are what Feynman uses only or not and in order to summarize my questions here, I have to just quote the contents of the book. However, one thing I notice is that "base state" that Feynman explains seems to be "basic orthonomal state vector"... With a pair of hamiltonian matrix equation $i\hbar \frac{d{C}_{1}}{dt} = {H}_{11}{C}_{1} + {H}_{12}{C}_{2}$ $i\hbar \frac{d{C}_{2}}{dt} = {H}_{21}{C}_{1} + {H}_{22}{C}_{2}$ where ${C}_{x} = <x|\psi>$ , $\psi =$ arbitrary state, the book set the states 1 and 2 as "base states". There are only two base states for some particle. Base states have a condition - $<i|j> = {\delta}_{ij}$. I think the "kronecker delta" means that once the particle is in the state of j, we will not be able to find the state i, so if we suppose all the components of hamiltonian are constant, we can say ${H}_{12}$ and ${H}_{21}$ should be zero. ..............(1) However the book says that states 1 and 2 are base states and ${H}_{12}$ and ${H}_{21}$ can be nonzero at the same time (if you have the book, refer equ. (9.2) and (9.3) and page 9-3.). There can be probability to transform from state 1 to state 2 and vice versa..... Then, the relationship that I think like (1) between the "Kronecker delta" and the components of hamiltonian is not correct at all?? Or, is my thought that ${H}_{12}$ and ${H}_{21}$ should be zero true?


title: Mechanism of Pair Production and Annihilation of Matter

text: Pair production is where an energetic photon on its interaction with strong electric field surrounding a nucleus produces electron-positron pair. Annihilation of matter is its converse where an electron-positron pair produces two photons on interaction with nucleus. But how do these happen? I mean, Why do they require strong electric field and nuclei..? (Does this process requires Strong Nuclear force?)


title: Carnot efficiency, coefficient of performance for a refrigerator

text: I was reading that a new type of refrigerator might reach a coefficient of performance (COP) of 10. This seems quite the achievement and the authors state that their approach might achieve a Carnot efficiency of 75%. (I could not get hold of the original source as it is a part of an expensive book series "Zimm C, Jastrab A, Sternberg A, Pecharsky V K, Gschneidner K Jr, Osborne M and Anderson I 1998 Adv. Cryog. Eng. 43 1759".) Now I am a bit confused. Wikipedia states that the COP should be the inverse of the Carnot efficiency but than these numbers do not fit together or am I missing something?


title: Tsallis entropy and other generalizations

text: If I am given a system, which I might have to describe using a generalized entropy, like the "q-deformed" Tsallis entropy, do I have to fit q from experiment or might I know it beforehand? How do I know the parameter q and/or how can I possibly obtain the degree of non-extensitivity via experiment? How can I measure the entropy of a part of the system, if the system is non- extensive? http://en.wikipedia.org/wiki/Tsallis_entropy Edit: After some browsing I think the answer might be related to the fact, that for q-deformed entropy, the most probable distribution is not the Gaussian, but seems to be the q-deformed Gaussian: http://en.wikipedia.org/wiki/Q-Gaussian Then I played around a bit: http://img835.imageshack.us/img835/5841/qgaussian.png Maybe one applies such an entropy concept if one comes across a distribution of such type, but that's only a guess. And I don't see why one would/could conclude non-extensitivity from a distribution?!


title: What exactly is "pressure", and what's its relation to force?

text: I have looked around, and I will admit that I'm a bit lost on the definitions. Wikipedia's opening reads: > Pressure (symbol: P or p) is the ratio of force to the area over which that > force is distributed. That's a bit hard to understand at first, I'll have to say. Is vaguely saying "pressure" relative to "air resistance", or drag? Because they seem to be mixed around often. Drag > refers to forces acting opposite to the relative motion of any substance > moving in a fluid. Sounds unrelated to pressure, so let's move on. If pressure is the ratio of force to an area which the force is distributed, wouldn't anything be force then? Wind's force is covered over an area that it's distributed from. A punch would count too, based on how I understand pressure. I have always envisioned pressure as a uniform body of gaseous force or energy acting versatilely upon everything, such as below illustrates: ![enter image description here](http://i.stack.imgur.com/5kOg0.jpg) Higher pressure of the atmosphere is like a tighter "squeeze" on everything within it. People often say pressure in exchange for force, such as when someone shakes someone's hand very hard (with a strong grip), one will say that the force is pressure. Is this correct? If so, why, and how do force and pressure coincide?


title: Why do 3d spheres and gravity tend to rotating discs on one plane?

text: Whether is it our solar system or a whole galaxy, there is usually a massive object (star or black hole) at the centre with gas and objects rotating around it. The gravitational effect of the star/black hole extends uniformly (more or less) in every direction in 3d. Why does matter tend towards a single plane? Furthermore, what happens to matter that approaches after the "disc" is formed when it is pulled in from anywhere off the plane, why does it join the plane rather than forming another plane? I suspect angular momentum has something to do with it, but would appreciate a "pop science" explanation. Many thanks Andrew


title: Stretching a rod?

text: The stress applied on a rod is linearly proportional to its strain. But shouldn't the opposite be true? I mean if you pull particles further apart doesn't the force they apply on each other decrease because the distance between them increases? Kinda like gravity?


title: TKNN invariant changes due to continuous deformation of parameter space

text: Naively, I would assume that a topological invariant remains invariant under continuous deformations of whatever space the invariant belongs to. In the case of topological insulators, this space is the space of parameters characterizing the material and its environment. If we consider the magnitude of a magnetic field incident onto a material supporting metallic edge states a parameter, then why does the Chern number (TKNN invariant) change when $|\textbf{B}|$ is increased smoothly and slowly? Since the Hall conductivity experiences sharp increases due to slow increase of magnetic field, the Chern number must be increasing sharply (by integer jumps), signifying sharp changes in the topology of the parameter space. What parameter is changing so drastically in response to the increasing magnetic field?


title: What is the Bremsstrahlung or dipole radiation mechanism involved in the emission of radiation by the rotating or moving charge?

text: > A rotating charge such as the electron classically orbiting around the > nucleus, will constantly lose energy in form of electromagnetic radiation. I asked my teacher about how this radiation is created from the charge. My teacher said that energy is usually lost by the mechanisms such as _Bremsstranhlung_ , _Dipole radiation_ , etc. But, I don't have these mechanisms in my text book and I need to know how these mechanisms are carried out, so that I would understand the concepts better. So, I thought Physics Stack Exchange will help me.


title: Probability distribution of two particle types system

text: Suppose that particles of two different species, A and B, can be chosen with probability $p_A$ and $p_B$, respectively. What would be the probability (and distribution) $p(N_A;N)$ that $N_A$ out of $N$ particles are of type A? I'm trying to apply the Binomial distribution here but am bothered by the fact that it applies for N trials (whereas here we only do 1).


title: Our universe the surface of a 4-dimensional sphere?

text: The cosmic microwave background that we observe uniformly around us is usually explained by assuming that our universe is the surface of a four dimensional sphere. That way the uniformity makes sense since there is no center. My question is if this is true then what is the explanation that describes the fact that the farther we look into space, the further we look back in time. I can't perfectly picture this and see how it would coexist. Help me out.


title: Acceleration is zero, for non-zero net force

text: I understand a problem like this has already been asked, but I have not found an answer that makes it clear. A force is applied to a box on a table(lets ignore friction), and the box moves with some constant velocity. In this case, acceleration is equal to zero. The net force does not. How can we explain this? $$ \sum F = F_{app} = m a $$ The acceleration $a = 0$, but $F_{app} > 0$. Can someone clarify this for me?


title: Potential energy for different states

text: While studying thermal physics at school, I have been taught that solids simply have more potential energy than the liquids and gases. Note that it was said that this potential energy is due to the intermolecular bonds between the atoms. However, my intuition makes me doubt this, why would there be more potential energy in a solid??? Lets see: the internal energy is the sum of the kinetic and potential energies of a body. Well liquids and gases have more kinetic - pretty clear. As for, our original point, the potential: well, if the body has potential to do work, via lets say, chemical reactions, then its solid form will have a higher activation energy, as the intermolecular bonds have to be broken first - so the energy then released from combustion or what not, is less than that of its gaseous equivalent! Then how is it, that solids are somehow said to have more "potential" energy than the more energetic states. I, upon questioning this, have been told that this is due to the forces of the molecular bonds, but really I think, these forces signalize a lack of potential energy!


title: Energy band diagram for solid state: what is the meaning of $k$?

text: I am having trouble with the meaning of the $k$ vectors in an energy diagram. If I want to populate some band, let say using a laser, what will be the significance of $k$? Does it correspond to the polarization angle with respect to particular planes?


title: Why the free energy is called 'free'?

text: The free energy, $F$ of a thermodynamic system at a given temperature $T$, is defined as, \begin{equation} e^{-\beta F} = \mathcal{Z} = \sum_{\\{configuration\\}} e^{-\beta E(configuration) } \end{equation} where $\beta = 1/k_BT$ and $E$ is the energy of a certain configuration. Now my question is: **Why the free energy is called 'free'? I mean it is free in which sense?**


title: Average speed question

text: Find your average speed if you go 15m/s for 1000 m and then 30m/s for 1000 m I understand that average speed is total distance / total time. The distance for this problem would be 2000m travelled. How would I figure out the total time?


title: Principle of relativity and motion perpendicular to the direction of light beam

text: consider you are in a train that moves with a speed of $c/4$. A bulb is hanged from the ceiling. A light beam leaves the bulb downward. Exactly under the bulb on the floor is a detector. According to the principle of relativity the light will not strike the detector in the reference frame of a non-moving observer (standing out of train), because the **velocity** of light must be constant (light should continue it's motion vertically). But obviously the observer on the train should see that the beam strikes the detector. But both can not be correct. (The beam is whether detected by the detector or not) Whats wrong?


title: How to concisely explain apparent retrograde motions of planets?

text: Some planetary orbits occasionally can appear to move backwards to an observer on the Earth? Does anyone know concise, clear, web-based visualizations, animations or tutorials that clearly show how this might come about? I have difficulties imagining it "in my head". A link to an animated solar system simulation would be fine. Here is a good static picture,but does anyone know even better ones? And yes, I know that Wikipedia.com has some links, too. ![Retrograde Motion](http://i.stack.imgur.com/cuX4u.png) http://wiki.astro.com/astrowiki/en/Retrograde_Motion


title: Can't understand a thing about emissivity concept

text: I can't understand something about emissivity. Emissivity is defined as the ratio of the radiant energy of an object, to radiant energy of blackbody at a temperature $T$. So, The vegetation has approximate emissivity of $0.99$ and concrete smaller than this like $0.95$. But in thermal images vegetation appears darker than concrete. If its emittance is much more; so why is it dark? If the vegetation emits more energy why does it look dark in thermal imagery?


title: PNMR, Pulsed Nuclear Magnetic Resonance, Experiment

text: I am writing a lab report for class on PNMR experiment that we did. _How come in this experiment we don't worry about the electron spins in our sample?_ Aren't the electrons affected by the PNMR machine as much as the protons? Isn't there electron proton spin interactions? For some reason, the lab only talks about the proton spins and does not mention the electron spins at all. Since we did this at room temperature, does this mean that the electrons have so much energy that their spins doesn't even align with the permanent magnetic field, or are they all paired up with other electrons and therefore do not contribute much? I am very confused. Please help!!!


title: Does Standard Model confirm that mass assigned by Higgs Mechanism creates gravitational field?

text: I am not comparing _passive_ gravitational mass with rest inertial mass. Is there an evidence in Standard Model which says that _active_ gravitational mass is essentially mass assigned by Higgs mechanism.


title: Remaining Potential Experimental Particle Physics Discoveries at the TeV Scale?

text: With the discovery of the Higgs Boson, some have been calling it the end of experimental particle physics for our generation, due to the fact that all of the particles predicted by the standard model have been found (the W and Z Bosons in 1983 and the Top Quark in 1995) and no new physics is expected at the energy ranges we can probe. I have also generally heard from my friends who work in experimental particle physics (or, at the very least, accelerator physics) a similar story, that there isn't too much left to be discovered. Is this an accurate assessment? What about dark matter particle models, like the axion/axino? Can the LHC (or SLAC, etc) probe enough of a range to perhaps shed light on this cosmological problem? Can we hope for any more insights into beyond the standard model theories from current accelerators?


title: How can a pion have a mass, given it's a "field mediator" and created/destroyed continuously?

text: Maybe some of my assumptions here are basically wrong, but isn't it true that   * pion is the "mediator" for the strong force field.    * the quantum field theory basically says that there are no fields, instead all forces are caused by interchanging of mediator particles all the time. So to me this look like two particles in a nucleus, which are hold together by the strong force, are "sending" these pion particles all the time. But all references say that these particles have mass! Why doesn't this mean that mass is created/destroyed all the time? Is mass of the whole nuclei affected in some way by the constant stream of pions? If the question is basically irrelevant, what are the most important parts that I am missing here?


title: Temperature dependence of resistivity in metals

text: We know that in high temperature, resistivity in metals goes linearly with temperature. As temperature is lowered, resistivity goes first as $T^5$ due to "electron-phonon" interaction, and then goes as $T^2$ as temperature is further lowered due to "electron-electron" interactions. My question is, is there an **intuitive** way of understanding the above power-law dependence at each temperature range **physically**?


title: Getting Slavnov-Taylor identity

text: Let's have generating functional in path integral form for gauge $SU(n)$ theory with interaction: $$ \tag 1 Z[J] = \int DB D\bar{\Psi}D\Psi D\bar{c}Dc e^{iS}. $$ Here $$ S = S_{YM}(B, \partial B) + S_{M}(\Psi , D\Psi) + S_{ghosts} + \int d^{4}x \left[-\frac{1}{2\alpha}(\partial_{\mu}B^{\mu}_{a})^{2} + \sum_{\varphi}(J_{\varphi} \cdot \varphi ) \right], $$   $$ L_{ghosts} = \bar{c}^{a}(\delta_{ab}\partial^{2} + gc_{acb}B_{\mu}^{c}\partial^{\mu})c^{b}. $$ I want to get Slavnov-Taylor identities for nonabelian case by using the fact that $(1)$ doesn't change under translation substitution $\varphi \to \varphi^{\omega} = \varphi + \delta_{\omega}\varphi $. In abelian case identity which acts role of Slavnov-Taylor identity (it shows that full propagator of EM field has transverse structure) can be derived by using the result of invariance of $\left(\frac{\delta}{\delta J_{\mu}^{A}(x)} Z[J]\right)_{J =0}$ under sets of transformations $$ \Psi \to \Psi^{\omega} \approx \Psi (1 + i\omega ), \quad \bar{\Psi} \to \bar{\Psi}^{\omega}\approx \bar{\Psi}(1 - i\omega ), \quad A_{\mu} \to A_{\mu}^{\omega} \approx A_{\mu} + \partial_{\mu}\omega . $$ The result is $$ \left(\frac{\delta}{\delta \omega (z)}\frac{\delta}{J_{\mu}^{A}(x)} Z[J]\right)_{J, \omega =0} = \int \prod_{\varphi}D \varphi \left(\frac{\partial }{\partial x_{\mu}}\delta (x - z) - \frac{i}{\alpha}A_{\mu}(x)\partial^{2}\partial^{\nu}A_{\nu}(z)\right) e^{iS} = 0 $$ (you can read about it, for example, on page 99 here). But when I have tried to get Slavnov-Taylor identity (I mean this one; I need to variate it with respect on $J_{\mu}^{b}$ and then set it to zero) bu using the invariance of $(1)$ under transformations $$ B_{\mu}^{a} \to (B_{\mu}^{a})^{\omega} = B_{\mu}^{a} + g\omega^{c}B_{\mu}^{b}g_{cba} + \partial_{\mu}\omega_{a}, \quad \Psi \to ... , $$ but I haven't got it. Is this way incorrect?


title: Liquid film in a narrow slit

text: If a thin sheet of some material in which a slit is cut is immersed in a liquid and removed, a liquid film (bridge) stays in the slit. Is it possible to calculate the shape of the film if the properties of the liquid and the dimensions of the slit are known?


What is wrong with this form of the Maxwell-Faraday equation? $$\oint \vec{E}\ \partial \vec l= \bigcirc \hspace{-1.4em} \int \hspace{-.6em} \int \frac{\partial \vec B}{\partial t}$$ "Line integral of the electric field is equal to the double integral of partial derivative of magnetic field with respect to time". So far as I remember the correct form used to be "Line integral of electric field is always (negative) surface **integral of partially derived magnetic field...** "


title: Electromagnetic waves and group velocity

text: I have three questions about electromagnetic waves and was wondering whether anybody here could comment on these things: Wikipedia says that there are no longitudinal EM waves, although TM and TE waves exist. How can I see that an EM wave can never be purely longitudinal? Does it always hold that the E-field is perpendicular to the B-Field in an EM wave? Wave packets have a group velocity that can be different from the phase velocity. Is it correct that due to $\omega= vk$, the phase and group velocity for a plane wave are always the same?


title: Unitary gauge in $Z_2$ lattice gauge theory with matter field

text: A $Z_2$ gauge theory with Ising matter field on a 2-dimensional square lattice has the Hamiltonian \begin{equation} H=-t\sum_{\vec r,j}\sigma_j^x(\vec r)-g\sum_{\vec r}\sigma^z_1(\vec r)\sigma^z_2(\vec r)\sigma^z_3(\vec r)\sigma^z_4(\vec r)-\lambda\sum_{\vec r}\tau^x(\vec r)-\mu\sum_{\vec r,j}\tau^z(\vec r)\sigma_j^z(\vec r)\tau^z(\vec r+\hat e_j) \end{equation} where $\sigma$'s are Pauli operators for the gauge field living on the links and $\tau$'s are the Pauli operators for the matter field living on the sites. $\vec r$ denotes the position of a site and $j$ denotes the link attached to the site which is in the $\hat e_j$ direction. The local gauge transformation is induced by \begin{equation} \tau^x(\vec r)\sigma^x_{\hat e_x}(\vec r)\sigma^x_{-\hat e_x}(\vec r)\sigma^x_{\hat e_y}(\vec r)\sigma^x_{-\hat e_y}(\vec r) \end{equation} In chapter 9.10 of the book _Field theories of condensed matter physics_ by Fradkin, he said we can choose a unitary gauge defined by \begin{equation} \tau^z(\vec r)=1 \quad\forall\ \vec r \end{equation} so that the last term becomes \begin{equation} -\mu\sum_{\vec r,j}\sigma_j^z(\vec r) \end{equation} On the other hand, gauge invariance implies the Hilbert space we are considering is the one that consists of vectors which are invariant under the local gauge transformation, so the third term becomes \begin{equation} -\lambda\sum_{\vec r}\sigma^x_{\hat e_x}(\vec r)\sigma^x_{-\hat e_x}(\vec r)\sigma^x_{\hat e_y}(\vec r)\sigma^x_{-\hat e_y}(\vec r) \end{equation} so the Hamiltonian becomes \begin{equation} H=-t\sum_{\vec r,j}\sigma_j^x(\vec r)-g\sum_{\vec r}\sigma^z_1(\vec r)\sigma^z_2(\vec r)\sigma^z_3(\vec r)\sigma^z_4(\vec r)-\lambda\sum_{\vec r}\sigma^x_{\hat e_x}(\vec r)\sigma^x_{-\hat e_x}(\vec r)\sigma^x_{\hat e_y}(\vec r)\sigma^x_{-\hat e_y}(\vec r)-\mu\sum_{\vec r,j}\sigma_j^z(\vec r) \end{equation} My question here is: after choosing a gauge, why do we still have gauge invariance so that the gauge transformation generators act trivially? In this example, the question is that after choosing the unitary gauge $\tau^z(\vec r)=1$, why do we still have $\tau^x(\vec r)\sigma^x_{\hat e_x}(\vec r)\sigma^x_{-\hat e_x}(\vec r)\sigma^x_{\hat e_y}(\vec r)\sigma^x_{-\hat e_y}(\vec r)=1$?


title: Efficiency of Metropolis algorithm

text: Context is 1D Ising model. Metropolis algorithm is used for simulate that model. Among all possible spins configurations (states) that algorithm generates only states with the desired Boltzmann probability. Algorithm chooses spin at random and makes a trial flip. If trial satisfies certain conditions related to Boltzmann probability, flip is accepted. Otherwise flip is rejected and system is unchanged. Define "acceptance ratio" as a percentage of accepted trials. Simulation shows that acceptance ratio is higher on higher temperature (behave as increasing function of temperature). Questions:   1. Why Metropolis algorithm is not efficient at low temperatures?    2. Is efficiency of algorithm related to acceptance ratio?


title: How to calculate the amount of night time during a flight?

text: I have been asked to find a way to calculate the amount of time that a flight takes during night time. So far, I have the departure latitude and longitude and the time of takeoff, the arrival latitude and longitude and the time of landing. I can easily calculate the sunset and sunrise time at the departure location, and at the arrival location. But I don't know how to proceed in order to find at which moment of the flight the sunrise and sunset will happen at the location where the aircraft is at that precise moment. I'm just asking for some guidance on how to resolve this problem. **PS:** The calculation does not have to be very precise, and estimation is more than enough. We are just looking for the simplest solution to obtain an acceptable result.


Why do we have so many dualities in string theory? Is there a reason for that?


title: Scalar product of coherent states

text: We suppose for semplicity to have a 1D oscillator, but this is a question abaout the general CCR algebra in oscillators, second quantization, quantum field theory etc. We know coherent states are a non orthonormal base and are a overcomplete base. We know them are given from $A |\alpha>=\alpha |\alpha>$. where $|\alpha>=e^{-\frac{|\alpha|^2}{2}} e^{\alpha A^+} \Psi_0$ How can we calculate generical scalar product between two different coherent states with eigenvalue $\alpha$ and $\beta$? $$ e^{-\frac{|\alpha|^2+|\beta|^2}{2}}\Psi_0 e^{\beta A} e^{\alpha A^+} \Psi_0 $$


title: Does MOND make good predictions?

text: Well, it does according to this preprint for certain scales. What would be a simple way to explain MOND to a layman? Does it ignore mainstream physics? How much?


title: Why are so many condensed matter phenomena so dependent upon impurities?

text: Why are so many condensed matter phenomena so sensitive to impurities? In fact, quite a number of them depend upon impurities for their very existence!


title: Schwarzschild metric

text: Why, if the Schwarzschild metric is a vacuum solution ($T_{\mu\nu}=0$) , do textbooks state that $T=\rho c^{2}$ when approximating Poisson's Equation from the Einstein Field Equations? Thank you.


title: Why is moonlight less colourful than yellow light?

text: Just what the title states; and this isn't an original question ... On full-moon, moonlight is almost as bright as mid-dawn ; yet there is hardly any colour visible. This is not the case with sunlight. I'm curious to know why this is the case


title: Fluid Mechanics Problem - Ball in Reservoir

text: I'm having a problem starting and solving the following problem: ![Fluid Mechanics](http://i.stack.imgur.com/QX76Y.png) My attempt at the solution was to realize you need to balance the upward forces with those of the downward forces in which case you have a buoyant force, for the upward force. And downward is the weight of the ball and the pressure of the ball at that height it's at. So then to solve it I assume you equate both and then you can find the specific gravity? Thanks,


title: The implications of the holographic principle on string theory S-matrices

text: The S-matrix of superstring theory will have finitely many massive particles which decouple in the asymptotic future/past and massless particles like gravitons, photons and dilatons. Typically, there will be an infinite number of massless "soft quanta". In the far future/past, all of these dressed particles will decouple and essentially travel along straight worldlines. So, we can draw a spatial boundary enclosing all the massive dressed particles but none of the massless particles. In the far future/past, that is. According to the holographic principle of quantum gravity, the massive particles would then be completely determined by the massless particles. Everything about them can be deduced in principle from the massless particles. This isn't as implausible as it may seem because we have infinitely many soft quanta. ''Does this mean the entire concept of an S-matrix has to be overthrown in superstring theory?''


title: Where did the idea of a neutrino come from?

text: Exactly what paper by W. Pauli introduced the idea for the existence of a neutrino and how was its existence confirmed experimentally (who did that and in what paper)?


title: Current in purely LR DC circuit at t=0

text: I was told by my professor that the current in a purely LR Circuit with at initial state (JUST after closing the switch) is zero. Can you please help me understand why it that so?


title: Limit of human eye flicker perception?

text: I am designing a LED dimmer using software-controlled Pulse Width Modulation, and want to know the minimum PWM frequency that I must reach to make that LED dimming method indistinguishable from variable DC current. To dim to X% power, the led will be fully on for X% of the PWM period, and fully off for the rest, with X from 0 to 100. I am assuming that the observer can compare with nearby LEDs driven by DC current at various power (but not exactly X% power), and can move the eye (in particular look at the LEDs in peripheral vision, which, I'm told, is the most sensitive to flicker). I disregard purposefully moving one's head relative to the LEDs, which (I guess) could be an effective distinguishing method; and use of a rotating slit disc or similar technique. I read movie projectors use 48 or 72 Hz shutter frequency (twice to thrice the frame rate) to reduce flicker to something tolerable (but quite noticeable in peripheral vision, I guess). I also read that the 50 or 60 Hz AC frequency has been chosen to avoid flicker, suggesting that that 100 or 120 Hz flicker is perceptible (but the limit could be much higher, because thermal inertia of the filament must damp variations). Another reference point is that CRT with 100 Hz scan rate have been designed, presumably with some benefit (but again that is not an indication of the upper limit). Edit: Found this relevant post, Effects of high frequency lighting on human vision Edit: My LED is "white", that is really blue with a luminescence converter consisting of an inorganic phosphor material.


title: Optical fibre modes and intensity pattern

text: I shine laser line in front of one side of multimode optical fibre.Light coming out of other side is projected into wall.Whether the intensity pattern seen on wall is same as the mode of propagation of light pattern drawn on text books?


title: Classical vs. Quantum use of the spin 4-vector

text: I have a few basic questions about the Pauli-Lubanski spin 4-vector S.   1. I've used it in quantum mechanical calculations as an operator, that is to say each of the components of S is a matrix operator that operates on an eigenvector or eigenspinor. But my question is about the utility of S in a classical sense, that is to say it represents the physical spin angular momentum. For example, in an electron's rest frame, is the spin 4-vector for the case spin-up along the z-axis given by S = (0, 0, 0, h/2) and for spin-down along x we have S = (0, -h/2, 0, 0) etc?   2. I know that in the particle's rest frame S = (0, Sx, Sy, Sz) where the spatial components are the spin angular momentum 3-vector components. However, when we Lorentz boost S, the time component is no longer zero. In this boosted case, do the 3 spatial components still give the spin angular momentum 3-vector (analogous to the case for 4-momentum where the 3 spatial components always give the 3-momentum), or do the spatial components now mean something else? The reason I'm not sure is that some 4-vectors, e.g. 4-velocity, have spatial components that do not represent 3-velocity at all since they may be superluminal, etc.


title: Planetary Gravity and its effects

text: This is my first question on the Physics portion of Stack Exchange. I was hoping to get some light on the topic of gravity. I don't have much background knowledge of physics so I might as well start here. I have a few planet and gravity related questions: **1).** Does planetary gravity depend on size or density? Or both? **2).** Is it plausible for a large (very large), terrestrial planet to have tall mountains like Mount Everest or even like on Mars? **3).** Is there a way, given an average density per cubic meter along with the average radius of the planet, to calculate the pull of gravity? I've been scouring the internet for these things with no luck so far. The idea behind this is to make a planet rendering program which takes all of this into effect. If this forum doesn't answer this sort of thing, feel free to point me in the right direction. Thanks!


title: Is the induced electric field due to time varying magnetic flux always circular?

text: According to Faraday's law, changing magnetic flux induces an electric field. Is that electric field always circular?


title: is this generalized Hawking radiation formula right?

text: Look at equation 11.2.17 in this page. The expression is: $$ T = 10^{-5} \text{K m} \frac{\xi}{\frac{GM}{c^2} \lbrace \frac{GM}{c^2} + \xi \rbrace - e^2 }$$ where $$ \xi = (r_s^2 - a^2 - e^2)^{1/2}$$ and the usual parameters $$ r_s = \frac{GM}{c^2} $$ $$ a^2 = \frac{L^2}{M^2 c^2}$$ $$ e^2 = \frac{Q^2 G}{4 \pi \epsilon_0 c^4}$$ This formula is supposed to describe the temperature of a black hole with angular momentum $L$, charge $Q$ and mass $M$ > **Question:** is the above formula correct? I'm trying to find the limit temperature for $a=0$ and $e=r_s$ The temperature expression could be simplified as $$ T = 10^{-5} \text{K m} \frac{\xi}{\xi^2 + a^2 + \frac{GM}{c^2} \xi }$$ if $a=0$, $$ T = 10^{-5} \text{K m} \frac{1}{\xi + \frac{GM}{c^2} } $$ so when the black hole has extremal charge, $\xi=0$ and the temperature looks like the normal black hole temperature for the Schwarzschild black hole, which looks very wrong Any idea where is the mistake? I was expecting the temperature of the charged extremal black hole with $a=0$ to be infinite


title: Ball Rolling in a Parabolic Bowl

text: I encountered a physics problem which inquired about a ball rolling inside a parabolic bowl (i.e. a bowl where any cross section through the vertex would make a parabolic shape given by $y = kx^2$). The ball was to be released from some initial height $z_0$ and allowed to roll back and forth inside the bowl. At first, I thought that the motion would be simple harmonic for any displacement, since $U(x) = mgh = mgkx^2$, and simple harmonic motion is characterized by a potential energy which is proportional to the square of the distance from some equilibrium point. However, the answer key said that the motion was not truly simple harmonic (and the rolling ball only appeared to be so for small amplitudes of motion). Why is motion with the a potential energy proportional to the square of displacement from equilibrium not simple harmonic in this case? EDIT: To clarify, the "ball" should have really been a point mass.


title: Uniform chain falls off table Diff EQ

text: I really need some assistance setting up this problem. any assistance would be a Godsend: a uniform heavy chain of length `a` initially has length `b` hanging off of a table. The remaining part of chain a - b, is coiled on the table. Show that if the chain is released, the velocity of the chain when the last link leaves the table is $\sqrt{2g\frac{a^3 - b^3}{3a^2}}$ Okay, so this is a variable mass problem, so momentum is constantly changing: $$F_{ext}=m(t)g=\frac{dP}{dt}= \frac{d(m(t)v(t))}{dt}=ma +v\dot{m}$$ $$mg = ma + v\dot{m}$$ Gravity acts on the mass hanging off of the table, mass can be written as a function of length, as can velocity (where $\lambda$ is the linear mass density) $m(t) = \lambda l(t)$   $\dot{m(t)} = \lambda v(t)=\lambda \dot {l(t)}$   $v(t) = \dot {l(t)}$   $ a(t) = \ddot{l(t)}$ $$\lambda l(t)g=\lambda l(t) \ddot{l(t)} + \dot {l(t)} \lambda \dot {l(t)}$$ Assuming this all to be correct $\implies$ $0 =l(t)(g -\ddot{l(t)}) +\dot{l(t)}^2$ I've tried solved this DE, but I don't know many methods for non linear DEs.


title: why does water do that?

text: I found recently that water fills a bottle quicker through only observation if I hold the tube away from the filling water than if I leave the tube inside the bottle. why does that happen? does potential energy have anything to do with this? ![enter image description here](http://i.stack.imgur.com/VUXJU.jpg)


title: Is traveling at the speed of light the same thing as teleportation?

text: If I were on one side of the room and moved at the speed of light to the other side of the room, to an observer it would appear that I teleported. If time stops at that speed, it would be instantaneous but we know that light is not instantaneous (see light years). So I'm curious what the difference between traveling at the speed of light and teleportation is. I think teleportation has to do with deconstructing objects and then reconstructing them which allows them to travel through things etc., but is there any other noticeable difference?


title: Fundamental particles with spin > 1

text: I am in undergraduate quantum mechanics, and the TA made an off-hand comment that currently no one knows how to describe fundamental particles with spin > 1 without supersymmetry. I was curious and tried to look up info on this, and wikipedia does make some comments about troubles with spin 3/2 http://en.wikipedia.org/wiki/Rarita%E2%80%93Schwinger_equation So my questions are   1. Is there an easy to understand reason why photons aren't a problem, but a hypothetical particle with spin 3/2 doesn't work?   2. Does this also mean there are troubles explaining 3/2 composite particles in a low energy regime where we can treat the composite as strongly bound / 'fundamental'?   3. How does supersymmetry help here?


title: Question about exterior derivatives

text: I know from Carroll that the integration in GR is basically a mapping from n-form to the real number. And it's given that $$d^nx=dx^0\wedge\ldots\wedge dx^{n-1}=\frac{1}{n!}\epsilon_{\mu_1\ldots\mu_n}dx^{\mu_1}\ldots dx^{\mu_n}$$ Now, I have an expression that is given in spherical coordinate system, where I have $$\int_\Sigma f(\theta,\phi) d\theta\wedge d\phi$$ when I want to integrate this (the epsilon part is already computed), do I just have $\int\int d\theta d\phi$ to integrate, or do I need to put the part from integration in spherical coordinate system $\int\int\sin\theta d\theta d\phi$? I haven't done much integration that involved forms before, so any help is appreciated :) EDIT: From the article I have: $$k_\xi[h,\bar{g}]=k^{[\nu\mu]}_\xi[h,\bar{g}](d^{n-2}x)_{\nu\mu}$$ $$(d^{n-p}x)_{\mu_1\ldots\mu_p}:=\frac{1}{p!(n-p)!}\epsilon_{\mu_1\ldots\mu_n}dx^{\mu_p+1}\ldots dx^{\mu_n}$$ $$k_\xi^{[\nu\mu]}[h,\bar{g}]=-\frac{\sqrt{-\bar{g}}}{16\pi}\ldots$$ where $\ldots$ is an expression. Does this mean, since I have an $n-2$ form and I'm in 4 dimensional space, I need to include this $\sin\theta$ after all? EDIT2: I need to add the $\sin\theta$. I got it. Thanks :D


title: Newton's second law of motion in terms of momentum

text: I am reading a document and in answer to the question **State Newton’s second law of motion** the candidate answers that **The force acting on an object equals the rate of change of momentum of the object.** While this is not a complete answer, the examiner picks on on a word _equal_ and says that it is _proportional_ instead. Now I understand that $F=\frac{dp}{dt}$ where both $F$ and $p$ are vectors — what is "proportional" about it? I checked my Good Old Ohanian (2nd edition) and it says explicitly "The rate of change of momentum equals force" in section 5.5 _The Momentum of a Particle_. What is this examiner talking about?


title: About $\phi^4$ model

text: In many books the $\phi^4$ model can produce a topological soliton called kink. Are they right? In the case of sine-Gordon model you can have a topological soliton due to you can express the Lagrangian in terms of two scalar fields taking values in a circle. As the field is constrained on a manifold the model could have stability. I am not sure if $\phi^4$ model could be viewed as a field constrained on a manifold also. So my questions are:   * Is it right to talk about topological solitons or kinks in $\phi^4$ model?   * Has the $\phi^4$ model an alternative formulation as a constrained field with target space a circle or any other manifold?


title: Why is electric field strong at sharp edges?

text: I learned about the coronal discharge, and the common explanation is because the electric field is strong where radius of curvature is small. But I haven't found anything yet that explains why electrons like to crowd at the peaks, and and escape from the holes. My intuition suggests electrons try to distribute on the surface as uniform as they can, but they don't. Why?


title: closure phase measurement in astronomy for vlbi technique?

text: I have read this page on closure phase and could not understand the following: \begin{align} \psi_1 & = \phi_1 + e_B - e_C \\\ \psi_2 & = \phi_2 - e_B \\\ \psi_3 & = \phi_3 - e_C. \end{align} If the error in $\phi_3$ is $e_C$ and the error in $\phi_2$ is $e_B$, then why should the error in $\phi_1$ be connected with the errors of $\phi_2$ and $\phi_3$ in the way given?


title: What is the difference which leads to attraction in e+e- scattering and repulsion in e-e- scattering in QFT

text: What is the theoretical difference between the physical elementary interaction that causes an e+ to attract an e− when they exchange a virtual photon? Why is this exchange different from an e-e- scattering (which produces repulsion)?   This question is related to but more specific than the recently posted The exchange of photons gives rise to the electromagnetic force. It has not been answered there. It would be good to see the answer somewhere.


title: 2 Event Horizons come into contact

text: Just a question out of curiosity, what would happen if the event horizons of 2 black holes of the same mass were to come into contact? Would both gravitational accelerations be canceled where the event horizons overlap?


title: How is partial trace simplified?

text: I'm currently reading papers about the effect of repeated measurements, such as "Purification through Zeno-Like Measurements", arXiv:quant-ph/0301026 (DOI: 10.1103/PhysRevLett.90.060401). It says > After N such measurements have been done, the survival probability of > finding system A still in its initial state is represented by > $$P^{(\tau)}(N)=\mathrm{Tr}\left[\left(Oe^{-iH\tau}O\right)^{N}\rho_{0}\left(Oe^{iH\tau}O\right)^{N}\right]=\mathrm{Tr}_{B}\left[\left(V_{\phi}(\tau)\right)^{N}\rho_{B}\left(V_{\phi}^{\dagger}(\tau)\right)^{N}\right] > $$ Notice that the quantity > $V_{\phi}(\tau)=\left\langle\phi\right|e^{-iH\tau}\left|\phi\right\rangle$ > is an operator acting on the Hilbert space of system B. where $O=\left|\phi\right\rangle\left\langle\phi\right|\otimes I_B $ is the projection operator onto an eigenstate of subsystem A, and $\rho_{0}=\left|\phi\right\rangle \left\langle \phi\right|\otimes\rho_{B} $ is the initial density matrix. My question is: how the trace is reduced to the partial trace with respect to system B?


title: Potential energy of a charged ring

text: Consider a ring of radius $R$, and charge density $\rho$. What will be the potential energy of the ring in its self field? The best I can do: $$dq = \rho R \cdot \, d \alpha $$ $$E_p = 2 \pi R \cdot 2 \int_{0}^{\pi - \delta \alpha} \frac{\rho^2 R^2}{r(\alpha)} \, d\alpha$$ where $$r(\alpha) = \sqrt{2} R \sqrt {1+ \cos (\alpha)}$$ **Edit** : This is wrong - $E_p$ doesn't even have the correct dimensions.


title: Should annealed disorder be characterized by the average of the partition function?

text: Most of the literature says that for a quenched average over disorder, an average over the log of the partition function must be taken: \begin{equation} \langle \log Z \rangle, \end{equation} while for the annealed average, it's \begin{equation} \langle Z \rangle. \end{equation} But a while ago, I came across a book that said that the annealed average is not $\langle Z \rangle$, though I don't remember what it said should be calculated instead. Does anyone know which book this is, or what they might want to calculate instead of $\langle Z \rangle$ for the annealed average?


title: Can a photon passing by an open space barycenter of a system of masses  be modeled as if all the system's  mass is at the barycenter?

text: To be clear, this example can't apply to the Solar System, since the barycenter is within the Sun, similarly the Earth/Moon system's barycenter is within the Earth. But, given a system of gravitationally attacting masses revolving about a barycenter that is not contained within any of the masses, i.e. a barycenter in open space, would a photon passing near that point behave in the same manner as a photon passing a point mass of the same mass as the system itself?


title: Air insulation cavity depth: rule of thumb to avoid convection

text: I want to know more about optimising an air gap between two surfaces, such as window panes or a building's walls. I was taught that as a general rule, a 6cm air gap is the optimum. Any larger and convection counters the effectiveness. But something tells me this is hardly an accurate figure! Can anyone improve on this rule of thumb? What are the most significant variables which would alter the optimum separation distance? Examples of what I would want to apply this to are:   * A building's window panes of different sizes: glass, perspex or combinations   * Oven / furnace surfaces of different sizes: between two ceramic refractory walls, or refractory walls and steel. (Temperature differentials and absolutes of perhaps up to 1000 degrees C) Perhaps there are some approximated formulas to calculate optimum separation distance? And if the effective thermal conductivity of the air changes, a formula for that too. Thanks! For interest to other readers, this question is related.


title: What is the true electric potential at the terminals of a battery?

text: I recently studied Laplace's equation and numerical methods for solving it, and so I'm trying to simulate an electron gun. ![Schematic of an electron gun](http://i.stack.imgur.com/Mwsk2.png) (assuming cylindrical symmetry on the Wehnelt, anode, and beam.) The setup is simple. The filament releases electrons, the Wehnelt cylinder is the cathode and helps focus electrons, and the electrons accelerate towards the anode. This was enough for me to set everything up and solve for the electric potential, with potential=0 as distance goes to infinity. There's a problem though: There are an infinite number of ways to set the potential of the cathode/anode that leave the voltage difference the same. For example, with a voltage of 1V between the plates... Cathode set to -1V: ![Cathode set to -1V](http://i.stack.imgur.com/T7iav.png) Cathode set to 0V: ![Cathode set to 0V](http://i.stack.imgur.com/rcU0l.png) (the visualization is a slice through the potential. It is cylindrically symmetric. Space has been compactified so that the midpoint is at the origin but the edges represent infinity, according to the equation $x_{\text{world}}=x/(1-x^2)$, where $-1<x<1$ are the coordinates in the image and both the radial/axial directions behave the same.) The gradients of the potentials are different, so these two potentials represent physically different situations. I realized that I don't know which one is more correct/reasonable. I understand that potentials are relative, but for example if a lab frame is taken to have a potential of 0... which picture is the true picture if I brought in a 1v battery? Are the terminals at $-0.5$V and $0.5$V? Why?


title: Why don't astronauts in orbit get stuck to the "ceiling"?

text: When a shuttle is in orbit, it is essentially rotating around the "centre" of the Earth at a great speed. So why does there seem to be no centrifugal force sticking them to the 'ceiling' of the shuttle? Is it because the shuttle is not actually being rotated around a point but rather continuously falling or something else? Thanks


title: Heisenberg's Uncertainty Forms

text: Can Heisenberg's Uncertainty principle be rewritten in terms of energy density writing $$\Delta E \Delta T \geqslant \hbar/2$$ in factors of energy density $\Delta \sigma \text{ }= \frac{3\Delta E}{4\pi r^3}$ I get $$\Delta \sigma \frac{3\text{$\Delta $T}}{4\pi r^3} \geqslant \hbar/2$$ does $$\frac{3\text{$\Delta $T}}{4\pi r^3}$$ represent something? Or does it make more sense to write $$\text{$\Delta $x} \frac{\text{$\Delta $E}}{c^2} \geqslant \frac{\hbar}{2}$$ EDIT after noting Willie's comment I've corrected the form to $\frac{\hbar}{2}$ However, now I see c is constant and can be moved to the right side - is it mathematical correct to write $$\text{$\Delta $x} \text{$\Delta $E} \geqslant \frac{\hbar c^2}{2}$$ or does this violate the Cauchy–Schwarz inequality? This is the question I was trying to understand at the beginning.


title: Whats left at the center of the Universe after Big bang?

text: If you consider big bang. According to that What's left at the center of universe where the Big bang occured?


title: Relativistic fomulae for energy and momentum?

text: I know that the relativistic formulae for energy and momentum are: $E = \gamma mc^2$ and $\textbf{p} = \gamma m\textbf{v}$; Can we derive these formulae? If yes, where from?


title: Eigenfunction associated with the $\hat{x}$ operator

text: Consider the following operator $\hat{x}=i\hbar \frac{\partial}{\partial p}$. I am trying to show that the eigenfunctions of $\hat{x}$ are not square- normalizable. I am interested in doing so since theoretically, we notice that the eigenfunction of the momentum operator $\hat{p}=\frac{\hbar}{i}\frac{\partial}{\partial x}$ is not square- normalizable. How do we set up the equation for the $\hat{x}$ operator?


title: Fourier series of single tone modulated wave

text: When a single-tone continuous modulating signal modulates a sinusoidal carrier, isn't the modulated wave periodic? If so, can't we apply fourier series and determine the harmonic frequency components instead of finding the frequency components using fourier transform?


title: How should I clean a part before installing it in a vacuum system?

text: What are proven procedures for preparing a part that comes fresh out of the workshop for ultra-high vacuum (UHV) and extremely-high vacuum (XUV)?


title: Monotherm cycle, negative $Q_{net}$

text: In the demonstration of the Clausius theorem in classical thermodynamics my book uses the fact that $Q_{net}\le0$ is negative in a cyclic monotherm (meaning with one heat reservoir) transformation. This fact, however, left me a bit uneasy. Taking, as usual, a Carnot cycle as an example, and calling the adiabatic and the isotherm expansions done, couldn't I just do an adiabatic compression to bring it back to the initial state? The second law wouldn't be violated: the heat absorbed wouldn't _entirely_ be converted into work. The expression of the first principle, though, is indeed a bit strange: since $\Delta U$ is $0$ in any cycle, then $Q = W$, and my reasoning breaks down. I can't really see what's going on here, would it be theoretically possible a process similar to the one I described? If not, why so?


title: How does a speaker produce sound?

text: What I read is that a speaker produces sound by the movement of a coil attached to a cone which moves back and forth. So, If I try to move the coil by hand, would it produce sound? If not, why? or Why would it produce sound only when the current moves the coil?


title: Some questions on observables in QM

text: 1-In QM every observable is described mathematically by a linear Hermitian operator. Does that mean every Hermitian linear operator can represent an observable? 2-What are the criteria to say whether some quantity can be considered as an observable or not? 3-An observable is represented by an operator via a recipe called quantization if it has an analog in classical mechanics. If not, such as spin since it has no classical analog, do we use data from experiment to guess what this operator could look like? are there any other methods for finding that? 4-Are there other observables, besides spin, which also have no classical analog?


title: resonance frequency

text: A string has a mass per unit length of 9 10–3 kg/m. What must be the tension in the string if its second harmonic has the same frequency as the second resonance mode of a 2m long pipe open at one end? The length of the string is 1.5 m. this is my solution http://screensnapr.com/e/u4kvtH.png i just want to ask if my second resonance mode understanding is correct.


title: Identifying hue, brightness and chroma of color and reaction time

text: If someone knows how identify hue, brightness, and chroma of color, please let me know. I am a PhD student at Educational Linguistics UNM.


What is the Jost function in scattering theory? Is it an operator or some kind of determinant? How is it obtained?


title: Zero-zero (lower indicies) term for affine connection ($\Gamma_{00}^\lambda$), why do some terms dissapear?

text: More simply a tensor algebra question, but in General relativity I have the following when I calculate $\Gamma_{00}^\lambda$:- $$ \Gamma_{00}^\lambda = \frac{1}{2}g^{\nu\lambda}\left( \frac{\partial g_{0\nu}}{\partial x^0} + \frac{\partial g_{0\nu}}{\partial x^0} - \frac{\partial g_{00}}{\partial x^\nu} \right) = -\frac{1}{2}g^{\nu\lambda} \frac{\partial g_{00}}{\partial x^\nu} $$ Why are the $\frac{\partial g_{0\nu}}{\partial x^0}$ terms zero?


title: Charged Black holes - What is the Interest in those?

text: Here in this forum (and elsewhre) I read about "charged" black holes. What is the background for this questions? Is this more a question of theroretical interest? In "practice" :=) black holes should be neutral, because matter is neutral globally. A star is charged a little bit, depending on surface temperature and plasma physics in corona. But this charge will be blast away (with surface material) when the interior of the star collapses to form a black hole. Again, when a black hole "eats" matter from around, charge separation might occur, but only to a limited extent, until the voltage developed would cancel the separation mechanism. (As in case of a star)


title: How to introduce generating function in Hamiltonian formalism for field theories?

text: Let's have hamiltonian $$ H(\psi , P ,\partial_{i}\psi ) = P\partial_{0}\psi - L(\psi , \partial_{\mu}\psi ), \quad P = \frac{\partial L}{\partial (\partial_{0}\psi)}. \qquad (.0) $$ I tried to introduce generating function for it. It will be easy if hamiltonian would be covariance, $$ H = P^{\mu}\partial_{\mu}\psi - L: $$ for $$ S{'} = \int L(\psi {'}, \partial_{\mu}\psi {'}){'}d^{4}x = \int (P{'}^{\mu}\partial_{\mu}\psi {'} - H(\psi {'} , P{'}, \partial_{i} \psi {'}))d^{4}x, \qquad (.1) $$ $$ S = \int (P^{\mu}\partial_{\mu}\Psi - H(\psi, P , \partial_{i}\Psi ))d^{4}x \qquad (.2) $$ the principle of invariance of stationarity of action leads to connection of $(.1), (.2)$ through $$ \int \partial_{\alpha}F^{\alpha}(\Psi, P, \partial_{i}\Psi)d^{4}x = \int F^{\alpha}dS_{\alpha}, \quad \delta \int F^{\alpha}dS_{\alpha} = 0. $$ But in the case of $(.0)$ I can only write the connection in form $$ \delta F_{t_{a}}^{t_{b}} = \delta \int \partial_{0}G(\psi , \psi{'}, t)d^{4}x = 0 , $$ where $G$ is some function which is continuous on all space. Is it correct?


title: Are magnetic forces medium-dependent?

text: I am learning the **Magnetic Effects of Electric Currents** at school, and I was given an equation. I was told it would be elaborated on next year. But since I can't seem to bring myself to wait, here it is: > B = µoI/2πr Now, **µ o** is supposed to be the **permeability of vacuum**. This suggests that there exist other values of permeability, which would effect the overall Magnetic force. Now, I still have a lot to learn, so please forgive any silliness that may arise in my question, But is Magnetism, then, medium-dependent? Does the power of a magnet change when in a vacuum so as to then when in air or water or any other **matter** , for that matter?


title: Semiclassical description of EM waves reflection from metallic surfaces

text: Imagine an EM wave impinging on a metal. Fresnel's formulas tell us that no wave can propagate through the metal, or that the transmitted field is an evascent wave with some penetration depth dependent on the refraction index of the metal. If we now would like to zoom into the microscopic physics of the reflection, we would have to take into account the crystalline structure of the metal, which imposes a certain electronic band-structure, which in turn determine the way the electrons respond to the external EM perturbation. We should calculate the reflected field from the back action of the electron motion on the incident field. Do you know of a semiclassical way of solving this problem? Is it feasable to compute numerically a set of coupled Maxwell-Schroedinger equations?


title: What's The Minimum distance?

text: My Friend gave me a question today. The question was.:: > We have a point A. At a distance of $x_0$ from the point. There is a > particle$(P_1)$. Also, a particle is present at the point A $(P_2)$.The > Particle $P_1$ moves with a velocity of $u_1$ towards Point A. At the same > instant the particle $P_2$ moves making a angle $\theta$ with $P_1A$ with a > velocity $u_2$. So what will be the minimum distance between the two > particles when both of them start at the same time. So can anybody help me with this problem. I tried decomposing it into vectors, used $\tan\theta$, but it did not give me a correct answer.


title: Are Mach Diamonds radially symmetrical?

text: Videos like this show a form that suggests a radial symmetry in the gas jet. The same is reinforced by schematic illustrations. But other images, like this, look like there is a twisting structure to the flow. (e.g. the flow is following field lines, or gaining some other accumulated bias due to them) Specifically, I mean that **_it could be seen as a ribbon_** (a flat strip, twisting; i.e. a _helix_ ) ![](http://i.stack.imgur.com/vz7Ze.jpg) * * * In the latter case (as detailed here), I wonder if the "plasma" qualifier has a significant relationship to a (possible) helical flow. Or, maybe this is a picture of a helical flow, but it's not dependent on being a plasma. I am by no means qualified in this area, just curious about what I see and well-read enough to read proper journals. I have turned up some potentially relevant information, here, but haven't digested it, yet


title: What does adding a visible light blocking IR filter do with heat?

text: Lets say I have a strong halogen light source which has a lot of IR radiation (e.g. a construction lamp). If I change the transparent front glass from the enclosure with a IR filter that blocks visible light, what will it do with the heat inside the enclosure?


title: How to detect gravitational waves?

text: Gravitational waves just have a little interaction with other particles. How can we detect such little influence on mass?


title: Why can't helicopters reach mount everest?

text: Is there a reason why people can't just take the helicopter to mount Everest? Why is it that helicopters can't reach that high?


title: What is the ion drag mechanism in dielectric heating?

text: While reading about dielectric heating on Wikipedia, I read about the ion drag mechanism but there wasn't enough information about. I know there is another Phys.SE question talking about the ion drag in heating but the answer isn't explaining the mechanism.


title: Doppler effect- will frequency continually decrease?

text: ![enter image description here](http://i.stack.imgur.com/dUapj.png) ![enter image description here](http://i.stack.imgur.com/3dGdQ.png) My problems:   1. I know that when a person is moving away, the perceived frequency will be lower than the frequency of the source. However, in the question, if the person is moving away, the perceived frequency WILL be lower by why would it continually decrease, would it just remain as $f'$   2. I also want to understand how the Doppler shift comes about... I know if the person is moving away, the successive wave fronts will be further apart. However, in the case of this question, will the person who is moving away instantly hear a difference in frequency as he starts moving or is there somewhat of a gradual transition from high to low frequency?


title: Wheatstone bridge galvanometer error

text: We had to measure the resistance of $R_x$, we balanced the Wheatstone bridge and did calculations. My question is: we didn't include galvanometer error into calculations. Why is that? I read that it's very precise, but that doesn't seem like a good enough explanation in exact sciences :/ Edit: The precision is not the case as I was told, I need to go into more detail. When I measured, I set a value on the potentiometer, then I adjusted the adjustable resistor, so that the galvanometer would show zero. Is the error somehow compensated? ![enter image description here](http://i.stack.imgur.com/yODpD.png) ɛ - electromotive force. K - switch. R - adjustable resistor. C - potentiometer.


title: How does the specific frequency of EM Radiation relate to displacing electrons from their orbits?

text: I've only a general grasp on how all this works, so it could be I'm asking this poorly or misunderstanding what happens. With that said: The energy of EM radiation is a function of its frequency. With enough energy (a high enough frequency) I understand it's possible to knock an electron out of its orbit(al). Let's say I had a neutral hunk of iron, and I want to try to knock out as many 3d6 electrons from it as I can. Beyond simply having a frequency with enough energy, would there be specific frequencies which were best able to knock out those electrons? Or, asked differently, does anyone have a graph of the initial current produced when different frequencies of light hit some substance? If so, does it have local maxima and minima?


title: How does the gluino change the strong force?

text: Given that the gluino is in the same representation of SU(3) that the gluon and so the same kind of interaction vertexes with quarks, I guess that it could have a role in the strong force. But there are obvious differences: being a fermion, there is no a classical field, and as you can not accumulate fermions in the same state surely its role in the formation of a QCD string field at long distances is minor, but it is still possible to have pairs of gluino/antigluino bubbling out from the vacuum. So, how is it? How different is the strong force in a meson with QCD from a meson of sQCD? Can sQCD have mesons of spin 1/2? Has sQCD still a coloured string between quark and antiquark, and if so, has it still the same tension that the QCD string? Is there some remarkable difference between QCD meson and baryons and sQCD composites?


title: Doppler effect and acceleration's impact

text: ![](http://i.imgur.com/KRhpgF2.png) Can anyone explain why they say Doppler effect does not depend on acceleration? Would having acceleration not affect the frequency? If the source emits the first circular wave, moving at 50 m/s towards you and speeding up towards you, by the time it sends out the second circular wave, the velocity will have increased which will affect the displacement of the source. So is f5 really equal to f6 and f7?


title: How similar is comparison between the principle behind Newton's Cradle versus AC Current moving down a transmission line?

text: Anyone who has taken high school physics has seen the following assembly. You drop one ball from the left hand side and the ball from the farthest right hand side gets knocked away. This is to illustrate the conservation of momentum and energy. ![enter image description here](http://i.stack.imgur.com/yXEun.png) I am reminded of this as I read an article on wikipedia on AC current. (See animation here: http://en.wikipedia.org/wiki/Transmission_line) ![enter image description here](http://i.stack.imgur.com/IUVbL.png) In this animation, the electrons are moving on the transmission line just like the motion of the balls on the Newton's Cradle. Can someone elaborate:   1. Whether the motion of the electron displayed in the animation is accurate for AC current moving down a transmission line (are electrons really getting pushed back and forth and why)   2. Principle behind this cyclic motion   3. Is there any correlation between the motion of the balls on the Newton's Cradle and the motion of the electrons?


title: When can photon field amplitudes be written as field operators?

text: Suppose I have some classical field equation for two photon fields with amplitudes $A_1(z),A_2(z)$ (plane waves) given as ${A}_1=\alpha f(A_1,A_2) \\\ {{A}_2}=\beta g(A_1,A_2) $ Under what conditions can I make the replacement $A\rightarrow \hat{A}$ ? I am uncomfortable with the Bohr correspondence principle because the correspondence principle seems to be a very weak argument. Edit: $z$ is position, $\alpha$,$\beta$ are real numbers. $f$ and $g$ are functions of the amplitudes.


title: How can I make a dry dilution refrigerator quiet?

text: The pulse tubes are really loud. We've wrapped the helium lines in sound absorbing foam, and we have a cage around the top of the cryostat also lined with foam, but it's still pretty loud. Are there any really good and practical ways to damp the sound?


title: Which areas of physics are most demanding of good intuition or physical insight?

text: I'm a beginning graduate student, and I have 2 years before I have to pick a specialization. I would like to know which areas of physics call for the most intuition or physical insight. If this sounds like too vague a question, let me give an example. See the problem below: "A rubber band with initial length L has one end tied to a wall. At t = 0, the other end is pulled away from the wall at speed V (assume that the rubber band stretches uniformly). At the same time, an ant located at the end not attached to the wall begins to crawl toward the wall, with speed u relative to the band. Will the ant reach the wall? If so, how much time will it take?" I wasn't able to solve this problem completely, but I could get good qualitative insight into the behavior of the system simply by imagining a few things happening as the rubber band was stretched. Is this possible in just about any area of physics or are there specific areas which contain problems that can be tackled using pure intuition? I really want to be in an area which I enjoy so your answers will be much appreciated.


title: Normal distribution of x, xdot

text: I have some real measurements from a process and I happened to look at the mutual distribution of (x(t), xdot(t)). I found that they seem to follow 2d normal distribution around (mu, 0). See image, the y-axis is xdot. ![x-xdot-histogram](http://i.stack.imgur.com/rQfWk.png) Is there some well known process which exhibit this property? What kind of dynamic equations on x(t) cause this?


title: Is existence of cosmic dark energy falsifiable yet?

text: Is "existence of cosmic dark energy" falsifiable yet? And what about dark matters? Let me rephrase it again(pardon me for my previous poor expression)- Is there any condition discovered which can be used to determine the "existence of cosmic dark energy" is true or false without any ambiguity? _I'm not a physics guy. Just being curious._


title: Are there any reasonable attempts at explaining the OPERA result

text: as there's so many papers on the opera result and it's a struggle flicking through them all combined with my limited knowledge meaning I cannot well differentiate silly nonsensical papers from ones with much more substance; could anybody sum up for me what papers or theories put forward which could decently explain how neutrinos could be superluminal (assuming the result is correct)


title: Find equations of motion from given Lagrangian density

text: Could someone help me solve this probably not very hard problem? Given Lagrangian Density: $\mathcal L=\bar{\psi}(i\gamma^\mu\partial_\mu-g\gamma^5\phi)\psi+\frac{1}{2}(\partial_\mu\phi)(\partial^\mu\phi)-\frac{1}{2}m^2\phi^2$   * Find the equations of motion for the wave-functions $\psi$ and $\phi$   * Demanding parity-invariance of the Lagrangian, show how $\phi$ transforms under parity


title: Grand canonical ensemble: 0 particle state

text: The partition function of the grand canonical ensemble can be generally stated as $$ \mathcal{Z} = \sum_{r} e^{-\beta(E_{r} - N_r\mu)}\tag{1}$$ where $E_{r}$ is the energy of the micro-state $r$ of $N_r$ particles, $\beta$ the usual inverse temperature, and where $\mu$ is the chemical potential. This is not the only way to state $\mathcal{Z}$. Quoting Gould and Tobochnik, > [...] it is possible to distinguish the subset of all particles in a given > single particle microstate from the particles in all other single particle > microstates. For this reason we divide the system of interest into > subsystems each of which is the set of all particles that are in a given > single particle microstate. Because the number of particles in a given > microstate varies, we need to use the grand canonical ensemble and assume > that each subsystem is coupled to a heat bath and a particle reservoir > independently of the other single particle microstates. From this definition it arises that one can _also_ compute $$\mathcal{Z} = \prod_{k} Z_{k}\tag{2}$$ with $Z_k$ the partition function for a subsystem (one particle microstate). And lastly, one can make the dependencies on the number of particle explicit by using the variation (Gould and Tobochnik sec. 6.11 ) $$ \mathcal{Z} = \sum_{N=1}^\infty e^{\beta\mu N} Z_c(N)\tag{3}$$ where $Z_c(N)$ is the _canonical_ partition function of the system of $N$ particles. * * * Now my question is , shouldn't equation (3) be summed from $0$ to $\infty$ rather than 1 to $\infty$? This would yield $$ \mathcal{Z} = \sum_{N=0}^\infty e^{\beta\mu N} Z_c(N)=1+\sum_{N=1}^\infty e^{\beta\mu N} Z_c(N)\tag{*}$$ Stated more clearly: _should we exclude the 0 particle state from the summation_? The rational _for_ excluding it is that   1. the textbook says so,    2. it will end up being a _very_ important term in most situation, since everything else is a decreasing exponential.    3. The interpretation of $Z_c(0)=1$ is rather ambiguous. The reason _for_ including it are:   1. it makes sense physically (all particle in the reservoir), and   2. it is essential for the coherence between eq. (2) and (3). i.e. in the fermion case, $Z_k = 1 +e^{-\beta(e_k-\mu)}$ and one has $\mathcal{Z}=1+....$


title: How do we know that the nucleus isn't a quark-gluon plasma?

text: The standard picture of the nucleus of atom is that is several distinct nucleons, which themselves are composed of quarks. However, it seems to me like a much simpler picture is that the nucleus is directly made out of quarks, without having nucleons as substructures. That is, that the nucleus is the ground state of a quark-gluon plasma. I can't find any evidence that this is false. For instance, if the distances between the nucleons was significantly larger than the size of the nucleons, there would be a difference in size between hydrogen and other atoms which can't be explained by this new framework, but according to wikipedia the diameter of all of the nuclei is the same order of magnitude. What is the evidence that large nuclei have nucleons as substructures?


title: Is solar wind positively charged?

text: Many times I heared that the solar wind consists of protons and alpha particles. The both are positively charged, but are there electrons in solar wind?


title: Small angle approximation for coupled pendulum

text: I am looking at a coupled plane pendulum problem where the pendula are identical with mass $m$ and length $r$, and connected by a spring with spring constant $k$ and an equilibrium seperation $d_0$. If $\theta_1$ and $\theta_2$ are the angular displacement with the vertical then in these coordinates then the Lagrangian (I believe) is $$L=\frac{1}{2}mr^2 \left( \dot{\theta_1^2} + \dot{\theta_2^2} \right) -mgr \left( \cos \theta_1 + \cos \theta_2 \right) -\frac{1}{2}k \left[ \sqrt{\left(r \sin \theta_1 - r \sin \theta_2 \right)^2 + \left(r \cos \theta_1 - r \cos \theta_2 \right)^2} -d_0\right]^2.$$ I'm trying to rewrite the approximate Lagrangian for small oscillations about the bottom in terms of $\eta_1$ and $\eta_2$ where $\eta_{\alpha}$ represents the horizontal displacements. I'm not sure how to go about doing this. Am I supposed to assume that $\theta_1=\theta_2=0$ is an equilibrium point and then expand the potential to second degree? In order to do this, can I assume that my $-mgr\left(\cos \theta_1 - \cos \theta_2 \right)$ can be approximated as $$-mgr\left(\cos \theta_1 - \cos \theta_2 \right) \approx mgr\left(\frac{\theta_1^2 + \theta_2^2}{2} \right)$$ and then for the spring terms write $$\frac{1}{2}k \left[ \sqrt{\left(r \sin \theta_1 - r \sin \theta_2 \right)^2 + \left(r \cos \theta_1 - r \cos \theta_2 \right)^2} -d_0\right]^2=P(\theta_1, \theta_2)$$ $$\frac{1}{2}k \left[ \sqrt{\left(r \sin \theta_1 - r \sin \theta_2 \right)^2 + \left(r \cos \theta_1 - r \cos \theta_2 \right)^2} -d_0\right]^2 \approx \frac{\partial^2 P}{\partial \theta_{\alpha} \partial \theta_{\beta}}\bigg|_{\theta_1=\theta_2=0} \theta_{\alpha} \theta_{\beta} $$ Then finally combine all like terms? Or before doing this can I simplify the root by writing $\sin \theta_{\alpha} = 0$ before doing the partial derivatives? I'm not sure which process is correct here. Any help would be appreciated. Thanks. **\------------------------EDIT----------------------------** If I write $$V(\theta_1, \theta_2) = mgr \cos \theta_1 + mgr \cos \theta_2 + \frac{1}{2}k \left[ \sqrt{\left(r \sin \theta_1 - r \sin \theta_2 \right)^2 + \left(r \cos \theta_1 - r \cos \theta_2 \right)^2} -d_0\right]^2$$ then according to my calculations $$\frac{\partial^2 V}{\partial \theta_1^2} = -\frac{1}{2}kr(2r-d)-mgr = \frac{\partial^2 V}{\partial \theta_2^2}$$ $$\frac{\partial^2 V}{\partial \theta_1 \partial \theta_2} = -\frac{1}{2}kr(2r-d_0)$$ Can I therefore write the Lagrangian as $$L \approx \frac{1}{2}mr^2 \left( \dot{\theta_1^2} + \dot{\theta_2^2} \right) + \frac{1}{2} \left[ \frac{1}{2}kr(2r-d_0) + mgr \right] \left(\theta_1^2 + \theta_2^2 \right)+\frac{1}{2}kr(2r-d_0) \theta_1 \theta_2$$ and then proceed to find the vibrational frequencies and normal modes using the K-M matrix procedure (after making the $\eta$ substitutions)? I'm wondering if there are any simplifications I can make to the potential (i.e. treat the approximation of the potential in multiple separate parts where I take derivatives of each individually) before I start taking partial derivatives to simplify the problem. Or does a multidimensional potential such as this always have to be treated as one massive beast?


title: Effective mass of a particle

text: I was reading about concept of effective mass on Wikipedia and came across the line that effective mass of a particle can be negative and even zero. How is this possible?


title: Is there a formula to work out how much the fabric of spacetime bends?

text: From my knowledge, a big mass (planet star etc) can bend the fabric of spacetime. Is there a formula that we can use to work out how much it bends?


title: Ising Ferromagnet: Spontaneous symmetry breaking or not?

text: In explaining/introducing second-order phase transition using Ising system as an example, it is shown via mean-field theory that there are two magnetized phases below the critical temperature. This derivation is done for zero external magnetic field $B=0$ and termed spontaneous symmetry breaking The magnetic field is then called the symmetry breaking field. But, if the symmetry breaking occurs "spontaneously" at zero external field why do we need to call the external magnetic field the symmetry breaking field? I am confused by the terminology.


title: Finding Max Radius of Propeller (angular velocity)

text: I am looking at a question from University Physics ![enter image description here](http://i.stack.imgur.com/loQo7.png) The given answer ![enter image description here](http://i.stack.imgur.com/cgP7p.png) Whats the intuition behind using the below diagram ![enter image description here](http://i.stack.imgur.com/BX14V.png) of finding $v_{tip}$? I was looking the the $v_{tan}$ not knowing how to continue. Also $v_{tip}$ will rotate and turn positive and negative


title: Angle of rays leaving a light source

text: I'm working on ray tracing and I'm trying to understand the impact of the angle at which a light-path intersects the surface of a light source on the amount of light that source delivers to the path (per unit time, of course). My assumption for an ideal diffuse light source is that light rays leaving a particular point on the surface of the light source are uniformly distributed, in which case the angle of the path is irrelevant. I think that in essence, this assumption models each point on the diffuse surface of the light as an individual point-light source. Is that a fair assumption for something like a frosted light bulb, where I'm modeling the light as coming directly from the surface of the bulb, not the filament? Or alternatively, if I wanted to actually model the filament itself, would the same assumption be relatively valid? Are there any light sources for which this assumption would not work? I'm assuming lasers would not work this way, with most of the light leaving in one particular direction, but I'm not really worried about modeling laser light. For spot lights, I believe this is still just an ordinary "diffuse" light source which uses mirrors, an aperture, and lenses to direct the light once it has already left the light source so I could model it this way (though I suppose a more efficient model would encapsulate all of this and take the angle of the path into account).


title: Andrew's experiment

text: In Thomas Andrew's experiment, consider the dome shaped saturation region. If we increase the pressure at constant volume until we reach the critical point, why does the density of vapours rise and the density of liquid fall? Moreover, why will the meniscus or the separation boundary of two phase slowly become less distinct and finally at critical point disappear?  ![enter image description here](http://i.stack.imgur.com/LXo52.gif) Will vapour pressure increase or decrease as we move vertically up in the P-V plane in the saturation region?