title: Force of a solenoid on a ferromagnetic material

text: As I was trying to find an answer to another problem, I was informed that the equation I was using would not work. The equation I was using was $$F = (NI)^2\mu_0\frac{\text{Area}}{2g^2}$$ If this is indeed wrong, could someone please explain how it is wrong? I have spent some time trying to figure out what's wrong, and I can't seem to find the issue. It seems to me, like my only problem is the one I asked about in my original question. TIA, to anyone who can clarify this problem. **Edit:** I may have just solved my own problem. I think I may have written the equation in my other question wrong. Looking at it, I don't know how I missed this before, it looks like I wrote it as $$F = (NI)\mu_0\frac{\text{Area}}{2g^2}$$ and not the correct equation of $$F = (NI)^2\mu_0\frac{\text{Area}}{2g^2}$$ While I believe this is correct, could someone please verify? I am going to be rearranging it to find the number of coils I require in my solenoid, and I really don't want to get this wrong. **Edit:** So, in order to help this along, I am going to do an example. As an aside, I was asking my other question because I was getting some funny numbers out of this when I used a value of Area taken from plane **b**. Setup: So, if I am presented with a cylinder 25mm long, and 5mm around, and I am asked to find the number of turns a solenoid that exerts a total force of 1 Newton/second on the cylinder at a distance of 5mm, and that uses 3 Amperes current must have, I would start by rearranging my solenoid force equation. $$F = (NI)^2\mu_0\frac{\text{Area}}{2g^2}$$ * * * $$\frac{F}{\mu_0} = (NI)^2\frac{\text{Area}}{2g^2}$$ * * * $$\frac{F}{\mu_0\frac{\text{Area}}{2g^2}} = (NI)^2$$ * * * $$\sqrt{\frac{F}{\mu_0\frac{\text{Area}}{2g^2}}} = NI$$ * * * $$\sqrt{\frac{F}{\mu_0\frac{\text{Area}}{2g^2}}}/I = N$$ Where F is the force exerted on the cylinder in Newtons, N is the number of turns, I is the current passing through the coil, μ₀ is 4π x 10^-7, Area is is the area of the cylinder in plane **b** (See photo), and g is the distance separating. the cylinder from the coil. Example of planes passing through a cylinder. ![Planes](http://i.stack.imgur.com/LMkGq.png) * * * $$\sqrt{\frac{(1)}{(4π · 10^{-7})\frac{\text{Area}}{2(15)^2}}}/(3) = N$$ * * * $$\sqrt{\frac{1}{4π · 10^{-7}·\frac{\text{(19.634375)}}{2(15)^2}}}/3 = N$$ And we know area is _about_ 19.634375 because area is π·r^2 and the radius was 2.5mm. * * * $$\sqrt{\frac{1}{4π · 10^{-7}· 19.634375}}/3 = N$$ * * * $$\sqrt{\frac{1}{0.000024673283}}/3 = N$$ * * * $$\sqrt{40529.66927830399}/3 = N$$ * * * $$201.3198183942753/3 = N$$ * * * $$N = 67.1066061314251$$ * * * $$N = 67.1 \text{ Turns}$$ I don't believe I missed anything but if I did something wrong, please tell me. I hope this helps determine if this is the correct equation or not. **Edit:** I don't suppose anyone can check my math?


title: Skin depth of current density in magnetic conductor at boundary between two different materials

text: Imagine a magnetic conductor with a cylindrical cross section, surrounded by a coil with a time varying current of $$I = I_0\cdot \cos (2\pi f t)$$ The conductor is split into two parts, the first with a conductivity and a relative permeability of $\kappa, \mu$, the second with $4\kappa, \mu$. There is a magnetic field $B$ through the conductor, which is caused by the current and therefore time varying as well: $$B = B_0\cdot \cos (2\pi f t)$$ ![enter image description here](http://i.stack.imgur.com/RPy2b.png) The change of this magnetic field induces a voltage inside the material and causes a current density $J$. This current density has the value $J_1$ on the surface of the left conductor and $J_2$ on the right side. The skin depth $\delta$ is defined by the distance from the surface where $J = 0.37 \cdot J_1$, respectively $J = 0.37 \cdot J_2$ , with $0.37 = 1/e$ and also: $$\delta = \frac{1}{\sqrt{\pi f\kappa\mu}} = \frac{\sqrt{2j}}{\alpha}$$ where a $\alpha$ is the propagation constant. I found out by simulation, that at the boundary between both materials, the blue one, and the orange one, applies: $$\frac{1}{\delta_{12}} = \frac{1}{2}(\frac{1}{\delta_{1}}+\frac{1}{\delta_{2}})$$ and therefore $$\alpha_{12} = \frac{1}{2}(\alpha_1 + \alpha_2)$$ But I'm really struggling to prove that. Can someone give me some hints, how I could get these relations analytically? * * * Here another plot: The upper one shows the current density at the surface. The second one shows the contour line where the current density decreased about $63\% = skin depth$. At $z=0$ is the boundary between both materials. Though the current density is a step function, the skin depth is continuous and has the value $\delta_{12}=\frac{2}{\frac{1}{\delta_{1}}+\frac{1}{\delta_{2}}}$ at $z=0$. ![enter image description here](http://i.stack.imgur.com/1ra0t.png)


title: Information Loss in annihilation

text: The concept of information loss is usually discussed with respect to a black hole. My understanding is that whatever matter you put into the black hole, it has only 3 "hairs" and so one doesn't know, just by determining the properties of the black hole, the mechanism by which the black hole was formed. There have been many developments with many people now believing that information is not really lost but gets mangled, etc. Why is this loss of information not discussed in a far more pedestrian context? If you have a particle and an anti-particle annihilating into two photons, say; by observing the photons, you cannot reconstruct the velocities of the two particles. Have you lost information in this case? Is this concept of information identical to the Shannon definition? If annihilation is unitary, with entropy being conserved, I understand that Shannon information is also conserved. But, we cannot reverse-evolve to a unique initial state, can we? (Velocity isn't Lorentz-invariant, but, let us say that everything is carried out in a single inertial frame.) More generally, I don't understand how information is not lost in so many processes that are many-to-one because of the nature of particle physics and why this is different from the scenario with black holes.


title: How to calculate this functional derivative?

text: How can I calculate the functional derivative of this functional? $$F[x(t)] = \int_{0}^{t}x(t_1)a(t_1)\left \\{ \int_{0}^{t_1}x(t_2)b(t_2) \,dt_2\right \\} dt_1 .$$ Where $a(t)$ and $b(t)$ are real fixed functions. Following the usual method, I'm getting $$\frac{\delta F\left [ x \right ]}{\delta x(s) } = \int_{0}^{s}\left [ a(t')b(s)+a(s)b(t') \right ]x(t')dt'.$$ which seems odd since that is also a functional, and I think that I should be getting a function.


title: Is there a minimum energy content of information, other than 0 Joules?

text: Lets say I want to send the bit string 010110 to someone. Is there a theoretical lower bound on the energy needed to do this?


title: Polarization vectors in scattering amplitudes

text: The annihilation ($e^+ e^-\rightarrow 2\gamma$) diagrams are ![Annihilation \(time goes upwards\)](http://i.stack.imgur.com/3nUwx.png) I'm just wondering if the amplitudes should be $$(-ie)^2\left[\bar v(p_2)\gamma^b\epsilon_b^*\frac{-i(-(\gamma^\mu p_{1\mu}-\gamma^\nu p_{3\nu})+m)}{(p_1-p_3)^2+m^2-i\epsilon}\gamma^a\epsilon_a^* u(p_1)+ \\\ \bar v(p_2)\gamma^a\epsilon_a^*\frac{-i(-(\gamma^\mu p_{1\mu}-\gamma^\nu p_{4\nu})+m)}{(p_1-p_4)^2+m^2-i\epsilon}\gamma^b\epsilon_b^* u(p_1)\right]$$ In particular, are the polarization vectors for outgoing photons correct? In general, do external photons always give a polarisation vector and an $\gamma^a$ contracted together? Also, if $\epsilon^a\rightarrow\epsilon^a+\alpha k^a$ for a constant $\alpha$, why is the amplitude unchanged and what does this mean physically?


title: How to figure out the magnets needed to repulse a given weight?

text: Given a weight. Is it possible to determine what strength of magnet and what width, length and thickness would be needed in order to repulse that weight a given height? If so, how? To be clear: Two magnets, one on top of the other, repulsing. ||=========||   ^^^^air^^^^   ||=========|| Now I'll place an object of a certain weight on top. I want the object to float in mid air at a certain height. I am trying to figure out how to calculate what strength magnet I'd need in order to make the object float at a given height, given a weight of said object. Pick any type of magnet.


title: No well-defined frequency for a wave packet?

text: There are similar questions to mine on this site, but not quite what I am asking (I think). The de Broglie relations for energy and momentum $$ \lambda = \frac{h}{p}, \\\ \nu = E/h .$$ equate a specific frequency and wavelength to a particle, yet we know that a wave packet is a linear combination of an infinite range of frequencies and wavelengths. How is it that we (or nature) choose one frequency and wavelength out of the range? Does this have to do with the collapse of the wave packet when measured? And if so, is the resulting measured frequency a random outcome? Similarly, when an electron jumps from one energy level to another in an atom, it emits a photon of frequency $$ \ \nu = \Delta E/h .$$ Since the photon is not a pure sinusoidal wave, how can a single frequency be ascribed to the photon?


title: Can facial surgical mask effectively filter particles of smoke pollution?

text: Singapore is having an episode of its regular hazy season because of smoke from forest fires. Some people have been using surgical masks like the the one shown at the bottom of this post, presumably for health reasons. I know the mask has to stop unwanted particles from getting through. But at the same time, it has to let in enough air to breathe. I understand how it could work to keep people from getting sick - after all, bacteria are much larger than air molecules, so you could have little holes that let air through but keep bacteria out. **But aren't smoke particles too small to be stopped in this way?** I think of smoke as a gas (it certainly looks that way, e.g., coming off a candle or a piece of burning incense). Are smoke particles as small as the molecules that I need to breathe in to survive? If they are, how can this mask differentiate between, say, oxygen and smoke? ![surgical mask](http://upload.wikimedia.org/wikipedia/commons/1/1c/Blossom_3_Ply_Bacterial_Filter_Masks_Elestic_Earloop_Face_Mask_Surgical_Masks_a.jpg)


title: Charging by induction (and grounding)?

text: The very familiar phenomenon of charging by induction includes bringing a charged object near an uncharged sphere to induce a separation of charges in it and then grounding the charge (same as the charge of the body brought close), which is at the end away from the one where the object is brought close. But my question:- The fact, is the resultant electric field in any conductor is zero and it is an equipotential surface. Therefore before the grounding, the field due to the charged body brought near (say +), the induced negative charge(say q-), and that due to the positive induced charge(q+) at the other end gave a null field within the sphere. But when the q+ is grounded, wouldn't the resultant field of the two remaining charges turn out to be non-zero as the charge which was grounded had some contribution in making the field zero.   Next question, the "grounding" occurs as the earth is almost at zero potential and therefore when a charged body at some higher or lower potential is attached to earth, the charge flows in and out to ultimately bring the object at zero potential. But in the induction example above, the positive charge got grounded due to potential difference but the negative charge didn't. why? Shouldn't the negative charge too get grounded?


title: Why does a smaller acoustic impedance mismatch produce lower transmission?

text: I'm attempting to solve a relatively trivial problem, but cannot seem to convince myself the answer I'm getting is true. I have two scenarios. **Scenario 1.** _An acoustic wave propagates from Medium W to Medium G._               Medium W    ¦    Medium G      **Scenario 2.** _An acoustic wave propagates from Medium W to Medium P to Medium G._               Medium W    ¦    Medium P    ¦    Medium G      Now **W** denotes water, **P** denotes some polymer and **G** denotes a type of glass. I would think that the transmission from water to glass would be low because of the high acoustic impedance mismatch. I would also think that the transmission from water to the polymer would be high, because of the low acoustic impedance mismatch. However, using $$T = \frac{2Z_1}{Z_1 + Z_0}$$ where Z1 is the medium into which the field is propagating, I get that the transmission in scenario 1 is 88.56% and in scenario 2 it is 42.35%. I using an acoustic impedance (units of Pa.s.m-1) of:               water 1,480,000      polymer 2,758,460       glass 23,676,565      Would anyone care to hazard a guess as to why the transmission in scenario 2 is so much lower?


title: Integral equations contradict The Uncertainty Principle?

text: I was reading about Integral equations, and I found this excerpt in Portuguese Wikipedia: > "integral equations serve to determine the position in all instances of an > object, if known, its instantaneous velocity at all times" But reading about the Uncertainty Principle, I found this excerpt: > For instance, in 1927, Werner Heisenberg stated that the more precisely the > position of some particle is determined, the less precisely its momentum can > be known, and vice versa. What's wrong in my conclusion? Really the two principles contradict each other?


title: Quantum levitation (locking) 3mm thick disk could carry 1000kg small car claim

text: I watched this TED presentation: http://www.ted.com/talks/boaz_almog_levitates_a_superconductor.html It is about superconductivity and quantum levitation. It tells that super- thin, three-inch disk can levitate something 70,000 times its own weight. In exact terms, presentation @9:40 says: 3mm thick disk could carry 1000kg small car. And author says "I can hold small car in my hand". Is he implying that if holds superconducting wafer with a car locked in using Meissner effect he will not feel weight of a car on his hand?


title: Lorentz transformations of the polarization vector

text: ![enter image description here](http://i.stack.imgur.com/jt63k.png) Let $\bf{n}'$ be a unit vector in the direction of a wavevector in the plasma rest frame and $\bf{B'}$ be a unit vector along the magnetic field in the plasma rest frame. The electric field of a linearly polarized electromagnetic wave is directed along the unit vector $\hat{e}=\bf{n'} \times \hat{B'}$, and the magnetic field of the wave is along the unit vector $\hat{b'}=\bf{b}\times \hat{e'}$, such that the Poynting flux along is $\hat{e'}\times \hat{b'}$ directed along $\bf{n'}$. We give a Lorentz boost to the explosion frame to find the electric field $e$ there, normalize it to unity, and project e$$ on some given direction (e.g. , along the projection of the flow axis on the plane of the sky). Fields in the wave expressed in terms of the direction of a photon in the explosion frame $\bf{n}$ is \begin{equation} \hat{e'} ~=~\frac{\bf{b} \times \hat{B'}}{\Gamma (1-\bf{n}\cdot\bf{v})}+\frac{1+\Gamma (1-\bf{n}\cdot \bf{v})}{(1+\Gamma(1-\bf{n}\cdot \bf{v}))} \end{equation} where $\Gamma=\frac{1}{\sqrt{1-v^2/c^2}}$ is the bulk Lorentz factor. Would some one help to deduce the above equation?


title: Can one make an axial mode circularly polarized radio wave filter with purely conducting materials?

text: Can one construct a filter out of nothing but pure metal that will pass only right circular polarized radiation and reflect left circularly polarized radiation?   What would it look like? A helical spring in a tube?   What about a twisted metalic ribbon inside a circular conducting tube? An axial mode helical antenna transmits circularly polarized radiation.   If I put an axial mode antenna (perhaps with a resistive load?) in a waveguide, can I create a filter that passes only say left circularly polarized radiation?   I would ideally like to do it with only conductive (i.e. not resistive) material.   I would also like it to reflect (not absorb) radiation of the opposite circular polarization.   Can someone tell me how to do this?


title: Higgs Boson mass in Electron volts?

text: Im no physics genius here, I was just interested in the Higgs Boson so I was reading this article : How the Discovery of the Higgs Boson Could Break Physics I came across this > Furthermore, all indications are that scientists will find that the Higgs > **weighs 125 gigaelectronvolts (GeV)** – or about 125 times more than a > proton – which means that it sits exactly where the Standard Model expected > it to be. How can weight be measured in electron-volts?


title: How many stabilised qubits have been achieved in Quantum Computing?

text: The latest I read is 3 but that was in Oct. With Lene Hau of Harvard's "frozen light" and with quantum donuts, newer strategies for stabilization are appearing, but the problem of keeping the qubit in superposition for long enough to sample it seems boggling. http://www.sciencedaily.com/releases/2009/03/090309105026.htm http://www.seas.harvard.edu/haulab/publications/HauPublications_All.htm


title: Question on dielectric sphere

text: The question is: Suppose the field inside a large piece of dielectric is $E(0)$ so that $D(0) =$$\epsilon(0)E(0)$ + $P$. Now a small spherical cavity is hollowed out of the material. Find the displacement($D$) in the center of the cavity in terms of $D(0)$ and $P$. Now, the question had a hint: carving out a cavity is the same as superimposing an object of the same shape but opposite polarization. Using that hint the field at center of sphere = $E(0)$ - field due to uniform polarization of sphere. So we get, $E=E(0)+P/(3\epsilon(0))$ To get $D$ it should be $\epsilon(0)E$ and not $\epsilon(0)E$ + $P$. Why is it so? We have a field $E(0)$ which gives rise to polarization $P$ of the dielectric material. So we get $D(0)$. So now while calculating $D$ at center why don’t we consider $P$? Is it because that, if I do consider $P$, then I would somehow consider its contribution twice?


title: What is the effect of surface tension on an air bubble inside water?

text: When an air bubble is formed inside water, and while it's still inside, does a force due to surface tension act on it? Why and where does this force act? I read somewhere that a bubble would detach from the bottom of a container filled with a liquid if the buoyant force exceeds the force due to surface tension. If surface tension is a property of a liquid surface, why does it work at the bottom, well inside the liquid? I think it follows from something very basic, but I just can't grasp it. EDIT : I think I figured it out. Is it beacuse the liquid pulls the layer of liquid surrounding the bubble?


title: Gravity force between two objects with different mass

text: We know thanks to Newton that: $$F=G\frac{m_1\cdot m_2}{r^2}$$ Where $G$ is the Gravitational Constant that is about $6.673\cdot10^{-11}$ $m_1$ and $m_2$ are the masses of two different objects $r$ is the distance between them. We also know that $F=ma$ so can we write this? : $$ma=G\frac{m_1\cdot m_2}{r^2}$$ If so what is $m$ in the first term of the equation? $m_1$ or $m_2$? Thanks a lot for help!!!


title: The difference between The Dilaton and The Radion?

text: I have read this question on the Dilaton, but I am a little confused with the distinction between the Dilaton and the Radion. I definitely have the feeling that these two scalar fields are different particles. I am fimiliar with the Diliton being related to the coupling constant in String Perturbation Theory. $$g_s = e^{\langle \phi\rangle}$$ Moreover, the Dilaton is related to the size of a compactificated dimension. This was covered in our Bosonic String Theory lectures (I cannot link from outside the network). On the other hand, the Radion is usually the name given to the $g_{55}$ (or $g_{zz}$ in the notes referenced) component of the metric tensor in a Kaluza- Klein theory, and too is related to the size of the compactified dimension $$ \hat{g}_{zz} = \exp(2 \beta \phi) $$ giving a four-dimensional effective field theory $$ \mathcal{L} = \sqrt{-\hat{g}}\hat{\mathcal{R}} = \sqrt{-g}\left(\mathcal{R} - \frac{1}{2}(\partial \phi)^2 \frac{-1}{4} \exp \left(-2 (D-1) \alpha \phi \right) \mathcal{F}^2 \right) $$ Furthermore, in Randal-Sundrum model I have seen the scalar field called a Radion, even though here we explicitly avoid compactification. Finally, in the Cyclic Model of the Universe I have heard the moduli scalar field which 'measures' the distance of seperation between the two branes the Radion. I have been studying the original paper on the Alternative to Compactification and a review on the Cyclic Model of the universe, as well the lecture notes on Kaluza-Klein Theory by C. Pope to try to learn about these things. User1504 mentions that they are the same in M-Theory and Type IIA string theory, but I am afraid that I have not studied Superstring theory or beyone yet. So to reiterate, my question is, can anyone give me a discription of the difference between the Dilaton and the Radion?


What happens when molten metal cools within a strong magnetic field? I don't know what more say, don't remember anything relevant to this in my uni text books.


title: Calculating an energy of an electron with known De Broglie wavelength (why can't we calculate it similar than we do it for a photon)

text: Lets say we have an electron with known De Broglie wavelength $\lambda$. Can anyone justify or explain why we calculate its energy $E$ using **1st** the De Broglie relation $\lambda = h/p$ to get momentum $p$ and **2nd** using the invariant interval to calculate $E$: \begin{align} p^2c^2 &= E^2 - {E_0}^2\\\ E &= \sqrt{p^2c^2 + {E_0}^2} \end{align} Why we are not alowed to do it like we do it for a photon: \begin{align} E=h\nu = h\frac{c}{\lambda} \end{align} These equations return different results.


title: What does it mean that Bruno can hit $6\,\rm{kN}$?

text: Is it the momentum? But momentum is $\rm{Ns}$, not $\rm{N}$. http://www.skepticforum.com/viewtopic.php?f=32&t=14624 > I'm not sure that 3,430 N is necessarily beyond human capability. Boxer > Frank Bruno's punch was in excess of 6,000 N. Unit of momentum is $\rm{Ns}$, not $\rm{N}$. Does it mean that Frank Bruno punch can excert force of $6\,\rm{kN}$? ![enter image description here](http://i.stack.imgur.com/q1l0z.jpg) That means with one hand, he can excert force enough to lift $.6$ tons of weight? The problem with this interpretation is that the amount of force yielded by a hand is very dependent on how the momentum of the punch distributed through time. An iron hand would generate huge force while a rubber gloves would generate a very low force even though the momentum is the same. Okay, back to the question. What does it mean anyway by a $6\,\rm{kN}$ punch force?


title: Are all natural processes that are bigger than a molecule also deterministic?

text: Is there something bigger than a molecule and still random in nature? I know that the decay of an unstable atom cannot be predicted, only the probability of decay can be calculated. Molecules moving in a fluid also could be random. But is there anything bigger than that that could be non-deterministic? Does randomness stop at the molecular level?


title: Why does global supersymmetry commute with gauge transformations?

text: In particular, I would like to understand the following quotation from a paper by Witten: Nucl.Phys. B188 (1981) 513 (p. 515 at the top) His statement: > This is so because in global supersymmetry the superalgebras always commute > with gauge symmetries; the commutator, if not zero, would be a supersymmetry > transformation depending on the infinite number of parameters of the gauge > symmetry, so would have to be local supersymmetry. Could someone please explain the last statement about the infinite number of parameters in a more explicit way?


title: How does spin appear in QFT?

text: In QFT, as I read, it appears naturally. It is connected with Poincare algebra, doesn't it? __ As explanation of the main part of the question. Operator of relativistic orbital angular momentum 4-tensor and 4-impulse operator creates Poincare algebra. It follows that eigenvalues of vector of angular momentum operator are expressed through the whole or half-integer values (in $\hbar $ units). But using only orbital momentum operator causes the possibility of having only integer values. We can artificially add operator, which implements an irreducible representation, so it doesn't connected with coordinate representation and may have half-integer values. But this method is an artificial, because without experimantal proof of existing of spin we can easily operate only with orbital angular momentum. In contrast, in quantum field theory, the spin occurs more naturally (?).


title: Why is glass much more transparent than water?

text: There is a related question (Why glass is transparent?) but I am coming at it only from Maxwell's equations. One can determine the skin depth $δ$ for poor conductors like (pure) water and glass using (see Wikipedia) $$δ =2ρ \sqrt{\frac{ϵ}{μ_0}}$$ If I ignore the frequency dependence of the permittivity (only to get a board range for the skin depth of glass), using appropriate values for the resistivity ρ (water = $2.5×10^5$ Ω∙m and glass = $10^{10}−10^{14}$ Ω∙m), electric permittivity ($ϵ=ϵ_0ϵ_r$) and magnetic permeability ($μ ≈ μ_0$), I calculate that $$δ(water) =10^4m$$ $$δ(glass) =10^8-10^{12} m$$ Maxwell’s equations determine the behavior of electromagnetic waves in conductors (as well as poor conductors), so if glass and water have such larger skin depths, then this is the reason why light is transparent for these two mediums – right? If so, I then have two related questions:   1. Mathematically, it’s fairly straight forward to show that the skin depth is independent of frequency. However, is there a physical explanation why the skin depth is independent of frequency for poor conductors but not for good conductors?   2. At least at optical frequencies, the skin depth is mainly dependent on the resistivity of the material. Since glass has a higher resistivity (is a poorer conductor) than water, electromagnetic waves penetrate farther through glass. So the key to understanding why glass is more transparent than water is physically understanding why δ ∝ ρ?  I have looked through the books of Griffiths and Jackson for help on this, and found nothing. Thank you in advance for any help on these questions. **Correction and edit due to Johannes’s comment below for question 2**


title: Uniqueness of the number of degrees of freedom

text: As per my knowledge, degrees of freedom of any physical system are the number of independent quantities(coordinates) which need to be specified in order to specify the state of a system uniquely. However, my question is, why is there a unique number of degrees of freedom for any system? Or, to put it differently, if a set of $n$ independent coordinates describes the state of the system completely, why is it true that any other set of $n$ independent coordinates also describes the state of the system completely? And why can there not be a set of $m$ independent coordinated which also describe the state of the system completely, where $m>n$ ?


title: Were the physical constants chosen randomly by the nature?

text: Were the physical constants chosen randomly by the nature or it was determined by some source of… _matter_ (?). I really don't have idea. If you could help me it'd be awesome.


title: What will happpen between a proton and an anti-neutron with unmatched color charges?

text: Maybe a stupid question because my search didn't return anything. Assuming in low energy state, a proton (made by a red up, a blue up and green down quarks) meets an anti-neutron (made by an anti-green up, an anti-red down and an anti-blue down quarks). Will they stay together without annihilation because of unmatched color charges? Or gluons will be emitted to change colors so annihilation happens?


title: How to solve bound states of 2D finite rectangular square well?

text: I want to solve bound states (in fact only base state is needed) of time- independent Schrodinger equation with a 2D **finite** rectangular square well \begin{equation}V(x,y)=\cases{0,&$ |x|\le a \text{ and } |y|\le b$ \\\ V_0,&\text{otherwise}}.\tag{1}\end{equation} $$\Big[-\frac{\hbar^2}{2m}(\partial_x^2+\partial_y^2)+V(x,y)\Big]\psi(x,y)=E\psi(x,y)$$ At first glance, this problem is simple. It seems that the solution is variable-separable and can be written as $\psi(x,y)=f(x)g(y)$. Then $$ \frac{f''(x)}{f(x)}+\frac{g''(y)}{g(y)}+\frac{2m}{\hbar^2}(E-V)=0.$$ Let $E=E_x+E_y$ and $V=V_x+V_y$, then the problem is reduced to two 1D problems $$\cases{f''(x)+\frac{2m}{\hbar^2}(E_x- V_x)f(x)=0\\\g''(y)+\frac{2m}{\hbar^2}(E_y-V_y)g(y)=0}.$$ However, how to determine $V_x$ and $V_y$ in the 2D space? A definitely wrong method is making $$ V_x=\cases{0,&$|x|\le a$\\\V_1,&$|x|>a$}\text{ and }V_y=\cases{0,&$|y|\le b$\\\V_2,&$|y|>b$}\tag{2}.$$ In fact, the potential Eq. (2) is equivalent to two independent "1D finite square well" problems in $x$ and $y$ direction respectively. However, a careful reader will note that the potential Eq(2) is DIFFERENT from Eq(1), which means that the potential Eq(2) is NOT what we want. It's not a rectangular well, but as following ![Potential of Eq\(2\), but NOT a 2D square well.](http://i.stack.imgur.com/IO5Xu.png). Then, I find that a variable-separable bound state for finite 2D square well does not exist. Although analytical solutions exist in each region with a constant potential, problems occur when matching boundary conditions to keep the continuity of $\psi(x,y)$. Unlike matching boundary condition at descrete points in 1D, in 2D we have to match boundary conditions along lines, e.g., $$ f_1(a)g_1(y)=f_2(a)g_2(y)$$ in the boundary between $x<a$(region 1) and $x>a$ (region 2). This leads to $$ g_1(y)/g_2(y)=f_2(a)/f_1(a)=constant.$$ Matching all boundaries this way will lead to that $\psi(x,y)$ have to be 0 outside the well. But this cronsponds to the case of INFINITE well. It's not the solution of finite well. Then I think no solutions exist under the separating-variable method. Then, the question is, beyond separating-variable method, how to solve this problem? BTW: Does anyone know that what kind (shape) of 2D well is solvable for bound states and how? (Potential with circular symmetry is excluded, because I know how to solve it. I want to find another shape of 2D well which is solvable.)


title: How much faster is the fusion we make on earth compared to the fusion that happens in the sun?

text: I have come across many websites that states that the proton-proton fusion which is the dominant type of fusion that powers the suns, is extremely slow and that is why the sun is still burning to that day. But also I have read that the sun fuses 620 million tons of Hydrogen every second, so that is considered fast for us on the earth scale. So my question now is : comparing the time it takes to release the same amount of energy, how much faster the D-T fusion for example is compared to proton- proton fusion ?


title: Two point function for massless boson in 2 dimension

text: Is it true that two point function for massless boson theory in 2 dimension is a constant? That is to say it is independent of the distance between the two points?


title: Path Integral on Einstein Cartan Manifold

text: In condensed matter, crystal with disclination and dislocation has both curvature and torsion. I am looking for a reference in which path integral quantization of Dirac equation on manifold with curvature and torsion was discussed.


title: Obtaining velocity or acceleration vector of a point on a rigid body?

text: If I have a cube that is moving at a velocity of $v$ and spinning at an angular velocity of $\omega$, how can I determine the instantaneous velocity vector of one of the vertices of the cube? What if the cube is accelerating? And what if it had angular acceleration, $\alpha$? Would this change the method of calculation? Would it be possible to get an acceleration vector?


title: How to see timelike excitation has a negative norm from the "old covariant quantization"

text: I have a question in reading Polchinski's string theory vol I p 123, about the "old covariant quantization". It is said > ... $\langle 0;k | 0; k' \rangle = ( 2\pi)^D \delta^D (k-k') \tag{4.1.15}$ > as follows from momentum conservation. The timelike excitation has a > negative norm. How to see "the timelike excitation has a negative norm"? Is Dirac delta function in the RHS of Eq. (4.1.15) either $0$ or $\infty$?


title: Is there a quantity measured in kilogram seconds?

text: I'm trying to get a full grasp on the relationship between many of the units that are used in kinetic physics. I've found that it is possible to make a venn-diagram that shows the factors of many of the units: Impulse in the centre, Force and Velocity between two circles each and Mass, Acceleration and Time on the Outside. The only space in the diagram that is missing is mass multiplied by time. Is there a such thing as a kilogram-second? Is it used for anything in physics?


title: Inductor Charging and Discharging

text: Why does the inductor discharges in the same path ( same direction of current ) as that of its charging unlike a capacitor which discharges in the direction opposite to its charging(current direction is reversed in discharging).? What could be its physical reason (for Lenz's Law)


title: Poynting vector plain wave

text: I'm calculating the poynting vector for a plain wave and I have some doubt. $$ \bar S = \frac 1 2 \bar E \times \bar H^* = ... = \frac {| \bar E|^2} {2 \zeta} \hat i_k $$ Now if I consider a cylindrical volume and apply the divergence theorem I get $$ \int_{s_1} Re \,\, \bar S \,\,\hat i_n dS = - \frac {| \bar E|^2} {2 \zeta} A$$ $$ \int_{s_2} Re \,\, \bar S \,\,\hat i_n dS = \frac {| \bar E|^2} {2 \zeta} A$$ $$ \int_{s_l} Re \,\, \bar S \,\,\hat i_n dS = 0$$ where $s_1$ is the left face, $s_2$ is the right face and $s_l$ is the lateral face of the cylinder. So I should have $$ \int_{S} Re \,\, \bar S \,\,\hat i_n dS = - \frac {| \bar E|^2} {2 \zeta}A +\frac {| \bar E|^2} {2 \zeta} A = 0$$ Is it possible?


title: Foucault pendulum initial velocity

text: This is a basic question I can't solve and it seems it is not addressed on the web. Sorry for my lame painting skills. ![foucault pendulum mspaint](http://i.stack.imgur.com/lNYaA.png) Assume we have a foucault pendulum suspended in the north pole. the pivot is just above the the axis of rotation (allowing the pendulum's plane to be in any direction relative to the apparatus). Before it starts swinging, the pendulum is attached to one of the supporting bars (the black lines) by a small string. To begin the motion of the pendulum you burn that string (this is the mechanism Foucault used). While the pendulum is connected to the supporting bars it is moving with the apparatus, thus posessing both a velocity and an acceleration, the moment it is released, it has an initial velocity in the direction of rotation. _what happens to this velocity??_ All explanations and illustrations I see, assume from the begining that the plane of the pendulum in the north pole is fixed relative to the "fixed stars" (I am aware of the phillosophical debates), somehow neglecting that initial velocity. I guess that I lack the understanding of the pivot mechanism itself, so if any one can shed light on my question and on the mechanism of the pivot, I would be very Thankful! # EDIT: I think I'm not so clear about my question and explanation of the situation. I found an incredible wiki article: Foucault pendulum Under this section: Relative motion of the plane of the pendulum swing to the surface of the Earth , At the North Pole : > Once the bob is displaced from the central axis of the pendulum and then > released there no longer is a force acting on the bob that causes it to > revolve about the central axis of the pendulum and rotate (turn) with the > Earth. As observed from an end-view of the swinging bob, the swing of the > bob will always line up or swing towards one star (just like the axis of the > Earth points at one star for the time periods considered) as the bob swings > through the central axis of the pendulum. There can be a slight ellipsoid > swing if the initial conditions of angular motion are not cancelled but > there is no longer a force acting on the bob causing it to have an angular > velocity after the bob is released. The plane of the swing of the pendulum > bob is now independent of the surface of the earth which was imparting a > force to the bob before it was released (through the holding point). As > noted previously, the bob is still spinning with the Earth (a spot of the > bob will spin with the Earth), even though the bob is no longer turning with > the Earth. Thus the Earth continues to turn underneath the swing of the > pendulum while the swing of the pendulum remains in a fixed plane that > doesn't rotate (turn). The point of significance is that the force imparting > an angular velocity to the pre-released bob is no longer acting on the > swinging bob. At the North Pole, this force takes one day for the direction > of the force to complete a full circle since it takes one day for the Earth > to rotate. The initial angular momentum is clearly mentioned here, and it should be cancelled, hos did foucault cancel it??


title: Total angular momentum - single electron

text: I have been dealing with total angular momentum of the single electron which is outside the closed shells in which sum of the angular momentums is zero. My book says that total atomic angular momentum is $\mathbf{J}=\mathbf{L+S}$ and its magnitude $J=\sqrt{j(j+1)}\hbar$ where quantum number $j$ can have half-integer values on an interval $|\ell-s|\leq j \leq|\ell+s|$. There is allso a quantum number $m_j$ which can have half-integer values on an interval $-j \leq m_j \leq j $. So lets say I have a single electron in an orbital $\scriptsize\boxed{\ell=1}$ _(p orbital)_. In this case $j=\tfrac{3}{2},\frac{1}{2}$ while $m_j=\tfrac{3}{2},\frac{1}{2},-\tfrac{3}{2},-\frac{1}{2}$ and I can calculate two magnitudes for $J$: \begin{align} J&=\tfrac{\sqrt{15}}{2}\hbar\\\ J&=\tfrac{\sqrt{3}}{2} \hbar \end{align} Now there is this weird vector sum image that I don't understand completely: ![enter image description here](http://i.stack.imgur.com/vvecP.jpg) On the left image we have $J=\tfrac{\sqrt{15}}{2}\hbar$ while on the right we have $J=\tfrac{\sqrt{3}}{2}\hbar$. It seems that vector $\mathbf{L}$ together with its magnitude $L=\sqrt{\ell(\ell+1)}\hbar = \sqrt{2}\hbar$ is the same in both cases - this makes sense as quantum number $\ell$ doesn't change. But how do we explain the change in $\mathbf{S}$?


title: Is the cosmological redshift caused by the Planck mass increasing?

text: The standard explanation for the cosmological redshift is that photons emitted from far away galaxies have their wavelengths lengthened as they travel through the expanding Universe. But perhaps the photons do not lose energy as they travel but rather the atoms in our detectors are more energetic in comparision with the atoms that emitted those photons a long time in the past leading to an apparent redshift effect? **Addition** (having had a comment exchange with @rob, see below) : My hypothesis is that the Planck mass $M_{pl} \propto a(t)$ where $a(t)$ is the Universal scale factor. **Addition 2** Of course if the Planck mass $M_{pl}$ is changing then $G=1/M^2_{pl}$ is changing so that we no longer have standard GR! I've asked this question before, see Cosmological redshift interpretation, but this time I'm including a little bit of theory to back up my hypothesis. For simplicity let us assume a flat radial FRW metric: $$ds^2=-dt^2 + a^2(t)\ dr^2$$ Consider the null geodesic path of a light beam with $ds=0$ so that we have: $$dt = a(t)\ dr\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$ Now at the present time $t_0$ we define the scale factor $a(t_0)=1$ so that we have: $$dt_0 = dr\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$$ Substituting equation (2) into equation (1) we have: $$dt = a(t)\ dt_0$$ In order for the interval of time $dt$ to stay constant as the scale factor $a(t)$ increases we must have the corresponding interval of present time $dt_0$ varying inversely with the scale factor: $$dt_0 \propto \frac{1}{a(t)}$$ Thus as cosmological time $t$ increases, and the Universe expands, equal intervals of cosmological time $dt$ correspond to smaller and smaller intervals of present time $dt_0$. Now the energy of a system is proportional to the frequency of its oscillation which in turn is inversely proportional to its oscillation period: $$E(t) \propto \frac{1}{dt}$$ The corresponding energy of the system in terms of the present epoch $t_0$ is given by $$E(t_0) \propto \frac{1}{dt_0}$$ $$E(t_0) \propto a(t)$$ Thus an atom at time $t$ is a factor $a(t)$ times more energetic than the same atom at time $t_0$. As the energy scale is ultimately set by the Planck mass then the Planck mass must be increasing as the Universe expands: $M_{pl} \propto a(t)$. This effect alone would account for the gravitational redshift of distant galaxies without the assumption that photons travelling from those galaxies lose energy due to wavelength expansion. Addition: I believe this hypothesis leads to a linear cosmological expansion $a(t)\propto t$ (see comments below).


title: Faster than light in plasma

text: Some plasmas have a refraction index of less than 1. In these plasmas the phase velocity of light can be faster than light-speed. But the phase itself won't transfer information, so no paradox occurs here. But what if I constructed a tube filled with plasma with a torch on one end and a photon sensor on the other. When I see a beam of red light travelling past me, I lit up the torch, and some guy at the other end can know there's red light coming at him before the red light arrives. Isn't that an FTL occasion?


title: Is there any usage of $x^x$ in physics?

text: I am so excited about the function: $x^x$. I would like to know if there is any usage in the physics world of this function. Is there any formula that uses $x^x$?


title: Why does the Standard Model predict Neutrinos are massless?

text: Why are neutrinos massless in the Standard Model? Is it connected with experimental fact that neutrinos always have only one direction of projection of spin on motion direction?


title: A Competitor for General Relativity?

text: GR stands alone in its ability to pass both weak and strong field tests of gravity fields. From 1905 to 1915, there was renewed interest in a somehow modified scalar field theory. Here is the action of a scalar gravity model, with $z^{\mu}$ a worldline parameterized by $\lambda$: $$ \begin{align*}I_{sg} =& \- m\int d \lambda \sqrt{-\left(e^\Phi \frac{d z_0}{d \lambda}, e^\Phi \frac{d z_1}{d \lambda}, e^\Phi \frac{d z_2}{d \lambda}, e^\Phi \frac{d z_3}{d \lambda}\right)\left(e^\Phi \frac{d z_0}{d \lambda}, e^\Phi \frac{d z_1}{d \lambda}, e^\Phi \frac{d z_2}{d \lambda}, e^\Phi \frac{d z_3}{d \lambda}\right)\eta} \\\ =& \- m\int d \lambda \sqrt{e^{2\Phi} \left( \frac{d z_0}{d \lambda}\right )^2 - e^{2\Phi} \left( \frac{d z_1}{d \lambda}\right )^2 - e^{2\Phi} \left(\frac{d z_2}{d \lambda}\right )^2 - e^{2\Phi} \left(\frac{d z_3}{d \lambda}\right )^2} \end{align*} $$ The terms under the square root look just like a metric (completely flat in this case). Physicists knew the $g_{00}$ term from Newton's law. Physicists completely guessed at the $g_{uu}$ terms [see footnote], and that guess was wrong. When a path is varied in the presence of a gravity field, the changes in $g_{uu}$ should be approximately inverse of $g_{00}$. I am exploring an action (scalar gravity coupling) that at the very least is consistent with experimental tests _unknown to physicists_ prior to 1919: $$ \begin{align*}I_{sgc} =& \- m\int d \lambda \sqrt{-\left(\frac{1}{e^\Phi} \frac{d z_0}{d \lambda}, e^\Phi \frac{d z_1}{d \lambda}, e^\Phi \frac{d z_2}{d \lambda}, e^\Phi \frac{d z_3}{d \lambda}\right)\left(\frac{1}{e^\Phi} \frac{d z_0}{d \lambda}, e^\Phi \frac{d z_1}{d \lambda}, e^\Phi \frac{d z_2}{d \lambda}, e^\Phi \frac{d z_3}{d \lambda}\right)\eta} \\\ =& \- m\int d \lambda \sqrt{e^{-2\Phi} \left( \frac{d z_0}{d \lambda}\right )^2 - e^{2\Phi} \left( \frac{d z_1}{d \lambda}\right )^2 - e^{2\Phi} \left(\frac{d z_2}{d \lambda}\right )^2 - e^{2\Phi} \left(\frac{d z_3}{d \lambda}\right )^2} \end{align*} $$ The terms under the square root look like the Rosen metric which will pass all weak field tests (take the velocity equal to square root of $g_{00}$ to $g_{uu}$ to calculate light diffraction, or get the equations of motion, and use the constants of motion to get the same value). Rosen's bi- metric tensor theory fails strong tests of GR, specifically it allows for dipole modes of gravity wave emission which are not consistent with observations of energy loss in binary pulsars. In the scalar gravity coupling model, the $g_{00}$ must be exactly the inverse of $g_{uu}$. It predicts slightly more bending that GR to second order PPN accuracy (11.7 versus 11.0 $\mu$arcsecond). New proposals for gravity are usually pretty easy to shoot down. Do you see a flaw in this one? It looks like the Rosen metric and thus curves spacetime for weak fields (good), and is simpler than a bi-metric theory for strong field tests (good). It sure is simpler than GR. Does GR finally have a real competitor? [footnote]: Metrics can be written in an unending variety of coordinates. I have written this action presuming a point source and Cartesian coordinates. Had I chosen spherical coordinates, then there would be no gravity term in front of $z_2$ or $z_3$. Note added in response to an issue raised by @Trimok. This question has been put on hold. For the record, I certainly do believe I "have specific questions evaluating new theories in the context of established science are usually allowed". There is both an action, and a metric associated with the scalar gravity coupling model. That is the canonical method for doing gravity research. The outcome is specific down to 0.7 $\mu$arcseconds (details below), something established physics like work with strings never achieves. It is a rare, but should be a valuable bird of a model that could respect the strong equivalence principle, meaning that gravity is only about making the interval dynamic. Given a metric for any model, the post post Newtonian deflection can be calculated using a formula from Epstein and Shapiro (Phys. Rev. D, 22:12, p. 2947, 1980): $$ \begin{align*} d\tau^2 =& \left(1 - 2 \gamma \frac{G M}{c^2 R} + 2 \beta \left(\frac{G M}{c^2 R}\right)^2 \right) dt^2 - \left( 1 + 2 \gamma \frac{G M}{c^2 R} + \frac{3}{2} \epsilon \left(\frac{G M}{c^2 R}\right)^2 \right) dR^2/c^2 \\\ \theta_{_{ppN}} =& ~\pi (2 + 2 \gamma - \beta + \frac{3}{4} \epsilon) \left(\frac{G M}{c^2 R} \right)^2 \\\ \theta_{_{GR}} =& ~\frac{15}{16} \pi \left(\frac{G M}{c^2 R} \right)^2 \\\ \theta_{_{sgc}} =& ~4 \pi \left(\frac{G M}{c^2 R} \right)^2 \end{align*} $$ The difference between the two will be about 6%. A longer blog on this subject is online. Feel free to contact me if you any technical issues with the proposal, large or small.


title: Explain the infinitude of magnetic flux about a finite length of wire

text: Ok so it is clear the flux caused by an inductor is limited. If I integrate about a flat spiral inductor I will find that the flux lines going up are a fixed quantity. Since every flux line going up can be mapped to a flux line going down in the outer part of the inductor, I can say that I have found all the flux. Now suppose I open up the spiral (think of a polar flat spiral) to create a length of straight wire. Logically I should have flux conversion, since flux isn't destroyed or created (magnetic monopole rule). Therefore, I should caclulate an identical flux from a length of wire. However, after doing the calculation similar to the one found here,:http://dev.physicslab.org/Document.aspx?doctype=3&filename=Magnetism_BiotSavartLaw.xml except for limited values of theta (its a finite length spiral), I find the flux to be infinite! This cannot be right, because an infinte flux for a certain current results in an infinite inductance, crazy as it is. No one would admit a finite wire length should have an infinite inductance, right? The current would never be started because it would take infinite energy! My calculation can be found here: http://www.wolframalpha.com/input/?i=integral+of+b%2Fsqrt%28x%5E2%2Ba%5E2%29. This is the integral of the B field I found from the Biot-Savart for a limited length of wire. Bear in mind that the flux is found by taking the integral of the magnetic field by dA, which is dxdy. The integral of dy is just y, or l for length of the spiral (picture the straghtened spiral vertically with increasing r acting as the x axis). Excuse the wordiness of the question but I'm terrible at mathjax or latex, whatever is used. My goal is for someone to be able to show me how to calculate the flux created by a finite length of wire with a hopefully non-infinite result. Help would be helpfully appreciated.


title: Flow velocity out of a pipe

text: For a compressible homoentropic gas is the flow velocity out of the pipe the same as the velocity into the pipe?


title: Why is $\omega = \sqrt{K/m}$ valid for a quantum oscillator?

text: I'm working in the 3rd edition of Modern Physics by Serway, Moses, and Moyer. In 6.6, it talks about a quantum oscillator. I don't fully understand how the definition of frequency works. Now, we assume that a particle feels a force $F = -Kx$ like the classical version, and we define $K$ as the curvature of $U(x)$ at the point of stable equilibrium ($x = 0$). That is, $$K = \left.\frac{d^2 U}{dx^2}\right|_a$$ where $x = a$ is the general point of stable equilibrium. It then uses the formula $\omega = \sqrt{\frac{K}{m}}$, though this is a classical equation. I don't understand why this is technically valid. I understand that that formula comes from the classical description $-Kx = ma = m \frac{d^2x}{dt^2}$ where $x = A\sin(\omega t + \phi_0)$. However, with quantum mechanics, why can we say that it oscillates by that description? We're not even dealing with a particle; it has wave and particle properties described by the wavefunction, so why can we immediately say that $U(x) = \frac{1}{2}Kx^2 = \frac{1}{2}m\omega^2x^2$? When we solve the Schrodinger equation, we find that the particle isn't limited to the interval $[-A, A]$, so why is it valid to use a formula that is derived from assuming that? Any help is much appreciated!


title: Does a wavefunction interact with itself?

text: Considering the double slit experiment with a charged particle, after the particle passes through the slits, do the two portions of the wavefunctions feel the electromagnetic attraction of the other portion? More generally, can the potential term in the Schrödinger equation be dependant on the wavefunction itself? On the one hand it is easy to imagine an electron wave splitting up, each part carrying a proportion of the charge. On the other hand, the wavefunction is a probability distribution, which implies that there is no interaction.


title: Explicit solutions to 2-d Dirac Equation

text: The 2-d Dirac equation without any constants is represented usually as $$i*dt (\phi) = D (\phi)$$ where $D = m\sigma_2-i\sigma_1dx-i\sigma_3dy$. Where can I find explicit closed form solutions to this equation? If there are none, are there perturbation-like non-exact solutions? Thanks alot.


title: Can the Solar System be assumed a single body concentrated in the Sun?

text: This question springs from a comment against my question posted on the Space SE My questions may seem inane, or obvious to most of you real physics people too ... Any number of sources put the peg the Sun at approximately 99.8% of the total Solar System mass. If one were to consider any interaction of the Sun with a body exhibiting gravity perhaps **several orders of magnitude higher** , could the Solar System be assumed to be concentrated in the Sun? In other words, how much would planetary movement in orbit around Sol be relevant to any interaction of Sol, and another body **where the other body is (potentially) several orders more massive than Sol**? p.s. _The context for the other body being more massive is available in the original question on Space SE_ (in case it is relevant/required for an answer here)


title: Covariance of the Dirac equation and Infinitesimal Lorentz transformation

text: Okay, I've got really desperate now. I've spend 10 hours of work (for a few days) trying to prove 'trivial' equation in Quantum Electrodynamics. To anybody who want to write an answer for my questions - treat me like an idiot who only can multiplicate and sum equations. NO HYPERLINKS (I know how to use Google and sources, really!), NO SHORTCUTS. Please, it's very important to me. So, main question is: Let's suppose Lorentz transformation: $\Lambda^a_b=g^a_b+\Delta\omega^a_b$, where $\Delta\omega^a_b$ is just a (small) number ($\Delta\omega^2$ can be neglected). So, we can build an equation for transformation matrix S: $S^{-1}\gamma^iS=\Lambda^i_j\gamma^j$ where $\gamma^i$ are Dirac matrix (you know, these fundamental in Dirac Equation). It is now easy to show that S: $S=I+iS^{ab}\Delta\omega_{ab}$ $S^{-1}=I-iS^{ab}\Delta\omega_{ab}$ Something like that. $\Delta\omega$ is still a number, but $s$ is a matrix. Now black magic happens and we get: $S^{ab} = \frac{i}{4}\big[\gamma^a,\gamma^b\big]$ In some like 5 sources proof of last equation is described as 'trivial' or as 'exercise for reader'. If my explanation is too much complicated, here is a short one:   1. Read a http://physics.ucsd.edu/students/courses/fall2009/physics130b/Dirac_Lorentz.pdf script   2. Go to page 26 (in pdf, at page numerated by page 22)   3. Prove equation (46).


title: QFT: basic process which violates energy-momentum conservation

text: Why is the following process not possible? ![process](http://www.quantumdiaries.org/wp- content/uploads/2012/01/qed_eGammaVertex1.png) My book says it's because it violate energy and momentum conservation. Can someone explain me explicitly why? Why couldn't an energetic electron not radiate an photon and continue to propagate with a smaller momentum?


title: How to understand the QED, QCD and standard model Lagrangians?

text: How do you read the QED, QCD and standard model Lagrangians? What do all the symbols and tensors represent? And, how can you derive them by yourselves?


title: How to determine that the renormalization constant $Z_3$ must depend only on $g$ and $\Lambda/m$

text: In Le Bellac's book, Quantum and Statistical Field Theory, the renormalization constant $Z_3$ is introduced with the equation $$ \Gamma^{(2)}_R(k^2, m^2, g) = Z_3 \Gamma^{(2)}(k^2, m_0^2, g_0; \Lambda) $$ where $m_0$ and $g_0$ are the bare mass and coupling and $m$ and $g$ are the renormalized mass and coupling. Shortly after, the author writes that, by dimensional analysis, $Z_3$ can be a function only of $g$ and $\Lambda/m$ ( $Z_3 = Z_3(g,\Lambda/m$) ). It's not immediate to me why is it so; can somebody explain it?


> Sing Huat drives a car with initial speed 30 km/h and acceleration 2 km/h2 > toward Tee, whose displacement is 200 m away from him. At the same time, Tee > drives another car toward Sing Huat with initial speed 50 km.h and > acceleration 3 km/h2. Find the distance from the initial location of Sing > Huat, where these two cars will meet. http://i.stack.imgur.com/WlyV0.jpg I wonder is Tee drives two cars at the same time? If so, > Find the distance from the initial location of Sing Huat, where these two > cars will meet. which two cars the question wants to compare with?


title: Anomalies for not-on-site discrete gauge symmetries

text: If a symmetry group $G$ (let's say finite for simplicity) acts on a lattice theory by acting only on the vertex variables, I will call it ultralocal. Any ultralocal symmetry can be gauged. However, in general there are discrete symmetries that cannot be gauged. For example, Freed and Vafa in http://link.springer.com/article/10.1007%2FBF01212418 discuss how in 1+1d one needs the pullback of a certain class in $H^3(G,U(1))$ to have some trivial periods. So, is the converse true--that if $G$ can be gauged then there is a formulation of the theory where $G$ acts ultralocally? In other words, is a symmetry with no ultralocal action necessarily anomalous? And if so, can we see this anomaly as an explicit class in $H^3(G,U(1))$ for 1+1d theories, for example? It seems to me like the answer is yes. If we have no anomaly and go ahead and gauge $G$, then we can put the result on a lattice where the $G$ gauge field will live on edges. These edge variables will have the flatness condition that the start and end vertex variables differ by the action of the edge variable (an element of $G$). It seems like $G$ shouldn't act anywhere else, since in some sense gauging $G$ is a type of "fat quotient" of the theory by $G$. Thus, if we take this lattice formulation and forget the gauge field, we end up again with the original theory, but now with an ultralocal action of $G$. What remains is how to quantify this anomaly in the cohomology group.


title: Heat kernel expansion for entanglement entropy

text: Can somebody please let me know where I can find a reference for calculating heat kernel coefficients on a manifold with conical singularities? I am trying to compute the entanglement entropy for gauge fields using the heat kernel and the conical deficit angle method. I wanted to know how can I express the terms (in the entanglement entropy) beyond the leading UV divergent one in terms of invariants constructed out of the extrinsic and intrinsic curvature of the entangling region. Looking forward for a response.


title: Twisted Nematic liquid crystal: extent of degradation due to DC drive?

text: It is commonly accepted wisdom that a Twisted Nematic LCD must always be driven using AC instead of DC, because DC will supposedly denature the LC. I have two questions:   1. Does this effect only happen at typical drive voltages of 8V ~ 9V? If the LCD is driven with a lower voltage, around 3V, is the degradation not as bad?   2. At 3V, how many hours does it take to degrade the LC? I've run a test for 500 hours with no bad effects. Thanks!


title: Does the Earth spin because its surface has less mass than its core as lighter things travel faster in space?

text: Would heavy elements from space hitting the Earth slow down the earth's rotation as the difference between the masses of the core and surface would be less?


title: What is meant by "net charge"?

text: Lets consider a system of two opposite charges separated by a certain distance (dipole), if we ask what is the net charge for this system? the answer would be zero. The net charge (what I have come to know until now) is said to be addition of sign of charges irrespective of the distance between the charges. Lets consider the following case:   According to Gauss's law the total electric flux through a closed surface enclosing a charge is equal to $\frac{1}{\epsilon_0}$ times the magnitude of charge enclosed. In the below figure we have a $+q$ charge enclosed in the guassian surface, so there is a flux linked with the guassian surface, which will be equal to $\frac{1}{\epsilon_0}$ times the $+q$ charge. ![enter image description here](http://i.stack.imgur.com/eiu87.gif) Now, replace the $+q$ charge with the dipole in which charges are separated by a certain distance. Magnitude of electric field ($E$) due to an electric dipole at a distance $r$ from its centre in a direction making an angle $\theta$ with the dipole is given by the equation,$$E=\frac{1}{4\pi\epsilon}.\frac{p\sqrt{3\cos^2\theta+1}}{r^3}$$   where, $p=2aq$ ($2a$ is the distance of separation of the charges $q$). Electric field around the dipole will be zero if and only if distance of separation of the dipole charges is zero. Although we can't expect field lines to be perpendicular to the surface, we can expect some contribution due to the electric field components (perpendicular to the surface) of the enclosed dipole (with charges separated by a distance). From the dipole equation it follows that there exists electric field around dipole when charges are separated by distance, so there is flux through the surface which implies that there is charge enclosed in the surface (from gauss law). But net charge of dipole is said to be zero. By gauss law, net charge is not zero. Then what is net charge? From gauss law we can directly find the $q$. Can we say this as net charge? Then, is it that net charge can't be said directly just by adding the signs and is it a function of electric field? If I have misunderstood, pardon me and explain. Sometimes the problem might not be understood well due to communication error, if any correction required, comment on the part.


title: Conditions for characterizing a wave as plane wave

text: Given a wave equation, say for example $\Psi(x,y,z,t) = a \cos\left(\omega t -\vec{k}\cdot \vec{r} \right)$, what conditions should be met for $\Psi$ to represent a **plane** wave?


title: Are Maxwell Relations valid only out of saturation?

text: My thermodynamics book suggests that the Maxwell Relations must be used when dealing with a single-phase system. The equations come right after the section **Mathematical Relations for the Homogeneous Phase** , in which the following is stated: > In the following sections we'll develop many expressions to calculate > differences of properties in a single phase, compressible and homogeneous > (applicable to the vapor, liquid and solid phases). Perhaps that is only to make a clear distinction between them and the Clapeyron equation, introduced right before. In any case, I don't agree with this view. To me, since the equations come from definitions of thermodynamic potentials, which are valid at any state, they must be valid anywhere. I'll give a simple example of validity. There's this Maxwell Relation, of which I take the inverse: $$-\left(\partial s \over \partial p \right)_T = \left(\partial v \over \partial T \right)_p \Rightarrow -\left(\partial p \over \partial s \right)_T = \left(\partial T \over \partial v \right)_p$$ We know that, in saturation, $p = p(T)$ and $T = T(p)$. Therefore, both sides of the last equation are zero. Therefore, this Maxwell Relation holds in two- phase (saturation). I know that this one example of validity does not mean that all relations are valid in saturation. But are they?


title: Explicit degeneracy in SPT phases

text: In the wikipedia article on symmetry protected topological phases the author states: > If the boundary is a gapped degenerate state, the degeneracy may be caused > by spontaneous symmetry breaking and/or (intrinsic) topological order. I am wondering why the degeneracy cannot be caused by an explicit symmetry. Say, in the case of time-reversal symmetry, why can't we have a Kramers degenerate ground state? As a specific example: The K-Matrix of a "non-fractionalized" Topological Insulator reads $K=\mathrm{diag}(1,-1)$. The ground state degeneracy is given by $\lvert \det(K)\rvert=1$. Is this consistent with Kramers' theorem stating that the ground state should be at least two-fold degenerate?


title: How does the physical motion of atom lead to photon emission?

text: It's known that what we call a temperature is in fact molecular motion at microscopic scale. But at which point the emission of photons happens due to this physical motion, so that we can talk about the thermal emission? the Wikipedia page on thermal radiation says the motion of atoms "..results in charge-acceleration and/or dipole oscillation which produces electromagnetic radiation..", what is beyond the terminology I am competent of. May I ask for some explanation that doesn't require a degree in Physics to understand it?


title: quantum mechanics current operators

text: How to derive the charge current and the energy current operators in second quantized form in Quantum mechanics ? Also if you could comment in a similar way on the entropy current operator, that will be nice. The way I am proceeding is simply taking the time derivative of the number operator and the Hamiltonian operator. What is wrong in this approach and what is a more appropriate approach ? Consider a non-relativistic fermionic/bosonic Hamiltonian given in second- quantized form. For example, we can take the Hubbard Hamiltonian. But please start with a general Hamiltonian H to explain the process in general.


title: Equivalent resistance of an infinite square mesh

text: Consider an infinite square mesh of resistors each of resistance $1 \Omega$. Calculate the equivalent resistance between two diagonal points on a single square.![enter image description here](http://i.stack.imgur.com/xuOWa.jpg)


title: Original paper on Lorentz representation theory

text: Which was the original paper on the representations of the Lorentz group? Is there even one paper on this, or was this knowledge gained iteratively in a series of papers?


title: What object is quantized in quantum gravity?

text: In theories of quantum gravity, which object is it that is quantized? Working on field theories, I expect the quantization to mean the promotion of a classical field to an operator valued field that works on some kind of Fock space. I heard in QG the metric gets quantized. This would imply, by my above reasoning, that the metric and thereby spacetime itself is an operator working on a Fock space of spacetime excitations. While I can accomodate this picture well with my idea of graviational waves, how does it rely to spacetime in general? Especially, how do the other quantum fields relate to this operator-spacetime, since they have an intrinsic dependence on spacetime?


title: Waveforms for a given ideal inverters circuit

text: I have the following circuit. All the inverters are ideal CMOS. ![Circuit](http://i.stack.imgur.com/Zu0TZ.jpg) I need to draw the waveforms for each point (A, B, D, E, Vo), given that waveform at input.   What I want to ask is if what I did so far is correct? ![Waveforms](http://i.stack.imgur.com/XKBvr.jpg) Thank you for your time. (And sorry for the poor quality of the images) **Edit** : Thank you for your response. So, is this correct? (It is how I understood your response) ![enter image description here](http://i.stack.imgur.com/80qC8.jpg)


title: Can light affect gravity?

text: I have a mentor at work, who wrote a paper in the past regarding light and relativity. I am an aerospace engineer by training and a system administrator/programmer by trade, so I know a little bit about physics and relativity, but not enough to critique this paper. The paper in question is Hall Photon Theory, which is a rather unsettling paper, written looking at physics through the lens of the author's personal experience. It is clear that light is affected by gravity (gravitational lensing, etc.) and that light can influence matter (solar sails, light "pressure", etc.), so it seems that what he claims could be feasible, but I want to throw this out there to see what people who know a good deal more about this have to say. So, my question, can light affect gravity/matter? Edit: Since, I am getting answers all over the board, let me increase the precision of my question: Are the hypotheses presented in the aforementioned paper, namely that cleverly used light can produce anti-gravity/anti-intertial fields, provably false? And if so, can you provide that proof?


title: A potential perpetual motion machine?

text: If I were to freeze a superconductor, the ones made up of yttrium barium copper oxide, with liquid nitrogen and placed it on one of those strong neodymium magnets, it should float right? Now what would happen if I made a Mobius strip of steel and lined it up with those magnets and then place the superconductors on them and place the whole setup in some cold place (like Neptune or anywhere with temperature less than $4K$), would the moving superconductors make it a perpetual motion machine? PS: I know perpetual motion does not exist, so please tell me what would stop it from achieving perpetual motion.


title: Force on a magnet in a magnetic field

text: I was wondering what happens when i put a wire in which current flows parallel to a magnet, something like this: ![image](http://i.stack.imgur.com/Sm5F9.png) where the straight line is the wire, the circle is a cylindrical magnet(with outgoing magnetic field) and the red line is a line force for the magnetic field of the wire. In particular i was wondering if the force on the magnet is directly proportional to the magnetic field produced by the wire. Thanks.


title: An alternative, algebraic way to introduce interactions. Are there other ways out there?

text: An opening paragraph: > The usual approach to introducing interactions in quantum field theory is to > make the constraint on the amplitude of the field towards smaller values > more forceful than harmonic. For example, instead of the massive Klein- > Gordon field's harmonic constraint towards smaller field values, > $\partial_\mu\partial^\mu\phi(x)+m^2\phi(x)=0$, we introduce a more forceful > constraint, > $\partial_\mu\partial^\mu\phi(x)+\lambda_1\phi(x)+\lambda_3\phi(x)^3=0$. We > require $\lambda_3>0$; $\lambda_1$ may be tuned so that a non-zero value of > $\phi(x)$ is preferred, if $\lambda_1<0$. Here, we instead introduce a > constraint on the observed field by an alternative, algebraic method. We > take the observed field to be a bounded function of the field, > $\Phi(x)=\mathsf{gd}(\phi(x))$, where the Gudermannian, > $$\mathsf{gd}(z)=2\tan{\\!{}^{-1}\\!\left(e^z\right)}-\frac{\pi}{2}=\tan{\\!{}^{-1}\\!\left(\sinh{z}\right)},$$ > amongst other possible definitions, is complex analytic for $z\not=(2n+1)\pi > i$ and has bounded real component between $-\frac{\pi}{2}$ and > $\frac{\pi}{2}$. The use of the Gudermannian is inessential to the > development, but we will use it as a convenient explicit mapping of the > infinite range of measurement values of a free (quantum) field $\hat\phi(x)$ > into a bounded range of measurement values $\mathsf{gd}(\hat\phi(x))$, with > which we can make some analytic progress. In the usual terms of a potential > energy, this construction effectively imposes an infinite potential on the > field outside the range $(-\frac{\pi}{2},\frac{\pi}{2})$, because field > values outside this range cannot occur, while preserving an attractive > harmonic potential when the field is close to zero. It seems possible that > the process of renormalization in the context of Hamiltonian or Lagrangian > formalisms, by infinite adjustments of parameters, effectively constructs a > potential that is comparable to this analytic ideal. My doubts about this paragraph are a little worrying. It emerged from the mathematics that I've been pursuing for the last two months, building on other mathematics that goes back about a year. I'm quite embedded in a way of thinking that is somewhat different from most peoples' conventional. When in this state, something like the above always looks _convincing_ to me --I discover six months later why it's not, but right now I'm rather (too) pleased with what you see above. I'm asking here (1) whether people see the paragraph above as plausible enough at least to withhold judgment until they see the Math, or even that it seems enticing; or (2) that it looks hopeless (hopefully including some hint as to why, of course); and (3) whether there's anything that this reminds you of [Citations welcome, in the Math or Physics literature]. Answers that let me shortcut around that six month wait to seeing why this can't work will be welcome. I guess this is, as the sidebar recommends, "sharing my research", though I don't want to take the space to "provide details" beyond a certain point. I'm curious whether it's an effective question. In formulating this question around the above paragraph, it occurs to me to clarify that I'm unhappy with deformation of a Hamiltonian or Lagrangian formulation of a quantum field as the only way we have to construct a new quantum field, particularly because of renormalization, and I'm also unhappy with the nonconstructive tendency in axiomatic quantum field theory. I don't see anything else in the literature, but is there? EDIT: Luboš rightly focuses on the unlikeliness of the Gudermannian as a preferred choice of transform. As little as possible should depend on the choice of field coordinatization. The working abstract I have calls it a toy model, which I would hope would bring the comment above into focus, that > "The use of the Gudermannian is inessential to the development". Nonetheless, the Gudermannian is mathematically useful in the QFT context I am developing (1) because it has poles only on the imaginary axis, so just two series expansions are enough to evaluate it, effectively around $\pm\infty$, which is of course possible only because the Gudermannian is finite at $\pm\infty$. This makes analysis considerably more manageable. [Explicitly, we can express $\mathsf{gd}(z)=\mp 2\sum\limits_{j=0}^\infty(-1)^j\frac{1-e^{\pm(2j+1)z}}{2j+1}$ for $\Re(z)<0$ and $\Re(z)>0$.] (2) Using $\mathsf{gd}$ instead of $\arctan{}$ is also useful for analysis, because the exponential of the quantum field acts as a displacement operator instead of as a derivation. [Explicitly, $[a(x),f(a^\dagger(y))]=f\mathbf{\,'}(a^\dagger)C(x-y)$, where $[a(x),a^\dagger(y)]=C(x-y)$ is the commutator, acts as a derivation, whereas $e^{\alpha a(x)}f(a^\dagger(y))=f(a^\dagger(y)+\alpha C(x-y))e^{\alpha a(x)}$ acts as a displacement operator.] (3) $\tanh{}$ is a superficially simpler choice, but it's significantly less convergent. [Explicitly, we have to expand the denominator of $\frac{-1+e^{2z}}{1+e^{2z}}$.] $\mathsf{gd}$ is the simplest bounded function that satisfies these three criteria. I so far can't think of a way to get all this into an opening paragraph, unless there is a precedent I don't know of that sets up a context that is close enough to what I'm doing. I find even the little I can do with this toy model gives _a little_ insight into renormalization, though it's certainly allusive. There is a chaotic regime at time-like separation, for example. Perhaps I have to get into the paragraph above that the move to analysis might be worth trying just because perturbation theory is a non-convergent expansion. I'm already seeing things slightly different because of your comments. Many thanks. EDIT(2): Luboš asks for clarification of the general idea. At the moment I'm trying to see ways to combine multiple heterodox ways of constructing models. To me, the Hilbert space structure is paramount because it underlies any probability interpretation; positivity of the energy is less important than it is usually taken to be, because it is neither necessary or sufficient to ensure stability; a continuous mass spectrum arises in the Källén-Lehmann representation of the 2-point function of a renormalized theory, so it should not be ruled out as part of other types of models; field renormalization suggests some kind of transformation of the unbounded quantum field to bounded values. Latterly, I've thought that a more analytic approach to models would be better than the series expansions that are universally used. In a few weeks or years I may have dropped any or all of these constituents of the models I have been constructing and characterizing. The published papers that are referred to on my web-site gives some idea of the different kinds of heterodox models I've constructed, understood at some level, and dismissed in the past. There is an answer, by the way, to the question "Are there other ways out there?", which I came across only a few days ago. Detlev Buchholz, Gandalf Lechner, Stephen J. Summers, "Warped Convolutions, Rieffel Deformations and the Construction of Quantum Field Theories". This and its references claim to construct an interacting QFT in 3+1 in the Haag-Kastler axiomatic context, although I'm just now making my way through it.


title: Direct current and metallic conduction

text: Suppose we have a circuit with and EMF source and a resistor. We know that when electron moves from one terminal of a voltage source to another it encounters resistance, which is basically collisions with stationary positive ions. During the collisions some of the electric potential energy and kinetic energy that electron has is passed in form of kinetic energy to the atoms of the material of the conductor. That means they lose some of their own kinetic energy, i.e. they're being slowed down. It seems reasonable to think that the electric current (charge per unit time) is changing, as the electrons slow down due to the collisions. Why then we still get a constant current? What is the flaw in what I've described? * * * Electrons must accelerate and gain a bigger velocity in medium with lower resistance. It seems reasonable for me to think that electrons are moving faster where the resistance is lower and slower where it is higher. So if we have zones with different resistances, electrons will move with different speed and thus the current will vary.


title: How to get Hamiltonian of QED from lagrangian?

text: I have the QED lagrangian: $$ L = \bar {\Psi}(i \gamma^{\mu }\partial_{\mu} + q\gamma^{\mu}A_{\mu} - m)\Psi + \frac{1}{16 \pi}F_{\alpha \beta}F^{\alpha \beta} . $$ I tried to get hamiltonian by getting zero component of energy- momentum tensor: $$ T^{\mu}_{\quad \nu} = i\bar {\Psi}\gamma^{\mu}\partial_{\nu}\Psi + \frac{1}{4 \pi}F^{\mu \gamma}\partial_{\nu}A_{\gamma} - \frac{1}{4 \pi}J^{\mu}A_{\nu}\Rightarrow $$ $$ T^{0}_{\quad 0} = i\Psi^{\dagger}\partial_{0}\Psi + \frac{1}{4 \pi}F^{0\gamma}\partial_{0}A_{\gamma} - \frac{1}{4 \pi}J^{0}A_{0} = i\Psi^{\dagger}\partial_{0}\Psi + \frac{1}{4 \pi}F^{0\gamma}\partial_{0}A_{\gamma} - \frac{1}{4 \pi}\Psi^{\dagger}A_{0}\Psi = H_{density}. $$ But it seems that it's incorrect, because I never get by this expression term $\bar {\Psi} \gamma^{\mu}\Psi A_{\mu}$, which refer to interaction part. So how to find the true hamiltonian? Thank you. Added. Hmm, I find the mistake in expression of energy-momentum tensor. Fixed.


title: Conservation of classical linear momentum

text: Imagine a cat jumping off a roof onto a sledge on ice. Model this as simply as possible, e.g. no resistive forces, perfectly rigid cat, sledge and floor. Moreover, the cat and sledge move in a straight line. The cat will have a vertical component to its momentum when it lands, but the sledge will - more or less - move off horizontally. What happens to the vertical component of momentum?   * Should we consider the Earth as a third body in the collision, and conclude that the Earth moves away with an immersurably small velocity?   * When performing calculations, can we simply ignore the vertical component post-landing?   * In reality, roughly what factor of the vertical motion's kinetic energy goes into heat, sound, vibrations, etc? (I don't mean list them one-by-one, just what is "non-Earth moving"?)


title: theoretical questions of time-of-flight in ultracold atomic gases

text: Time-of-flight is an experimental method to measure the momentum distribution of atoms in the trap or optical lattice. At time $t=0$, one can suddenly switch off the trap and optical lattice, and let the atomic gas expand freely. Then after a long expansion time $t>0$, one can obtain the real space density distribution by absorbing measurements. Then one can get the information of initial ($t=0$) momentum distribution which is then corresponding to the final ($t>0$) real space density distribution. My question is why the momentum distribution before the expansion is related to the real space density distribution after expansion?


title: How many domains are there in a ferromagnetic material?

text: How many are there? Also what is the size of those domains? Finally, when domains align with exterior magnetic fields do they move or are they all fixed? I assume they are all fixed because the atoms in a materials that have there dipole moments are fixed?


title: Force of exploding pumpkin

text: Okay so in physics class we are going to explode a pumpkin by placing some kind of bomb inside of the pumpkin. Well us students have to come up with an apparatus to measure the force of the explosion, i am in desperate need of some help or some ideas of some solutions and what i need to measure to find the force of the explosion. I know that i need to find the mass of the chunks of pumpkin, the position of the pumpkin chunks from start to finish, and the time it took for the chunks to reach a certain point. What else do i need to know and/or find? PLEASE HELP


What are experimental techniques to probe band structure of solids? I am not looking for exhaustive list. Few simple to understand techniques will suffice.


title: Straight line on 8-axis

text: How would a straight line appear in a 8-axis co-ordinate system. Will it continue to look like a straight line or will it appear like zig-zag line. We can see that straight line appears like a dot in one-dimension, straight line in two-dimension graphs.


title: The speed of light during the inflationary period

text: **Introduction:** As a thought experiment, suppose I modified the value of $c$ (speed of light) in some local region and attempted to measure it with a clock placed in that same region. I will denote this new unmeasured speed $c'$. If I tried to measure $c'$ with a clock then (based on my understanding of physics) I would still arrive at the conclusion that $c=c'$ since the rate at which physical processes evolve in a region is dependent on the value of light-speed in that region. Consequently the rate at which the internal processes of a clock evolve will be dependent on the value of light-speed as well. This will effectively change the way I measure $c'$ and cancel out any measurable $\Delta c$. **Questions:** Based on my reasoning above, I feel it is necessary to ask... 1.) What do exactly do cosmologists mean when they assert that the inflationary period was a period where the speed of light was different than what we measure it to be today? 2.) How do cosmologists define a value of light-speed which is different from our current value? 3.) How could a cosmologist measure a value of the speed of light which is different from its current value? 4.) Were is my logic flawed? **Additional Note:** I understand they did this because it could explain the isotropic nature of the physical characteristics space-like separating events; that is, I understand _why_ they did it just not _how_ they model it theoretically.


title: About the definition of expectation value in quantum mechanics

text: In quantum mechanics, the expectation value of a observable $A$ is defined as $$\int\Psi^*\hat A\Psi$$ But in probability theory the expectation is a property of a random variable, with respect to a probability distribution:$$E(X):=\int X\;d\mu$$ I can't see how probability theory can be adapted to quantum mechanics. Observables are associated with linear operators, not measureable functions, so how can we talk about the expectation of a linear operator? And quantum mechanics textbooks use expectations and variances without mentioning underlying probability spaces. Does quantum mechanics use something other than ordinary probability theory?


title: What is a graviton?

text: We know that a graviton is a hypothetical particle that carries gravitational energy. I have no definite knowledge in string theory, being a student; taking classical gravity into consideration or Einsteins general theory of relativity into consideration, what significant change does theories associated with graviton show comparatively? # My question: What is a graviton actually?


title: Why do some bound states disappear in a discontinuous way?

text: Generally, we have the picture that as the parameter (say, the depth of a trap) of a system varies, the bound state gets more and more extended and disappears eventually at some critical parameter value. The point is that, the whole process is a continuous one. This is the case for the defect mode induced by a site-defect in the 1d tight binding model. However, Mattis discovered an exotic bound state which disappears discontinuously: http://journals.aps.org/prl/cited-by/10.1103/PhysRevLett.52.755 Personally, we had a similar finding: http://journals.aps.org/pra/abstract/10.1103/PhysRevA.87.023613 The point is that, even at the critical parameter value, the bound state is still of a FINITE size, and then, as the parameter changes by an infinitesimal value, it is gone. What is the mathematical reason behind some a discontinuous behavior? It has baffled me for a long time. I would like to draw an analogy of this behavior with the first-order phase transition, while the more common continuous behavior with the second-order phase transition.


title: Perturbative vs. non-perturbative approaches to a well-defined Yang-Mills theory in 4 dimensions

text: Another question regarding the Yang-Mills Existence and Mass Gap problem (http://www.claymath.org/sites/default/files/yangmills.pdf). Does the problem require that the "construction" of a four dimensional quantum Yang-Mills be non-perturbative? I get the feeling that this problem is to make notions such as the renormalization group rigorous, and thus is perturbative, but isn't lattice gauge theory already mathematically well-defined? If so, why can this not be used as an approach to this problem? Essentially, which is the preferable approach as specified by the problem: perturbative or non- perturbative?


title: Modeling an Adiabatic Expansion (EES)

text: How would one model an isentropic (reversible) adiabatic expansion using thermodynamic software in order to determine isentropic work? What I am looking for is an actual pressure-volume curve for the isentropic expansion of a gas. Approach using EES: 1) Start with a known gas state/scale (Mass, Volume, Pressure, Temperature, Enthalpy, Entropy, etc.)   2) Update gas volume by a small increase in volume dV   3) Work done = Pressure * dV   4) Keep entropy constant   5) Recalculate new gas state based on the new Specific Volume (Volume/Mass) and Entropy   6) Repeat until final state has been reached. Summing the work done in this method results in significantly less work (~85%) than is possible through isentropic expansion. The max isentropic work is calculated through   Wmax = Mass * (h_initial - h_final) Even with very small step sizes (thousands of steps), this approach doesn't work. Can anyone shed some light as to why these work values don't match up? Code:               Procedure Adiabatic Expansion(P, T, P_final : Work_max, Work)       Vol=1       v=Volume(CarbonDioxide,P=P,T=T)       s=Entropy(CarbonDioxide,P=P,T=T)       h_initial=Enthalpy(CarbonDioxide,P=P,T=T)       Mass=Vol/v       Vol_step=1/10000       Work=0            repeat       Vol:=Vol + Vol_step       v:=Vol/mass       Work_step:=(P)*Vol_step       Work:=Work + Work_step       P:=Pressure(CarbonDioxide,v=v,s=s)       until (P <= P_final)            h_final=Enthalpy(CarbonDioxide,P=P,s=s)       Work_max=mass * (h_initial - h_final)       End


title: How do electrons repel?

text: I understand the basics, protons have a positive charge, neutrons have no charge, and the electron has a negative charge. But looking at the lines of force from a proton, they flow outward and push each other away. But, the electrons flow inward or towards themselves. How does that make them repel? Wouldn't they be more neutral towards themselves? I understand that the lines of force cannot cross as well. I'm sure this is an easy answer for someone.


title: Does this statement make any sense?

text: I am asking this question completely out of curiosity. The other day, my roommate, by mistake, used 'Light year' as a unit of time instead of distance. When I corrected him (pedantic, much), he said the following: "Units are relative. And according to Fourier Transforms, units can be changed so Light year is a unit of time." That got me thinking and I read up Fourier Transforms on wikipedia but couldn't find anything about using a unit in one domain as a unit for another measurement. I do agree that units (particularly, base units are relative. eg: the meter), but does his statement make any sense? EDIT Thank you everyone for all the answers. It isn't so much to in it in or prove a point as it is to understand the concept better. Anyways this is his response after I showed him this thread. Any comments would be appreciated. His response: Nevermind, for the first time I accept I was wrong. BUT using lightyears to measure time is possible. My example didn't make sense bacause I was wrong when I meantioned that I'm still measuring dist. If you have a signal in time domain and ...take the FT, I get a signal which DOES NOT HAVE to be in frequency domain. Clarify this to the guy who posted last. Now the new signal is in a domain defined by me and so is its units. This signal although not equal to the original signal, still represents that if ya take an inverse FT. So, the idea of time will still be there. Now coming back to our case: lightyears here is not the lightyears you are used to read when dealing with distance. It represents time.


What is the difference between Raman scattering and fluorescence? Both phenomena involve the emission of photons shifted in frequency relative to the incident light, because of some energetic transition between states that the system undergoes. As far as I can tell, fluorescence is caused by ionization of the atomic electrons while Raman scatterings agitate phonon modes - but technically speaking, aren't they effectively the same?


title: Acceleration/tension of object. 1 on horizontal surface one hanging

text: > A 5.00-kg object placed on a frictionless, horizontal table is connected to > a cable that passes over a pulley and then is fastened to a hanging 9.00-kg > object, as in Figure. Find the acceleration of the two objects and the > tension in the string. ![enter image description here](http://i.stack.imgur.com/BiSxC.png) The given answer looks like: $$ mg-T=9a \\\ 9 \times 9.8 - T = 9a \\\ 22.2 - T = 9a \text{ --- (1)} $$ Where did 9a come from? Then it continues $$ T=5a \text{ --- (2)} \\\ \text{sub (2) into (1)} \\\ 22.2 - 5a = 9a \\\ 14a = 88.2 \\\ a = 6.3 \\\ T = 5a = 5 \times 6.3 = 31.5N $$


title: Meaning and application of convolution or deconvolution in physical sciences

text: In which real case scenarios a convolution or deconvolution operation is useful ?


title: Possibly naive question about quantized space-time

text: I beg your pardon in advance if this question is _naive_. In Quantum Mechanics, discrete values of measurements occur ~~only~~ in relation to _bound_ states. This is because of the general solution for the time-independent Shroedinger's equation. Similarly, in order to quantise space-time, do we need to make space-time compact? How would a _compact_ space-time look like? Would we simply need close it by including $\infty$? Is that physical (infinity doesn't sound very physical to me)?


title: How to find exponential values

text: I have a power equation for flow between two circular disks on a common axis as stator and rotor, which follows, $$P = \frac{n^a}{G}\tag{1}$$ Where:   * $P$ = power,   * $n$ = revolution (1200 rev/min),   * $G$ = gap between two plates (0-2.5 mm) I have experimental data showing Power vs Gap curve at a constant revolution of 1200 rmp. How do I get the value of $a$ and how do I plot Eqn (1)?


title: Hall Effect p-type Germanium holes density

text: I had to do some experiments with a Hall Effect Apparatus of Leybolds with a p-doped Germanium plate, this is the datasheet (pdf). **eq. (1)   ** $U_H=R_H \frac{IB}{d}$ $U_H$ = Hall voltage   $R_H$ = Hall Coeffiecient   $I$ = current through plate   $B$ = Magnetic Induction through plate   $d$=thickness of plate   **eq. (2)   ** $R_H=\frac{1}{e}\frac{p\mu_p^2-n\mu_n^2}{(p\mu_p+n\mu_n)^2}$      $p$ = total density of holes $\rightarrow p = p_e+p_s$   $e$ = Elementary charge of electron   $p_e$ = density of holes (intrinsic conduction) $p_s$ = density of holes (due to p-doping) $\mu_p$ = Mobility of holes   $\mu_n$ = mobility of electrons   $n$ = density of electrons   With eq. (1) and (2) they derive the $p_s$, eq. (3) but I dont really understand how they do this. How do they do this? **eq. (3)**   $p_s =\frac{B}{ed}\frac{I}{U_H}$ Thanks :)


title: What causes a ring-like image around light of the moon?

text: I just encountered an interesting image in sky. As you can see in following images there was a ring-like image around light of the moon. I don't know if it was clouds but it was looking like it is far above the clouds. I got curious and only place I know I could get some answers is here. It is much more clear in first photo, other ones are a bit blocked by clouds in between Here are some photos I took: ![Photo 1](http://i.stack.imgur.com/s4cBL.jpg) ![Photo 2](http://i.stack.imgur.com/zWijW.jpg) ![enter image description here](http://i.stack.imgur.com/6c4tf.jpg)


If acceleration causes relative time dilation does the eventual deceleration reverse it? For example: traveling to Alpha Centauri Based on me reading this site: http://www.convertalot.com/relativistic_star_ship_calculator.html -


title: Show that purity = 1 in a pure state

text: How can you show that for **any** pure state, the purity = 1? Pure state: $\rho^2 = \rho$ and $Tr(\rho^2)=1$ Mixed state: $\rho^2 = \rho$ and $Tr(\rho^2)<1$


title: Refraction and Reflection Seismology

text: So I am wondering if I got the difference right. Both methods use explosives to send waves into the earth's surface. Now reflection seismology tries to get information from the reflected waves; the ones, that do not get into another layer but get reflected from it r get directly through the layer and get reflected by the next. Thereby you would get a much smaller space in which waves would get back to the surface. In refraction seismology, you want the waves to get into the next layer and as you know that there are different velocities in different layers, that is how you get information out from there (detail here on how that works not of interest right now). So, you need to use different places for your seismographs in order to get the refracted waves, which are further away from the source than the reflected waves. I know there is more behind this, but I just want to know if this basically is the difference? Because I am not sure of that... Then, I would guess, you could always do reflection seismology when you do refraction seismology, just using the reflected waves to determine the structure of the first layer.   Is that somehow correct or Am I missing something?


title: Solving a statically indeterminate truss

text: I'm trying to solve the statically indeterminate truss shown below and I'm having a little trouble. ![truss](http://i.stack.imgur.com/7tLWa.png) $H$, $P$ and $\beta$ are given. The material is aluminum (density is 2700 ${kg/m^3}$) and has a cross section of 0.002 $m^2$. We are instructed to take the weight of each member into account by distributing $\frac12$ of each member's weight to its end joints. I can easily solve for $f_1$, $f_3$, $f_4$, $f_5$ and the support reactions using statics (method of joints), but I can't figure out how to go about solving for the remaining members. The main point of my question is, how do you solve statically indeterminate trusses? I assume I'll have to use some other property such as deformation. I've only posted the homework question to provide background as to why I am asking. Thanks for any help.


title: Calculating elastic energy constant

text: I ran into a kinetic physics problem: _"A spring is hanging on the ceiling. Let's place an object 'M' at the end of the spring. Yet hold 'M' so the spring doesn't stretch. The distance between the floor and 'M' is 'h'. Now let 'M' be free in the air. The spring stretches as much as 'x'. What is the value of the elastic potential energy constant 'k'? ( k from $F = kx$ and $E = \frac{1}{2}kx^2$ )"_ Here are constants for the problem. > $m$ : the mass of 'M' > > $g$ : gravity acceleration > > $h$ : The initial distance between the floor and 'M' > > $x$ : Stretched length of the spring after 'M' is attached. **My approach was:** $$m g h = m g (h - x) + 1/2 kx^2 $$ subtract $mgh$ $$-mgx + 1/2 k x^2 = 0$$ add $mgx$ $$ mgx = 1/2 kx^2$$ factor by $(1/2 x^2)$ $$k = (2mg)/x$$ **However the official answer says I'm wrong.** The official answer says I should get $k$ in the relation of Force. $$mg = kx$$ factor by $x$ $$k = (mg)/x$$ **Why do I get two different values of k?** I asked my school assistant and he doesn't know why but you can use two different methods separately depending on the situation. He was also puzzled because, $$d/dx (mgx - 1/2 kx^2) = mg - kx$$ So the my logic seems to make sense. Could someone kindly explain why please? Thank you!


title: Canonical momentum Velocity dependent Lagrangian

text: I have a homework problem wich I think I'm on the verge of solving but need help with some relations: > _Show that if the potential $U$ in the Lagrangian contains velocity- > dependent terms, the canonical momentum corresponding to a coordinate of > rotation $\theta$ of the entire system is no longer the mechanical angular > momentum $L_{\theta}$ but is given by_ > $$p_{\theta}=L_{\theta}-\sum_{i}\mathbf{n}.\mathbf{r_{i}}\times\nabla_{v_{i}}U.$$ This is what I have so far: I know that the position vectors $\mathbf{r_{i}}$ are functions of the $q_i$ generalized coordinates. On the other hand $U=U(\mathbf{r_{i}},\mathbf{\dot{r_{i}}})$ The canonical momentum with respect to a coordinate of rotation $\theta$ is given by: $$p_{\theta}=\frac{\partial{L}}{\partial\dot{\theta}}=\frac{\partial T}{\partial{ \dot{\theta}}}-\sum_{i}\bigg(\frac{\partial u}{\partial{{r_{i}}}}\frac{\partial r_{i}}{\partial \dot{\theta}}+\frac{\partial U}{\partial \dot{r_{i}}}\frac{\partial \dot{r_{i}}}{\partial \dot{\theta}}\bigg) $$ Using: $$\frac{\partial \dot{r_{i}}}{\partial \dot{\theta}}=\frac{\partial r_{i}}{\partial \theta}$$ We have: $$p_{\theta}=\frac{\partial{L}}{\partial\dot{\theta}}=\frac{\partial T}{\partial{ \dot{\theta}}}-\sum_{i}\bigg(\frac{\partial u}{\partial{{r_{i}}}}\frac{\partial r_{i}}{\partial \dot{\theta}}+\frac{\partial U}{\partial \dot{r_{i}}}\frac{\partial r_{i}}{\partial \theta}\bigg) $$ I know that $$\frac{\partial U}{\partial \dot{r_{i}}}$$ can be rewritten as the scalar product of gradient of U and the unit velocity but I don't see how a cross product can be made to appear.