title: Newton's color Disk

text: How does Newton's color disk work? Newton's disk - Take a circular white color disk, make 7 equal intersections and paint section with respective VIBGYOR colors, now when you spin the disk in certain speed, it appears white.


title: How do I correctly interpret $\rho = \psi_1^* \psi_2$?

text: > **Summary** : This turned out to be a rather trivial one indeed. As Marek > mentioned in the comment, the continuity equation is trivial. And it indeed > turns out be so. Godfrey Miller elaborates on this, showing that the > continuity equation merely shows that the transition amplitude is constant > with time. All this is only confusing as long as you stick to the belief > that the continuity equation is some exclusive indication of probability > density. $\rho = \psi_1^* \psi_2$ is the probability density for transition from state 2 to 1. I am having trouble interpreting the probability though $$dP_{1\leftarrow 2}=\psi_1^* \psi_2 d^3 r $$ does this mean the probability in a region of volume $d^3 r$ to go from state 2 to state 1? That does not make sense at all! **EDIT:** I more or less agree with Marek's answer. However, that makes it seem this is not the probability density. **How do we reconcile that with the fact that it satisfies the continuity equation.** If we have two states:   $$ i \hbar \partial_t \psi_{1,2}= -\frac{\hbar^2}{2m} \nabla^2 \psi_{1,2} + V\psi_{1,2}$$ After making standard manipulations (taking conjugate, multiplying by conjugate, subtracting) we would get $$\partial_t ( \psi_1^*\psi_2) +\frac{\hbar}{2mi} \nabla\cdot \left( \psi_1^* \nabla \psi_2 - (\nabla \psi_1^* )\psi_2\right)=0$$ This seems consistent with the continuity equation: $$\partial_t \rho +\nabla.\mathbb{j} =0$$ However, we seem to be changing its relation to the probability by making the probability $P=\int \int\rho^*(r,r') \rho(r,r') d^3r d^3r' $ instead of the regular $P=\int \rho(r) d^3r$ For a kind of intuitive "dimensional consistency" can we instead say $P=\int \int \; \sqrt{\rho^* \rho} \; d^3r d^3r'$ $$****$$ **Full steps:** $\psi_1$ and $\psi_2$ are two states of the same system, i.e both satisfy the SE of the same potential: Taking the SE for $\psi_2$ and multiplying by $\psi_1^*$ : $$ \psi_1^*\times\left( i \hbar \partial_t \psi_2 = -\frac{\hbar^2}{2m} \nabla^2 \psi_2 + V\psi_2\right)$$ Taking the SE for $\psi_1$ complex conjugating the entire equation and multiplying by $\psi_2$ : $$\left( -i \hbar \partial_t \psi_1^*= -\frac{\hbar^2}{2m} \nabla^2 \psi_1^* + V\psi_1^*\right)\times \psi_2$$ ($V$ is real, no decay processes etc) On subtracting the second equation from the first $$\partial_t (\psi_1^*\psi_2 )= -\frac{\hbar}{2mi} \left( \psi_1^* \nabla^2 \psi_2 - (\nabla^2 \psi_1^* )\psi_2\right)$$ which is the same as $$ \partial_t (\psi_1^*\psi_2 )= -\frac{\hbar}{2mi} \nabla\cdot \left( \psi_1^* \nabla \psi_2 - (\nabla \psi_1^* )\psi_2\right)$$


What is the resistivity coefficient of  graphene?


title: Quantum Space Simulation

text: There are a number of simulators, which simulate the macroscopic world around us (Space Engine: http://en.spaceengine.org/ comes to mind, but there is also Universe Sandbox). My question, does anything like this exist for the quantum world ? (I know the formulation of QFT is exactly saying that this isn't possible, but maybe an approximation is).


title: Why are laws of physics always of product forms?

text: A first observation is that all the extant laws of physics are of product forms. This phenomenon is somewhat intriguing. The question is: why do law of physics always take, instead of a sum of two terms, sum-of-multiple-term (i.e., product) forms? Counterexample exists in other disciplines of natural sciences. For example, biology has the White's formula (http://www.maths.ed.ac.uk/~aar/papers/eggar.pdf). Since mathematics investigates every possibility, certainly it has results involving sum of exactly two terms. I am seeking after a reason other than technical reasons. For example, the importance of Lorentz transformation in physics does not lie in its mathematical necessity but in its physical implications, as revealed by A. Einstein. H. Poincare, E. Mach, and several others had realized the concept of relativity before Einstein, but, as the paper ``The structure of thoughts'' (For now I cannot recall the exact title) published in Nature points out, discerning a concept is not equivalent to discerning its meaning. It is Einstein that discerns the meaning of relativity.


title: N level nucleon photon emission in a infinite potential well

text: Little question : Consider a nucleon in a infinite square potential well. At any given instant the nucleon goes from energy $E_n$ to $E_1$, emitting a photon with energy $3E_1$. In what circumstances can we be sure that before the transition the nucleon was on energy level $E_2$? And in what way will it be affected by the stationary solutions to the well? Considering $C \cos(n \pi \frac x a)$ for odd $n$ and $C \sin(n \pi \frac x a)$ for even $n$. (Square well from $-a/2$ to $a/2$)


title: In what sense do Goldstone bosons live in the coset?

text: Goldstone's theorem says that if a group, $G$, is broken into its subgroup, $H$, then massless particles will appear. The number of massless particles are given by the dimension of the coset, $G/H$. It is then often said that the Goldstone boson's live in the coset. In what sense is this statement true? The Lagrangian is not invariant under transformations of the coset so what does this "living" explicitly mean? To be explicit we can consider the linear sigma model: \begin{equation} {\cal L} = \frac{1}{2} \partial _\mu \phi ^i \partial^\mu \phi ^i - \frac{m ^2 }{2} \phi ^i \phi ^i - \frac{ \lambda }{ 4} ( \phi ^i \phi ^i ) ^2 \end{equation} We define, \begin{align} & \phi _i \equiv \pi _i \quad \forall i \neq N\\\ & \phi _N \equiv \sigma \end{align} and give $\sigma$ a VEV. The spontaneously broken Lagrangian is, \begin{equation} {\cal L} = \frac{1}{2} \partial _\mu \pi _i \partial ^\mu \pi _i + \frac{1}{2} ( \partial _\mu \sigma ) ^2 - \frac{1}{2} ( 2 \mu ^2 ) \sigma ^2 - \lambda v \sigma ^3 - \frac{ \lambda }{ 4} \sigma ^4 - \frac{ \lambda }{ 2} \pi _i \pi _i \sigma ^2 - \lambda v \pi _i \pi _i \sigma - \frac{ \lambda }{ 4} ( \pi _i \pi _i ) ^2 \end{equation} The Goldstone bosons, $\pi_i$, exibit a $O(N-1)$ symmetry, but this is not the coset group symmetry. So where in the Lagrangian do we see this symmetry?


title: What length should be the gap to block UV waves from the Sun?

text: The origin of the question is practical, but the theoretical part is interesting too. Let's rest a little bit on the roof on a sunny summer day. We don't want to get burned, so we usually place some cloth in between the Sun and our skin: an umbrella or a canopy. Given the length of the wave, how to calculate the minimum and the maximum length of the gaps (within cloth) and the distance from the cloth to a detector (human skin) such that waves get blocked or absorbed and doesn't make a significant trace on a detector? I may miss some details - the comments are welcome!


title: Geiger meter affected by plasma lamp

text: Whilst testing a geiger meter I took it near a plasma globe (the type sold in party/toy stores). The meter showed 0.13 mcSv/h. Then I turned the lamp on and low and behold the meter jumped to 0.89 mcSv/h. Shutting the down the the plasma globe caused teh reading go back to normal in about couple of minutes. I also tested by placing the meter a feet away from the plasma globe (set to on). No spike there. Just what could have caused this? The plasma rays affecting the electronics inside the geiger counter?


title: Why should photons obey the rules of causality?

text: This may have a quick answer, but it has been on my mind so here it goes: why, if according to special relativity, do photons obey the rules of causality, if they are inherently timeless. According to special relativity, wouldn't a particle travelling at light speed experience no time, and therefore, why would it be expected to obey the rules of causality as experienced by those not going the speed of light?


title: Dimensional Analysis : Thermodynamics

text: I was coming across some notes online for phase transitions. In one of the places, the author has written the Claussius-Clayperon equation in this form, $$ \frac{d(ln P)}{d(ln T)} = \frac{T\Delta S}{P\Delta V} $$ I am wondering if this equation is valid, since both P and T have dimensions and quantities inside log or exp cannot be dimensional quantities (since they are a power series).


title: Variation of Action with time coordinate variations

text: I was trying to derive equation (65) in the following review: http://relativity.livingreviews.org/open?pubNo=lrr-2004-4&page=articlesu23.html This slightly unusual then usual classical mechanics because it includes a variation of time also, $\delta t$. Usually one would define, $$\delta S~=~ \int \left[ L \left(\tilde{q}(t),\dot{\tilde{q}}(t),t\right) - L \left(q(t),\dot{q}(t),t\right)\right] dt , $$ where, $$\tilde{q}~=~q+\delta q. $$ We have then (before applying int by parts), $$ \delta S ~=~ \int \left(\frac{\partial L}{\partial q}\delta q +\frac{\partial L}{\partial \dot q}\delta \dot q\right)dt.$$ How does one proceed if both $q$ and $t$ vary and further that $q$ depends on a varying $t$? Is the definition now $$\delta S~=~ \int \left[ L \left(\tilde{q}(t+\delta t),\dot{\tilde{q}}( t+\delta t), t+\delta t\right) - L \left(q(t),\dot{q}(t),t\right)\right] dt~? $$ If so how does one proceed?


title: Peak at zero in one device and not the other

text: I was wondering if anyone could shed some light on this problem. I have placed two accelerometers on an animal one sampling at 50 Hz the other at 100 Hz. They were placed in the same position. I then computed the FFT on my results and found that the spectra from the device sampling at 100 Hz had a large peak at zero whilst the other device didn't display this large zero peak. Any ideas why this might be? I thought that it may be because of the increased sampling rate thus displaying more points that were the same or 'stationary'.


title: Methods for quantifying a network of coupled oscillators

text: I usually am more on the statistics part of things, so pardon my misuse of the terminology. I am simulating a network of non-pulse coupled non-linear oscillators ( I am not sure what the correct term for that is). The network topology is one of the parameters I'm investigating, but considering nearest d-neighbours and scale free networks. I am interested in finding a method for analyzing (or summarizing, like a statistic) the phase relations of the whole network, given time varying forcings applied to each oscillator. Preferentially, I'm looking for a model free approach, so that I don't have to simplify the dynamics of the oscillators.


title: Any difference between thermodynamic double-derivative and derivative "at constant" value?

text: Reading about the Maxwell relations has left me confused, and I want a basic sanity check regarding the notation. The Wikipedia article breezes over the following switch of notation without really describing it: $$ \left(\frac{\partial T}{\partial V}\right)_S = \frac{\partial^2 U }{\partial S \partial V} $$ Probably, it is also true to say: $$ \left(\frac{\partial T}{\partial V}\right)_S = \frac{\partial^2 T }{\partial S \partial V} $$ There are many examples, this is just the first case-in-point. I'm familiar with the subscripts being presented as "at constant x". In the above case, that would be saying, "dT/dV at constant entropy". Firstly, is that a correct interpretation? Secondly (and if it is), how are the above two things the same? For another example, how would the following two be the same things:   * differentiating pressure with respect to temperature at constant volume vs.    * differentiating temperature with respect to volume and _then_ pressure? While I ask this, I can already see that my mental picture is incomplete. It's nonsense to ask to differentiate temperature with respect with volume with no other specifiers because that doesn't respect the total degrees of freedom. So could you introduce another auxiliary variable, and then and then explicitly show the above equality? I'm sure this is often taken as trivial, but I find it non-trivial for myself.


title: Balloon, lighter than air and vacuum?

text: As I understand, in a balloon/air balloon, a gas burner is used to heat air or using some lighter atom like helium. Since helium or hot air is lighter and less dense than the cool air around the balloon, the heated air or helium causes the whole balloon to rise. And as I understand, vacuum is a space where there are no atoms inside. So no atom inside means it's lighter than to have atom an inside? Isn't it? So why we can't make something fly by giving it a vacuum container?


title: Fourier expansion of the Klein-Gordon field

text: Is there a reason(both physical and mathematical) why the Klein-Gordon field is represented as a fourier expansion in the second quantization as opposed to other mathematical expansions? Be gentle with the answers!


title: Why are these periods the same: a low earth orbit and oscillations through the center of the earth?

text: Related: Why does earth have a minimum orbital period? I was learning about GPS satellite orbits and came across that Low Earth Orbits (LEO) have a period of about 88 minutes at an altitude of 160 km. When I took a mechanics course a couple of years ago, we were assigned a problem that assumed that if one could drill a hole through the middle of the Earth and then drop an object into it, what would your period of oscillation be? It just happens to be a number that I remembered and it was 84.5 minutes (see Hyperphysics). So if I fine-tuned the LEO orbit to a vanishing altitude, in theory, I could get its period to be 84.5 minutes as well. Of course, I am ignoring air drag. My question is: why are these two periods (oscillating through the earth and a zero altitude LEO) the same? I am sure that there is some fundamental physical reason that I am missing here. Help.


title: How to wire a STRONG electromagnet?

text: I have a 12mm long hard ferrite core that I want to use as an electromagnet. I have wound it with 0.8mm copper as tightly as I can by hand, in a single layer, but the field is not strong enough. I have placed another coil over that first coil, and then a third coil, each in parallel - branching before the core, and then re-connecting afterward - wound concentrically in the same direction, to increase the number of windings. I'm still not getting the magnetic force I need. The electrical source is a car battery with 490 cold cranking amps (CCA) -- There is no overheating now that I have 3 branches. Am I doing something wrong? Is there a winding method that will work better? Do I need higher voltage or just higher current?


In classical mechanics, we usually try to find a set of coordinates by Hamilton-Jacobi method to transform the Hamiltonian to zero such that the coordinates are conservations. However, we never try by similar steps to transform the Hamiltonian to zero because what we can get is nothing from the zero-Hamiltonian. Why don't we use Hamilton-Jacobi method in QM?


title: How would an X-ray scanner identify a mirror?

text: A mirror is under normal circumstance used to reflect Electromagnetic radiation also known as photons (light) and in airport security or medical facilities, they use X-rays to detect anomalies inside objects or bodies to detect narcotics or injuries. However, I always wonder _what if I add a mirror inside the luggage or put a mirror in front of me during scanning?_ That in mind, how would an X-ray scanner see the mirror? Would it be invisible? I am sure I am not the first one to think of this, as a lot of security and criminals thought of this, however I never got an answer, so can someone tell me please? If there are X-ray reflecting mirrors? Why don't Airport security ban these items and Mirrors all together? Would X-Ray mirror look like a normal mirror? Do they reflect visible light spectrum as well?


title: Finding the moments of the Boltzmann/Gibbs Distribution

text: I am trying to compute the moments of the Boltzmann distribution using a moment generating function, by taking the Fourier transform of the distribution and then taking derivatives to find the appropriate moment. However I am unsure of the integration limits of the Fourier transform step: $$\int_{a}^{b}dE e^{-ikE} e^{-\beta E}$$ Its clear that this integral doesn't converge on the limits negative infinity to positive infinity (the usual limits for FT's). So my first thought is alright, lets rig it and integrate from zero to infinity. But I'm not sure this is justified in the FT step. The "rigged" limits give me an answer, but I think its wrong. I want to ultimately relate the specific heat, to the moments of the Boltzmann distribution. Perhaps there is a better way to go with this. Thanks ahead.


title: Laplacian in tensor

text: Find $\vec \nabla^2\phi $ when $$ds^{2}=-dt^{2}+a^{2}(t)[dx^{2}+dy^{2}+dz^{2}] $$ or $$g_{ij}=\begin{bmatrix} -1 & 0 &0 &0 \\\ 0 &a^{2}(t) &0 &0 \\\ 0&0 &a^{2}(t) &0 \\\ 0 &0 &0 & a^{2}(t) \end{bmatrix}$$ I am using this relation $$\vec \nabla^2\phi=\frac{1}{\sqrt{g}}\frac{\partial }{\partial x^{k}}[\sqrt{g}g^{kr}\frac{\partial \phi }{\partial x^{r}}]$$ to calculate the laplacian but I am struggling here since if I calculate the determinant (g), it will give me $-a^{6}(t)$, so if I calculate $\sqrt{g}$, it will give me a complex number which is impossible. Also, even though I calculate $g_{11},g_{22},g_{33},g_{44}$, I won't to be able to find $x^{1},x^{2},x^{3},x^{4}$ as I don't know which coordinate system is it and hence don't know its coordinates.


title: Physics behind a match performing a trick on center of mass

text: https://www.youtube.com/watch?v=Ucdw0DDI4n8 I've seen another variation where the whole match stick turned to ash. What's going on in this trick?


title: Brans-Dicke gravity and Black-Hole "vacuum"

text: The field equations of the Brans-Dicke gravity are $$\Box\phi = \frac{8\pi}{3+2\omega}T$$ $$G_{ab} = \frac{8\pi}{\phi}T_{ab}+\frac{\omega}{\phi^2} (\partial_a\phi\partial_b\phi-\frac{1}{2}g_{ab}\partial_c\phi\partial^c\phi) +\frac{1}{\phi}(\nabla_a\nabla_b\phi-g_{ab}\Box\phi)$$ In general relativity, the singularity in the Schwarzchild black hole seems to be considered either a 'delta function' like mass source term, or the "edge"/boundary condition of a manifold. Either way, the equations can't tell us how the "infinite curvature"/singularity evolves and therefore we are forced in some sense to "impose it" as a boundary condition either way. However, in Brans-Dicke gravity the effective gravitational coupling constant appears to be sourced by $T$. So it appears in a true vacuum $T \rightarrow 0$, $\phi \rightarrow 0$, and therefore no Schwarzchild solution is possible _regardless_ of the value of the parameter $\omega$. But I've heard that Brans-Dicke is equivalent to General Relativity at least in some limit, so what is going on here? Brans-Dicke seems to really care precisely how we deal with singularity, while General Relativity doesn't care what is behind the event horizon. **1) Can Brans-Dicke gravity support black hole vacuum + curvature singularity solutions?**   **2) Does it force us to interpret the singularity a certain way, and if so does the Brans-Dicke $\longrightarrow$ General Relativity limit tell us the singularity should be interpretted as non-vacuum to make sense?**


title: Singularity-free stationary electro-vacuum solution

text: Let's say we have a spherically symmetric fluid: $$ T^{\alpha \beta} = \begin{bmatrix} \rho & 0 & 0 & 0 \\\ 0 & p & 0 & 0 \\\ 0 & 0 & p & 0 \\\ 0 & 0 & 0 & p \end{bmatrix} $$ where $\rho$ and $p$ are functions of the radial coordinate r. If $p=-\rho/3$ this would be traceless, and possibly compatible with an electro-vacuum solution (Are there other necessary conditions for compatibility besides the stress-energy tensor being traceless?). While it would probably be unstable, **is there a singularity free electrovacuum solution of GR which is stationary?** It seems that this has enough symmetry that someone skilled with differential- geometry and GR could actually probably just solve this themselves. I'm interested in not just the answer, but also in seeing the process of how to solve this in GR. So please include any work in your solution, thank you!


title: Can a planet cover the whole sky as seen from its satellite

text: Sorry if this is a trivial question, but I do not know enough physics to answer this on my own. Can there exist a satellite that revolves around a planet so large such that the planet covers the entire sky as seen from at least one point on the satellite? **Added 1** I am sorry for being unclear earlier. But assume human eyes can't resolve less than $0.02^\circ$. **Added 2** Assuming both the planet and satellite (ignoring atmosphere to make things simpler) are spheres and since spheres are locally euclidean, for large enough radii a planet could cover the whole sky as it appears to the eye. Consider this picture: ![enter image description here](http://i.stack.imgur.com/h5Emi.png)


title: Is it possible to change units in order to simplify the value of an exponential?

text: I have the equation $$F=e^{E_0 i \pi}, $$ where $E_0$ is the time-independent electric field, and $F$ is just some important value I am trying to calculate. Obviously, it would be better if $F=-1$, but the $E_0$ is in the way. Therefore, is it permissible to take units where $E_0=1$ to get $F=-1$? I feel a little uncertain with this operation.


title: Why I like the DO data

text: I tried to post a good question with images, but I was stopped by the 10 pt rule. My question is now posted at: http://d0vscdf.blogspot.com/2011/06/maybe-i-just-dont-know-enough.html Just want to know if anyone can explain what I see.


title: What kind of phases nanoparticles have (gas-solid-liquid)?

text: If a phase transition requires a number of particles that is in the TD-limit, can nanoparticles (~10 atoms) have phase transitions? What kind of phases and transitions nanoparticles have?


title: What happens for the spins around the phase transition

text: Suppose we now consider a lattice of spin, say Ising model, and the phase transition at the critical temperature $T_c$. There are few scaling laws describe the regime around the critical temperature such as correlation length $\xi$: $\xi\sim|T-T_{c}|{}^{-\nu}$ where $\nu$ is the critical exponent. This equation describes the divergence of the correlation length as the temperature approaching $T_c$. However, it gives no direct physical picture of the spins near the $T_c$ because we need to compute the summation over each pair of spins to find the correlation. So the question is: If we really look at the lattice, what is the most obvious feature and scaling that we can obverse?


This is a question that was rised when we were discussing "what is melting actually". How many particles you need to form a liquid or solid. I have some remarks to point out what I want to know. Q: How many particles is needed to observe a phase transition? A:   1. just one, phase transition is not a collective phenomena.   2. 1 million, Ising model in 1000x1000 lattice produces a phase transition.   3. $N_a$, phase transition is a thermodynamic concept and you always need to be in TD limit.   4. The question is dependent on the process. For instance, in phenomena A you need N particles, but in B you need M particles.    5. The question is stupid, you should ask X instead. I would like to know which arguments are true and which are false (and why). Also, how many particles you need to form a liquid from solid (one particle with Gibbs energy above some limit X is a liquid?).


title: Teleportation by quantum isolation

text: This is a thought experiment, so please don't treat it too harsh :-) Short: If we could isolate two places A and B in the universe from all and any interaction with the surroundings, is there a physical law which states "if something is dropped in place A, it has to stay there"? Long version: Let's assume that the energy of the whole universe is fixed. Let's further assume that it is (by some trick) possible to completely isolate a box of 1m^3, say in the center of a planet (all gravitational and centrifugal forces cancel themselves out), the mass of the planet shields against radiation and we use a trick to shield against neutrinos or we ignore them since the rarely interact with matter). How does an object behave when it has no interaction with the rest of the universe whatsoever? If I put an object in a box described above and I have several such boxes, would it matter in which box the object is? Is there a law which says "even if no one knows, the object **has** to stay where it is"? Or is that just our expectation based on everyday experience?


title: Measuring the speed of light and defining the metre - absolute or relative?

text: If the metre is now defined as the distance light travels in vacuum in `1⁄299,792,458th` of a second and the speed of light is accepted to be `299,792,458 m/s`, doesn't this seem like the chicken and egg problem? I remember reading somewhere (in the context of uncertainity principles).. > ..the more precisely one property is measured, the less precisely the other > can be measured So, how then do physicists claim to have accurately measured the speed of light? Does this mean the definition of the metre is dependent on our ability to accurately measure the speed of light? After all, to determine speed (distance traveled/time taken) you must first choose some standards of distance and time, and so different choices can give different answers? What about so many other factors that affect these measurements. I do not claim to understand the theories of relativity entirely but what about the chosen frame of reference? Spacetime curvature? Does this mean that our measurements are only relative and not absolute?


title: Simple Harmonic Motion Question - Block on Platform

text: A platform is executing SHM in a vertical direction with an amplitude of $5$ cm and a frequency of $\frac{10}{\pi}$ vibrations per second. A block is placed on the platform at the lowest point of its path. $a)$ At what point will the block leave the platform? $b)$ How far will the block rise above the highest point reached by the platform? For part $a)$ I'm not even sure how to tackle it...From my understanding, the block would leave the platform when it has an acceleration that exceeds gravity, but I don't know when that would be? I'm just very confused about the entire problem, and i'm having a hard time visualizing it as well. Any help would be appreciated.


title: Determine whether the ground state is an eigenfunction of [p] and of [p^2]

text: Consider a particle confined in an infinite square well potential of width L, $$V(x)=\left\\{ \begin{array}{ll}\infty, &{\rm for}\ (x \le 0)\vee (x \ge L) \\\0, &{\rm for} \ 0 < x < L \end{array}\right. $$ The ground state solution of the time independent Schrödinger equation is given by $\psi$(x)=Asin(kx), where k = $\frac{\pi}{L}$ and A = $\sqrt{\frac2L}$. Determine whether the ground state is an eigenfunction of $[p]$ and of $[p^2]$. Discuss the implication of the results. So far I have guessed the expectation value to be $\frac{L}{2}$ and confirmed it by evaluating the integral (from an earlier part of the question). I'm stuck on how to do the last part though. From a previous second year quantum mechanics exam, any help would be greatly appreciated.


title: Can there be a phase transition without change in entropy and volume?

text: **Context** : I'm studying basic thermodynamics. My lecture notes pose a question, whether there can be a phase transition without a change in entropy or volume ? Do you have any hints on where to start from ? * * * **Idea 1** : I know that $dG = 0$ during the equilibrium ($G$ is the Gibbs free energy). If I take that route, I get: $dG = -SdT + VdP = 0$. Something that I thought was that I'm interested in $dV,dS$ but my equation has $dT,dP$ in it, so perhaps a Legendre transformation would help here ? I.e., $\widetilde{T} = G + TS - VP \Rightarrow d\widetilde{T} = dG + TdS + SdT -VdP - PdV = TdS -PdV$. How would I proceed from now on ? If I plug $dS=0,dV=0$ in the equation I get $d\widetilde{T}=0$. What if I claim that only $G$, from the various thermodynamic functions, is kept constant during the transition, therefore $\widetilde{T} \equiv G$, which contradicts $\widetilde{T} = G + TS - VP$ ? **Idea 2** : Another line of attack is to consider the Clapeyron equation which holds for the transition \begin{equation}\frac{dP}{dT} = \frac{\Delta S}{\Delta V}\end{equation}. If both entropy and volume do not change, then I get an indefinite form: \begin{equation}\lim_{\Delta S\to 0,\Delta V\to 0} \frac{\Delta S}{\Delta V} = \frac{0}{0}\end{equation} But how do I proceed from here on ?


title: What is average life in radioactivity and what is its significance?

text: By definition, average life of radioactive sample is the amount of time required for it to get decayed to 36.8% of its original amount. But what is the significance of 36.8% and why has that value been chosen?


title: can we charge a neutral glass rod by touching it with charged glass rod?

text: Can we charge a neutral glass rod by touching it with charged glass rod?Can we charge similar objects by contact of which one is charged and other neutral?


title: What is the physical interpretation of the Papadodimas/Raju mirror operators?

text: In this paper http://arxiv.org/abs/1310.6335, the authors discuss the firewall problem and contruct so called mirror operators appearing in the correlation function. The key part seems to be (2.6) where the horizon correlator is computed (using the field operators and their mirror images) only from 'one side' for a thermal, mixed, state (as if they traced over the interior). This ensures the smooth transition across the horizon (ignoring some subtle issues they discuss as well). So, the mirror operators have the nice properties that they act localy (inside or outside) and you can decide whether to calculate the correlator on the whole state ($\Psi_{tfd}$ being the Hawking state) or just on one branch (outside as in (2.6) or (2.7)). Here comes my question(s): They often talk about doubling the degrees of freedom (inside vs outside or no tilde vs tilde in their notation). Does their construction imply that in reality there is some redundancy in the description of the black hole interior? What I mean is that if I imagine an evaporating black hole, I get a Hawking photon outside (no tilde) and this has a counterpart inside (tilde). But by emitting the photon, the black hole (I want to consider those formed by a gravitational collapse) should lose its mass (as if a particle tunneled through the horizon) and there should be no _real_ counterpart inside. Is this the picture suggested by the authors of the paper when they talk about doubling the degrees of freedom? Is somebody (inside the black hole) actually holding the tilde degrees of freedom?


title: Charge of electrons and protons

text: I have a question which reads: > "A conductor has a charge of $9.6 * (10)^{-19}$ coulomb. The conductor will > have- " The problem I face in this question is due to not knowing the exact value of the charge of electrons. Is it $1.6 \cdot (-10)^{-19}$ or $-1.6 \cdot (-10)^{-19}$ or $1.6 \cdot (-10)^{19}$ ? In fact, since, the $1.6$ remains constant, I divided $9.6$ by $1.6$ and got $6$. But the options ((a) 6 electron less (b) 6 electron more) are confusing. I researched and found these things (1, 2 and rest are marked duplicate) here but did not find the value of charge in them. . Could you help me out?


title: Why in some cases $0\alpha$ component of stress-energy tensor don't form 4-vector?

text: In electrodynamics there is Poynting vector and energy density, which refer to $0\alpha $ components of stress-energy tensor, don't create 4-vector. Analogous situation with mass density and mass current density in general relativity. So why don't they form 4-vector (with what property is it connected)?


title: Understanding multiple instanton contributions to vacuum tunneling in a double potential well

text: I'm trying to understand the method of calculating the transition probability of one vacuum state to another in a double well potential, using instantons. The reference I am following is Sidney Coleman's book "Aspects of symmetry". The approach uses the path integral formulation of QM and we look at the semi- classical limit of small $\hbar$. In this limit our path integral is dominated by the stationary points of the (euclidean) action. Solving the equation of motion I find an infinite number of single instanton solutions (each different by a shift of the time origin of the instanton) starting at one vacuum and ending at the other. * * * For example, if I took a potential of the form $V = \lambda(x^2-a^2)^2~~~$ and define $~~~\omega^2 = 8\lambda a^2$ our solution to the E.O.M with $~x(-\infty) = -a~$ and $~x(\infty) = a~$ would be $x(t) = a\tanh\left(\frac{\omega}{2}(t-t_c)\right)$ where $t_c$, the center of the instanton is arbitrary. * * * Coleman then introduces other _"approximate stationary points"_ which are effectively instanton/antiinstanton chains which start at one vacuum and end at the other. Coleman then continues in his analysis by adding up all of these approximate stationary point contributions to the original single instanton stationary point term. **My question is the following:** in what sense are these multiple instanton/antiinstanton solutions "approximate stationary points"? (Is there some limit I can take where it becomes exact, or could someone explain why these paths don't have a negligible contribution to the path integral unlike other paths?)


title: Is light brightness subjective or can it be quantified?

text: What is the relationship between brightness perception and physical properties of light, such as light intensity?


title: Is my physics correct or torque correct?

text: ![enter image description here](http://i.stack.imgur.com/VPQaJ.png) I would like to put this into a differential equation. This is what I have. $$r \times F = I \ddot\theta + \mu \dot\theta + k \theta$$ What I need verified:   * $\text{Torque} = I\ddot\theta + \mu \dot\theta + k \theta$   *     * $I$ = moment of inertia (constant)   *     * $\mu$ = kinetic friction (constant)   *     * $k$ = spring constant ( in my application it will be 0 ) (constant)   *     * $\theta$ = rotation (function)   * Torque = $r F$   *     * $r$ = radius   *     * $F$ = force   * It's safe to set the two equations for Torque equal to describe the system.    * That a solution to the differential for $\theta(t)$ will describe the rotation with respect to time of the circle cylinder. Any insight would be much appreciated, thanks.


title: What are 'two way' magnets and how do they work?

text: Thomas & Friends Take-n-Play series connect to each other via something described as 'two way' magnets: > the new models use special 'two-way' magnets As a parent of an owner of two engines of the said series I can attest that indeed the magnets seem to ignore polarity. They attract each other in any combination and, furthermore, they attract 'classical' magnets both N pole and S pole. I suspect this is some alloy that is capable of rapidly re-aligning its magnetic domains, but is just speculation. I couldn't find any reference to something called 'two-way' magnets nor anything else describing the behavior I'm seeing.


title: How does one calculate calculate buoyancy with gases?

text: I have hot air flowing vertically down a circular duct into the ambient cold air. I'm trying to find the power $P$ that resists the flow. I know the:   * Volumetric flow rate $Q$ $[m^3/s]$   * Flow speed $v$ $[m/s]$ (calculated from flow rate and duct diameter)   * Density of the ambient air $\rho$ $[kg/m^3]$ * * * So far I have assumed that the air flowing from the duct is an object that displaces the ambient air - much like if you were to submerge a solid object in water. I've used Archimedes' Principle to calculate the buoyant force $F$ $[N]$ and multiplied it by the flow speed $v$ $[m/s]$ to get the power $P$ $[W]$. **Method 1 - Convert mass to force** $m$ $[kg]$ $=$ $V$ $[m^3]$ $*$ $\rho$ $[kg/m^3]$ ($V$ is gotten from the volumetric flow rate $Q$, assuming that we're looking at one second.) $F$ $[N]$ $=$ $m$ $[kg]$ $*$ $g$ $[m/s^2]$ $P$ $[W]$ $=$ $F$ $[N]$ $*$ $v$ $[m/s]$ **Method 2 - Impulse** $m$ $[kg]$ $=$ $V$ $[m^3]$ $*$ $\rho$ $[kg/m^3]$ $J$ $[kg*m/s]$ $=$ $m$ $[kg]$ $*$ $v$ $[m/s]$ (Once again $V$ is from the flow rate and $v$ is from the flow speed.) $F$ $[N]$ $=$ $J$ $[kg*m/s]$ $/$ $t$ $[s]$, ($t$ $=$ $1s$) $P$ $[W]$ $=$ $F$ $[N]$ $*$ $v$ $[m/s]$ * * * Method 2 produces a significantly larger power $P$ and I feel that it's closer to the truth. (This is an impulse, right?) However, I am no physicist and the whole assumption of the flowing air being comparable to a solid object may be entirely wrong. **Can you approximate the power with the Archimedes' Principle** or what should I attempt to get the resisting power $P$ caused by the less dense hot air flowing into the denser ambient air - is that buoyancy even significant? Thank you for taking the time to read this!


title: Density of states of a photon gas in volume V and temperature T

text: I have a question on the density of states for a photon gas: Suppose I have a photon gas in a box of volume $V$ at temperature $T$. If I enumerate the total number of states accessible to the system given that the system has a total energy $E$, and differentiate this with respect to the Energy, I should obtain the density of states in the energy space. However, when I do this I obtain: $$ \Gamma = \int d^{3}r_{1} d^{3}r_{2} d^{3} p_{1} d^{3}p_{2} \delta((\frac{|p_{1}|}{\hbar}+\frac{|p_{2}|}{\hbar})c-E) $$ where the two momenta are for the independent polarizations (modes of oscillation) of the gas. Then $\frac{d \Gamma}{d E} $ is not the "correct" density of states given in a textbook. What is wrong with this approach? **Edit:** Showing more work:   The answer I obtain is $\frac{d \Gamma}{E}=\frac{8 * (4 \pi )^{2} V^{2} \hbar^{6} E^{7}}{c^{6} 4*280}$ the textbook says that it is $\frac{E^{2}}{\pi^{2} c^{3} \hbar^{3}}$


title: Doubling the energy of an oscillating mass on a spring

text: From this question: > **Question 1.**   >  What do we need to change in order to double the total energy of a mass > oscillating at the end of a spring?   >  (a) increase the angular frequency by $\sqrt{2}$.   >  (b) increase the amplitude by by $\sqrt{2}$.   >  (c) increase the amplitude by $2$.   >  (d) increase the angular frequency by $2$.   >  (e) increase the amplitude by $4$ and decrease the angular frequency by > $\frac{1}{\sqrt{2}}$. The correct answer is given as B, but from my calculations, option A is correct. What am I doing wrong? Also, I can't find a relationship between amplitude and energy. $$\begin{align} KE_\text{max} &= \frac{1}{2} mv_\text{max}^2 \\\ &= \frac{1}{2}m(rω)^2 \\\ &= \frac{1}{2}mr^2ω^2 \\\ KE'_\text{max} &= \frac{1}{2}mr^2(\sqrt{2}ω)^2 \\\ &= mr^2ω^2 \\\ &= 2KE_\text{max} \end{align}$$


title: Estimating the fatigue life of elastomer diaphragm

text: _**Disclaimer_** : I have asked this question before, but it got put on hold because it was judged as off-topic. I happened to disagree and explained my reasons in the comments, but they have been ignored, so I decided to repost the question. **The question is not off-topic, do not put in on hold**. It is a very specific physics-related question. I have a diaphragm in a pump, which is a disc of some elastomer (probably either EPDM or Viton), clamped on its circumference, with a hole in the middle where a piston is moving up and down. There is an interference fit between the piston diameter and the hole diameter in the diaphragm, so that the disc stretches over the piston, and the vertical motion of the piston results in deflection of the diaphragm. I need to estimate (approximately) the number of cycles the diaphragm is capable of going through before failure, based on amplitude, frequency, thickness and diameter of the disc, material properties, etc... Can anybody point me in the right direction? I have found various in-depth studies on stress/strain and fatigue life, but not a simple way of estimating the fatigue life. Another compounding factor is the fact that the elastomer may be in contact with some aggressive chemicals, but I will ignore this for the time being.


title: Interval and proper time

text: Is the definition of $$d s^2=-d \tau^2$$ assuming that $c=1$, so that we always have $$\left({ds\over d\tau}\right)^2=-1$$? Is there a reason for this definition? Don't we get an imaginary ${ds\over d\tau}$?


title: Evaluate Commutator with Partial Derivatives

text: I need to evaluate the following commutator... $[x(\frac{\partial}{\partial y})-y(\frac{\partial}{\partial x}),y(\frac{\partial}{\partial z})-z(\frac{\partial}{\partial y})]$ i tried applying an arbitrary function $\psi(x)$ but I am unsure of how to evaluate the whole thing. My first step was: $=(x(\frac{\partial}{\partial y})-y(\frac{\partial}{\partial x}))*(y(\frac{\partial}{\partial z})-z(\frac{\partial}{\partial y}))\psi(x)-(y(\frac{\partial}{\partial z})-z(\frac{\partial}{\partial y}))*(x(\frac{\partial}{\partial y})-y(\frac{\partial}{\partial x}))\psi(x)$ This is where i am lost, I am unsure of where to go from here. Maybe there is something else I can do as my first step to simplify the question? please let me know what I can do. Thanks.


title: Effect of water on wobble of rotating structure

text: If we have a structure that rotates to create artificial gravity, then if the mass isn't perfectly distributed along the circumference the CM will be offset from the geometric center so there will be a wobble. I made this illustration, to show the wobbles that happens becuse a person goes to visit his/her friend on the opposite side of the structure. ![wobble](http://i.stack.imgur.com/3Cezf.png) Some artist conceptions include water in the forms of lakes, and are often contiguous bodies of water that circle the entire circumference (Bernal sphere example). If we assume there is a free-flowing body of water around the entire circumference like this (and is sufficiently deep), then it's clear that water will move as a result of a mass movement since the previous water surface is no longer equipotential. This will also move the CM. How would the presence of water affect the wobble?


title: Which was first, energy or matter in the creation of our universe?

text: Was it the Big Bang or was it something else that gives us our universe in its present condition? Did it all begin with just pure energy that eventually evolved into simple atoms of matter, that themselves evolved again into more complex structures similar to the way living organisms have evolved?


title: How can the nucleus of an atom be in an excited state?

text: An example of the nucleus of an atom being in an excited state is the Hoyle State, which was a theory devised by the Astronomer Fred Hoyle to help describe the vast quantities of carbon-12 present in the universe, which wouldn't be possible without said Hoyle State. It's easy to visualise and comprehend the excited states of electrons, because they exist on discrete energy levels that orbit the nucleus, so one can easily see how an electron can excite from one energy level into a higher one, hence making it excited. However, I can't see how the nucleus can get into an excited state, because clearly, they don't exist on energy levels that they can transfer between, but instead it's just a 'ball' of protons and neutrons. So how can the nucleus of an atom be excited? What makes it excited?


title: Reading Paul Dirac's "Principles of Quantum Mechanics"

text: I have a similair question to the question here, but regarding a different book. "Principles of Quantum Mechanics" is a 1930 work by British Nobel laureate Paul Dirac. The wikipedia article on this book states that there have been a couple of editions (with improvements and additions), the latest being the 4th edition, published in 1958 (6 years before the Feynman lectures were published). This is a long time ago and I'm wondering if parts of the book are outdated, and if so, which ones and why?


title: Should I use Coulombs law when magnets attract/repel?

text: When magnets attract to each other or repel. Should I use Coulombs law? If not, why not? Some would say that I shouldn't because: "Coulomb's law deals with static charges and force due to them. Whereas magnetism is force due to moving charges!" What do you all think?


title: Free-Free Modal Analysis of Vibration

text: I am guessing if it possible to use Modal Analysis to simulate the vibration on a Free-free object. By free-free, I mean an object which don't have any fixed points and vibrates freely. Directly in the simulator, Free-free means that all the tetrahedra can move freely. So, there isn't any fixed tetrahedra. In my experiments, every time I do this the simulation becomes numerically unstable and crash. So there is any method to use Modal Analysis for vibration, without specifying any fixed points ? or there is any method to give stability to a Modal Analysis without fixed points ?


title: Why the CFL light bulb flashes even if there is a power cut?

text: Today when there was a power cut I saw that the CFL light bulbs were flashing time to time at regular interval. I was amazed as there was no source of power to the CFL bulbs even they were flashing as if they were sucking energy from surrounding, collecting it and then releasing them with a flash. Why was it happening?


title: Chern-Simons degrees of freedom

text: I'm currently reading the paper http://arxiv.org/abs/hep-th/9405171 by Banados. I am just getting acquainted with the details of Chern-Simons theory, and I'm hoping that someone can explain/elaborate on the following statement: > _In a field theory with no degrees of freedom like Chern-Simons theory, the > only relevant degrees of freedom are holonomies or global charges._ I've also heard things like "Chern-Simons is topological" which sounds related. Is it?


title: Why does electric field remain constant in parallel plates when distance changes?

text: Given the formula E0 = ∆V/d, why does electric field stay constant when distance changes? If a property of parallel plates is that ∆V is constant, shouldn't E0 change as d changes?


title: What would be physics of a star totally consisting of electrons?

text: What is the minimum radius and mass (if any) for such a structure to be stable? What is the ultimate future of such structure? Will the electron repellation always be stronger than any gravitational attraction? Even at any density?


title: Potential at Center of Earth

text: If using the surface of the earth as a reference point how much work is needed for gravity to pull me to the center. Is it negative infinity or am I wrong? Also is a single value of potential meaningless. Does it represent anything? Do you at least need a potential difference for it to represent work done? I'm extremely confused about potential and potential energy.


title: Relation between component and algebraic definition of covariant vectors

text: I studied contravariance and covariance concepts in following way: For any vector if we get its components by parallelogram way we achieve contravariant components, and if we want to get its components by orthogonal projection we achieve covariant components. For latter case we need reciprocal basis in order we can expand the vector by its covariant components. But I read somewhere, in fact covariant vectors are elements of dual space and they expanded by dual basis. My question is: how is the latter approach to covectors equivalent to previous approach? Furthermore, I need some good explanation for latter approach to covariant vectors.


title: Using Jacobians to prove an identity

text: I have to prove the identity $$ \left(\frac{\partial U}{\partial V}\right)_T=T\left(\frac{\partial p}{\partial T}\right)_V-p $$ I know how to do this using Maxwell's relations, but the assignment specified that I have to use Jacobian transformations. Reading many websites, pdf's, etc. has brought me no closer to the solution. I think that I have to start from the equation for internal energy and then differentiate it using $dV$ - this produces almost the end result, but when I work with the Jacobian I'm always lost. Any explanations on how to use the Jacobian would be great. I know the basic rules, but cannot solve this. Could you help me please?


title: Resonant Frequency & Opera Singers

text: Would it be possible under math of strings to note the frequency of each string vibrations? And in doing so, in hand with using the technique opera singers use to shatter glass with their voice, would it be possible to destroy/aggravate each string accordingly?


title: Can magnets rotate infinitely?

text: There are many videos on youtube in which people arranged magnets in circle and rotated one placing in middle of that circle on a shaft, and the magnet (magnet motor) starts madly and continues its movement.   Do they really rotate infinitely?   If so, do they from where they get that extra energy to move so fast and infinitely? EDIT: This question is about real world scenario UPDATE: I am unable to find that particular video about which i talked about -- I added them to my favourites, but they are all now deleted :( -- there are hundreds of that that kind of videos. Some example videos links   http://www.youtube.com/watch?v=dT9s33X9D4I   http://www.youtube.com/watch?v=yei0NMqUaZ8   http://www.youtube.com/watch?v=5DvPLFX5gwA   http://www.youtube.com/watch?v=i0oUaPZ_wF8   http://peswiki.com/index.php/OS:Screw-Magnet_Motor   http://www.youtube.com/watch?v=aHVBu77jz4w


title: Are perpetual magnetic motors a scam?

text: I read articles and watched videos and more videos where people say they've built perpetual magnetic motors. I do not see any proof that this can work and also I do not see proof that it is wrong. I'm not really into physics. People tend to say this is scum and cannot happen. Can somebody explain what is wrong in the first video?


Why is absolute zero considered to be asymptotical? Wouldn't regions such as massive gaps between galaxy clusters have temperatures of absolute zero? I just do not see why our model must work the way that it does. I mean there have to be regions with no thermal energy out there, the universe is massive.


title: Condensate fraction and single-particle density matrix

text: In Bose–Einstein condensation (BEC), how to prove the largest eigenvalue of the single-particle density matrix $$\rho_{ij}=\frac{\langle\Psi|a_i^{\dagger}a_j|\Psi\rangle}{N}$$ is $$\frac{1}{N}\sum_{i}{|\langle \Psi|a_i|\Psi\rangle|^2},$$ which is the condensate fraction f of the total particle number N in the thermodynamic limit?


title: Can the potential difference produced in Van De Graff generator used to generate electric current?

text: If this machine can build up high voltages of order of millions of volts, can we use this volt the generate current of electrons? The building up of potential difference can result in electric field. Can we not use this electric field to generate current?


title: Ashcroft Mermin Solid State Eq. 22.15

text: I recently read a paper on Specific heat of a Classical Crystal: Dulong-Petit law. In Eq. `22.15`, I don't understand why the ionic displacement $\mathbf{u}$ and momentum $\mathbf{P}$ become independent of temperature after $\beta^\frac{-1}{2}$ is extracted. Where does this $\beta^\frac{-1}{2}$ come from?


title: Why does the wind direction vary locally?

text: If the air is flowing with the earth, as it rotates, how do we feel a breeze and strong winds at times, as everyone has experienced, the wind shifts as if from many directions.


title: What metals much harder than steel can be met in real life? Like pins in clothes security tags

text: Once I bought a piece of clothes and forgot to remove magnetic tag in the store. I found, that it consists of two disks, connected with thin pin (of some metal). I was sure it is possible to break this pin with some instruments. But I broke all my saws, rasps and nippers I was applying to the pin. I wonder, which material it is made of?


title: Lenz's law and bar magnet falling from some height above

text: I am reviewing my electromagnetism studies and I became confused with where energy of induced emf due to Faraday/Lenz' law come from. Suppose that there is a curved track that starts with some height above loop of wires. The path passes through the center of the loop at the lowest point of the track. The track is symmetrical and it also ends at the same height as the starting height. Now the bar magnet is placed at the starting point. As there is gravitation potential energy converted to kinetic energy, magnetic flux through the loop will change causing an induced emf. But this suggests that by conservation of energy, this electrical energy must have come from somewhere. Otherwise, we would be able to create electrical energy forever using a bar magnet. This is how my confusion arises. Where does the energy come from?


title: What are type system examples of local gauge transformation- and field strength-like objects?

text: This is essentially a follow up motivated by this answer to my question about the gauge transformation interpretation of identity types. A field $$\psi:\mathcal M\to\mathbb C^n$$ is a section of the $\mathbb C^n$-bundle over the spacetime manifold $\mathcal M$. We have a local gauge transformations $$\psi(x)\mapsto \psi'(x):=U(x)\,\psi(x)\ :\ (\mathcal M\to \mathbb C^n)\to(\mathcal M\to \mathbb C^n).$$ Now consider a language with type polymorphism and the class $M$ of all it's types whose elements can be put into a list. Let $\Psi$ be the polymorphic function which, for every type $X\in M$, maps an $X$-list to an integer, namely its length. For example, using Haskell syntax, if $X=\mathrm{Bool}$, then $\Psi_\mathrm{Bool}\left([\mathrm{True},\mathrm{True},\mathrm{False}]\right) = 3$. In System F notation, we have $$\Psi:\forall X.\left([X]\to\mathrm{Int}\right).$$ The gauge transformations should correspond to maps $$u\ :\ \forall X.\left([X]\to\mathrm{Int}\right)\ \longrightarrow\ \forall X.\left([X]\to\mathrm{Int}\right).$$ I could come up with some $u$'s, for example mapping the length function $\Psi$ to a map $\Psi':=u\,\Psi$ which instead returns 42 times the length of a list. But that would be, in physics terms, a global gauge transformation because it's not sensitive to the type $X$. I think, given that the only invariant of a finite dimensional vector space is its cardinality, it shouldn't be possible to construct a local transformation in this case. What would be a hands-on example for a local gauge transformation in this sense? Moreover, I wanted to draw an everyday life parallel to identity types. Well first there is the minor obstacle that the above transformation can't be given by an expression in most language, as types are usually not first class objects. I guess this design choice is made because otherwise type inference would be spoiled. In homotopy type theory you have realization of "types are terms too" (via n-categories?) and then it's possible. But in any case, I still can't quite pin down the specification when a type is an _identity_ type. I understand "identity" for homotopy equivalent spaces and gauge invariant Lagrangians, but are there non-geometric structures, specifically programming relevant ones, which behave identical before and after the transformation? **edit** : I now made two visualization of the example here and then: ![enter image description here](http://axiomsofchoice.org/_media/poly_bundle.png?cache=) The question then is what a sensible gauge of the section in the second pic would be. (I also made two more pictures going beyond what's asked: natural transformations and monads as in Haskell.) Btw., I know that HoTT does implement dependent types, not "just" parametrically polymorphic ones, but that should not be an obstacle.


I am trying to create a simple 2D Simulation of magnetic fields similar to this; showing the field lines or the effect of permanent magnets. I keep coming across equations like this: $\vec{F}=q(\vec{E}+\vec{v}\times \vec{B})$ which I know are correct. However i Don't know B and I was assuming E was negligible/non existent because its a permanent magnet. I have also looked for related questions on stack exchange and from that come across http://en.wikipedia.org/wiki/Biot%E2%80%93Savart_law#Point_charge_at_constant_velocity . However that seems to show the magnetic field generated by a moving particle, not a permanent magnet. My plan at the minute is to use test charges in the simulation to measure the effect at that point and then repeat at intervals to see the complete field.so how do I calculate the B field (strength and direction) on a point charge due to a permanent magnet? Also I keep seeing the B field mentioned as just use the right hand rule, but how is that adaptable to a 2D simulation? thanks!


title: Physics of Snow Globe

text: What is the physics behind snow flakes inside a 3D snow globe? If I were to implement a snow globe in computer graphics what kind of model do I need to the flakes motion like a real snow globe? What field of physics does this problem belong to?


title: Where does the extra force generated by a lever come from?

text: Given an object being pulled down to earth by a force of 1000 lb: If a machine were to lift it off the ground for me, and I grabbed on to the rope before it was released, there is no way in the world I could hold it in place. But, if I had a sufficiently long (and weightless) lever, I could achieve this with just a finger, with a force of less than a pound. So where do the other 999 lbs of force come from? What triggers it to be applied now when I apply my finger as opposed to before I did? (I understand the rules of torque and its equations; I am looking for and understanding of how & why. It does not help to be told that "by the rule, this must occur, and therefore the rule makes sense" - that is obviously circular. And saying, "well, observation bears it out" defends it but fails to explains it.) Edit 1: To clarify, I'm just asking assuming there is a better answer than "it is a basic law of nature: torque causes force to appear out of nowhere, and cannot be explained in terms of other more basic laws or attributed to some supplier of force". When I ask to understand, I mean as opposed to an axiom, i.e. a starting point of reasoning, accepted but not _understood_ or derivable from other accepted/proven rules. Of course, it's definitely possible that it is an "axiom". Edit 2: After seeing several answers, I think the following is what is really irking me: Shouldn't there be, intuitively, conservation of force? How can there ever be something from nothing? Unless we define a new type of spontaneous force 'lever force', outside of the known four, then it must arise from the known four. But the known four transfer exactly what was applied to them (along the same line), no more no less. (Applying 1lb of force to the top atom in a stack of atoms will never increase its force on the next atom by 2lb after equilibrium is reached - it will be increased by exactly the 1lb being applied to it.) Edit 3: In response to the suggestion that the fulcrum provides the force, please consider the following example: A Ferris wheel with a 1000lb load in one seat. It falls straight to the bottom. And ultimately, like is being suggested, it won't fall to the ground, because the center spoke supports it. However, it _will_ hang straight down - i.e. the center spoke is _not_ keeping the object an extra $r$ (where $r$ is the radius of the wheel) feet in the air! Clearly, any time you raise an object above where it was supported by another object, it requires a force _equal_ to that of gravity. A table-top will _never_ assist you in raising the object above the table top, though it _will_ keep it off the ground (should you release it). Think of the table-top as the seat-containing-the-object hanging straight down - any higher is like lifting the object off the table, and should require a force equal to gravity. Except it doesn't. Hence my question.


title: The physics of the deodorant spray bottle volume

text: I noticed that deodorant (or anti-transpirant) spray bottles (pressurized?) usually contain $150~\text{ml}$, max $200~\text{ml}$ (that I've seen; although _I think_ I spotted some $300~\text{ml}$ on the Chinese internet). Is there a _physics_ reason for this (not being larger)? ![enter image description here](http://i.stack.imgur.com/I2ZbS.png) (For example, and just guessing wildly, could it be that larger bottles are dangerous at certain mountainous heights?? Or larger bottles have a "spraying pressure" that is not constant enough??) PS: Note the quantum bit in the product name. :)


title: How to find the force to cause certain trajectory?

text: So I got a little board today and decided to do some physics. What I came up with is confusing me alot. Say someone throws a rock and when this happens you get the known's of. From here is there a way to find the force that took it to have that trajectory. (kind of like http://imgur.com/2ozNMgH) Angle it was thrown at. (45 degrees) The weight of the rock. (.1 kilo) The distance that the rock was thrown. (82.9056m) Missing Time in Air. Force it was thrown at. Trying to find Speed it was thrown at/force. So with this information we can't really get alot at first but I can almost feel a solution at the top of my tongue. I think that the most critical piece of information in this situation is the angle at which it was thrown (being 45 degrees). From this you can get that the force was split equally between the Y and X component of the situation. I have racked my head for a possible equation (not very advanced in physics of math only high school freshman). All I can think of is that there is some sort of calculus wizardry behind finding this (possibly integral?). I really really have a feeling this can work. Sense I don't know how to write the fancy equations on this and I don't know how to explain what I am thinking enough to explain well. Maybe this will help some more http://www.imgur.com/LHfQJhZ) I understand that because the normal D=1/2gt^2 that there would be two different answers due to it being a quadratic but with it being at 45 degrees they would both be the same. Any sort of help towards finding the answer would be greatly appreciated. Like the old saying,"the fun is getting there, not being there".


title: Loop-the-Loop with Friction

text: Let's consider a track that begins vertically becomes a 450 degree loop, and level off. (See diagram) ![enter image description here](http://i.stack.imgur.com/Kuy52.png) We drop a block from height $H$ that falls and goes around the loop. Ignoring air resistance, but taking into account a friction coefficient of $\mu$ against the track, find the smallest $H$ for which the block will complete the loop. By conservation of energy I have obtained the following equation: $$ mgH = \frac{m v(\theta)^2}{2}+mgR(1+\mu-\sin(\theta)) + m\mu R \int_0^\theta\left( g \sin(\theta) + \frac{v(\theta)^2}{R}\right) d\theta $$ Where $v(\theta)$ the the block's speed at $\theta$, and $\theta$ is an angle measured between the block's position (centred at loop center) and a horizontal line going left-wards. The left hand side is gravitational potential energy, the first term on the right is kinetic energy, the next one is also gravitational potential energy, and the last one - the work done by friction. Dividing by $m$, differentiating with respect to $\theta$, and using the chain rule on the first term on the right hand side we obtain: $$ 0=v(\theta)\frac{dv(\theta)}{d\theta} +gR(\mu\sin(\theta) -cos(\theta)) + \mu v(\theta)^2$$ Assuming the block doesn't make it, because it lost's its speed to friction, let's find $\theta$ for which $V(\theta) = 0$. Substituting we get: $$ 0 = \mu \sin(\theta) - \cos(\theta) $$ But it is totally counter intuitive (and incorrect by experiment), that this point depends only on $\mu$ and not on the initial height $H$. So where did I make the mistake? Could somebody please clarify.


title: How many earth-sized planets have been discovered outside the solar system?

text: How many earth-sized planets have been discovered outside our solar system? Is there a combined registry of them anywhere? Where might I look for more information?


title: After the Space Shuttle is retired, how will the new sections of ISS be launched?

text: The International Space Station (ISS) had many of its components launched by the Space Shuttle. After the Space Shuttle's retirement, what is the proposed launch vehicle for new ISS sections?


title: How many stars are in the Milky Way galaxy, and how can we determine this?

text: I have heard multiple estimates on the quantity of stars within our galaxy, anything from 100 to 400 billion of them. The estimates seem to be increasing for the time being. What are the main methods that are used to make these estimates, and why are there such large discrepancies between them?


title: Do all narrowband filters tint the field of view?

text: I observed color in a nebula for the first time this past weekend when looking at the Ring Nebula through my Orion XT8. I noticed a bit of red on the outer edges. I placed my Orion Ultrablock Narrowband filter on to attempt to observe more detail when I realized that it washes the field of view out in green, so I could no longer see the red. Is this normal? Are there any other narrowband filters that preserve color?


title: Which speed can an electric scooter reach on a given slope?

text: Electric scooters are always "given" as "working with slopes up to xx%", but what does it mean? Given motor torque and power, scooter+driver weight and wheels diameter, how can I determine at which speed it will run on a given slope, regardless of air friction?


title: Industry application of computational quantum mechanics?

text: I was wondering if anybody knew of an industry application of computational quantum mechanics. For example, the efficient placement of circuit elements on a PCB is in part motivated by classical FDTD simulations. I can make a long list of applications of classical methods and simulation, but I can't think of one where quantum mechanics is used. Additionally, I know of quantum mechanics simulators and their research applications, but I feel they are largely used to derive or very more succinct theories. What problems do people use computational quantum mechanics for?


title: CP-violation in SUSY QED?

text: I have just gone through the exercise of constructing the supersymmetrized QED action. In the end, I get a reasonable action which matches literature. But after a little analysis, I find that the supersymmetrized gauge interactions (electron-selectron-photino vertices) seem to violate CP. In the two-component Weyl notation, this is: $$\mathcal{L}_\text{int}=\ldots+\sqrt{2}e(\phi_Le_L^\dagger\lambda^\dagger+\phi_L^*e_L\lambda)-\sqrt{2}e(\phi_R e_R^\dagger\lambda^\dagger+\phi_R^*e_R\lambda)$$ Where $\lambda$ is the gaugino, $\Phi_L=(\phi_L,e_L)$ is a left chiral superfield with charge $-1$, and $\Phi_R=(\phi_R,e_R)$ is the Dirac-partner chiral superfield with charge $+1$. I am assuming, that under $CP$, the two chiral superfields transform into one another: $$CP:\qquad\Phi_L\leftrightarrow\Phi_R$$ **Question** : Is my analysis correct? Is is really true that supersymmetric QED violates CP? **Edit** : I just realized that in my construction I assumed that $m$ in the superpotential ($m\phi_R\phi_L$) is real. It seems to me that more generally, it is allowed to be complex...


title: Trouble following the Saclay method (spectral representation of thermal Green functions)

text: **Note:** I just answered my own mathematical question by writing it up, but I thought I'd share it anyway in case someone else has a similar difficulty. :) I'm still left with my real physical question: _why_ do the distribution functions appear in the thermal Green function the way they do? It is not intuitively obvious to me at all. Is there any way of understanding this on an intuitive or physical level? **Here follows the solved question:** I'm trying to follow a derivation in Parwani, "Resummation in a Hot Scalar Field Theory" and having trouble following the "Saclay method" he introduces for simplyfing the Green functions for finite temperature scalar field theory. (The reference he gives for this is his own work here but I can't seem to get the full-text through my university account for some reason.) If this is too much of a pure math problem I'm happy to move it over to the other stackexchange, but I'm really trying to teach myself thermal field theory. The spectral representation mentioned in this paper looked interesting to me because of the way the distribution functions come in and how it is supposed to simplify the Matsubara sums, but trying to reproduce the result is giving me trouble. I'm getting stuck between equations 2.3 and 2.4. Equation 2.3 is a propagator: $$ \Delta(K^2) = \frac{1}{K^2 + M^2} \equiv \frac{1}{(k^0)^2 + k^2 + M^2} \;\;\;\; (2.3) $$ and in equation 2.4 he gives a spectral representation for it: $$ \Delta(K^2) = \int_0^{1/T} \mathrm{d}\tau\ \mathrm{e}^{i k^0 \tau} \Delta(\tau, k) \;\;\;\; (2.4) $$ where $$ \Delta(\tau, k) \equiv \frac{(1+n_k)\mathrm{e}^{- E_k \tau}+n_k\mathrm{e}^{E_k \tau}}{2 E_k} $$ and $E_k^2 = k^2 + M^2$ and $n_k = [\exp(E_k/T)-1]^{-1}$ is the Bose-Einstein distribution. I'm trying to verify the formula by going from 2.4 to 2.3. The $\tau$ integral is trivial. I get: $$ \int{\cdots} = \frac{1}{ik^0 - E_k} \frac{1+n_k}{2 E_k} \left[ \mathrm{e}^{(i k^0 - E_k)/T} - 1 \right] + \frac{1}{ik^0 + E_k} \frac{n_k}{2 E_k} \left[ \mathrm{e}^{(i k^0 + E_k)/T} - 1 \right] $$ Combining denominators I need to show that: $$ f(k,k^0,T)\equiv (ik^0 + E_k)\frac{1+n_k}{2 E_k} \left[ \mathrm{e}^{(i k^0 - E_k)/T} - 1 \right] + (ik^0 - E_k) \frac{n_k}{2 E_k} \left[ \mathrm{e}^{(i k^0 + E_k)/T} - 1 \right] \overset{?}{=} -1 $$ I don't see how this could possibly be a constant. I've tried plugging in $n_k$ and simplifying but it doesn't work, either by hand or in Mathematica. In fact, putting in simple values like $M=0,\ T=1$ and plotting (the real part, and I've called $k^0\rightarrow\omega$) I get: ![non-constant function](http://i.stack.imgur.com/RKzjy.png) There's also a non-trivial imaginary part, which looks like that but shifted in $\omega$. So clearly I'm missing something basic or the method doesn't actually work. I've also tried changing signs in a few places in case there was a typo, to no avail. Anybody know where I'm going wrong? Or even better if you have a good technique that accomplishes the same thing this supposedly does. :-)


title: The theory of strings stretching between intersecting D-branes

text: I am trying to understand various aspects of intersecting D-branes in terms of the gauge theories on the worldvolume of the D-branes. One thing I'd like to understand is the worldvolume action for strings stretching between the D-branes. One thing I have considered is $M+N$ D-branes initially coincident but then an angle $\theta$ developing between $M$ and $N$ of them. The gauge symmetry is broken from $U(M\times N)$ to $U(M) \times U(N)$ with the off- block-diagonal terms of the gauge field becoming massive. I anticipate that $\theta$ is the vev of some Higgs field that mediates this transition. What is the action of this Higgs field and where in the string spectrum does it come from?


title: Experimental proof for Conservation of Total Energy

text: > In physics, the law conservation of energy states that the total energy of > an **isolated system** cannot change—it is said to be conserved over time. > (Source : wikipedia(conservation of energy) And > Truly isolated systems cannot exist in nature, other than **allegedly** the > universe itself, and they are thus hypothetical concepts only. (Source : > wikipedia (isolated system) Clearly depict a loophole in the law of conservation of energy as the law holds in case of **isolated systems** and apparently these do not exist. I thought of various practical experiments to inspect the actuality of the law, but in reality I could not think of a single experiment which does not send off energy in form of heat/sound/light out of the system. Even the experiments performed in UCLA here states that some energy will be uncaccounted for in the experiment ! Even a simple experiment of a ball falling down creates heat from friction with air. Clearly the law must hold if we conduct the entire experiment in an **isolated system** , but these simply do not exist. Magnetic levitation to experiment with conservation of energy experiment is not an option as the levitation requires energy and going back to source, we come up with the same heat release problem which makes this experiment only a complex version of the first one. Note : I am not talking about approximations Since we can not make any sort of isolated system without it leaking/absorbing energy into the surroundings. The discussion of the conservation of energy boils down to the conservation of energy of the entire universe. Through various texts I came to the conclusion that the total energy of Universe is not a constant. Those texts being (1, 2, 3) Even the ArxiV paper listed as 3 states only > Despite remaining hesitations about where additional mass could be coming > from, on the overall balance of consistency, we conclude that the energy > conservation law is better obeyed by means of increasing mass of the > universe with its radius Conserves only a type of energy as it suggests > increment in mass which is also a type of energy. I believe this looks extremely like challenging the law of conservation of energy which is supported by neother's theorem among many others. But I just wish to determine whether practically this law is actually valid or not ! I do not expect to get answers of 99.99% conservation with the negligible heat being lost neglected. I understand that GR does not say energy is conserved but the scales of day to day life and that of GR are different which makes energy conservation at earthly scales even if 99.99% a boon. But this just means that energy conservation depends on the working scale and experiment and ranges from nearly 100% in some to not conserved in various aspects of GR. Finally the bottom line of question is, can we experimentally prove that energy, total energy is actually conserved ?   (a Yes answer requires a detailed experiment with complete conservation and no loopholes)


title: Does natural unit of information and entropy, nat, play special role in the freebit picture?

text: Please refer this question to understand why I consider the freebit picture important. In short, it is conjectured, that for certain real systems the most complete physical description possible involves Knightian uncertainty so that some qubits in nature actually "freebits". So I decided to calculate entropy of a system that involves Knightian uncertainty. Lets define c-freebit (classical freebit) as a two-level system whose Knightian uncertainty is maximal. We know that entropy of a two-level system depends on the probabilities of the respective levels, if the probability of the state 0 is $p_0= x$, then the entropy (in natural units) is: $$S= -\sum_{i=0}^1 p_i \ln p_i = -x \ln x - (1 - x) \ln (1 - x)$$ So if $x=1/2$ then $S=\ln 2$ nat, equal to 1 bit. But when we introduce the Knigtian uncertainty $x\in[a,b]$, the total entropy becomes $$S=\int_a^b \frac{-(1-x) \ln (1-x)-x \ln (x)}{b-a} \, dx$$ For the maximum case, $a=0$, $b=1$ we get $S=\frac12$ nat. This is remarkable, because we got a rational fraction of natural units of information! The nat, which plays no role in probablistic description of anything suddenly appears in the evidential description of the Dempster–Shafer formalism! We obtained that a c-freebit contains 1/2 nat of information! Given this result I wonder whether something similar happens in quantum world. Can anybody please provide the calculation for entropy of a quantum freebit?


title: What is the difference between thermodynamical equilibrium and statistical equilibrium?

text: I am trying to understand what is the different between thermodynamical equilibrium and statistical equilibrium, for example, between photons and electrons at the early universe. (I read through paper http://arxiv.org/abs/astro-ph/9705101, where I have encountered this distinction at page 3)


title: Divergence of series

text: How do I understand whether a series will diverge at some point? For example, in the Legendre's diff. equation $$(1-z^2)y' '-2zy'+\ell(\ell+1)y~=~0,$$   does $y(z)$ diverge for $z=-1$ and $z=+1$?


title: Different age of universe

text: According to general relativity time runs slowly near a massive body. So, I have a question bothering me from some time, that if time for every body/object in universe runs at a different rate, then how can we determine the age of universe? i.e. 13.82 billion years? Near a blackhole as time runs slowly, which implies that the age of universe would be much much less. In fact, ever body/object in universe would have different age of universe according to it. And if we really want to calculate universe present age, we should put an upper limit to its age, by determining the time passed in empty space from big bang, which is away from gravitational source (as time passes the fastest here) and a lower limit by the oldest and massive blackhole in universe, as time there would have passed the least (i.e) the age of universe from its point of view would be even less than 0.5 billion years. Am I right?


title: Calculate water use and design parameters for closed loop wet cooling tower

text: I am looking to estimate water use at a power plant with a closed loop wet cooling tower system. I have done some reading on cooling towers, but am struggling to connect the equations to a final outcome.   * T1 - water temp that leaves condenser, and enters cooling tower (warm)   * T2 - water temp leaving tower and enters condenser (cool)   * T_dry - dry bulb temperature of the air   * T_wet - wet bulb temperature of the air My main interest is to model the changes in either water use (i.e. evaporation losses) or electrical output - depending on the climatic conditions (dry & wet bulb T, humidity). i.e. if it gets hotter and more humid - how will electrical output or evaporative losses decrease?   1. I can calculate the flow rate (m3/s) required through the closed system in order to remove the heat to the cooling water (CW) circuit - if I specify the permitted temperature increase between the condenser and the CW circuit (T1-T2). This comes from an equation relating carnot efficiency, the % of heat discharge to water, the specific heat capacity of water and the temp difference permitted.   2. This water temperature increase, I am supposing, is determined by maximum "range" achievable by the cooling tower system - the range being the difference between warm Cooling Water (warm at the top of the tower) and cold CW (cold at the bottom of the tower).... such that the water warms from T2 to T1 at the condenser, and then cools from T1 to T2 in the cooling tower? If we know that the maximum T_wet that the system will encounter is 20 degC, and with a desired Approach of 5 K - that gives us a minimum T2 of 25 degC. How do we determine the range, or T1? Is it reasonable to set this at the same level at the maximum flow rate permitted by the pumps or is there another suggested approximation? Given a certain combination of dry and wet bulb temperatures (T_dry & T_wet), and a constant - how can I determine the evaporation losses? Also, does the difference between the air temperature (dry bulb) and the warm water T1 affect the range of the cooling tower? Sorry, it's a lot of questions and some quite patchy understanding! Very grateful if someone can fill in some blanks - THANKS!


title: Why doesn't the color of overlapped region between opaque red and green objects become opaque yellow?

text: I post the question here several minutes ago. > In theory, additive mixing Red color and Green color produces Yellow color. In my mental model, additive mixing opaque Red and opaque Green will produce opaque Yellow. But the following attempt, I got a result that is different from my mental model. WHY? ![alt text](http://i.stack.imgur.com/nkONV.png)


title: Why is the equivalence principle so important to general relativity?

text: In its simplest form, equivalence principle states that the inertial mass and the gravitational mass should be the same. This is easy to understand. But why is it so important to the formulation of General Relativity? To be more specific, I don't understand how the gravitational field equation: ![alt text](http://i.stack.imgur.com/Ote28.png) can be derived from this principle.


title: Melting point, is it determined only experimentally?

text: I am interested in the underlying physics of the concept of a melting point, a temperature at which an object tends to gain enough energy to break the bonds that hold it together and be in a liquid state. Or the other way around, a freezing point. What I am interested is... Since temperature is basically a measure of the kinetic energy of particles within, when they actually start to vibrate enough they break apart. And the changes in temperature in the surrounding medium is due to them transferring heat during "impact" with the solid object, be it air or some other medium at a certain temperature. So, if we know all that, why do we depend upon experimental data? Why isn't there some logical way to deduce what is the melting point of some matter under some variable pressure?


title: Is there a concise-but-thorough statement of the Standard Model?

text: I’m a grad student in high-energy physics. I’m familiar enough with the Standard Model, but I’ve always wondered whether there existed a canonical statement of, effectively, “what we talk about when we talk about the Standard Model”. Obviously the SM didn’t spring fully-formed from the pen of one author, but since its inception surely someone has compiled our current understanding into one document? (The closest things I’ve found to this are the Wikipedia article and the explanatory chapters of the PDG.)


title: What is the 2-point correlation function of the electron field in QED?

text: The Feynman propagator for the free electron field is the Fourier transform w.r.t. $y$ of the time-ordered 2-point VEV $\left<0\right|\mathcal{T}[\hat\psi(x)\hat\psi(x+y)]\left|0\right>$, taking $\hbar=c=1$, $$\mathcal{F}[\left<0\right|\mathcal{T}[\hat\psi(x)\hat\psi(x+y)]\left|0\right>](k)= \frac{k\\!\cdot\\!\gamma+m_e}{k^2-m_e^2+\mathrm{i}\epsilon}.$$ In QED, after renormalization, we obtain, valid in the infrared, when $k^2-m_e^2$ is small (given in this neat form, representing the sum of _many_ perturbative integrals, in, for example, Appelquist & Carrazone, Phys.Rev.D **11** , 2856 (1975)), $$\mathcal{F}[\left<0\right|\mathcal{T}[\hat\psi(x)\hat\psi(x+y)]\left|0\right>](k)= \frac{k\\!\cdot\\!\gamma+m_e}{(k^2-m_e^2+\mathrm{i}\epsilon)^{1-{\alpha_{EM}}/{\pi}}},$$ where $\alpha_{EM}\approx 1/137$. This is often called an infraparticle propagator or a dressed particle propagator (I'm rehearsing this stuff, please bear with me, and feel free to comment on anything that seems to need some change). When we move to renormalization group methods, the coupling constant becomes a function of the renormalization scale. In this case, if we measure the Fourier transform of the 2-point VEV, using the operator $\mathcal{F}[\mathcal{T}[\hat\psi(x)\hat\psi(x+y)]](k)$, the measurement scale is determined by the wave-number $k$, so I take it we can write the 2-point VEV as $$\mathcal{F}[\left<0\right|\mathcal{T}[\hat\psi(x)\hat\psi(x+y)]\left|0\right>](k) {{?\atop =}\atop\ } \frac{k\\!\cdot\\!\gamma+m_e}{(k^2-m_e^2+\mathrm{i}\epsilon)^{1-{\alpha_R}(k)/{\pi}}},$$ where the running coupling constant $\alpha_R(k)$ is $\alpha_R(m_e)\approx 1/137$ when $k=m_e$, at about 0.5 MeV and $\alpha_R(m_Z)\approx 1/127$ when $k=m_Z$, the mass of the $Z$ particle, at about 90 GeV (see here). My understanding is that in QED the function $\alpha_R(k)$ is an increasing function of $k$, even to the extent that there is a Landau pole at finite inverse length $m_L$, $\alpha_R(m_L)=\infty$, but that the high-energy behavior of QED has only been calculated perturbatively, so that an analytic form for $\alpha_R(k)$ is not known. I would like, however, to look at a good article, perhaps a review, that gives as closed a form for $\alpha_R(k)$ as is currently known, in as neat a form as possible (this is an _implicit_ question that may be too much to ask of the literature, given that almost everyone has moved on to supersymmetry, noncommutative geometry, string theory, etc., etc., and QED is obviously not empirically useful at high energy). I would also like better to understand the relationship between the running coupling constant formalism and the Källén–Lehmann formalism. Can we equate the two, at least approximately, $$\frac{k\\!\cdot\\!\gamma+m_e}{(k^2-m_e^2+\mathrm{i}\epsilon)^{1-{\alpha_R}(k)/{\pi}}} {{?\atop \approx}\atop\ } \int\limits_{m_e}^\infty \frac{(k\\!\cdot\\!\gamma+m)f(m^2)\mathrm{d}m^2}{k^2-m^2+\mathrm{i}\epsilon}.$$ Here, $f(m^2)\ge 0$ is undefined for $m>m_L$ if there is a Landau pole in QED, but I suppose $f(m^2)$ would not have any poles and would approach zero fast enough for at least the integral $\int\limits_0^\infty f(m^2)\mathrm{d}m^2$ to exist in a well-defined theory? We could equate the two exactly if we were talking about a scalar Feynman propagator, because we could solve for $\alpha_R(k)$, but the change from $k\\!\cdot\\!\gamma+m_e$ to $k\\!\cdot\\!\gamma+m$ doesn't look good because the two expressions have different effects for different components of the Dirac spinor. Better understanding the 2-point VEVs would of course leave the $n$-point VEVs to think about. Uggh. Finally, there are three explicit Questions here (marked by question marks!), but I would also be interested in any ruminations from this starting point that include references.


title: Speed of light energy

text: Considering the amount of energy necessary to accelerate a particle to the speed of light (ie; half the energy in the entire universe) how could we have so many things already going the speed of light? Maybe there should be only two or three things (objects or particles etc.) in the universe already going that fast. Just a few photons for example, in the whole universe.


title: Symmetries for an inertial frame

text: According to Noether's theorem, a symmetry of space-time w.r.t. an observer, will yield a corresponding conservation law for a closed system w.r.t. that observer. Now if our space-time has 3 symmetries w.r.t. an inertial observer namely isotropy and homogeneity of space and homogeneity of time, so only 3 conservation laws are possible in classical mechanics , namely conservation of angular momentum,momentum and energy of a closed system. Aren't there more conservation laws possible ? What stops me from defining a quantity like $\vec{r} \cdot (\vec{r} $ **x** $ \vec{f})$ w.r.t. an inertial observer and take its derivative with time, and see when it is conserved w.r.t. time by equaing derivative to be $0$. If yes, how to find the corresponding symmetry ?


title: Why does an octave on a piano have the divisions of 8 white keys and 5 black keys?

text: An octave corresponds to a doubling or halving of the frequency. Each octave on a piano and in classical music score is broken downs into 8 white keys and 5 black keys. Is there a physics explanation for that division?


title: Robot controling pouring process from a bottle

text: I need to solve a problem within mechanic of fluids for a part of my thesis. Robot will pick up a bottle of beer, cola, julebrus or any other kind of beverage. And then it has to bring it to the glass and pour it in the glass. But now I came to some problems, because I have to pour only half of the volume in side of the glass, and pouring process will be controlled by robot controller. I have to find out mathematical models that are governing the process of pouring, within this I have to find dependencies between flow from the bottle, angle of the bottle. And with this I will use force sensor to control the pouring process. Some of the assumptions I made for the beginning:   1. The flow is laminar and there is no viscosity   2. Flow will be constant and intake of air will be constant But later when I started to do the math many problems appeared here. First is the gulping of the air into the bottle neck, in the beginning of the pouring it's a big problem, later when there is smaller volume of fluid in a bottle that's not a problem anymore. And I can't count that my flow will be constant because of the air intake and foam that is constantly forming. And also because the area from where fluid is going out is changing constantly how can you relate it to the angle of bottle tilt? And before I forget I have to take inertial and gravity compensation in account for this.


title: Entanglement Entropy

text: I needed some references which involves, the basic definition and motivation for Entanglement Entropy, and its one or two applications to many-body physics/black holes.


title: Can the choice of reflection angle for light can be derived from a minimality condition?

text: When the light hits on a surface, it reflects with the "same" angle as the one that hits the surface. I was wondering if this choice of angle can be explained by a minimality condition?


title: What's the outcome of two polarized electrons beam collide head on?

text: Let's say that I have two electron beams targeted towards each other along x-axis. Both beams are polarized on axis (z-axis) perpendicular to x-axis. Another beam's polarization is up and another down on z-axis. What would happen when those polarized electrons collide (with low energy)? Just plain elastic collisions? Has somebody ran such an experiment? Any references? **Edit:** Could somebody conduct described experiment? With electrons or protons.


title: The speed of electrons moving in different medium

text: Can one have a simple (dimensional-analysis) model to evaluate the speed of electrons moving in: (1) a copper wire (2) a silver wire (3) a gold wire (4) a semiconductor wire let's use meter over seconds: m/s as a unit of the speed.


title: Potential Around Suspended Particles

text: In my textbook, the classic example of a dielectric sphere suspended in a dielectric solution is solved, and a general form for the potential inside and outside of the potential is given without derivation. Now, I know that these general forms are coming from solving laplace's equation in both regions (you get solutions containing legendre polynomials for theta-component and $r^n$ where n is an integer for the radial component), but I'm not clear on how the constants are being defined. Could someone explain why that first constant has an $E_o$ and also why the legendre polynomial $\cos{(\theta)}$ is being chosen? See below: ![enter image description here](http://i.stack.imgur.com/e3LOL.png)


title: When a planet has a high gravity, is it impossible to build and launch a successful chemical rocket to space?

text: Just recently a a large rocky planet has been discovered. "Astronomers have discovered a new type of rocky planet beyond the solar system that weighs more than 17 times as much as Earth while being just over twice the size" http://www.reuters.com/article/2014/06/02/us-astronomy-exoplanet- idUSKBN0ED29V20140602 Then I was thinking, if aliens lived on this planet, could they get off the planet with a normal chemical rocket, or would the rocket have to be so massive that it could not be built? or so heavy that the amount of fuel required would push the weight too far so that it could never reach space? I wonder is there are sad planets where aliens never made it into space due to the large amount of gravity.


title: Field Tensor and classical limits

text: I would be very grateful if someone would kindly explain this generalization of the Lorentz force law to the special relativity domain. Please bear with me. > _Classically, the Lorentz force law is $$m\frac{d^2x}{dt^2}=q(E+v\times > B).$$ We want to reformulate this relativistically. Noticing that > $\frac{dt}{d\tau}\approx 1$, we could postulate that the relativistic > version of the equation has a factor of $\frac{dt}{d\tau}$ multiplying E. > But the force now depends on velocity linearly classically, so the only > possibility hat is consistent with both the classical limit and SR is > $$m\frac{d^2x^\mu}{d\tau^2}=qF^\mu{}_\nu \frac{dx^\nu}{d\tau}$$_ Here are some of the things I don't understand:   1. What does it mean to reformulate it relativistically? As far as I understand, it means we want an equation that holds at high velocities but reduces to the Lorentz equation in the Newtonian limit. OK. But what does that mean mathematically? How does one go about generalizing equations to fit SR?   2. Why does $\frac{dt}{d\tau}\approx 1$ in the classical case suggest multiplying $\frac{dt}{d\tau}$ in front of $E$?   3. How do they come about with $$m\frac{d^2x^\mu}{d\tau^2}=qF^\mu{}_\nu \frac{dx^\nu}{d\tau}~?$$ I suppose the linear-dependence of the force on velocity in the classical limit suggests the RHS has to be linear in $dx\over d\tau$. But what is so SR about it? How is it consistent with/accommodate SR?


title: The shape of the graph of the equation: $V= -r I + E$

text: I have recently collected data (for a school experiment) in order to measure the EMF and the internal resistance of a solar cell. The data complied with the equation: $V = -rI + E$, i.e. the voltage decreased as the current increased and I have a negative gradient. However, I am having some trouble as to explaining why the graph is the way it is, that is, why is there a negative gradient? Here's my rationale: Using Ohm's law, we can see that decreasing the (external) resistance increases the current flowing throughout the whole circuit (it is a simple series circuit with just a solar cell, ammeter, variable resistor and a voltmeter (in parallel, obviously)). Now, the voltage drop across the solar cell should increase as well (assuming its internal resistance in constant) because $v = V_r = Ir$, where $v$ is the 'lost volts'. For this reason the voltage drop across the load must decrease(which is what we see from the graph) because $E = V + v$. Thus, the voltage drop across the resistor decreases as current increases. I don't know to what extent this is correct or if it is correct at all but I would appreciate any and all help given. Also, as you can probably tell, the level of physics used here is very basic, so I would also appreciate it if the answers explained it in this way as well, although, where possible, do not sacrifice accuracy for the sake of simplicity.


title: Dense Spherical Black Hole Shell with a Region Inside

text: I'm going to propose a thought experiment, based on two ideas. One: A uniform spherical shell, by the Shell Theorem, does not exert any gravitational force on objects existing in the interior of the shell. Two: A black hole, created by matter dense enough to lie within its Schwarzschild radius, is inescapable, even by light. Suppose (don't ask how) we create a uniform spherical shell composed entirely of matter dense enough to form a black hole, such that we have a continuous event horizon that appears as two concentric spherical regions. One can think of this as a spherical shell with an infinite number of infinitesimally sized black holes (at least, from a macroscopic point of view, to avoid Pauli's Exclusion Principle), or simply with a very large mass density across the surface (which is allowed thickness in the radial direction, to remain a three-dimensional construct). The overlapping event horizons make this system appear as a single black hole from the outside. Obviously, this system is unstable, and will collapse into a messy crunch fairly quickly, but before it does, its properties seem contradictory. So, what happens on the inside? This question should probably be addressed for completely overlapping interior event horizons (such that no region inside the shell sits outside of the collective event horizon, and all of the black hole "cores" sit inside of the event horizon of every other black hole), and for some space existing between the event horizons of black holes on opposite sides of the shell, so that an event horizon free region exists within the shell. Will objects on the inside feel the effects of the gravitational force, or will it be a happy island of no external gravity (that is, simply flat space)? Does the answer vary based on what major theory is used to address it?


title: Energy of a system

text: I'm considering a ring of mass $m$ sliding along a cardioid with equation $r=a(1+cos\theta)$. We let the angle between the downward vertical and the radius vector be $\theta$. The ring is attached to the origin by an elastic string with natural length $a$ and modulus $2kmg$ where $k$ is a positive constant (and $g$ is the common one). I'm trying to write down the energy of the system but I can't get my head around it. Can you help me please? Right now I'm guessing there's a potential energy that I would say is $mgh$ where $h=a(cos\theta+1)sin(\theta-\pi)=-a(cos\theta+1)cos\theta$ and the elastic one that I would say is $\frac{2kmg(extension)^2}{2a}$ where the extension is $a(cos\theta+1)-a=acos\theta$. Is that right or am I completely mistaken?


title: Is there any case where one would use, snap, crackle or pop?

text: As we all know, if you differentiate distance with reference to time, you get speed, and likewise, differentiating speed you get acceleration. However, if you keep differentiating, to the rate of change of acceleration and so forth (more emphasis on the so forth): is there any point to this? Does it have any useful application in any industrial or practical industry, as opposed to theoretical physics? Also, what could we possibly learn about a situation by understanding the rate of the rate of the rate of change of the objects distance?