title: How to properly find equilibrium vectors when 3 or 4 vectors are present?

text: I'm having trouble regarding how to find the equilibrium of vectors. We were given tables with given information and had to fill them in; I'm fairly sure my initial attempts were incorrect. first example               Vector A: mass - .2kg, magnitude - ?, direction - 20deg, xcomp -?, ycomp - ?     Vector B: mass - .15kg, magnitude -?, direction 80deg, xcomp - ?, ycomp - ?     Vector C: mass - ?, magnitude -?, direction -?, xcomp ?, ycomp - ?      A and B are no problem, I'm just not sure if I add or subtract my components to get the components for C. The same goes for this one:               Vector A: mass - ?, magnitude - ?, direction - 0deg, xcomp - ?, ycomp - ?     Vector B: mass - ?, magnitude - ?, direction - 90deg, xcomp - ?, ycomp - ?     Vector C: mass - .3, magnitude - ?, direction - 240deg, xcomp - ?, ycomp - ?      and the oddest one:               Vector A: mass - .1, magnitude - ?, dir - 30deg, xcomp - ?, ycomp - ?     Vector B: mass - .2, magnitude - ?, direction - 90deg, xcomp - ?, ycomp - ?     Vector C: mass - .3, magnitude - ?, direction - 225deg, xcomp - ?, ycomp - ?     Vector D: mass - ?, magnitude - ?, direction - ?, xcomp - ?, ycomp - ?      NOTE: I was using f = ma, tan(angle) = a/b, and xcomp = fcos(angle) and ycomp = fsin(angle) I just didn't know what to do about the final vectors for sure. NOTE 2: I don't need help finding the first few vectors' information, just the last ones (e.g. Vector D). More specifically, when do I add or subtract the components of the first vectors in order to get the components for the final vector and then get the rest of the information for it?


title: Why are Topological Superconductors hard to make?

text: Topological insulators (TI) have already been made in lab. Topological superconductors (TSC), being close cousins of TI, seem harder to make. Why is that? It seems that materials in connection with majorana zero modes all have bad fate in lab. Are there any deep reasons behind?


title: How to determine your position underground?

text: Crossed my mind after random rant on wikipedia that lead me to articles about chronometers and measuring position. Let's assume I were trapped in the underground laboratory with lots of equipment but without any access to the surface. Would I be able to properly determine my position (latitude, longitude and altitude), and if so, what instruments are needed? (and mny what's the coolest way to do it :) I thought about measuring Coriolis effect, which could lead to latitude measurement, and earth's gravity map could give more hints, but it's still far too imprecise. @Edit Wow. You guys never cease to amaze me, so many good answers. If dmckee had posted an answer instead of a comment, it would be the one to accept, as secondary variable being 'the coolest way to do it :)'. Also John's idea of using Foucault's pendulum which I completely forgot to exist and determining Moon's position underground sounds neat.


title: What is a boundary condition for capacitors/dielectrics?

text: I am extremely confused about what boundary conditions are. One minute ago I was solving easy capacitor questions and the next minute I am being asked boundary condition questions and there is no such section in my textbook on boundary condition. I have some questions below if you could please explain from scratch/beginner level what boundary condition is and what the goal is. I have some questions below you could use to explain what these conditions are… I have looked all over google and search various videos and I still cant find any information on them. * * * ![enter image description here](http://i.stack.imgur.com/enZ48.png) * * * ![enter image description here](http://i.stack.imgur.com/qNbaO.png)


title: Maxwell in multiple dimensions: What happens to curl?

text: I read this answer a while ago, and while thinking about $\nabla$, I realized something. Since the cross product can be written as a determinant, in higher dimensions we require extra vector inputs. IIRC it's called the "wedge product" in higher dimensions. Alright, how does this work when we generalize Maxwell to higher dimensions? Curl can be written (abuse of notation, yes) as a cross product with $\nabla$. But, to generalise it to higher dimensions, we need multiple inputs. We need something like $\nabla_4(\mathbf{B_1},\mathbf{B_2})$ in four dimensions, and so on. So we have two ways to get out of this: We can either use a different way to write curl in multiple dimensions (The wikipedia page has stuff on this which I don't understand), or there are more than one $E$ and $B$ fields in higher dimensions. So which is it? How are Maxwell's laws generalised to higher dimensions? Just a note: I never understood the linked answer after the first sentence (didn't know enough), so if there's something obvious there that answers this question, I missed it. I know nothing of higher-dimensional analysis, so if complex notation is going to be unavoidable for the multiple dimensions, I'd be fine if you showed me what happens in four dimensions.


title: Energy Measurements in a Two Fermion Double Well System

text: This question is related but my question here is much more elementary than discussions of the Pauli principle across the universe. There has been a fair amount of discussion around at the moment on the double well potential model in quantum mechanics. The question I have here is just concerned with some very simple basic schoolboy quantum mechanics that you need to get right in order to progress with the discussions (I think these questions would be too elementary for a physicist to ask - fortunately I'm not a physicist, so my embarrassment is under control). The scenario is this: I have a double square finite depth potential well with a large distance separating the wells. Call them left hand and right hand wells. The model I want to use is elementary nonrelativistic quantum mechanics. The energy eigenstates of such a system are similar to those of a single well, but what was one energy eigenstate now becomes two with incredibly closely spaced energy levels (ref). Suppose I have two electrons in this system in the two very closely spaced lowest levels $E_1$ and $E_2$ . The two electrons are indistinguishable, so the state is $${{1}\over{\sqrt{2}}}(|E_1\rangle|E_2\rangle-|E_2\rangle|E_1\rangle)$$ The statement that has often been made in these discussions is something like "an observer in the left hand well now makes an energy measurement of an electron and gets the value $E_1$ ". In talking about this scenario, in the link I gave above Sean Carroll stated > "Imagine that we have such a situation with two electrons in two atoms, in a > mutually entangled state. We measure our electron to be in energy level 1. > Is it true that we instantly know that our far-away friend will measure > their electron to be in energy level 2? Yes, absolutely true." This is where I'm confusing myself: By assumption, our system _was_ in an energy eigenstate, and we've made an ideal energy measurement, so by the rules of quantum mechanics after the measurement it ends up still in an energy eigenstate after the measurement. But the observer knows that we have a double well two-fermion system and the correct system state is still the antisymmetrized one $${{1}\over{\sqrt{2}}}(|E_1\rangle|E_2\rangle-|E_2\rangle|E_1\rangle)$$. So for an idealized measurement, the system started in this state and remained in this state. So I ask question (1): If the measurement proceeds like this, without any extra entanglement-introducing mechanism, won't the observer think "if my far-away friend now makes an energy measurement, he'll have a 50/50 chance of getting $E_1$ or $E_2$" ? In other words: since this is non-relativistic QM, the particle could have moved to the other well immediately after the first measurement, so the state is still antisymmetrized over the particle identities. In other other words, the state I described isn't mutually entangled - just antisymmetrized over the electron identities. question (2): if this correct then how, typically, would I introduce some entanglement between the electrons, as in Sean's statement ? Finally, when we say "an observer in the left well makes an energy measurement", doesn't this introduce some locality into the picture for the following reason: What it implies is that the observer's measuring equipment is spatially localized in the left well, so that when an energy measurement is made, there is also an implied acquisition of information about the measured electron's location at that time. So I come to: question (3) Wouldn't this mean that after such a _local_ measurement, the measured electron _cannot_ be in one of the _system's_ energy eigenstates ? So the system state would be something like $${1\over{\sqrt{2}}}(|\Psi\rangle|E_2\rangle-|E_2\rangle|\Psi\rangle)$$.   where $\Psi$ is a spatially localized state, concentrated in the LH well. This doesn't sound right. Where is my reasoning going wrong ?


title: Internal energy according to the van der Waals equation

text: I am trying to derive the internal energy of a gas which obeys the van der Waals equation. I have however encountered some problems. I calculate the integral of $dU$ from $V=0,T=0$ to $V=V, T=\infty$ to $V=V,T=T$. I can calculate the work: $$\left(p+\left(\frac{a}{n}\right)^2\right)(V-nb)=nRT \implies p=\frac{nRT}{V-nb}-\frac{an^2}{V^2}$$ For the second part of the path $V$ is constant so $W=0$. $$W=-\int\limits_\infty^V p\textrm dV=\int\limits_V^\infty \frac{nRT}{V-nb}-\frac{an^2}{V^2} \textrm dV\\\=nRT\ln(V-nb)|_V^\infty+\frac{an^2}{V}|_V^\infty\\\= \infty-\infty +\frac{an^2}{\infty}-\frac{an^2}{V}=-\frac{an^2}{V}$$ I know I haven't been mathematically rigorous but that is not really important to me at the moment. I think this is right. I can't however think of how i should calculate the heat involved in following this path. Any help on how to do this is appreciated. **EDIT** : I see now that the work I calculated is wrong as well as $$nRT\ln(V-nb)|_V^\infty\neq\infty-\infty$$


title: How can I subtract Spectra?

text: I am writing an application that should allow to view and manipulate spectra ( **Raman** ). I am only given one normal spectrum consisting of 1024 coordinate points. Plus I have a background spectrum that is supposed to be subtracted from the first spectrum. f.e.: The first point in spectrum_1 is: **-55.988** , **9373** The according point in background spectrum: **-55.988** , **9382**. After subtracting the two in a third-party software, the first point of the resulting spectrum is: **-55.988** , **-641.968** So I asked myself how this is calculated. According to the Documentation of this software, it is using the following formula: > result spectrum(n) = multiple component spectrum(n) > > – Factor1 * single component spectrum (n) > > – Factor2 * single component spectrum (n) Unfortunately I can't do anything with it. What is meant by "component spectrum(n)"? Is it the Spectrum equation? Also what could the both factors be and what does the (n) mean? Here is an image of all three spectra that are involved in the calculation. Here is an image of all three spectra that are involved in the calculation. http://imageshack.us/photo/my-images/546/spectrumsubtractionresu.jpg/ From Top to bottom:   1. First spectrum   2. Background spectrum   3. The resulting spectrum after the subtraction I simply can't imagine or reproduce the calculation that leads to this result. Any help and/or advice is welcome. I am highly willing to learn and understand so please don't hesitate to share some links or other references that could help. Thanks in advance, BC++


title: Is $E=mc^2$ reserved to nuclear physics?

text: I was wondering, while putting a log in my fireplace, how much energy the piece of wood would give. The most famous formula poped into my head: $E=m \cdot c ^ 2$! Is this formula applicable to a burning object or is in only applicable to a nuclear reaction?


title: Why does foam dull knives?

text: I have recently taken up prop making and just started my first foam-built costume from a video game. These kinds of costume armour builds are often built out of the various foam floor mats you can buy in say Home Depot or BJ's or Five Below for around $1 for 4 sq. foot tile. The one thing that puzzles me is why foam dulls a sharp steel blade so easily and so quickly. I have tried x-acto knives, those snap-off knives, even surgical scalpels. Regardless, after cutting several linear feet (maybe in the neighborhood of 5-10?), the knife blade begins to dull rapidly and no longer produces nice cuts, but begins to tear the foam instead. You can always sharpen your blade on a stone, but in short order it will again dull and tear. I generally do all my cutting on one of the green, self-healing cutting mats. This _could_ explain the tip dulling, but it is actually the whole length of the blade exposed to the foam that gets dull. It happens across types, thicknesses and densities of foam. The floor mats are about 0.5 " and medium density. The craft foam from Michaels is 6mm, high density and rigidity. The roll I have is 0.25" low density, low rigidity It absolutely baffles me how soft squishy foam just _destroys_ the edge of a metal knife. I am **guessing** it has something to do with the molecular organization of the foam or something, but would love to know if anyone has a good explanation for why and how this happens.


title: Photonic. vs Plasmonic

text: I hear a lot about photonic and plasmonic structures being the future in many different applications, but what are the differences between them? Why would one want a material with both as opposed to just one?


title: Why does the sun "shine brighter" some days?

text: Today, the sun seems extremely bright; more dazzling than usual, and even the roads seem to be brighter so it's not just when you look up in the sky. Is more light actually getting through (perhaps there is less moisture in the air than usual to refract the sun's light?) or is it just because, being winter, the sun is lower in the sky and I notice it more? But then, why wouldn't I notice it the same every day in winter?


title: How is it possible the universe expanded faster than the speed of light during inflation?

text: In a documentary written in collaboration with Stephen Hawking, the narrator (supposedly Stephen Hawkings) says that by the time the cosmos was 10 minutes old, it had already expanded thousands of light years in diameter. How is it possible that matter was traveling faster than light?


title: Where is the candle hottest?

text: Where is the hottest spot above a lit candle? In the flame or just above the flame tip or some cm above the flame or other? **Update** My question more precisely concerns the heat energy I can get from a candle. I can see that isn't necessarily the hottest part. So to make the question more accurate: From which part/point of the candle can I get the most heat energy pr. time unit? E.g. if I was to warm something up, which part the candle would then be the most efficient heater?


title: Construct the Hamiltonian of electrons on a graphene sheet ( in xy plane)

text: Graphene is a two-dimensional material formed by carbon atoms in a honeycomb lattice. Because of the symmetry of the honeycomb lattice, the electrons in graphene obey a linear dispersion relation $E\propto|\vec{p}|$ (as in Dirac case) rather than a quadratic one $E\propto|\vec{p}|^2$ (as in Schroedinger case). Nevertheless, the speed of light c appearing in linear dispersion relation (found for Dirac case) is replaced by the Fermi velocity $\vec{u_F}$ of electrons. In explicit terms, dispersion relation of electrons on graphene is given by $E =\vec{u_F} |\vec{p}| \times20$ where $ \vec{p} = p\hat{x}x + p\hat{y }y$. (A proportionality constant is absorbed in the Fermi speed υF .) I was trying to construct this Hamiltonian in xy-plane and i don know how i can do for honeycomb lattice .


title: Question on the Gell-Mann Low equation

text: Question on the Gell-Mann Low Equation. In this paper, http://arxiv.org/abs/1205.3365, page 21, the author argues that if: t →∞(1-iϵ), all the terms in equation (193) goes to zero, except the first term. Can anyone explain this to me?


title: Asking for references on the variational treatment of spin wave

text: My idea is the following: We have a system with Hamiltonian $H$, and we know that there is spin wave in this system by some symmetry-breaking arguments. Now we start from the ground state $\left|\uparrow\uparrow\uparrow\cdots\right\rangle$, and I hope to express the Goldstone mode by an exponential of hermitian operator $\hat{O}\left(\mathbf{k}\right)$, where $\mathbf{k}$ is the wave vector: $ \left|\mathbf{k}\right\rangle \sim e^{i\hat{O}\left(\mathbf{k}\right)}\left|\uparrow\uparrow\uparrow\cdots\right\rangle $ I wish to get the explicit form of $\hat{O}\left(\mathbf{k}\right)$ by making analogy to the quantum mechanical spin rotation $e^{-i\phi\hat{\mathbf{n}}\cdot\frac{\mathbf{\sigma}}{2}}\left|\uparrow\right\rangle$. Furthermore, is it possible to have some result like $\left\langle \uparrow\uparrow\uparrow\cdots\right|e^{-i\hat{O}\left(\mathbf{k}\right)}\mathbf{S}\left(\mathbf{r}\right)e^{i\hat{O}\left(\mathbf{k}\right)}\left|\uparrow\uparrow\uparrow\cdots\right\rangle \sim\mathbf{S}_{0}e^{i\mathbf{k}\cdot\mathbf{r}}$? Could someone offer some hint here, or reference containing the explicit expression? Thx :D All the $\left|\uparrow\right\rangle$ kets are $\sigma_{z}$ eigen vectors, i.e. spin-z up.


title: Double-Slit Experiment Timed

text: I read an article online about someone somewhere timing the length of time it took a particle to travel from the source to the detector but I have lost all my bookmarks. :( Could someone point me to it?


title: How do spin orientations change during adiabatic demagnetization of paramagnet?

text: Suppose a paramagnetic solid is taken on the following cycle, where $B_1 > B_2$ ![enter image description here](http://i.stack.imgur.com/UQEeK.png) (1)->(2)->(3) I'm trying to explain what happens to occupation numbers in magnetic levels. **Step (1)** As $B_1$ increases to $B_2$, more spins get aligned due to the stronger field and entropy which is a measure of disorder falls. The occupation levels in +spin direction increase. **Step (2)** Opposite happens? Some spins revert back to random orientation. **Step (3)** As temperature is increased while B is fixed, there are more thermal fluctuations in the occupation levels?


title: Expectation value calculation for a weird operator

text: In the paper Fundamental monopoles and multimonopole solutions for arbitrary simple gauge groups.\- E weinberg I am not being able to see one of the calculation. The author states (eqn 3.26) $$\langle x\mid \frac{1}{[-\nabla^2 + v^2(\alpha \cdot h)^2+M^2]^2}\mid x\rangle=\frac{1}{8 \pi}\frac{1}{[v^2(\alpha.h)^2+M^2]^{\frac{1}{2}}}.$$ I have no idea how he gets that answer. How does the power change to half? In the denominator, everything except $\nabla^2$ are scalars independent of $x$. Here $v, \alpha, h, M$ are all scalars independent of the position.


title: Why is this identity an if, rather than if and only if?

text: A recent question (Product of exponential of operators) asked who to proved that the exponentials of operators multiply in same manner as those of scalars if and only if the commutator of the operators is zero. As nervxxx's answer there points out, the only if part of that theorem isn't true. This is clearly correct, as nervxxx gives an example of two operators where $[A, B] \neq 0$, but where $e^{A+B} = e^A e^B = I$. I remain confused by one point. nervxxx gives a proof of this theorem using a differential equation approach. I understand that approach. I can see clearly that it relies on the operators having well-defined inverses. The counterexample works because neither of those operators is invertible. What confuses me is that I have also worked out this derivation by expanding the exponentials in their Taylor series, and matching terms of equal order in each operator. When you do this, the $I, A,$ and $B$ terms all cancel out regardless. Equating the terms containing $AB$ and $BA$ gives the condition for the commutator. When I try to equate higher order terms, I don't get any new information. As far as I can tell, every step in that derivation can be reversed, so I should be getting an if and only if for the original proposition. But that's clearly false. So what's going wrong? I can think of two things. First, there might be an implicit assumption that the operators are invertible. I don't see where that would be, but I might be overlooking it. Second, it could be the expansion and term-by-term equating of the infinite series. Just because they are equal term-by-term doesn't mean they are equal. That doesn't make a whole lot of sense to me either, unless I am missing something really basic about infinite series.


title: How deep is the Great Red Spot?

text: The Great Red Spot (GRS) is a very persistent storm system that's easily visible through a telescope on the surface of Jupiter. But what is the three- dimensional structure of the GRS, and how deep into the planet does it extend? When I first wrote this question I assumed that the GRS must be the end of a linear vortex, which leads to the question of why we can't see the other end elsewhere on the surface, as I discuss next. However, as hwlau pointed out in an answer, this isn't the only possibility, so I've added a section discussing another, more probable, case below. **If the Great Red Spot is the end of a vortex** According to my possibly-flawed understanding of fluid mechanics, you can't have a vortex with only one end, at least in invsicid flow. On the surface of Jupiter, the flow has an enormous Reynolds number on the scale of the GRS, so it's very nearly inviscid. If this is also true in the planet's interior then the other end of GRS vortex should terminate against a surface - but there is no solid surface on Jupiter (at least not anywhere near the upper part of the atmosphere that we can observe), so what plays that role in this situation? Is it just that the viscosity of Jupiter's atmosphere rapidly increases with depth, allowing the Red Spot to be only a shallow surface feature, or does it actually extend deep into the planet's interior? Wikipedia has some information about deep models of Jupiter's interior. It doesn't mention vortices in this context but it does say these models predict that Jupiter's flow is organised into concentric cylinders parallel to the axis of rotation. A friend pointed out to me that this means the Red Spot vortex should extend into the interior parallel to the rotation axis as well, like this: ![enter image description here](http://i.stack.imgur.com/BbaQK.png) As shown in the figure, if the flow were completely inviscid (and this idea is correct), we should expect to see the other end of the vortex as a second storm, rotating in the opposite direction in the north equatorial belt. This doesn't match the observations, so presumably the vortex terminates before this point - but I'd like to know how far into the planet it extends before doing so. **If the Great Red Spot is a closed vortex** hwlau pointed out that the GRS needn't be the end of linear vortex at all, but could instead be a closed vortex oriented parallel to the surface, like a flattened smoke ring, as I've attempted to draw below: ![enter image description here](http://i.stack.imgur.com/TmL8k.png) This probably makes a lot more sense than my linear vortex assumption, because the GRS is presumably driven by thermal convection, and this geometry makes sense for a rotating convection cell. (Large storms on Earth have a similar structure.) Intuitively, if the structure of the GRS is like this, it seems like it should be less deep than it is wide - but since the GRS is wider than the Earth, that could still be quite a substantial distance, and I'm interested in whether a number can be put to it. If the structure of the GRS looks like this, I would also be interested in anything that might be known about the structure of the flow beneath it, and how the two flows might interact.


title: Recipe for computing vertex factors in Feynman diagrams

text: I am currently studying quantum field theory from Srednicki. In class we have covered till chapter 14 and then skipped to IR divergences. So my knowledge of quantum field theory is limited to those sections. I just mentioned it in case it helps you guys to calibrate your responses to my level! Anyways, my question is related to determining vertex factors for any arbitrary interaction. We were introduced to vertex factors in chapter 9. For interacting theories such as: $$\mathcal{L}_I = \frac{g}{3!}\varphi^3$$ and $$\mathcal{L}_I = \frac{\lambda}{4!}\varphi^4$$ the vertex factors are $ig$ and $i\lambda$ respectively. Ignore the sign; I can never remember it. Also, for this discussion it probably won't be a big deal. Anyways, getting back to the main point, I only remember being told that the vertex factors are the ones given above but no one ever told me how they were obtained. Srednicki never formally defined them. I just assumed that you just read off the coefficient from the interacting part. Also, physically I have an intuition for the vertex factor. It represents the strength with which the interactions couple free fields. And when you compute Feynman diagrams more interactions means higher powers of this coupling parameter (or vertex factor). In some later chapters Srednicki defined this vertex function $V_n(k_1,k_2,\dots,k_n)$; but he only formally talks about it in the context of vertex corrections. For the moment I'm not interested in corrections. I just want to know how you can compute this function for an arbitrary theory (say): $$\mathcal{L}_I = f_n(\varphi,\partial_\mu \varphi)$$ where the subscript $n$ the maximum number of fields and/or their derivatives defined in the function $f_n(\varphi,\partial_\mu \varphi)$. In general the vertex factor will not be just a constant like the two cases above. It will be a function of the incoming and outgoing four-momenta. What recipe should I follow to get that expression?


title: Divergence calculation of a lie algebra valued quantity having spinor indices

text: I am reading this paper by E. Weinberg - Fundamental monopoles and multimonopole solutions for arbitrary simple gauge groups. I am having a problem with a calculation. I don't have much experience with such calculations, and have never come across the gamma matrices mentioned here. So, it would be great if someone could help. M is an position independent quantity. The author claims that the divergence of the following quantity $$J_i=\frac{1}{2}tr<x| \gamma_5 \gamma_i(\gamma \cdot D) \frac{1}{-(\gamma.D)^2+M^2} |x>$$ The divergence of $J_i$ is the expression given below $$tr <x| \gamma_5 \frac{M^2}{-(\gamma.D)^2+M^2}|x>$$ tr is the trace over group and spinor indices. $\gamma$ are the gamma matrices forming the Clifford algebra. $D_4=\Phi$ where $\Phi^{ac}=f_{abc}\phi^b$, $\phi$ is the scalar higgs field. and $D$ is the usual covariant derivative in Yang-Mills theory given by $(D_{\mu}\phi)^a=\partial_{\mu}\phi^a + f_{abc}A_{\mu}^b\phi^c$ **Attempt at a solution** $$\partial_iJ_i=\frac{1}{2}tr<x| \gamma_5 \gamma_i \partial_i(\gamma \cdot D) \frac{1}{-(\gamma.D)^2+M^2}|x>-\frac{1}{2}tr<x| \gamma_5 \gamma_i(\gamma \cdot D) \frac{-2(\gamma.D)\partial_i(\gamma.D)}{(-(\gamma.D)^2+M^2)^2}|x>$$ Now in the second expression I can combine the two $\gamma \cdot D$. After taking $\partial_i(\gamma \cdot D)$ common from both the terms, you can add and subtract $M^2$ from the second term. Divergence=$$\partial_iJ_i=\frac{1}{2}tr<x| \gamma_5 \gamma_i \partial_i(\gamma \cdot D) \frac{1}{-(\gamma.D)^2+M^2}|x>-\frac{1}{2}tr<x| \gamma_5 \gamma_i[(\gamma \cdot D)^2 + M^2 - M^2] \frac{-2\partial_i(\gamma.D)}{(-(\gamma.D)^2+M^2)^2}|x>$$ The first part of the second term would reduced the power of the denominator by one, and then combine with the first term, and $M^2$ in the numerator would get stranded with the inverse squared term. Divergence = $$\frac{1}{2}tr \ \gamma_5 \gamma_i \partial_i(\gamma \cdot D) <x|- \frac{1}{-(\gamma.D)^2+M^2} + \frac{2M^2}{(-(\gamma.D)^2+M^2)^2}|x>$$ I don't know how to proceed after this. The problem is the denominator is squared in the second term, where is should have power 1. Help will be very much appreciated.


title: Kinetic energy with no velocity

text: When a body is displaced against the gravitational field of force it gains potential energy. When we drop the body it begins to move downward with a certain amount of acceleration, and the potential energy turns into kinetic energy. Kinetic energy is given by the equation: $$ E_k = \frac{1}{2}mv^2$$ When the body falls off the ground, and it stops moving then its velocity is zero, and therefore the kinetic energy is also zero in this case. In which form of energy does this kinetic energy usually convert into?


title: How to construct the charge conjugation matrix for any given dimension?

text: Generally, Gamma matrices could be constructed based on the Clifford algebra. \begin{equation} \gamma^{i}\gamma^{j}+\gamma^{j}\gamma^{i}=2h^{ij}, \end{equation} My question is how to generally construct the charge conjugation matrix to raise one spinor index in the gamma matrix. In even dimensions (D=2m), consider complex Grassmann algebra $\Lambda_{m}[\alpha^{1},...,\alpha^{m}]$ with generators $\alpha^{1},...,\alpha^{m}.$) Namely, we define $\widehat{\alpha }^{i}$ and $\widehat{\beta}_{i}$ as multiplication and differentiation operators: \begin{equation} \widehat{\alpha}^{i}\psi=\alpha^{i}\psi, \end{equation} \begin{equation} \widehat{\beta}_{i}\psi=\frac{\partial}{\partial\alpha^{i}}\psi. \end{equation} According to the Grassmann algebra, we have \begin{equation} \widehat{\alpha}^{i}\widehat{\alpha}^{j}+\widehat{\alpha}^{j}\widehat{\alpha }^{i}=0, \end{equation} \begin{equation} \widehat{\beta}_{i}\widehat{\beta}_{j}+\widehat{\beta}_{j}\widehat{\beta}% _{i}=0 \end{equation} \begin{equation} \widehat{\alpha}^{i}\widehat{\beta}_{j}+\widehat{\beta}_{j}\widehat{\alpha }^{i}=\delta_{j}^{i} \end{equation} This means that $\widehat{\alpha}^{1},...,\widehat{\alpha}^{m},\widehat{\beta }_{1},...,\widehat{\beta}_{m}$ specify a representation of Clifford algebra for some choice of $h$ (namely, for $h$ corresponding to quadratic form $\frac{1}{2}(x^{1}x^{m+1}+x^{2}x^{m+2}+...+x^{m}x^{2m})$). It follows that operators \begin{equation} \Gamma^{j}=\widehat{\alpha}^{j}+\widehat{\beta}_{j},1\leq j\leq m, \end{equation} \begin{equation} \Gamma^{j}=\widehat{\alpha}^{j-m}-\widehat{\beta}_{j-m},m<j\leq2m, \end{equation} determine a representation of $Cl(m,m,\mathbb{C})$ For example, in $D=4$, we can obtain $$\Gamma^{1}=\begin{pmatrix}0& 1& 0& 0\\\ 1& 0& 0& 0\\\ 0& 0& 0& 1\\\ 0& 0& 1& 0\\\ \end{pmatrix}$$, $$\Gamma^{2}=\begin{pmatrix}0& 0& 0& 1\\\ 0& 0& {-1}& 0\\\ 0& {-1}& 0& 0\\\ 1& 0& 0& 0\\\ \end{pmatrix}$$, $$\Gamma^{3}=\begin{pmatrix}0& {-1}& 0& 0\\\ 1& 0& 0& 0\\\ 0& 0& 0& 1\\\ 0& 0& {-1}& 0\\\ \end{pmatrix}$$, $$\Gamma^{4}=\begin{pmatrix}0& 0& 0& {-1}\\\ 0& 0& 1& 0\\\ 0& {-1}& 0& 0\\\ 1& 0& 0& 0\\\ \end{pmatrix}$$ My question is how to generally construct the charge conjugation matrix C, so that we could have $$C\Gamma C^{-1}=+/-\Gamma^T$$


title: Atomic clocks and how to synchronize them

text: During teaching measurement section in the class, Our teacher told us about atomic clocks. I have two questions:   1. What is exactly an atomic clock?    2. and how do we synchronize two atomic clocks far far away, e.g. in New York and I Beijing?


title: deriving differential equation of a cart moved by a motor

text: This is homework and I'm having some trouble getting started. How do I go from whats given to the form that they ask for? Normally in something like this I would try to balance torque, but I'm not quite sure how to set it up in this case with voltage. ![enter image description here](http://i.stack.imgur.com/pIANH.png)


title: Quantum Mechanics in terms of *-algebras

text: I'm currently trying to find my way into the geometric description of Quantum Mechanics. I therefor started reading: Geometry of state spaces. In: Entanglement and Decoherence (A. Buchleitner et al., eds.). Lecture Notes in Physics 768, Springer Verlag, Berlin, New York, 2009, 1-60. A document that can also be found as a manuscript via: http://www.physik.uni- leipzig.de/~uhlmann/PDF/UC07.pdf Even though I thought that I have a solid background in abstract algebra I somewhat got lost in Chapter 2 when he's trying to classify all the *-algebras that represent actual physical systems (starting at page 24 in the document). Do you have some recommendations for texts that introduce the *-algebra language in Quantum Mechanics in a more 'detailed' way. Because I kind of have the feeling that at a certain point Uhlmann just keeps skipping steps and I also lack some of the physical intuition concerning partial traces, canonical traces, purification and all that. From time to time I'd also be happy to see a concrete example. I'm looking forward to your responses. Best regards.


title: Spaghettification of humans near black holes

text: A few months ago I was discussing the spaghettification phenomenom with my wife, just for the fun of it. This was when the mass of the super massive black hole from M87 hit the news. The black hole was meant to have 6.7e+9 solar masses, so I thought it would be fun to figure out at which distance the tidal forces become so great, that the forces between your head and feet start getting... uncomfortable. I defined the distance from feet to head to be 2 m, the head having a mass of 10 kg and one foot having a mass of 2.5 kg (I didn't research the numbers, I just did some wild guesses). So, the force between head and foot should be: $$G*m_B*m_H \over (d+2)^2$$ $$-$$ $$G*m_B*2*m_F \over d^2 $$ where $d$ is the distance from foot to the black hole, $m_B$ is the mass of the black hole, $m_H$ the mass of the head and $2*m_F$ the mass of the feet. Assuming you fall with the feet first. This is one of the plots I made: ![Spaghettification: Human vs. M87](http://i.stack.imgur.com/5NFf0.png) _(note: the plot only takes the mass of one foot into account.)_ I made up this short octave script to plot the forces on this values with the distance in AU as the free variable and thought "ouch, that's gonna hurt pretty soon". I haven't checked which part of the body will rip off first at what forces though... Anyway, what I wanted to know is: Do my calculations make sense? Is my math correct? Thanks, Alex.


title: Calculating Systematic Error for capacitor charge time constat

text: I need to calculate systematic error for $\tau$ in capacitor's charging formula( $V_c(t)=V_s(1-e^{-t\over\tau})$ ) I converted it to : $\tau=-{t \over ln(1-{V_c \over V_s})}$   and continued by doing: $ln(\tau)=ln(-t)-ln(ln(1-{V_c \over V_s}))$   then tried to derivative: ${d\tau \over \tau}={dt \over t}- ...$   I can't go ahead any more! What about discharge equaltion ( $V_c(t)=V_0 e^{-t\over\tau}$ )?   How should i continue and get result for $d\tau \over \tau$?


title: Gas pressure equalisation: Where does the excess energy go?

text: Assume for purposes of discussion a closed container with a concentration of gas on one side and near-vacuum on the other. We could let the gas pressure equalise naturally, or we could construct a barrier with a turbine such that the gas moving from the concentrated side to the vacuum side spins the turbine and produces electricity. In both cases, the gas reaches equilibrium after some time, but in one of the cases the system performs work. My question is, what happens to the excess energy _not_ used to produce electricity in the first example (with no turbine). My intuition tells me that the answer has to be one of the following: Either the gas that drove the turbine is colder than the gas that did not, or that the electrical energy produced is actually less than the energy required to build the turbine so the electrical energy was actually input into the system by action of building the turbine. Is either of my answer right? What else have I not accounted for? Thanks.


title: Will the positive ions in an aqueous solution be attracted to a charged body ?

text: If I had a negatively charged body , say an electret , and i put it in a container of NaCl solution . Will the positive ions of sodium be attracted to it ? and why? If no, why do the positive ions attract to the cathode during electrolysis?


title: Renormalization and the Hierarchy Problem

text: The hierarchy problem is roughly: A scalar particle such as the Higgs receives quadratically divergent corrections, that have to cancel out delicately with the bare mass to give the observed Higgs mass. I have a couple of related questions about that: Why is this a problem, _isn't this just ordinary renormalization_? Other particles receive similar divergent corrections - not quadratically divergent, but still. The regulator gives you a parameter $\Lambda$ that you'd like to take to infinity, but can't, because the corrections would blow up. Renormalization adds a counterterm that cancels the divergent term, and stuffs it into the bare mass. Now $\Lambda$ is gone from the expression for your measurable parameter, and you can let it go to infinity. I know you can choose a finite value of $\Lambda$, and regard the theory as an effective field theory, valid up to that scale. But that doesn't seem necessary if the divergence vanishes into the bare parameter. Framed differently: Why is the quadratic divergence in case of the Higgs a problem, but not the logarithmic one in QED? If you insert a value for $\Lambda$, say $m_\mathrm{Pl.}$, OK, then $\Lambda^2 \gg \log \Lambda$. But if we don't and keep $\lim_{\Lambda\rightarrow\infty}$ in mind, then... infinity is infinity... and haven't we got rid of $\Lambda$ by renormalizing anyway? The second part was touched in another question: _Why worry about what value the bare mass has, isn't it unphysical and unobservable anyway?_ I always thought that it is just a symbol, like $m_0 = \lim_{x\rightarrow\infty} x^2$, and it is meaningless to ask how many GeV it is. Just like it's meaningless to ask about the value of a delta function at zero (while it is well-defined if you integrate over it with a test function). But according to this comment by Ron Maimon, the bare mass _is_ experimentally accessible. _Is it?_ I thought you can keep pushing and pushing to higher energies, but will principally never observe the bare mass, just as you cannot observe a bare electron charge (you'll hit the Planck scale or the Landau pole first). (Apologies that I put two questions in one, but I have a strong feeling that they might share the same answer.)


title: Dirac equation in QFT vs relativistic QM

text: How does the Dirac equation in quantum field theory solve the existing problems in the interpretation Dirac equation (as a single-particle wave equation) in relativistic quantum mechanics? EDIT: The question was not clear. So I ask again. Why is the Dirac equation not meaningful as a single particle wave equation? Does many-particle interpretation of Dirac equation make it meaningful? If yes, how?


title: Toroidal Planets

text: I have read online in numerous places about the possible existence of toroidal planets, and I most people seem to believe that they could exist, but they also have no evidence to support this claim. I recently did a project in which I calculated the gravitational field around one-such planet, and I would like to continue working on it to either prove or disprove the possibility of these planets. Before I continue, I would like to know if any mathematically rigorous investigations into this question have already been conducted. I have searched, but I haven't found anything on planets, only black holes.


title: Lorentz force from velocity-dependent potential and Lagrangian

text: There is something i'm missing. I am at page 22-23 of Goldstein Classical Mechanics 3rd ed. Lorentz force can be derived from a potential $$U=q\phi-q\mathbf{A}\cdot\mathbf{v}$$ Where $\phi(t,x,y,z)$ is a scalar potential and $\mathbf{A}(t,x,y,z)$ is a vector potential. The Lagrangian $L=T-U$ Where $T$ is the potential energy. I'm trying to solve this with Lagrange equations for the x coordinate, and for the term $$\frac{d}{dt}\big(\frac{\partial{L}}{\partial\dot{x}}\big)$$ I'll show you how I'm trying to compute this with detail: $$\frac{d}{dt}\big(\frac{\partial{L}}{\partial\dot{x}}\big)=m\ddot{x}-q\frac{d}{dt}(\frac{\partial{}}{\partial{\dot{x}}}\phi)+q\frac{d}{dt}(\frac{\partial\mathbf{A}}{\partial{\dot{x}}}\cdot\mathbf{v}+\frac{\partial\mathbf{V}}{\partial{\dot{x}}}\cdot\mathbf{A})$$ Now, here's my big problem, I'll may have some trouble understanding the condition in wich i can interchange the differentiation with respect to $\dot{x}$ and $t$, i.e for example, $$q\frac{d}{dt}(\frac{\partial{}}{\partial{\dot{x}}}\phi)=q\frac{\partial{}}{\partial{\dot{x}}}(\frac{d}{dt}\phi)$$ This is done some pages before when Goldstein is deriving the lagrange equations, and I solved a problem in this chapter usig this. So now I have: $$q\frac{\partial{}}{\partial{\dot{x}}}(\frac{\partial{}}{\partial{x}}\phi\frac{dx}{dt}+\frac{\partial{}}{\partial{y}}\phi\frac{dy}{dt}+\frac{\partial{}}{\partial{z}}\phi\frac{dz}{dt}+\frac{\partial{}}{\partial{t}}\phi)$$ Now I have looked now for many hours for many ways in wich this derivation is made and the the expression above vanishes, but i don't understand why, is it because I'm assuming x,y,z in the scalar potential are functions of time themselves? If not i can see why it would vanish. Also this vanishes because i can't exchange the order of differentiation? * * * I totally see what you mean, I was applying it wrong, after you answered me I encountered that final term $\frac{\partial \phi}{\partial q_i}$, I had to substract it because it was actually: $$q\frac{d}{dt}(\frac{\partial{}}{\partial{\dot{x}}}\phi)=q[\frac{\partial{}}{\partial{\dot{x}}}(\frac{d}{dt}\phi)-\frac{\partial f}{\partial q_i}]$$ Thanks a lot you're awesome!


title: Shallow water wave question from Acheson's book

text: I am learning Fluid mechanics by reading Acheson's book entitled "Elementary Fluid Dynamics". Below is from problem 3.1. Consider the Euler equation for an ideal fluid in the irrotational case. We are studying two-dimensional water waves, so the velocity vector is on the form: ${\bf{u}}=[u(x,y,t),\,v(x,y,t),\,0]$. Because of ''irrotationality'', there is a velocity potential $\phi$ such that ${\bf{u}}=\nabla \phi$. By the incompressibility condition, $\phi$ satisfies the Laplace equation. Let $y=\eta(x,t)$ be the equation of the free surface. Note that the bottom of the water is at $y=-h$, where $h$ is the depth. Then \begin{equation} \frac{\partial\eta}{\partial t}+u\frac{\partial\eta}{\partial x}=v,\;\;{\mbox{on}}\;\;y=\eta(x,t).\end{equation} Then the Euler equation on the free surface (once integrated) gives (note that both the pressure at the free surface and the density are considered constant and can be absorbed in the constant of integration): $$\frac{\partial\phi}{\partial t}+\frac{1}{2}(u^2+v^2)+g\eta=0,\;\;{\mbox{on}}\;\;y=\eta(x,t).$$ We now consider ${\bf{u}}$ and $\eta$ small (in a sense to be determined later) and linearized the equations above: $$\frac{\partial\eta}{\partial t}=v,\;\;{\mbox{on}}\;\;y=0.$$ $$\frac{\partial\phi}{\partial t}+g\eta=0,\;\;{\mbox{on}}\;\;y=0.$$ We look for sinusoidal traveling wave solution of the form $\eta=A\cos{\left(kx-\omega t\right)}$. With the condition that $v=\phi_y=0$ at $y=-h$ (the bottom), with the Laplace equation and the two linear equations above, we find that $$ \phi=\frac{A\omega}{k\sinh(kh)}\cosh{(k(y+h))}\sin{(kx-\omega t)}$$ and the dispersion relation $\omega^2={g}{k}\tanh(kh)$. Now, to determine the sense in which ${\bf{u}}$ and $\eta$ are small, we need to compare $u^2+v^2$ to $g\eta$. The term $u^2+v^2$ is of order $A^2\omega^2=A^2{g}{k}\tanh(kh)$ and the term $g\eta$ is of order $gA$. Then one gets the condition $A\ll\lambda$ by asking that $u^2+v^2\ll g\eta$. One gets the same condition by comparing the other nonlinear terms to the linear ones in the two nonlinear equations above. My question is this: I should be able to also deduct that the displacement of the free surface is small w.r.t. the depth, i.e. $A\ll h$. How is this condition obtained?


title: What is the difference between angular speed and tangential speed in a circular motion?

text: I was looking a long time for the way the equations of this two speeds are obtained, and i found pretty much nothing important, so can someone explain how are those obtained, and which is the difference between them?


title: Is light red shifted in optical tweezers?

text: This is a question I put to my supervisor during my PhD many years ago, and never really got a satisfactory answer to. In an optical tweezers, assume that a beam of light is used to move a glass bead. My question is whether the outgoing light is red-shifted. If it is not I cannot reconcile how energy is conserved in the system, as work has been done to move the bead? Answers appreciated as always...


title: Maximum separation when 2 unlike charges shot apart

text: I want to compute the maximum separation when 2 unlike charges are shot apart from each other                        <--- (+)     (-)--->     (+)                                 (-)      |----------------- d ---------------|      How do I do it? Its actually in my lecture notes: $$E_i = KE_i + U(r_i)$$ $$E_f = U(r_{max})$$ And to compute them, I equate them $E_i = E_f$. And solve for $r_{max}$, but there are 2 charges, so my $KE_i = 2 \times \frac{1}{2} m v^2$? $U(r_i)$ will be just 1 value or also 2? I am thinking 2 since there are 2 separate charges, each should have their own PE/KE? Hmm, ... or maybe I can get away with using 1 value if they have the same KE/PE at any point in time? * * * **UPDATE** : Another similar question > An electron and positron are $100 fm$. Whats the min speed each particle > must have to escape from each other?                   <---(-)        (+)--->              |- 100 fm -|     (-)                         (+)      The answer looks like: $$KE_i + KE_i - U = 0$$ So my question is:   * Why 2 KE but only 1 PE (U)?    * I suppose reason why its $-U$ because they are unlike charges?   * Is the reason why $KE_f = 0$? because its asking for min escape velocity? So I assume all KE are used up?


title: Calculating uncertainty in the final result (combining uncertainties)

text: I'm struggling to determine the uncertainty in $F$ so it would match the textbook answer. The problem statement is: **A force F is obtained using the equation: $F = \frac{mv^2}{2\pi(x_2 - x_1)}$. The readings taken were: $m = 54.0 \pm 0.5\ \mathrm{kg}$, $v = 6.3 \pm 0.2\ \mathrm{ms}^{-1}$, $x_2 = 4.7 \pm 0.1\ \mathrm{m}$, $x_1 = 3.9 \pm 0.1\ \mathrm{m}$. Calculate the value of F and determine the uncertainty in your value.** Calculating the force: $F = \frac{(54.0\ \mathrm{kg}) \times (6.3\ \mathrm{ms}^{-1})^2}{2 \pi \times (4.7\ \mathrm{m} - 3.9\ \mathrm{m})} \approx 426.388\ \mathrm{N} = 430\ \mathrm{N} \text{(to 2 s.f.)}$. This agrees with the textbook answer. Now, let $X = x_2 - x_1$. Then $X = 4.7\ \mathrm{m} - 3.9\ \mathrm{m} = 0.8\ \mathrm{m}$. The uncertainty in $X$ is $\delta X = \sqrt{(\delta x_2)^2 + (\delta x_1)^2} = \sqrt{(0.1\ \mathrm{m})^2 + (0.1\ \mathrm{m})^2} \approx 0.1414\ \mathrm{m}$. Thus, $X=0.8 \pm 0.1\ \mathrm{m}$ and the formula becomes $F = \frac{mv^2}{2\pi \times X} = \frac{1}{2 \pi} \frac{m v v}{X}$. This means I can now use another standard formula to calculate the uncertainty in $F$: $\delta F = \sqrt{(\frac{\delta m}{m})^2 + (\frac{\delta v}{v})^2 + (\frac{\delta v}{v})^2 + (\frac{\delta X}{X})^2}$. And with values $\delta F = \sqrt{(\frac{0.5\ \mathrm{kg}}{54.0\ \mathrm{kg}})^2 + 2 \times (\frac{0.2\ \mathrm{ms}^{-1}}{6.3 \ \mathrm{ms}^{-1}})^2 + (\frac{0.1414\ \mathrm{m}}{0.8\ \mathrm{m}})^2} \approx 0.133$ or about 13%. But the bloody textbook says it is 40% and quotes the answer as $430 \pm 180\ \mathrm{N}$. I've tried some calculations with various values within the uncertainty and my result was always within 13% (or about 60 N) of 430 N, just as I would expect. Where have I got wrong?


title: Silly bet about the best way to walk up a hill

text: I have a silly bet with a friend of mine. One of us argues that walking up a hill whilst leaning forward requires less exertion, due to your centre of mass being further forward, and therefore gravity would assist and less muscle energy would thus be required to climb the hill. The other of us argues that this cannot possibly be true, as the amount of work required to climb the hill is the same regardless. I know this all sounds ridiculous but we'd really like to settle it! TIA!


title: Moving "virtual" singularity?

text: Imagine two close, really big black holes rapidly spinning around each other. That setup would emit a terrible amount of gravitational waves. My question is, could those gravitational waves, if big enough, temporarily collapse space around them into a singularity? That is, not the black holes themselves forming a singularity, but the ripples they form on space-time, constructively interfering to form a singularity in a place where there is no actual mass.


title: How long in advance can dates of moon phases be accurately predicted?

text: When researching an article on Chinese New Year I found a few websites that predict the dates of Chinese New Year (CNY) tens - or sometimes even hundreds - of years in advance. I´m wondering how many years in advance the date of CNY can be predicted. As the date of CNY is based on the phases of the moon, my question is: Up to how many years in advance can the dates of new moon be predicted with 100% accuracy?


title: equivalence of wave equations

text: I wonder if the following 2 PDEs are equivalent:   1. $$\frac{\partial^2}{\partial t^2}\psi(\vec{r},t)-c(\vec{r})^2\nabla^2\psi(\vec{r},t)=s(\vec{r})\delta'(t)$$ subjects to zero initial conditions $$\psi(\vec{r},0)=0, \quad \left.\frac{\partial}{\partial t}\psi(\vec{r},t)\right|_{t=0}=0,$$ where $\delta'(t)$ denotes the derivative of Dirac delta function.   2. $$\frac{\partial^2}{\partial t^2}\psi(\vec{r},t)-c(\vec{r})^2\nabla^2\psi(\vec{r},t)=0$$ subjects to initial conditions $$\psi(\vec{r},0)=s(\vec{r}), \quad \left.\frac{\partial}{\partial t}\psi(\vec{r},t)\right|_{t=0}=0$$ I don't how to prove or disprove it. Any help will be appreciated. Thanks!


title: Electric field due to nonconducting plastic sheets

text: > Two very large, nonconducting plastic sheets, each 10.0 cm thick, carry > uniform charge densities $\sigma_1, \sigma_2, \sigma_3$ and $\sigma_4$ on > their surfaces (the four surfaces are in the following order $\sigma_1, > \sigma_2, \sigma_3$ and $\sigma_4$ going from left to right). These surface > charge densities have the values $\sigma_1 =-6, \sigma_2 = +5, \sigma_3 = > +2$ and $\sigma_4 = +4$ all in C/(m*m). > > A) Use Gauss's law to find the magnitude of the electric field at the point > A, 5.00 cm from the left face of the left-hand sheet. > > B) Find the magnitude of the electric field at the point B, from the inner > surface of the right-hand sheet. > > C) Find the magnitude of the electric field at the point C, in the middle of > the right-hand sheet. So in the book the author derives an equation for this kind of situation, and there is actually an example in the text, but the example is for a plate that has only one charge and not two like this one. Anyways, the equation is $E=\frac{\sigma}{2\epsilon_0}$. I got question A) right by just adding up the electric field due to all the charges, namely $E_{net}=\frac{\sigma_2}{2\epsilon_0} +\frac{\sigma_3}{2\epsilon_0}+\frac{\sigma_4}{2\epsilon_0}-\frac{\sigma_1}{2\epsilon_0}$, but I am not sure about B) and C), would they be the same? When I drew the electric field vectors, going towards the negative side and away from the positive one, I couldn't really figure out what direction the net vector would be pointing to. ![diagram of problem](http://i.stack.imgur.com/MVB9K.jpg)


title: A different type of Gaussian Grassmann Integral

text: Let $A$ be an antisymmetric matrix. Usually, one proves that for a Grassmann integral of the type, $$\int d\psi d\theta \exp( \psi^T A \theta) = \det(A)$$ where $\psi$ and $\theta$ are vectors of Grassmann variables. I have a different problem which I feel should trivially reduce to the same, but it can't figure it out: $$\int d\psi d\theta \exp( 1/2 \psi^T A \psi + 1/2 \theta^T A \theta)$$ I think it should be possible to show that my expression is equivalent to the upper one, but the necessary variable substitution escapes me.


title: Question on Total, Orbital and Spin Angular momentum

text: I am reading about the total, orbital and spin angular momentum, and I am not clear as to what these generators actually do after exponentiating. Could you give me a physical picture of what happens to the quantum set, after being acted upon by the operator obtained after exponentiating these? For example does the total angular momentum rotate the ket in a circle, about the normal? What about the other two?


title: Difference between Quark-Gluon Plasma and Color-Glass Condensate?

text: From the news. Apparently they are very similar, yet different.


title: Question on inflation

text: I have two particular questions regarding the inflationary scenario. They are: 1.) What is the physical origin of the inflaton field? 2.) Why has the potential of the inflation field its particular form?


title: is it possible to flow current in open circuit?

text: First , i don't know much about chemistry and physics. I'm just a graphic designer but i have this question in my mind. I'm sorry if this question is too basic and use 'generic' language. As i know a battery is an example of a closed circuit where it can then produced electricity , electron will flow from negative pole to positive. A chemistry representation of this battery is for example i got this link : http://www.youtube.com/watch?v=0TvYlJ06MXo&feature=related so it's a closed circuit. Where one of the side will release (lose) electron while another side will gain electron. That's how the circular process will occure (the liquid act as a connector or bridge to close the circuit) . My question is what if we 'open' the circuit , so i mean separate the liquid. Of course the chain process will stop , right ? But what about if we can create somekind of instrument where i can supply electron to the liquid and onn the other side using another tool/instrument to grab electron from another liquid. Does it make any sense , that we can still produce electricity using open circuit? Am i wrong ? Sorry for this basic question. I just need some enlightment :-)


title: Can we use only the observables of Fermion fields?

text: There are legion ways to consider fermionic Dirac spinor fields, but is it possible to consider the asymptotic free field only in terms of observables, which in the case of the Dirac spinor field must be $U(1)$ invariant? For a specific construction, to give some context, in a 4-dimensional formalism we might consider normal-ordered objects of the form \begin{equation} \int:\hspace{-0.2em}\overline{\hat\psi_{\xi'}(x)}F_{\xi'\xi}(x)\hat\psi_\xi(x)\hspace{-0.2em}:\mathrm{d}^4x, \end{equation} smeared by a Dirac matrix-valued test function $F_{\xi'\xi}(x)$. As we will see below, it's better if we instead first construct a _2-point_ formalism, for which we can then consider the limit as the two points approach each other. So, if we construct a 2-point Dirac operator as \begin{equation} \hat\Phi_F=\int \overline{\hat\psi_{\xi'}(x_1)}F_{\xi'\xi}(x_1,x_2)\hat\psi_\xi(x_2)\mathrm{d}^4x_1\mathrm{d}^4x_2, \end{equation} where the test function $F\in\mathcal{F}$ is Dirac algebra- valued for each pair of points $x_1,x_2$, we find that for the commutation relation $\left[\hat\Phi_F,\hat\Phi_G\right]$ we have \begin{equation} \left[\hat\Phi_F,\hat\Phi_G\right]=\hat\Phi_{[F,G]}, \end{equation} where $[F,G]=F.\mathrm{i}\\!\mathsf{S}.G-G.\mathrm{i}\\!\mathsf{S}.F$, with $\mathrm{i}\\!\mathsf{S}(x,x')_{\xi\xi'}=\left(\mathrm{i}\gamma^\mu\frac{\partial}{\partial x^\mu}+m\right)_{\xi\xi'}\mathrm{i}\\!\Delta(x-x')$ as a Dirac spinor Green's function, and $$\left(F.\mathrm{i}\\!\mathsf{S}.G\right)_{\xi'\xi}(x,y)=\int F_{\xi'a}(x,x_1)\mathrm{i}\\!\mathsf{S}_{ab}(x_1,x_2)G_{b\xi}(x_2,y)\mathrm{d}^4x_1\mathrm{d}^4x_2, $$ so that the 2-point field $\hat\Phi_F$ generates a Lie algebra (and a justification of sorts for introducing a 2-point formalism). What we have constructed is effectively a matrix algebra that is a tensor product of both space-time indices and Dirac indices. It's not too difficult to use Wick's theorem to show that the generating function for the time-ordered VEVs for these operators, in terms of the Feynman propagator $\mathrm{i}\mathsf{S_F}$ and using a Trace that contracts over both space-time points and Dirac indices, is \begin{equation} \left<0\right|\mathrm{T}[\mathrm{e}^{\mathrm{i}\lambda\hat\Phi_F}]\left|0\right> =\exp{\left[\sum\limits_{n=1}^\infty \frac{(\mathrm{i}\lambda)^n}{n}\mathsf{Tr}{[(\mathrm{i}\mathsf{S_F}.F)^n]}\right]} =\exp\hspace{-0.4ex}{\Bigl(-\mathsf{Tr}[\ln{(1-\mathrm{i}\lambda\mathrm{i}\mathsf{S_F}.F)}]\Bigr)}, \end{equation} where the last equality is essentially formal, with the result being a quantum analogue of an infinite system of independent exponential distributions (without time-ordering, the result is straightforward but cannot be expressed so compactly). We can also introduce a normal-ordered form, $\hat\Phi_F^{\hspace{-0.3ex}:\hspace{-0.1ex}:}$, for which $$\left[\hat\Phi^{\hspace{-0.3ex}:\hspace{-0.1ex}:}_F,\hat\Phi^{\hspace{-0.3ex}:\hspace{-0.1ex}:}_G\right]=\hat\Phi^{\hspace{-0.3ex}:\hspace{-0.1ex}:}_{[F,G]}+\mathsf{Tr}[F.\mathrm{i}\mathsf{S}_+.G.\mathrm{i}\mathsf{S}_- - G.\mathrm{i}\mathsf{S}_+.F.\mathrm{i}\mathsf{S}_-],$$ where $\mathrm{i}\mathsf{S}_\pm$ are the positive and negative frequency parts of $\mathrm{i}\mathsf{S}$. This construction changes the lowest degree VEV to $\left<0\right|\hat\Phi_F^{\hspace{-0.3ex}:\hspace{-0.1ex}:}\left|0\right>=0$ but all higher connected VEVs are unchanged; for this 2-point field operator, the limits of the VEVs as the two points approach each other exist. The commutator algebra of such 1-point objects is not closed, because $\hat\Phi_{[F,G]}^{\hspace{-0.3ex}:\hspace{-0.1ex}:}$ is a 2-point object, but the algebra of 2-point fields is closed. There is a curiosity associated with this object, that $[F,G]$ is non-zero only when the test functions $F$ and $G$ have time-like separated supports, so that if we construct field operators using only _merged_ 2-point field operators (operators such as $\hat\Phi^{\hspace{-0.3ex}:\hspace{-0.1ex}:}_F$, where the test function is $F(x,y)=\delta^4(x-y)F(x)\hspace{0.3ex}$), we can only obtain 2-point operators for which the two points are at time-like separation. By taking superpositions of terms of the form $\hat\Phi^{\hspace{-0.3ex}:\hspace{-0.1ex}:}_{[F,G]}$, we can construct terms of the form $\hat\Phi^{\hspace{-0.3ex}:\hspace{-0.1ex}:}_F$ for any $F$ for which $F(x,y)$ is zero whenever $x$ and $y$ are space-like separated. Consequently, gauge invariance amounts to requiring, in particle terms, that when constructing states, creation of an electron (or positron) must always be paired either with annihilation of an electron (or positron) at time-like separation or with the creation of a positron (or electron) at time-like separation. Equivalently, measurement operators cannot create a charged particle without introducing a compensating charged particle creation or annihilation at time-like separation. This allows us more-or-less to reconcile the Lorentz and translation covariant 2-point formalism constructed above with the usual Lagrangian formalism, which is only barely covariant because it requires two space-like hyperplanes for its formulation, by taking all the test functions $F(x,y)$ to have support only on two time-like separated space-like hyperplanes. This is enough different from anything I've ever seen on the Dirac spinor field that I can't believe that there's something out there already that's like it, but, if there is, I'd like to know about it! So this Question is partly a reference request. An obvious Answer to the Question as asked would be _no_ , or indeed _of course not_ , because we need to be able to construct states using the action of an odd number of fermion field operators on the vacuum vector, however I take that objection to be nullified by the 4-space and two-time construction given above, which allows a state to be constructed in which an odd number of fermion field operators act on the 4-dimensional vacuum state at each of two times. Feel free to comment that the formalism as given doesn't include a means to introduce interactions. I know, and I'm working on it, but comment anyway. If you think there's some other reason to save me wasting my time on this approach, feel free to be as substantively insulting as you like.


title: How to find the distance of the source of sound in this problem?

text: On a cloudy day, the sound of thunder was heard 4.5 s after the flash of light was seen. How far was the cloud? Given that speed of sound = 340 m/s **This is what I've tried:** speed of light(c) = 10^6 * speed of sound( _v_ ) therefore, speed of light=340*10^6 m/s If _x_ is the distance of the cloud, then               x/(340*10^6)  -  x/340 = 4.5          =>(340*10^6  -  340)/x =  1/4.5     => x=1529998470`      but the correct answer is 1530m . Where am I going wrong?


title: Variational principle

text: In the LMTO method, the interstitial region is approximated by plane waves and the muffin tin region of the potential by solutions to the radial Schrodinger equation. In using the variational method for calculating the ground state energy a linear combination of the two solutions is used. The Hamiltonian consists of $V_0$ (interstitial), $V_1$ (muffin tin-spherical part and the kinetic energy term) and $V_2$ (muffin tin non spherical part). Now how do these various parts act on the linear combination of wavefunctions? Will they act separately as in $V_0$ (interstitial) acts only on the plane waves?


title: Vector Nature Of Angular Velocity

text: I am currently reading about angular position, angular velocity, and angular acceleration. I came across this paragraph that was particularly confusing, and was wondering if someone could perhaps help me. Here is the paragraph: "If a particle rotates in the $xy$ plane, the direction of $\vec{\omega}$ for the particle **is out of the plane of the diagram when the rotation is counterclockwise and into the plane of the diagram when the rotation is clock- wise.** To illustrate this convention, it is convenient to use the right-hand rule demonstrated in Figure 10.3. When the four fingers of the right hand are wrapped in the direction of rotation, the extended right thumb points in the direction of $\vec{\omega}$The direction of $\vec{\alpha}$ follows from its definition $\vec{\alpha}=d\vec{\omega}/dt$. It is in the same direction as $\vec{\omega}$ if the angular speed is increasing in time, and it is antiparallel to $\vec{\omega}$ if the angular speed is decreasing in time." The above text **in bold** is what I am having most difficulty with. What do they mean? Also, how do I color words?


title: What is the maximum mass of planet HD 40307 g

text: Recently discovered planet HD 40307g around HD40307 Star system is claiming that its inside a habitual zone of a solar system. Besides It has water where life could be evolved. I tried to calculate `g` of that planet. But failed! This is because the true mass of the planet could not be found. What we have is Minimum Mass which is `7.1` times Earth mass. My question is how do I know the Maximum Mass?


title: Force on a bar magnet

text: A bar magnet is freely falling vertically inside a conducting ring placed in horizontal plane. Will it fall with an acceleration equal to g, less than g or greater than g? ![magnet](http://i.stack.imgur.com/KsRlC.png)


title: Why minibands are formed in superlattices?

text: In a single, finite quantum well, there are energy levels defined by the eigenstates - the solutions of the Schroedinger's Equation. The corresponding wavefunctions leak to the barrier because of its finite height. If we bring two identical QWs together, the wavefunctions of corresponding states will mix and, as far as I understand, the energy levels will have to split in order to satisfy the exlusion principle. If we then add another QW, the levels will split again, and so on. A structure containing many adjacent QWs with thin barriers is called a superlattice and instead of discrete energy levels it has "minibands", where many levels are very close to each other. **What is the mechanism behind the creation of minibands?** I think I understand a similar situation in a homonuclear molecules: if two atoms of Hydrogen are brought together, their atomic orbitals mix in a process that is called LCAO-MO (Linear Combination of Atomic Orbitals into Molecular Orbitals), leaving a bonding and an antibonding $\sigma$ molecular orbital. When we bring together many atoms and create a crystal, their valence orbitals also mix, giving rise to valence and conduction bands (altough here I find it somewhat more confusing than in a H-H case). Should I therefore think of miniband creation as the "next step of mixing"? I.e. the already mixed orbitals (valence and conduction band) mix once again and create just another configuration? I will be grateful for an explanation.


title: Determining if a decay is due to strong or weak interaction

text: How do we determine if a decay is due to weak or strong interaction. Is it true that as long as any one of the 6 flavour numbers are not conserved, then it must be due to weak interaction? Is there a good general rule of thumb to determine this?


title: Exponential or logarithm of a dimensionful quantity?

text: I have a unit measure, say, seconds, $s$. Furthermore let's say I have a dimensionful quantity $r$ that is measure in seconds, $s$. What is the unit measure of $e^r$? ($1/r$ is in $Hz$.) My question is general, how to find the unit measure of a transformation function $y=f(x)$ where $x$ takes some known unit measure. I give above two functions $f(\cdot)=e^\cdot$ and $f(.)=1/\cdot$.


title: Are comets known to exist in other star systems?

text: Are comets a feature unique to our Solar System? Or, are comets/cometary clouds detected around discovered/observed extra-solar systems too? If they were detected elsewhere, how do such cometary clouds affect discovery by perturbation of planets in that system ?


title: Is the cosmic horizon related to the Big Bang event?

text: The Universe expands according to the Hubble's law: velocity is proportional to distance. There must be some distance, therefore, at which the velocity reaches the speed of light. This defines the horizon. The Doppler shift is so huge that the length of waves becomes infinitely long: We can not see beyond that horizon. Is this correct? On the other hand, when we look in the sky we look back in time. Since the Universe has finite age we can not see beyond some distance. So these are two horizons. Are they related to each other?


title: How does an aeroplane maintain balance during maneuvers?

text: I understand the principle behind flight, how the lift is generated etc. What I don't understand is when there are maneuvers made where the plane flies such that the wings are in vertical plane, how does the plane not lose its balance? How are the forces acting to keep it steady and from falling? Same doubt about helicopters doing maneuvers. Fun question: Did an engineer or a scientist figure this on paper or did a maverick pilot got them thinking and sent them back to analysis?


title: Find the max projection angle so that distance from point of projection always increases

text: We want to launch a projectile from the surface of earth so that its distance from the point of projection is always increasing. What is the maximum angle of projection for which this is possible? One way to visualize this would be that there should not be a component of velocity opposite to the position vector of the particle, the point of projection being the origin, at all times of motion. So following this line of thought I wrote : $$\overrightarrow v \cdot \overrightarrow r > 0$$ Which did not yield a very nice expression in the angle $\theta$. There should be better methods, as is often the case with physics, please suggest some of them.


title: Definition of a year

text: Is there an acceptable definition of a year (in number of days)? Google Calculator: https://www.google.com/search?q=seconds+in+1+year returns 3.15569e7 seconds and then https://www.google.com/search?q=seconds+in+1+year#q=3.15569e7%2F(24*60*60) returns 365.241898148 What is the rationale behind choosing 365.242 as the number of days in an year? Is it defined by SI?


title: Erogodicity in a Monte Carlo simulation

text: **Q1:** What is the **ergodicity** and **ergodicity breaking** in a Monte Carlo simulation of a statistical physics problem? **Q2:** How does one ensure that the ergodicity is maintained ?


title: How far does typical view of clouds/atmosphere extend?

text: The specific "sub questions" I'm asking are:   1. When you are looking at clouds just on the horizon, how far away would they be?   2. How wide (in km) is that total field of vision at roughly cloud height.     * Let's assume that the clouds being viewed are of the high family form which range from 5-12.2km (16,500 to 40,000 feet) in altitude.     * Let's assume flat or nearly flat ground at an elevation of 0-100 metres (328 feet) above sea level.     * Let's also assume the viewer is human and approx 170cm tall (5'7"). I understand if the answers can only be approximates. Looking for both a general way to think about this question as well as how to measure the field of view in question. If it's difficult to take all the variables into account then can anyone explain what the absolute maximum distance would be in a best case scenario? Similar but not the same question: How much sky do we see at any one moment?


title: image distortion when taking a picture from space

text: I'm trying to understand if there is a distortion of an image taken from space (i.e. from a satellite), and if there is, then how to model it mathematically (depending on the angle in which we take the photo). So, is there such a distortion (I assume there is, and it increases with the angle of the camera relatively to the nadir angle), and why (mathematically)?


title: What would gravity be like on a hemispherical planet?

text: On a hemispherical world half the size of Earth (ignoring that the planet should break apart and become a smaller sphere) what would the gravity be like? How would it change as you traveled from pole to pole along the curved side, as well as down the flat side? What other interesting or distinguishing facts would this planet have? I suspect that if you 'fell' off the edge on the flat side it would at first be like falling off a cliff and then transition more and more into a slope and finally be level in the center. If so, how far would you fall before the slope feels like a 45 degree angle? A 30? A 60? I also suspect that gravity at the poles would be greater than gravity at the curved side on the equator opposite the flat side. I'm not sure what percentage of Earth's gravity would be experienced at any given point.


title: Entropy, Mass and Brane gravity

text: Does string theory state as vibrational entropy increases, mass increases? Related: What is a D-brane?   Reference: Cambridge Relativity


title: What counts as a measurement?

text: In quantum mechanics, an elementary particle does not have a well defined position until a measurement is performed on it (right?). Such a "measurement" is any sort of interaction with other particles that gives those particles information about the position of the first particle. However, systems of elementary particles, such as atoms or buckyballs, can also cumulatively lack a well-defined position, which is why the double slit experiment can be performed with atoms and buckyballs. It thus seems that interaction with just any other elementary particle is not enough to collapse the wave function. If this is the case, it would seem that an arbitrarily large system of known elementary particles could lack a well-defined position because they would be insufficient to collapse each other's wave functions. Thus it would seem that perhaps consciousness is necessary to cause the collapse. Does this seem reasonable or correct? (If there are any errors in this question, please tell me . . . not that you wouldn't if I didn't ask you to.)


title: Understanding Heat

text: Heat or thermal energy as understood is nothing but motion of molecules of the matter. If the molecules are tightly bound (in case of solids), it is to-and- fro molecular vibrations, otherwise it is continuous random motion of molecules (in case of liquids/gases/plasma). Sound, being waves, is also vibrations in matter. Why, then, if we heat one end of a solid rod, assuming rod is at least few feet in length, does it take ages for the heat to reach the other end, whereas sound reaches in no time ? (sound travels at 1400 m/s in solid) Doesn't it show that heat is more an intra-atomic feature rather than an atomic or molecular motion? Observation that electrical good conductors are also good conductors for heat, can we assume that heat is chaotic motion of electrons (the "electron gas")? (Of course, all the known phenomena related to heat will be needed to be explained with this hypothesis.)


title: Ashcroft Mermin Eq. 17.47ff

text: In "Solid State Physics" by said authors, Eq. 17.46 is $$ \rho^{ind}(\textbf{r}) = - e[n_0(\mu + e\phi(\textbf{r})) - n_0(\mu)]$$ and then the authors write > In the present case we assume that $\phi$ is small enough for Eq. 17.46 to > be expanded to give in leading order $$\rho^{ind}(\textbf{r}) = -e^2 \frac{\partial n_0}{\partial \mu} \phi(\textbf{r}) \hspace{1cm} \mbox{[Eq. 17.47]}$$ I wonder how to formally exact write this expansion. To understand this, I would like to know   1. What is the independent variable in Eq. 17.46? Is it $\phi$, $\mu$ or $(e\phi + \mu)$?   2. Does it make sense to write it as $$\rho^{ind}(\textbf{r}) = -e \left[ n_0(\mu) + \left . e\phi(\textbf{r})\frac{\partial n_0(e\phi+\mu)}{\partial \mu}\right|_{e\phi=0} - n_0(\mu) \right]$$ But then, if I set $e\phi = 0$, would the middle term not collapse to zero?


title: Quantum entaglement and the arrow of time

text: I have seen several claims to that quantum mechanics is required to explain the arrow of time which I take to mean the macroscopic irreversibility of physical systems. This is presumably to resolve Loschmidt's paradox. A recent example is a recent Simons Foundation article Time’s Arrow Traced to Quantum Source. Another example from is A quantum solution to the arrow-of-time dilemma. There is also an older expository article that contains the following rather strongly worded paragraph: > The application of classical mechanics to the explanation of thermodynamics > resulted in a complex and disgusting mess that explains nothing, though > great efforts were expended along this line. This is not surprising, since > thermodynamics depends sensitively on mechanics at an atomic scale, for > which classical mechanics fails spectacularly. ... with the introduction of > quantum mechanics, it has been possible to understand entropy clearly and > completely ... However, as I understand, coarse grain irreversibility has been been demonstrated for theoretically and by numerical simulations for purely classical idealized systems like "billiards" and "gasses" in 2 and 3 dimensions. I am particularly thinking of Simanyi's results on ergodicity and the numerical investigations of Orban & Bellemans, Levesque & Vertlet and Komatsu & Abe. These would suggest that macroscopic irreversibility comes from the geometrical structure of the vector field on the phase space representing the evolution of the systems. States on entropy decreasing trajectories are unstable and the smallest perturbations result in trajectories that become entropy increasing after time. This means that unless systems can are prepared with infinite precision, then any system prepared is almost certain to be on an entropy increasing trajectory unless it is at equilibrium already. What is required is that the dynamics allow trajectories in phase space to diverge nonlinearly. This would imply that, although there are deep connections between quantum mechanics and irreversibility, quantum mechanics is not strictly required for irreversibility as some articles seem to be claiming. What is still left to explain why the universe is in a far from equilibrium state, but the articles cited at the beginning do not seem to be addressing this aspect either. What am I missing?


title: Does first quantization perturbation theory imply a large scale web of electron entanglement?

text: My question may seem quite esoteric given the title, but I think it's relatively straightforward when explained properly. Imagine a relatively simple situation of 2 hydrogen atoms (numbered 1 and 2), which we treat semi classically (first quantization QM). If we look at each atom individually and ignore the other one, we have a clear cut analytic solution that describes the electron orbital. On the other hand, if we consider the impact that proton 2 has on electron 1 (and vice versa), we need perturbation theory. Correct me if I am mistaken, but I believe we will end up with an oscillating solution whereas the electrons will trade places over time. If this is indeed the case, then these electrons' wave function will be greatly correlated at some point (in particular when it is equally likely for either of them to be orbiting proton 1 or 2). Here then are my questions : 1) Despite the approximate nature of the perturbation theory solutions, is it fair to say that such significant entanglement, or at least some reasonable measure of it, is likely ubiquitous to the exact description of such systems ? Is the back and forth "switch" from proton 1 to proton 2 for electron 1 (and vice versa) also a ubiquitous behavior ? 2) If this is the case, and if the timescale of such oscillations for typical separation size between proton 1 and 2 (say the distance between protons in the air) is much smaller than say the age of the earth's atmosphere, wouldn't it lead to all the electrons of the earth's atmosphere to be entangled in a giant mess resembling some nightmarish correlation web ? If so, shouldn't such entanglements be readily observable ? I limited the case to the earth's atmosphere almost as a gimmick to help visualize my question, though obviously the conclusion (if correct) would apply on global scales. 3) Does a QFT theoretical description substantially change the conclusions arrived at above ? My understanding of QFT is shaky at best, and breaks down completely when I try to apply it to this question.


title: Why can't we do some basic algebra in tensor calculus?

text: I have a very, very stupid question on the basics of tensor calculus. Consider $R_{ij} = 0$. 1)If I expand the ricci tensor $R_{ij}= g^{lm}R_{iljm}=0$. Now, my question is that, **_why_** can't we divide this by either the riemann tensor or the metric tensor to get either, $R_{iljm}= 0$ (which is wrong) and $g^{lm}$ = $0$ (which is meaningless). 2)Also, if we "transpose" a covariant / contravariant tensor to the other side of the equation, does the position of the index change? Or is is it just wrong to transpose a tensor? More specifically, for ARBITRARY tensors, if, $A_{bc} L^{mn}$= $K^{mn}_{bc}$, Then which of the following is correct: $A_{bc}$= $K^{mn}_{bc}L_{mn}$ (The index becomes covariant for L) $L^{mn} = K^{mn}_{bc}A^{bc}$ (The index becomes contravariant for A) Or $L_{mn}= \frac{K_{bc}^{mn}}{A_{bc}}$, And the similar expression for $A_{bc}$. I know this is an idiotic question, and even I get annoyed thinking about it, but please bear with me.


title: Absorption lines in the context of identifying elements in far away celestial objects

text: I understand that absorption lines are used to identify elements but how are individual absorption spectrums identified in the light that is received by a telescope?


title: Black holes attracting each other

text: I was watching the tv series called, through the wormhole, and they mentioned a theory about multiple universes that are separated by a very small distance, like parallel membranes. They mentioned that there is also a theory that says that black holes are in fact connections between these 2 universes and that the matter travels between them (goes into a black hole in this universe, goes out from a white hole on the other) since the gravity is so strong it manages to pull the other universe's membrane close enough for stuff to travel between them. Also I remember another episode mentioned that the big bang could in fact be these two universe membranes colliding into each other. So this got me thinking, wouldn't many black holes attract each other and become a super gigantic point of matter that is so strong that it pulls the other universe strongly enough to touch it in which makes the universe with the black hole finally blow up and expel the matter accumulated there as if it was the big bang? This kind of effect would also account for the way the universe expands (if you visualize it as a membrane that has been pulled inwards, when released it grows uniformly in every direction similar to a drop of water creating ripples. But then i was told black holes would not attract each other because they are drifting away from each other, the speed at which both of them separate from each other is faster than the speed of light which is said to be the speed the gravity force travels at. But wouldn't they still attract other black holes that are not as fast? (even if they are just slowly moving towards each other) and thus decelerating the speed they get away from each other. And these black holes could be affecting other black holes close to each other, and at the end everything is really affecting something else because regardless of the speed they are within something's gravitational pull which in turn will lead to an implosion generating a super massive black hole (like i described before) making another big bang?


title: A question about atomic clocks

text: I have a rather simple question about atomic clocks. I have read that: > _Microwave radiation with a frequency of exactly 9.192.631.770 cycles per > second causes the outermost electron of cesium-133 atom to reverse its spin > direction. An atomic clock uses this phenomenon to tune moicrowaves to this > exact frequency. It then counts 1second for each 9.192.631.770 cycles._ So does that mean that in an interval of 1 second the outermost electron of the cesium-133 atom changes its spin direction 9.192.631.770 times? And let 1 and 2, be the two possible directions of spin. Does one change mean a change from 1 to 2 , or a change 1 to 2 and back to 1?


title: Can we directly measure vectors' quantities?

text: Can we perform some kind of experiment that will give us, for example, the $p_x$, $p_y$ and $p_z$ of a particle in a single measurement? I'm aware that they commute so one measurement will not disturb the others, but I want to know if it is possible to obtain all three components with one single measurement. The real question I'm trying to get my head around is: Can we directly measure anything that is NOT a scalar after all?


title: Charged particle close to a charged black hole - what happens?

text: Let's assume the Reissner–Nordström metric (charged black hole, non-rotating), for simplicity. The black hole is charged with a powerful electric charge. There's a particle nearby, of non-zero mass, let's say an electron, its charge being the same sign like the black hole's charge. The particle's initial speed, relative to the BH, is zero. The particle is close to the event horizon, but still outside of it. The question is - what happens? Are there any combinations of parameters where the particle starts falling in, but stops before being swallowed? Or even repulsed outright? If so, any quantitative, intuitive examples? I'm asking because I know how easily intuition can get deceived by General Relativity, and doing the math involving the R-N metric is probably not feasible for me now. :)


title: Boundary Conditions Invariant Under Conformal Transformations in Electrostatics?

text: in **two dimensional** electrostatics it is assumed that the whole physical system is translationally invariant in one direction. Here, the two- dimensional Laplace equation $$\Delta \phi(x,y) = \frac{\sigma(x,y)}{\epsilon_0}$$ holds in free space and **solutions** can be **mapped on solutions** if space is transformed conformally \- the metric changes from $$g\rightarrow \Omega^2 g\ .$$ This approach has a lot of nice applications like the possibility to calculate the field around a complicated conductor or, in aerodynamics, the flow around an airfoil as discussed earlier. **Aubry et al.** have in a number of papers (see e.g. Interaction between Plasmonic Nanoparticles Revisited with Transformation Optics (PRL) and Plasmonic Light-Harvesting Devices over the Whole Visible Spectrum (Nano Lett.)) used this technique to calculate the eigenfunctions of "crescend" structures and coupled cylinders: ![Plasmonic Light-Harvesting Devices over the Whole Visible Spectrum](http://i.stack.imgur.com/5kWTr.png)(taken from the Nano Lett.) I think this is a very **nice approach** but I am not sure why it can be done in this way. My problem is that as far as I know usually **only Neumann** and **Dirichlet boundary conditions** are **invariant under conformal transformations**. In electrodynamics, however, we have that both $E_t = -d\phi(\mathbf{t})$ (tangential) and $\epsilon\cdot E_n = -\epsilon\cdot d\phi(\mathbf{n})$ (normal) are continuous at a boundary where the permittivity $\epsilon$ changes. In the papers I could not find a discussion of this issue so I might ask here: > ### Are the boundary conditions for the electrostatic potential $\phi$ > invariant under conformal transformations? Thank you in advance   Sincerely


title: Work done by Internal Forces

text: Why is it that internal forces do not change the potential energy of the system? Assume that a man stands on earth with a dumbell. If the dumbell is the system, when it is raised a height h, the work done by the man is +mgh and the work done by the earth is -mgh. If the system is the earth and the dumbell, then the man will increase the potential energy of the system by +mgh. Where does the gravitational force of the earth on the dumbell go?


title: Toroidal magnet in Tonomura et al. demonstration of Aharonov-Bohm effect

text: What does the magnetic field look like in Tonomura's toroidal magnet coated with a superconductor? ![Tonomura's toroidal magnet coated with a superconductor](http://i.stack.imgur.com/Q0e6x.jpg) See this paper to find results showing conclusive proof of the A-B effect and flux quantization. Context: For the A-B effect to work, there needs to be a magnetic flux in the area between the alternate paths which will produce the interference fringe patterns (but not necessarily along the paths themselves). The paper shows fringe patterns due to interference between the center of the toroid and outside the toroid, which means that there needs to be a flux in the plane of the toroid for A-B effect to be observed. Doesn't the superconductor coating quantize the flux penetrating the center of the toroid rather than the in- plane flux?


title: How to find the electric field due to a point charge in 3 dimensions?

text: A point charge with charge $+q$ is situated at $(x,y,z)$. How do I find the electric field at $(p,q,r)$? $E=k\frac{q}{r^2}$, right? So why isn't the electric field, $E=k\frac{q}{(x-p)^2+(y-q)^2+(z-r)^2}$ ??? Question: A charge $q=1\,\mu C$ is placed at a point (1m, 2m,4m). Find the electric field at point P (0m,-4m,3m). I end up with $E$ = 2.38x10$^2$ N/C; the given answer is 1.46x10$^3$ N/C


title: Using heat energy to increase temperature

text: It's been a long time since I studied thermodynamics in college years ago, but I was just wondering this: I know that a large bathtub of 10 C water has more heat energy than a small cup of 20 C water, and that heat naturally flows from objects of higher temperature to objects of lower temperature, but is there any way to use the heat energy of the bathtub to get the temperature of the cup water higher _on a small scale level like this_? I purposefully chose number values that would make it hard to use the heat energy of the bathtub water to do something conventional like drive a turbine to generate electricity to heat the cup water because I'm specifically curious about possibility and efficiencies without huge amounts of energy.


title: Radial Schrodinger equation with inverse power law potential

text: Recently I read a paper about solving radial Schrodinger equation with inverse power law potential. Consider the radial Schrodinger equation(simply set $\mu=\hbar=1$): $$\left(-\frac{1}{2}\Delta+V(\mathbf{r})\right)\psi(\mathbf{r})=E\psi(\mathbf{r}).$$ A well-known substitution gives a one-dimension equation: $$-\frac{1}{2}D^2\phi(r)+\left(V(r)+\frac{1}{2}\frac{l(l+1)}{r^2}\right)\phi(r)=E\phi(r),$$ where $D=\dfrac{d}{dr}$, and $l$ is the azimuthal quantum number. If we only consider the ground state, then $l=0$, so $$-\frac{1}{2}D^2 \phi(r)+V(r)\phi(r)=E\phi(r).$$ We want to find eigenvalue $E$ such that $\phi(0)=\phi(+\infty)=0$. The central potential discussed in the paper is of this form: $$V(r)=\alpha r^{-\beta}.$$ It states (see page 4) that if $\beta>2$ then the potential is repulsive (i.e. $\alpha>0$). My questions are:   1. Is this conclusion(i.e. If $\beta>2$ then we must have $\alpha>0$) generally valid in physics?   2. What would happen if $\alpha<0$ and $\beta>2$? Is there 'ground state' in this condition?   3. What about the condition $\alpha>0$ and $\beta=1,2$? I have tried to solve the equation numerically with $\alpha=1,\beta=1,2$, and the ground state energy in this two conditions seem to be $0$, is my result correct? P.S. When I try to find ground state when $\alpha=-1,\beta=2$, the energy seem to be $-\infty$, which is qualitatively different from $\alpha=-1,\beta=1$.


title: Car horn in the cold

text: I live in Chicago, where it is EXTREMELY cold right now. A couple of mornings ago it was -20C in the morning, and as I was driving through a parking lot a car pulled out of a space right in front of me. I swerved and hit the horn hard to warn the other driver. Fortunately I managed to avoid a collision, however, my car horn did not make a sound. Nothing, nada. As I was driving to work, after the engine and car had warmed up, all of a sudden the horn started working again. The fact that it worked tells me there was not a connection problem or such. The only variable was the temperature, but I can't understand why warm air would vibrate, and cold air wouldn't. Can anyone offer an explanation?


title: The error of the natural logarithm

text: Can anyone explain why the error for $\ln (x)$ (where for $x$ we have $x\pm\Delta x$) is simply said to be $\frac{\Delta x}{x}$? I would very much appreciate a somewhat rigorous rationalization of this step. Additionally, is this the case for other logarithms (e.g. $\log_2(x)$), or how would that be done?


title: How do you find a particular representation for Grassmann numbers?

text: This question is more general in the sense that I want to know how one finds a particular (say matrix) representation for any object. For the case of Grassmann numbers we have from Wikipedia the following representation: > Grassmann numbers can always be represented by matrices. Consider, for > example, the Grassmann algebra generated by two Grassmann numbers $\theta_1$ > and $\theta_2$. These Grassmann numbers can be represented by 4×4 matrices: > > $$\theta_1 = \begin{bmatrix} 0 & 0 & 0 & 0\\\ 1 & 0 & 0 & 0\\\ 0 & 0 & 0 & > 0\\\ 0 & 0 & 1 & 0\\\ \end{bmatrix}\qquad \theta_2 = \begin{bmatrix} > 0&0&0&0\\\ 0&0&0&0\\\ 1&0&0&0\\\ 0&-1&0&0\\\ \end{bmatrix}\qquad > \theta_1\theta_2 = -\theta_2\theta_1 = \begin{bmatrix} 0&0&0&0\\\ 0&0&0&0\\\ > 0&0&0&0\\\ 1&0&0&0\\\ \end{bmatrix}. $$ How do one find these matrices? Do you guess them or is there a procedure? What about finding differenct matrix-representations for **Dirac $\gamma$-matrices? How do you find them?**


title: Topologically distinct Feynman diagrams

text: Are these two diagrams topologically distinct? ![enter image description here](http://i.stack.imgur.com/wIdP7.jpg) I consider $\phi^4$ theory and use MS-scheme. A vertex corresponding to counterterm $-\imath \frac{m^2 \lambda}{32 \pi^2 \epsilon}$ is denoted by cross and $-\imath \lambda$ vertex is denoted by bold point.


title: When was the cosmic background radiation in the visible spectrum?

text: From what I understand about the Cosmic Microwave Background Radiation is that it was from the big bang, and since space has streched and become bigger since then the wavelength of the CMBR has increased over time, which means that when the universe stopped being opaque it wasn't microwaves but gamma rays, that must mean that it went from gamma rays to x-rays to ultra violet to visibile light (blue then all the way to red) then infra red and now microwave. So the question is simply, at what point in time could someone have looked up in the sky and seen light from all directions? And how intense was this light?


title: Experiments which show the phenomenon of closed electric field lines

text: In german physics introductory texts often the following experiment is cited which "shows" that a changing magnetic field can induce an electric field with has closed field lines. Here is a picture of the setup: ![](http://i.stack.imgur.com/iUV4v.jpg) A high frequency ac-voltage is applied to a coil around a glass sphere which is filled with a gas. Then a circular lighting effekt occurs as shown on the image. I have some questions about this:   1. How is this experiment called in english? Do you have references of textbooks or papers which discuss this in detail?   2. Why is it reasonable to say that this experiment "shows" that there "must" be a circular electric field? Let general epistomology aside, I think that the rough theory about this is, that the electrons wiggles back and forth because of the changing induced electric field. Since the magnetic field changes in sign the mean lorentz force due to the magnetic field is zero, so the lorentz force doesn't matter here. But if you don't know that the field lines are circular, especially that the field vectors are tangential to the circle, how conculde this from the experiment? For example it could be conceivable, that the field changes from radially invards to radially outwards and we get a radial wiggling of the charges which produce the lighting!? What (and what not) can one reasonable "conclude" from this experiment about the nature of the induced electric field and why?   3. How can one describe this experiment mathematically in detail?   4. Are there other, perhaps better or clearer experiments which demonstrate the existance of closed electric field lines produced by induction?


title: Structure of Hilbert Space in Bosonic String Theory

text: My question is about the canonical quantization of free bosonic string theory as described by Green, Schwarz & Witten. There they use spurious states to calculate a value for the ambiguity parameter $a$ of the Virasoro Generator $L_0$ and the dimension $D$ of the Spacetime. They argue as follows: Since there is a regime of values of $a,D$ where the physical Hilbert Space has negative norm states and there has to be a regime of these values where the Hilbertspace is positive definite there will be a border between these two regimes on which we can find an increase in the number of zero normed states. So my question is: How can we know that the Hilbertspace really has such a simple structure? Couldn't there be various borders which then meant that the regime for which the physical Hilbertspace really is physical is not simply connected?


title: How to calculate how many can a stick hold?

text: I got two boxes, and one stick, two boxes are high enough for a human body. I put one stick between two boxes, like this: ![enter image description here](http://i.stack.imgur.com/LukcF.png) The boxes are yellow, and the stick is brown. So, I would like to put something on the stick, but how can I know how much things I can put on the stick? Except from trial and error, is there any scientistic way to calculate it? Thanks.


title: Proca Lagrangian manipulation

text: How can I show that the Lagrangian density $$\mathcal{L} = -\frac{1}{2}\partial_\alpha \varphi_\beta \partial^\alpha \varphi^\beta + \frac{1}{2} \partial_\alpha \varphi^\alpha \partial_\beta \varphi^\beta + \frac{\mu^2}{2}\varphi_\alpha \varphi^\alpha$$ for the real vector field $\varphi^\alpha$ leads to the field equations $$[g_{\alpha\beta}(\square+ \mu^2)-\partial_\alpha\partial_\beta]\varphi^\beta=0$$ I have done through lot of work, but I was wondering if there are some tricks or efficient ways to do it? And second question, how can I show that the field satisfies the Lorenz condition $\partial_\alpha \varphi^\alpha = 0$? Source: Mandl Shaw QFT problem 2.3


title: Since Synchronous machines are constant speed machines how are their speeds varying with variation in load?

text: I understand that the power output depends on torque angle. So how does an increase in torque angle cause the rotor speed to reduce? Aren't the synchronous machines supposed to be running at constant speed?


title: What's the difference between helicity and chirality?

text: When a particle spins in the same direction as its momentum, it has right helicity, and left helicity otherwise. Neutrinos, however, have some kind of inherent helicity called chirality. But they can have either helicity. How is chirality different from helicity?