title: A complicated question about $E=mc^2$

text: I know this is a little outside the normal question and there may not be a direct answer, but it is an interesting thought experiment. Starting with a supermassive black hole, if you were able to feed in mass without limit, at what point (total mass) would gravity tear apart all the particles in the singularity to the point that all the mass would be converted to energy? I am assuming that matter is considered energy when it is broken down to its simple building blocks.


title: How would I explain Ohm's Law in terms of Electrical Fields and Force?

text: In terms of current, resistance, and voltage, it's easy: Ohm's Law is the relationship between current, voltage, and resistance of a circuit. Boom, simple as that. How could I put this in terms of $E$ and $F$? I can sort of see a way to do it by relating the formulas $E=F/q$ and $I=q/t$ to Ohm's Law, $V=IR$, but I'm not entirely sure how I could explain this in words.


title: What are brain waves?

text: I watched a documentary showing that humans could alter the results of an experiment by thinking about it, and the data these people collected supported their hypothesis. And the reason these people gave was that brain waves were interfering with the experiment. The experiment involved the tossing of a coin done by computers. People first registered what do they want to happen more heads or more tails and surprisingly the results of the tossed matched their needs.


title: What is the current science on entropic gravity?

text: In the past couple of years there has been some buzz about 'entropic gravity', including here on stackexchange in Jan 2011. Many say that there is no way that gravity can be entropic, while others think it is an interesting idea and bears more research. Has a consensus been reached on entropic gravity? Is the discussion exactly the same as it was a year ago when the previous question was asked?


title: If : V(Phi) : is nonlocal in space, does that mean interacting quantum field theory is nonlocal?

text: Free field theories are definitely local in . In the interaction picture, we can decompose the fields into creation operator modes and annihilation operator modes. The product of operators can be regulated to finiteness by Wick ordering. In Wick ordering, all creation operators are moved to the left of all annihilation operators. To introduce interactions, we can add a Wick ordered interaction term $:V[\Phi(x)]:$ to the interaction Hamiltonian, but this term is nonlocal in space. To Wick order, we have to Fourier transform to momentum space, subtract some zero point terms, and then Fourier transform back to position space. The end result when expressed in terms of the original spatial fields is a nonlocal convolution with a spatially extended kernel. Let me clarify a bit. For potentials which are quadratic in the fields, the zero point subtraction is just a c-number, and this is trivially local. But for quartic couplings, for example, the zero point subtraction we get from Wick ordering is still quadratic in the fields with nonlocal point splitting. Does that mean interacting quantum field theory is nonlocal?


title: Lagrangian to Hamiltonian in Quantum Field Theory

text: 1. While deriving Hamiltonian from Lagrangian density, we use the formula $$\mathcal{H} ~=~ \pi \dot{\phi} - \mathcal{L}.$$ But since we are considering space and time as parameters, why the formula $$\mathcal{H} ~=~ \pi_{\mu}\partial^{\mu} \phi - \mathcal{L}$$ is not used? Is there any particular book/lecture notes dealing with these kind of issues in theoretical physics, I would love to know them?


title: Critical temperature difference between Ising and XY model

text: The following formula gives the critical coupling (more precisely the ratio of the spin-spin coupling over the temperature) for $O(n)$ models on a triangular lattice: $$\text{e}^{-2K}=\frac{1}{\sqrt{2+\sqrt{2-n}}}$$ with $K=\beta J$ Numerically, it says that: Ising model (n = 1) has $K \approx 0.27$ XY model (n=2) has $K \approx 0.17$ Thus, the critical temperature for the XY model is higher than the Ising model. I've been thinking about it but I can't come out with a reason of why allowing the order parameter to take continuous values means that we need to go higher in temperature to destroy order. Is there a (semi) intuitive reason for that?


title: Number of unique 2-electron integrals

text: Consider 2-electron integrals over real basis functions of the form $$(\mu\nu|\lambda\sigma) = \int d\vec{r}_{1}d\vec{r}_{2} \phi_{\mu}(\vec{r}_{1}) \phi_{\nu}(\vec{r}_{1}) r_{12}^{-1} \phi_{\lambda}(\vec{r}_{2}) \phi_{\sigma}(\vec{r}_{2})$$ I am told that for a basis set of size K=100, there are 12,753,775 unique 2-electron integrals of this form. Symmetry considerations mean that we have less than $K^{4}$ unique integrals, since we can exchange electrons and also exchange the basis functions for each electron without changing the value of the integral. How could one work out the number of unique integrals? My method is: Find the number of unique integrals of the forms $(\mu\nu|\lambda\sigma)$, $(\mu\mu|\lambda\sigma)$, $(\mu\mu|\nu\nu)$ and $(\mu\mu|\mu\mu)$ (where in these integrals, each index is unique unless repeated) and sum these together. My working gives the wrong answer, though: $$\frac{4!}{8}{100 \choose 4}+\frac{3!}{4}{100 \choose 3}+\frac{2!}{2}{100 \choose 2}+1!{100 \choose 1} = 12,011,275$$ My rationale is this: for the integral form $(\mu\nu|\lambda\sigma)$, there are ${100 \choose 4}$ unique unordered combinations of basis functions. There are $4!$ ways of arranging these unique basis functions. We can exchange the electrons, basis functions on electron 1 and basis functions on electron 2 without changing the value of the integral, thus halving the number of unique integrals 3 times ($\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}=\frac{1}{8}$). Therefore the number of unique integrals of form $(\mu\nu|\lambda\sigma)$ is $\frac{4!}{8}{100 \choose 4}$. Where am I going wrong?


title: how is the reference for the star maps in google determined?

text: If I go to google sky - http://www.google.com/sky/ Ihere is a longitude and latitude measure at the bottom of the page. Iow are these coordinates determined and who sets it? Do those coordinates move at all over time?


title: Dual conformal symmetry and spin networks in ABJM

text: In this question, I would love to hear some independent opinions on an issue I asked Juan Maldacena, Nathan Berkovits, Dan Jafferis, and others, but all the physicists may be missing something. The question has 2 main parts:   1. Is there a dual superconformal symmetry in 2+1-dimensional ABJM theories? In what limit does it apply? (For $N=4$ gauge theory, it's the planar limit, but what is the counterpart in 2+1 dimensions?)   2. Are there spin network states in this extension of Chern-Simons theory? Can their expectation values be related to scattering amplitudes in the ABJM theory, much like the piecewise null Wilson loops are related to scattering amplitudes in $N=4$ gauge theory? Concerning the first question, there's some confusion what $N$ should go to infinity and what should be kept fixed, especially when it comes to the scaling of the level $k$ in the limit. In string theory, the dual superconformal symmetry (and the Yangian) is very constraining - but that's because it still constrains a lot of the "stringy stuff". Should one necessarily go to the limit with the stringy stuff - e.g. a high- level, a $CP^3$ compactification of type IIA - to see some dual superconformal symmetry? Or is there a dual superconformal symmetry that only works in the planar limit which simply means that only the (leading?) SUGRA amplitudes are constrained? Concerning the second question, the ABJM theory secretly contains membranes, and their discretization - analogous to the piecewise null Wilson loops - could involve faces (pieces of membranes?) surrounded by open Wilson lines. Are they gauge-invariant?


title: Does increasing diameter of a water pipe increase speed or volume?

text: Let's say water is flowing from an arbitrary `Source` to some arbitrary `Sink` via a small pipe: ![enter image description here](http://i.stack.imgur.com/Mno0k.png) I'm curious as to what actually happens when we change the diameter of the pipe: ![enter image description here](http://i.stack.imgur.com/WvFrq.png) If all other things are held constant, does the **speed** of the water increase through the pipe increase, or does only the **volume** increase? BTW, this is not homework I'm a 30-yr old software developer/homeowner trying to figure out a problem with my plumbing.


title: What does library number 539 mean in physics books?

text: Every book I borrow have have this number in them. Even my library categorizes them with this number. Is it a universal particle physics book number?


title: Prove EM Waves Are Transverse In Nature

text: Why we say that EM waves are transverse in nature? I have seen some proofs regarding my question but they all calculate flux through imaginary cube. Here is My REAL problem that I can't here imagine infinitesimal area for calculating flux because em line of force will intersect (perpendicular or not) surface at only one point so $E.ds$ will be zero so even flux through one surface of cube will always be zero. I am Bit Confused. I DON'T KNOW VECTOR CALCULUS BUT KNOW CALCULUS.


title: Is the plane wave model always valid in reflection and transmission?

text: my question is related to another one I asked, but I foolishly made that question about several things (experiment, computation, theory) at once so it was confused. I was talking to my boss about theoretically modeling the reflection and transmission of incident light on a thin film, like this (warning: PDF). Essentially they use a plane wave model with propagation matrices through "slab" (thin film) and "matching matrices" to match the fields across boundaries. In each region (except the final transmitted region) there are waves going in both directions because both are solutions to MW's equations. That seems to be pretty standard and straightforward to me. There's no inherent length scales in it, so intuitively it would seem to me like it'll work at all dimensions. However, he took one look at it and said that it's wrong to use plane waves here. I didn't entirely understand it at the time (hence, why I'm asking here), but I think he was saying:   * A plane wave is just one part of the solution of dipole radiation, the far field part (true, but why is this important here? Is it because the material is really just dipoles being oscillated and re-emitting?)   * He also mentioned something about the "amplitude not being well defined" I think, if the wave is decaying very quickly. This confused me because I thought if you have a notable decay index, you just find $n = n_r + i n_i \rightarrow k = k_r + i k_i \rightarrow E \propto e^{ikz} = e^{-k_i z}e^{ik_r z}$ and then your amplitude is determined by the exponential decay term at any point. Can anyone shed some light on this? Thank you!


title: Power counting with a cutoff

text: In Effective Field Theory video lectures found here, the professor explained power counting in effective field theories and the difficulties of power counting associated with loop diagrams. He then mentions that introducing a cutoff ($\Lambda_{UV}$) to regulate our divergences does not preserve power counting due to the new scale that we are introducing. To see this he uses four-fermi theory with the diagram, $\hspace{6cm}$ ![enter image description here](http://i.stack.imgur.com/VwG04.png) We do our power counting (i.e., Taylor expansions) in powers of $m^2/M^2$ and then go on to consider the mass correction through, $\hspace{6cm}$ ![enter image description here](http://i.stack.imgur.com/xkHEw.png) Using a cutoff this gives a mass correction, \begin{align} a\frac{ m }{ M ^2 } \int _0^{\Lambda_{UV}}\frac{ \,d^4k _E }{ (2\pi)^4} \frac{1}{ k _E ^2 + m ^2 } & = a\frac{ m }{ ( 4 \pi ) ^2 } \left[ \frac{ \Lambda _{ UV } ^2 }{ M ^2 } + \frac{ m ^2 }{ M ^2 } \log \frac{ m ^2 }{ \Lambda _{ UV } ^2 } - \frac{ m ^4 }{ M ^2 \Lambda _{ UV } ^2 } + ... \right] \end{align} If I understand correctly this breaks the power counting because even if $\Lambda_{UV} \sim M$, the first term is an order 1 correction since its not proportional to $m^2/M^2$. So far so good. However, then the professor says that you can still use power counting with a cutoff if you fix the power counting order by order and that this can be done by introducing an intermediate scale, $\Lambda$. But I don't how this fixes anything... With an intermediate scale ($\Lambda$) we have, \begin{equation} a\frac{m}{M^2}\int _{ \Lambda } ^{ \Lambda _{ UV }} \frac{ \,d^4k _E }{ (2\pi)^4 } \frac{1}{ k _E ^2 + m ^2 } = \frac{a\,m}{ (4\pi)^2M^2 } \left\\{ \left(\Lambda ^2 + m ^2 \log \frac{ m ^2 }{ \Lambda ^2 + m ^2 } + ... \right) + \left( \Lambda _{ UV } ^2 - \Lambda + m ^2 \log \frac{ \Lambda ^2 + m ^2 }{ \Lambda ^2 _{ UV }} \right) \right\\} \end{equation} but how does this fix anything? For more context see my lecture notes here under Effective Field Theory (starts around equation 4.6)


title: What is Photoproduction

text: I wonder what photoproduction means in the context of pion decay and vector meson dominance? What is the reaction formula, Feynman diagram for such a photoproduction thing? Is it simply a reaction where the final state is a bunch of photons?


title: Why can you have shiny black objects?

text: Knowing black is supposed to be the "color" (I don't want to get into the color/hue/shade debate, please) that absorbs light. how does one manage to have shiny black surfaces? I know about "gloss black" versus "matte black" finishes, but shouldn't the light passing through the gloss (if they didn't pass through the gloss, you wouldn't see the black, right?) be absorbed by the underlying black object? Then there are black gemstones like jet and opal. How do black objects shine?


title: experimental technique for measuring temperature of an ant

text: I am taking a course on thermodynamics. I have a question from my text(halliday & resnick,physics-1). They asked me to measure temperature of an ant or an insect or a small body,like a small robot. If I build a thin thermometer then it is probable that surface tension would have greater influence than thermal expansion. Then How can I measure the temperature of an ant?


title: A Short way to show Conservation of Quantum Laplace–Runge–Lenz Vector?

text: I had been asked to prove the conservation of Quantum Laplace–Runge–Lenz Vector: $$\hat{A}=\frac{i}{\hbar}\left[\hat{p},\,\frac{1}{2}\hat{L}^{2}-k\left|\hat{r}\right|\right]=\frac{1}{2}\left(\hat{p}\times\hat{L}-\hat{L}\times\hat{p}\right)+k\frac{\hat{r}}{\left|\hat{r}\right|}.$$ But unlike the case of classical mechanics, even after a lot of tries I found no beautiful/short way to show it's commutation with Hamiltonian, the calculation are quit lengthy opposite to my first expectations, but something tells me that there is a shorter way, so dose anybody managed to find a simple way or trick to show that $$\left[\hat{H},\hat{A}\right]~=~0?$$


title: How is the Dirac adjoint generalized?

text: I am wondering how one can generalize the Dirac adjoint to flat "spacetimes" of arbitrary dimension and signature. To be more specific, a standard situation would be to consider 4 dimensional Minkowski space (signature $-+++$) and the 4 (complex) dimensional Dirac representation of $\text{spin}(3,1)$ via $\gamma^a= \left( \begin{array}{ccc} 0 & \sigma^a \\\ \bar{\sigma}^a & 0 \\\ \end{array} \right)$. Then we would define the Dirac adjoint as $\bar{\psi}=\psi^\dagger \gamma^0$. Let us denote the representation by $D:\text{spin}(3,1) \rightarrow \text{Lin}(\bf{C^4})$. The fundamental property of the Dirac adjoint of $\psi$ is that under a Lorentz transformation, $\bar{\psi}$ transforms to $\bar{\psi} D^{-1}$ which ensures, for instance, that $\bar{\psi} \psi$ is a Lorentz scalar. So now to my concern. It seems to me that this construction of $\bar{\psi}$ is super specific to 1+3 dimensions and maybe also to the chiral form of $\gamma$. What if we consider a Diracy representation of $\text{spin}(n,m)$? That is, we take a collection of $d$ dimensional matrices $\left\\{ \gamma^a \mid 1\leq a \leq n+m \right\\}$ such that $\left\\{ \gamma^a,\gamma^b \right\\} = 2 \eta^{ab}$ ($\eta$ is the diagonal matrix with $n$ 1's and $m$ $-1$'s) and use these to construct a $D$ dimensional representation. Then, if $\psi \in \bf{C}^D$, I'd like to find a (hopefully real) function $\psi \mapsto \bar{\psi} \psi$ that is invariant under the action we just (kind of) constructed.


title: How does a vacuum oven work?

text: I am doing some research on vacuum oven, but I cannot find out any website/book to tell me the theory. To my knowledge, it works better than regular oven. However, I have no idea why it is. Please help


title: Why does it require such little energy to create the fastest thing in the universe?

text: I have noticed when I turn on the light switch in my house light comes from the bulb. How is this light created?(process occurring in the bulb) and why is this small amount of electricity enough to create light?


title: Why Counterclockwise starts from East?

text: In vector lessons, we have counterclockwise angle of rotation of the vector starts from East. "The direction of a vector is often expressed as a counterclockwise angle of rotation of the vector about its "tail" from due East" http://www.physicsclassroom.com/class/vectors/u3l1a.cfm Why doesn't start from north or south or any other direction? Or is it just one of common rule between physicist.


title: Kubo Formula for Quantum Hall Effect

text: I'm trying to understand the Kubo Formula for the electrical conductivity in the context of the Quantum Hall Effect. My problem is that several papers, for instance the famous TKNN (1982) paper, or an elaboration by Kohmoto (1984), write the diagonal entries of the conductivity tensor in the form $$ \sigma_{xy}(\omega \to 0) = \frac{ie^2}{\hbar} \sum_{E^a < E_F < E^b} \frac{\langle a|v_x|b \rangle \langle b|v_y|a \rangle - \langle a|v_y|b \rangle \langle b|v_x|a \rangle}{(E^a - E^b)^2} .$$ This is the static limit $\omega\to 0$ and low temperature $T\to 0$. The sum goes over all eigenstates $|a\rangle$ and $|b\rangle$ of the single-particle Hamiltonian. $E_F$ is the Fermi energy. $v_x$ and $v_y$ are the single- particle velocity operators. However, these papers don't derive this equation, which is unfortunate because the Kubo formula is usually not presented in this form. I have found (and succeeded in rederiving) the following variation instead $$ \sigma_{xy}(\omega+i\eta) = \frac{-ie^2}{V(\omega + i\eta)} \sum_{a,b} f(E^a) \left( \frac{\langle a|v_x|b \rangle \langle b|v_y|a \rangle}{\hbar\omega + i\eta + E^a - E^b} + \frac{\langle a|v_y|b \rangle \langle b|v_x|a \rangle}{-\hbar\omega - i\eta + E^a - E^b} \right).$$ This is formula (13.37) from Ashcroft, Mermin, though they don't actually prove it. $f(E)$ is the Fermi distribution. A nice derivation is given in Czycholl (german). Now, my question is, obviously > How to derive the first formula from the second? I can see that the first equation arises as the linear term when writing the sum as a power series in $\omega$, but why doesn't the constant term diverge?


title: Do generalized Pauli Operators generate SU(n)?

text: A commonly used generalization of Pauli Operators is the "clock" and "shift" operators summarized here: http://en.wikipedia.org/wiki/Generalizations_of_Pauli_matrices Pauli Operators are generators of SU(2). Are these generalized Pauli Operators generators of SU(n)?


title: Work and energy problem

text: I am trying to solve the problem (see image), my concern is about the 2 factor, can anyone please help me understand this. ![enter image description here](http://i.stack.imgur.com/nYxHc.png)


title: Everyday example of diffusion unobscured by advection, wetting etc

text: Diffusion is an important concept in elementary science education, especially because it supports (or seems to support) the notion of matter consisting of very small everyday particles (as opposed to a continuous substance). A colleague of mine is writing an introductory science text for 8th graders. She is struggling to find a good way to introduce the reader to diffusion using an everyday example. We do not like the canonical example of propagation of kitchen smells or perfume, because it is not obvious how to rule out spreading by air convection (i.e. advection of the smelly substance). So, **what's the best example to explain diffusion to an 8th grader?**


title: Do low frequency sounds really carry longer distances?

text: It is a common belief that low frequencies travel longer distances. Indeed, the bass is really what you hear when the neighbor plays his HiFi loud (Woom Woom). Try asking people around, a lot of them believe that low sounds carry longer distances. But my experience isn't as straightforward. In particular:   * When I stand near someone who's listening loud music in headphones, it is the high pitched sounds that I hear (tchts tchts), not the bass.   * When I sit next to an _unamplified_ turntable (the disc is spinning but the volume is turned off), I hear high pitched sounds (tchts tchts), not the bass. So with very weak sounds, high frequencies seem to travel further? This makes me think that perhaps low frequencies do not carry longer distances, but the very high amplitude of the bass in my neighbor's speakers compensates for that. Perhaps also the low frequencies resonate with the walls of the building? Probably also the medium the sound travels through makes a difference? Or perhaps high frequencies are reflected more by walls than low frequencies? I found this rather cute high school experiment online, which seems to conclude that low and high frequencies travel as far, but aren't there laws that physicist wrote centuries ago about this?


title: DC current in ideal conductor and skin effect?

text: I know that the skin depth derived for AC current goes as $$\delta \propto \sqrt{\frac{1}{\omega \sigma}}$$ where $\omega$ is the angular frequency of the field and $\sigma$ is the conductivty. Now: for a conductor with **infinite conductivity** but with **DC current** , we find a $0\times\infty $ in the denominator and I do not know how to treat it. That said: 1) In a conductor with infinite conductivity and DC or AC current, does the current only flow on the surface? 2) What is keeping charge inside a real conductor? Shouldn't they always try to eliminate the electric field inside?


title: Relative perimittivity of conductor?

text: Why is the relative permittivity $\epsilon_r$ infinite for a conductor? Where does it come from mathematically and does it make physical sense?


title: Find a central force given the orbit

text: I've been trying to solve the following problem for a long time. Let's consider a particle of mass $m$ in $\mathbb{R}^3$ with polar coordinates $(r,\theta,\phi)$. The particle moves on the orbit described by the equation $r=2a\sin\theta,\,\,a>0$, with velocity $|\mathbf{v}|=k/\sin^2\theta,\,\,k>0$. The task is to show that the force is central and to determine such a force. My idea has been to use the two equations given to prove the two components \begin{equation} F_\theta=m\left( r\,\ddot\theta + 2\dot{r}\,\dot\theta - r\,\dot\varphi^2\sin\theta\cos\theta \right)\\\ F_\phi=m\left( r\ddot\varphi\,\sin\theta + 2\dot{r}\,\dot\varphi\,\sin\theta + 2 r\,\dot\theta\,\dot\varphi\,\cos\theta \right) \end{equation} to vanish; unfortunately, I wasn't able to get to any interesting result. Can anybody give me some help on this?


title: Charging one thing with static electricity by using battery

text: How can I make one thing to get statically charged by using batteries but not friction ? For example charge a rod so it can attract shredding ?


title: length, width, and time

text: is it possible to have length, width and time. A 2D or 1D object with the 4th dimension (although then there would not be 4 dimensions so would it b 3D?) .


title: Entanglement entropy for U(1) lattice gauge theory

text: Can someone please let me know if there is some reference for the calculation of entanglement entropy of U(1) lattice gauge theory? I have seen a few references where Z2 lattice gauge theory has been dealt with. Also these references suggest that since in a lattice gauge theory the degrees of freedom live on links hence the Hilbert space of states can't be decomposed into a direct product of states belonging entirely to a region, say A and its complement B because of links which cross the boundary of the region of interest. My question is can I not gauge fix the link variables crossing the boundary of the region of interest so that link variables belonging solely to the complement region can be traced over unambiguously to obtain the reduced density matrix. Also how do I write down a gauge invariant ground state wave function for a U(1) lattice gauge theory in order to define the density matrix?


title: Raindrops on a fence. Why a regular pattern?

text: I observed a fence composed of plastic-coated metal wires running vertically and horizontally at intervals of around 3 cm. After it had rained, drops of water remained on the horizontal wires. I was surprised to see that the drops on each horizontal wire were situated just below the drop on the wire above. The drops seemed to form vertical columns. The fence is similar to that shown here. What forces are responsible for this unusual regularity?


title: Why are four-vectors needed in the Dirac equation, when there are 4 linearly independent 2D matrices?

text: I was taught that for the Dirac-equation to "work", you need matrices of the following form:   * $Tr(\alpha^i) = 0$.   * Eigenvalues +1 or -1   * 2 previous points together: equal number of negative and positive eigenvalues, thus even dimension.   * $\alpha^i$ and $\beta$ meet the anti-commutator relation. My Advanced QM professor told me convincingly that there are only 3 linearly independent 2D matrices, the Pauli matrices, which can be used in 2D. That's one short, so you'll need at least 4D matrices. Now my QFT professor, who excells at confusing me (and not only me, mind you), stated that it is perfectly possible to use only 2D matrices, by extending the Pauli matrix set with another one. I think he's wrong, but he seems too old (and "wise") to be wrong, and I didn't remember all conditions above at the time, so I didn't want to argue without arguments. Is it possible to 2D-ify the Dirac equation? (and not only in a 2D system like graphene, but in general) Thanks


title: Analytically transform a spherical wave into a planar one

text: Is it possible to transform a light spherical wave front or another wave front with a known behaviour into a planar one? Say I have the position of a point light source, and I can approximate it by an isotropic spherical model (just as an example), what kind of operations do I need to do to make it a planar wave. More specifically, if I take a plane perpendicular to the light axis from an unknown distance to the light, I want to simulate a constant lighting on that plane. I know a spherical wave can be transformed to a planar (or almost planar) using different optics, however, I am not sure how/if this can be done analitically. Is it at all possible?


title: The bar tender says we don't serve Tachyons around here?

text: ![enter image description here](http://i.stack.imgur.com/iEOWe.jpg) What's the physics behind this joke?


title: Looking for a good introductory-level review of pseudopotential methods

text: I'm looking for a good introductory-level review of pseudopotential methods. In particular, I'd like to understand how the self-consistent pseudopotential methods work.


title: Stochastic processes and wavefunction collapse

text: Some time ago I had an idea that, as the unitary evolution of the wavefunction is described by a deterministic equation (PDE, simply), could be the collapse of it be described by some kind of a stochastic differential equation? Was this question posed before (I suppose yes...), and what kind of an SDE could it be?


title: How much lead is there in the universe?

text: I know there is around 10^80 atoms in the observable universe, but is there any estimate the amount of lead in the universe (within a couple magnitudes of 10)?


title: Is it possible to build up holography in a closed manifold, i.e., in a manifold with a mathematical boundary?

text: I was wondering about the AdS/CFT correspondence basics. It is constructed on the idea of conformal compactification, in which a open manifold $M$ is homeomorphic related to a closed one $N$ through a conformal transformation of both metrics. The open manifold acquires a conformal boundary in which the conjecture is stabilized. If one thinks on the Index Theorem, a single issue appears: a pseudo differential operator (e.o.m) on the topological boundary does not have a well defined index, so would not be possible to extend the idea of holography to closed compact spaces?


title: Normalizable wave functions?

text: 1. How can I test whether a wave function is normalizable?   2. If you apply an operator to a wave function, sometimes the result will not be normalizable. But how can I find these wave functions that do not correspond to normalizable eigenstates of this operator?   3. The Hamilton operator for the harmonic oscillator, I am told, has both a discrete and a continuous spectrum. The discrete spectrum is the eigenvalues, but how do I find the continuous spectrum? It's relevant because I am also told these are the non normalizable values of the spectrum.


title: Global Chern-Simons forms and topological gauge theories

text: I am reading the classic Dijkgraaf and Witten paper on topological gauge theories and something struck me that I didn't understand. For a trivial bundle $E$ on smooth 3-manifold $M$ with compact gauge group $G$ the Chern- Simons form is of course $$S(A)=\frac{k}{8\pi^2}\int Tr (A\wedge dA + \frac{2}{3}A\wedge A\wedge A).$$ The confusing sentence is the following: "If the bundle $E$ is not trivial, the formula for the action does not make sense, since a connection on a non- trivial bundle cannot be represented by a Lie algebra valued 1-form as in that formula." My concern is that this is wrong - of course we can represent a connection in a non-trivial bundle with a Lie-algebra valued 1-form! In fact, some texts define the connection in exactly that way (my trusty Choquet-Bruhat and DeWitt-Morette has that as one of their three equivalent definitions for a connection)- locally, a connection is represented as a linear map $\omega_p:T_p(E)\to \mathfrak{g}$. So are we supposed to read the "as in that formula" to mean "globally"? Since of course, if the bundle is not trivial there isn't a _global_ Lie-algebra valued 1-form that we can use for the Chern-Simons action. So that seems like a fine argument, but in QFT we are _always_ writing down actions with sections of bundles (fields $\phi(x)$, say) which are not trivial - we just make sure that the action is invariant under the gauge transformations. So, what is going on with that statement? Is it really saying we cannot represent a connection in such a way, or is it just stating the action is now local, and the rest of the paper is going to go on and tell me how to fix this with an element of $H^3(BG,\mathbb{R})$, etc.?


title: What would it take for a physical phenomenon to be telekinetic?

text: I've just watched an episode by MinutePhysics called "Real World Telekinesis". In it, Neil Turok (I wonder if that is his actual name; I remember playing a game called "Turok: Dinosaur Hunter" on Nintendo 64) explains that before Faraday and Maxwell, there was a real problem describing things such as magnets affecting each other or the sun warming the earth without resolving to telekinetic explanations. He then explains that when Faraday and Maxwell introduced the concept of fields, and that changes in this field traveled at the speed of light, thereby dissolving the problem that these phenomena would comprise "spooky action at a distance" explanations. ![Look! A Field!](http://i.stack.imgur.com/IHz7I.png) However, this got me thinking. What would it have taken for these phenomena to _actually_ be telekenetic? Would it be if the electromagnetic wave traveled to all points in space instantly (and was not bounded of the speed of light)? Would it be if only certain points were affected, but not intermediate points? I guess no one, before Faraday and Maxwell, believed that if the Moon got in between the earth and the sun, it wouldn't be heated up. **So, in physics, what would a definition of something telekinetic look like?**


title: Path integral on matrix model

text: I was looking at a 0-dimensional matrix model, where the variables are $N\cdot N$ Hermitean matrices. It had a gauge symmetry, e.g. $U(N)$. And in the path integral, the Faddeev-Popov trick was used. So instead of integrating over the set of matrices, one was reduced to integrating over the $N$ (real) eigenvalues. Or, I tried counting the number of degrees of freedom and I as puzzled:   1. N by N Hermitean matrices have $N^2$ real independent variables ($N(N-1)$ values for the triangular part (non counting the diagonal ones), plus _$N$_ real variables for the diagonal ones, giving us a total of $N^2$   2. $U(N)$ has real dimension $N^2$, classic result from group theory   3. At the end, I arrive at $N^2 - N^2 = 0$ variables instead of $N$ real eigenvalues. I tried seeing it from another point but I can't derive what I'd like to. What I am thinking/doing wrong?


title: Derivation of Schrodinger equation for a system with position dependent effective mass

text: How to derive the Schrodinger equation for a system with position dependent effective mass? For example, I encountered this equation when I first studied semiconductor hetero-structures. All the books that I referred take the equation as the starting point without at least giving an insight into the form of the Schrodinger equation which reads as $$\big[-\frac{\hbar^2}{2}\nabla \frac{1}{m^*}\nabla + U \big]\Psi ~=~ E \Psi. $$ I feel that it has something to do with conserving the **probability current** and should use the **continuity equation** , but I am not sure.


title: Does the black hole only increase?

text: I don't understand this: > _The more mass a Black Hole has, the more gravity it create! The more > gravity it has, the more mass it can get!_ With this lines, someday, all the universe will be a black hole. Is this correct?


title: Simultaneous diagonalization of Hamiltonian and momentum operator

text: I'm looking at a translationally invariant problem with 3 atoms arranged in a circle each with one valence electron capable of tunelling to either of its two neighbors. With a tunelling rate of $-|A|/\hbar$, we have the Hamiltonian $$H = \begin{pmatrix} E_a & \- |A| & \- |A|\\\ \- |A| & E_a & \- |A|\\\ \- |A| & \- |A| & E_a \end{pmatrix}$$ which can be shifted by $+|A|$: $$\begin{pmatrix} E_a + |A| & 0 & 0\\\ 0 & E_a + |A| & 0\\\ 0 & 0 & E_a + |A| \end{pmatrix}.$$ Because of translational symmetry, $p$ is a good quantum number and $[H,\hat{p}]=0$. This means that we can diagonalize $H$ as well as $\hat{p}$ simultaneously and construct momentum eigenstates out of the Hamiltonian eigenstates. How do I do this? **EDIT:** Okay so I've shifted the matrix as follows (simply $-E_a$ on the diagonal): $$\begin{pmatrix} 0 & -|A| & -|A|\\\ -|A| & 0 & -|A|\\\ -|A| & -|A| & 0 \end{pmatrix}.$$ Eigenvalues are $-2|A|$ corresponding to the eigenstate (1,1,1), $|A|$ corresponding to (-1,0,1) and |A| corresponding to (-1,1,0). So $|A|$ is degenerate. Now I need to diagonalize $p$. I guess what my problem is, is that I'm not quite sure of the matrix representation of $p$ here. How do I then diagonalize it, if I can't find its matrix representation? I know each matrix element is $\langle \psi_i | p | \psi_j \rangle$, but I don't know what the wave functions look like. How do I proceed?


title: Do I understand impedance correctly?

text: $$Let\quad \vec { Z } =R+Xj$$ If $X>0$,then the impedance is lagging (current lags behind voltage).If $X<0$, then the impedance is leading (current leads voltage or more accurately, voltage lags behind current). If $X=0$, the current and voltage are in phase and the load is called purely real. If $X\neq 0$, the current and voltage are not in phase and the load is called complex. Some load is purely real if and only if the load is purely resistive. Some load is purely imaginary if and only if the load is purely reactive. Questions:   1. Does it make physical sense for a load to be purely imaginary or purely reactive?   2. How do we know that when $X>0$, current lags behind voltage and not the other way around?   3. Is everything I have said correct?


title: Precision of spectroscopy for astronomy

text: How precise can the measurements be when looking at spectral lines in astrophysics? For example, suppose I have a telescope in orbit, and I am looking at $H_\alpha$ lines coming from a star at 613 pc. Is it realistic to observer the lines move from 626.282 nm to 656.282002854 nm? this would correspond to a radial velocity of about 1.3 m/s (I am not an astronomer, and I just want to get a feel for whether this is realistic to measure, or out of the question)


title: Understanding moments as forces?

text: I was watching this lecture on analysis of stress for mechanics of materials. At time 7:20, the lecturer says that in equilibrium, the sum of forces and "moments" in each direction (x,y,z) must be zero. What exactly is meant by "moments" in this context? Does this mean the "twisting" forces? If so, why are twisting forces counted separately from the other forces in the x,y, and z directions?


title: What does this sort of notation mean?

text: I'm revising for my exams at the moment and am seeing this notation everywhere ![enter image description here](http://i.stack.imgur.com/21qfN.png) Can anyone tell me what these mean?


title: In a non auto-ranging digital multimeter, what do the increasing numbers in the Ohm section represent?

text: For my science fair project I am measuring the resistance of nichrome wires under different conditions. I am using an Innova 3300 digital multimeter to measure the resistance. Other websites have told me to move the pointer to the Ohms section. However, there are multiple numbers in that section. What number should I use and how does changing the number affect the results?


title: Concurrence in higher dimensions?

text: Does anybody know of any calculation of the concurrence for some mixed state other than the qubit-qubit case (which was solved by Wootters)?


title: What does adding a scalar field component to the Einstein field equations mean for black holes and string theory?

text: If a scalar field component has to be added to the Einstein field equations (see below) to solve dark matter/energy, then how would string theory need to be modified and do black holes still exist? The proposed modified equations are (ignoring physical constants) $$ R_{ij} - \frac12 R g_{ij} = T_{ij} + \nabla_i\nabla_j \varphi $$ where $\varphi$ represents some kind of scalar potential. The conservation law for energy- momentum is proposed to be $$ \nabla^i(T_{ij} + \nabla_i\nabla_j\varphi) = 0 $$ Refs:   * Dark Matter, Dark Energy, and the Fate of Einstein’s Theory of Gravity http://mathgradblog.williams.edu/dark-matter-dark-energy-fate-einsteins-theory-gravity   * UNIFIED THEORY OF DARK ENERGY AND DARK MATTER, TIAN MA, SHOUHONG WANG http://www.indiana.edu/~fluid/paper/report.pdf


title: The magnitude of the resultant?

text: The two structural members, one of which is in tension and the other in compression, exert the indicated forces on joint O. Determine the magnitude R of the resultant R of the two forces and the angle θ which R makes with the positive x axis (measured counterclockwise from the x axis). ![enter image description here](http://i.stack.imgur.com/wBHT5.png)


title: Physics of Fireworks

text: This evening I saw the 4th July classic Fireworks in San Diego, and I was wondering about the most physical picture of what was happening. Does anyone have a good way to explain the detailed physical features of the phenomenon?


title: Realtivistic explanation of forces between two conducting wires

text: The formula for the force between two infinite conducting wires per unit of length, $F_l = \frac{\mu_0}{2\pi} \frac{I_1 I_2}{r}$, is quite well known. It was used for definition of ampere and was revealed to me in high school, along with the right hand rule, which implies that two wires with currents flowing in the same direction should attract. Now I'm trying to get my head around relativistic explanation of things. Consider two parallel conducting wires, with electrons moving in both of them, in the same direction with the same velocity. Let the linear charge density in the stationary frame of reference be $\tau$. 1.) What is the linear charge density $\tau'$ in the frame of reference connected with a moving electron? I think it should be $\tau' = \frac{\tau}{\gamma}$. However, I have seen somewhere that it is $\tau - \gamma \tau$. 2.) Also the forces are transformed $F' = \gamma F$, lengths are transformed $l' = \gamma l$. Therefore, forces per unit of length should be the same. In the moving frame of reference, there are only electric forces in play, since the electrons are stationary relative to each other! 3.) Furthermore, if the lengths between electrons contract, their density is higher then if they were stationary, so the wire becomes charged. However, the wires have the same charge, since the current goes in the same direction, so they should _repel_. I am really confused. What am I missing? Is the right hand rule and formula from the beginning of the question even true? Thanks for any response!


title: where we can find higgs boson particle

text: This weeks greatest news is the discovery of higgs boson like particle. My question is where higgs boson is found; whether it is found in both matter and antimatter or it is found only in matter, if only in matter then there is any theoretical anti higgs boson like particle present in antimatter or not?


title: Center of rotation and trajectory of a rigid body in a plane with applied *fixed* forces

text: This is my first question so please excuse me if my format is a bit off. Given a 2D rigid body with forces applied to it in such a way that the angle the force vector makes with the surface of the object remains constant (think of a spaceship with fixed rockets attached to it), I have problems mapping out it's trajectory and angle it has rotated respect the starting position over a certain period of time. The net force from the rockets applied on the centre of mass of this object is easy to calculate on a local reference solitary to said object, but given the existence of external torque, it is a reference possessed by angular acceleration, which unsettles me as I do not know if this requires any extra considerations when translating that to a general reference. Determining the centre of rotation is also difficult, as from what I have researched, it's bound to be the centrer of mass, wherever the forces I've applied are at, although this seems unintuitive to me.


title: Maximum Weight Sustained by wire / Breaking limit of wire

text: I know that breaking stress depends on material of wire, not on the diameter or the length of wire. and also that Breaking load depends on the area of cross section of wire. But still i have a confusion in this question. > a steel wire can sustain at the most 100 kg weight without breaking.if the > wire is cut into equal parts , each parts can sustain at the most a weight > of a) 400 kg b)100 kg c) 50 kg d) 200kg Here the area of cross section remains the same so the breaking load also remains the same. And The lenght is halved but breaking stress does not depend on lenght of wire so it also remains the same. Does it mean that Both the halved wires can still sustain 100kg? Please correct me if i am wrong.


title: Rotate the phase of a wavelet

text: Let's take a zero-phase Ricker wavelet which is given by: $$ \psi(t)=\frac{2}{\sqrt{3\sigma}\pi^{1/4}}\left(1-\frac{t^2}{\sigma^2}\right)e^{-t^2/2\sigma^2} $$ in the time domain which is often used in exploration seismology. How can I determine the phase of this given wavelet and how can I rotate the phase of the wavelet by an angle $\theta$ ? **UPDATE** So if I rewrite the equation in terms of frequency I get this: $$ x(t)=(1-2 \pi^2 f^2 t^2) e^{-\pi^2 f^2 t^2} $$ How can I apply a phase rotation by $\Phi$ degrees? Like this? $$ x(t)=(1-2 \pi^2 f^2 t^2) e^{(-\pi^2 f^2 t^2 - \Phi t)} $$ Is this so difficult? I mean shifting a sine is pretty easy...


title: Y Intercept of photoelectric effect graph?

text: In a graph of KE=hf-W (the photoelectric effect equation), why is the work function equal to the y intercept? I mean I can see why it is using the equation (y=mx-c) but conceptually, why is the work function - the minimum energy required to cause emission of electrons - the y intercept of this graph?


title: About thermionic emission

text: Does thermionic emission have any limits? If we continued heating the metal plate can it reach a charge of 1 C ? Will the work function increase as the charge of the metal plate increase?


title: On a frictionless table, what is the angle at which a cue ball should hit another ball for it to go in?

text: Assume there are 2 balls on the table. Cue ball and another ball. What is the angle at which the cue ball should hit another ball for it to go in?


title: How to calculate time evolution of a wave function in an 1D infinite square well potential?

text: A particle in an infinite square well has an initial wavefunction $$\psi (x,0) ~=~ Ax(a-x) \qquad \mathrm{for}\qquad 0\leq x\leq a.$$ Now the question is to calculate $\psi (x,t)$. I have normalised it and calculated the value of A in terms of $a$. Now I do not know how to proceed. Any help please? P.S. I am from a non-physics background.


title: QM -group reps and transforming wavefunctions

text: QM texts seem to have many ways of motivating the angular momentum operators and deriving the l and m quantum numbers . But the connection between physical rotaions in 3 dim space and an operator in hilbert space that is called " angular momentum" is still somewhat unclear to me I found a set of notes on Lie Groups in Physics by Nobel prize winner Veltman - that seems to be particualry clear explanation of irreducible reps of SO(3) and angluar momentum operatiors see on page 10 of www.phys.uu.nl/~thooft/lectures/lieg07.pdf He begins with discussion of general linear diffrential equations that have a set of solutioins . He then says that when a 3 dim rotation takes place a solution of the DE is then transformed into a linear mixture of other solutions Is this true for all linear diffrential equations -?? or are these type of transformations amoung solutions in response to 3 dim rotations an assumption that is made about wavefunctions or about states in a Hilbert space as part of the assumptions of QM ? THanks Len


title: Difference between $|0\rangle$ and $0$ in the context of isospin

text: I know this has been asked before, but I am confused having read it in the context of isospin, where the creation operators act on the "vacuum" state (representing no particles) $$a^\dagger_m|0\rangle =|m\rangle$$ to create nucleon labelled by $m$ (m can be proton/neutron). And the action of annihilation operator is $$a|0\rangle =0$$ As $|0\rangle$ was already the vacuum state with no particles, what exactly did the annihilator annihilate? In the linked question, I could understand David's and Ted's answer in the context of the Harmonic oscillator, angular momentum and such. But in this case, it is hard to imagine $|0\rangle$ as a normalized non-zero vector , which does not contain anything whose energy is _not_ zero.


title: Question about streamline in boundary layers

text: I was wondering what the streamlines look like in a boundary layer. I found a picture that shows this: ![Streamline in boundary layer](http://i.stack.imgur.com/pgnq0.png) I would like to know if this presentation of the streamline is correct. I don't think that it is correct in the turbulent region, but I'm more interested in the laminar region. The streamline seems horizontal to me in that region, so it looks like there are no velocities in the y-direction, so $v_y=0$. I wonder if this is correct, because this seems to contradict the boundary-layer equations in which they use a $v_y$. I feel like that the picture isn't correct, but that the streamlines tend to follow the angle of the dotted line. So I wonder: is that feeling correct and do the streamlines have more or less the same slope as the dotted line? I also feel that the more we go in the y-direction the more the tendency to follow the dotted line becomes, so close to the plate they are horizontal, can anyone confirm or contradict this?


title: Given the spring constant & maximum kinetic energy; length of spring extension?

text: I need to understand the following question before i right my exam tomorrow. A body attached to a spring with spring constant 100 N/m executes simple harmonic motion. The maximum kinetiv energy of the body is 2J. Calculate the spring extension (in m) when its potential energy is equal to the kinetic energy. I understand the concept of total mechanical energy, i just think im missing something small thats got to do with springs in order to calculate this.


title: Is there a difference in handwritten nabla $\vec{\nabla}$ with an overset arrow and typeset nabla $\nabla$?

text: According to some physicist at KIT it is usual to write the following when using pen and paper: ![enter image description here](http://i.stack.imgur.com/cDBVO.png) whereas in typeset texts you write $\nabla$. Is that true? Are there sources for this convention?


title: Calculating height and evaluating the vicinity of a peak in a multiplicity function for a two-state paramagnet

text: I have to calculate the height and width of the multiplicity function for a two-state paramagnet. I'm supposed to do this analogously to the method in the text book, which I will try to outline below. The text book uses a lot of approximations: for instance, the first derivation considers a limit of $q\gg N$. Hopefully it's still clear what I'm asking. * * * # **_Calculating the multiplicity of an Einstein solid for the limit $q\gg N$_** First I'll consider the example from the text book for an Einstein solid of $q$ units of energy and $N$ oscillators. Starting with the exact formula for multiplicity: $$\Omega\left(N,q\right)=\binom{q+N-1}{q}=\frac{\left(q+N-1\right)!}{q!\left(N-1\right)!}\approx\frac{\left(q+N\right)!}{q!N!}.$$ Taking the natural logarithm and applying Stirling's approximation: $$\ln\Omega=\ln\left(\frac{\left(q+N\right)!}{q!N!}\right)$$ $$\ln\Omega=\ln\left(q+N\right)!-\ln q!-\ln N!$$ $$\ln\Omega\approx\left(q+N\right)\ln\left(q+N\right)-\left(q+N\right)-\left(q\ln q-q\right)-\left(N\ln N-N\right)$$ $$\ln\Omega=\left(q+N\right)\ln\left(q+N\right)-q\ln q-N\ln N$$ Using $q\gg N$, we can manipulate $\ln\left(q+N\right)$ using a Taylor series of only the first term: $$\ln\left(q+N\right)=\ln\left[q\left(1+\frac{N}{q}\right)\right]$$ $$\ln\left(q+N\right)=\ln\left(1+\frac{N}{q}\right)$$ $$\ln\left(q+N\right)\approx\ln q+\frac{N}{q}$$ Plugging this result into the previous expression derived through Stirling's approximation, we find: $$\ln\Omega\approx N\ln\frac{q}{N}+N+\frac{N^2}{q}$$ of which the last term is approximately zero in the limit $q\gg N$, to find a final expression of: $$\Omega\left(N,q\right)\approx e^{N\ln(q/N)} e^N=\left(\frac{eq}{N}\right)^N$$ This is the example in the text book. * * * # **_Calculating the multiplicity of a two-state paramagnet_** The next step is to apply the same methods to a two-state paramagnet, for which the multiplicity function is given to be: $$\Omega\left(N_\uparrow\right)=\binom{N}{N_\uparrow}=\frac{N!}{N_\uparrow!N_\downarrow!}.$$ Taking the logarithm on both sides: $$\ln\Omega=\ln\left(\frac{N!}{\left(N-N_\downarrow\right)!N_\downarrow!}\right)$$ $$\ln\Omega=\ln N! -\ln\left(N-N_\downarrow\right)!-\ln N_\downarrow!$$ $$\ln\Omega=N\ln N - \left(N-N_\downarrow\right)\ln\left(N-N_\downarrow\right)-N_\downarrow\ln N_\downarrow$$ Approximation of the logarithm $\ln\left(N-N_\downarrow\right)$ can be done by using the Taylor expansion about $a=0$: $$\ln\left(1-x\right) \approx x + \frac{x^2}{2}+\frac{x^3}{3}+\dots$$ Manipulating the logarithm like so: $$\ln\left(N\left(1-\frac{N_\downarrow}{N}\right)\right)=\ln N+\ln\left(1-\frac{N_\downarrow}{N}\right)$$ Allows to set $x=\frac{N_\downarrow}{N}$ (granted $\ln\left(1-x\right)\approx x$) to acquire a final expression (after cancelling some terms) of: $$\ln\Omega\approx N_\downarrow\ln \frac{N}{N_\downarrow}+N_\downarrow+\frac{N_\downarrow^2}{N}.$$ Which consequently results in an expression for the multiplicity of: $$\Omega\approx e^{N_\downarrow\ln \left(N/N_\downarrow\right)}e^{N_\downarrow}=\left(\frac{eN}{N_\downarrow}\right)^{N_\downarrow}$$ Similarly, the multiplicity as function of $N_\uparrow$ is: $$\Omega\approx e^{N_\uparrow\ln \left(N/N_\uparrow\right)}e^{N_\uparrow}=\left(\frac{eN}{N_\uparrow}\right)^{N_\uparrow}$$ (In both cases the limit $q\gg N$ still applies, so as to take $\frac{N_\downarrow^2}{N}$ to be negligible in the final result.) * * * ## _Intermezzo: the height of the peak of the multiplicity function for two large Einstein solids_ Furthermore, in the text book example, the height of the peak in the multiplicity function of two large Einstein solids is calculated through: $$\Omega=\left(\frac{eq_A}{N}\right)^N\left(\frac{eq_B}{N}\right)^N=\left(\frac{e}{N}\right)^{2N}\left(q_Aq_B\right)^N$$ And consequently setting $q_A=q/2$ to find: $$\Omega_\mathrm{max}=\left(\frac{e}{N}\right)^{2N}\left(\frac{q}{2}\right)^{2N}$$ * * * # **_Applying to the same methods to a two-state paramagnet_** I've tried doing the same thing, but thing are starting to not add up now. Filling in $\Omega=\Omega_\uparrow\Omega_\downarrow$ (multiplicity of a combined system - using a combined system is a guess, but things are not adding up either if I calculate the multiplicity of a single solid), I find: $$\Omega=\left(\frac{eN}{N_\uparrow}\right)^{N_\uparrow}\left(\frac{eN}{N_\downarrow}\right)^{N_\downarrow}=\frac{\left(eN\right)^{N}}{N_\uparrow^{N_\uparrow}N_\downarrow^{N_\downarrow}}.$$ Filling in $N_\uparrow=N_\downarrow=N/2$, I find: $$\Omega_\mathrm{max}=\left(2e\right)^N$$ * * * Even though I believe this is the wrong answer, the expression itself is not such a big issue until the next exercise, where I'm suppose to calculate the multiplicity function in the vicinity of the peak, by setting $N_\uparrow=\frac{N}{2}+x$ (from which automatically follows $N_\downarrow=\frac{N}{2}-x$). So far so good, I plug that into the general equation I found for the multiplicity of the combined system. After rewriting a little, I find: $$\Omega=\left(\frac{eN}{\sqrt{\left(\frac{N}{2}\right)^2-x^2}}\right)^N\left(\frac{N/2-x}{N/2+x}\right)^{x}$$ Apart from being able to plug in $x=0$ which checks out for my previous answer, the result seems very fishy, since I cannot work with it to estimate the height of the peak (third problem - follows now). * * * # **_Calculating the width of the peak in the multiplicity function of combined system of two Einstein solids_** To consider what the graph looks like near the peak, plug in the expressions for up and down-state $N$: $$\Omega=\left(\frac{e}{N}\right)^{2N}\left[\left(\frac{q}{2}\right)^2-x^2\right]^N$$ Manipulating the logarithm again: $$\ln\left[\left(\frac{q}{2}\right)^2-x^2\right]^N=N\ln\left[\left(\frac{q}{2}\right)^2-x^2\right]$$ After two steps this check out to be: $$\ln\left[\left(\frac{q}{2}\right)^2-x^2\right]^N=N\left[\ln\left(\frac{q}{2}\right)^2-\left(\frac{2x}{q}\right)^2\right]$$ Plugging this back into the original equation for multiplicity, I find $\Omega$ to be: $$\Omega=\left(\frac{e}{N}\right)^{2N}e^{N\ln(q/2)^2}e^{-N(2x/q)^2}=\Omega_\mathrm{max}\cdot e^{-N(2x/q)^2}.$$ The solution in the text book example is great! It allows to calculate an approximate peak width of $\Delta x=q/\sqrt{N}$ (at which point the multiplicity falls off by a factor of $1/e$). But my equation doesn't check out so nicely with my original expression of $\Omega_\mathrm{max}$, and I'm not sure what next step to take. Any suggestions welcome. * * * # **_EDIT_** Perhaps I wrongly took the low temperature limit for my two-state paramagnet ? Can anyone clarify and explain _why_ I should take the high temperature limit?


title: Simple Conduction Simulation

text: What is the simplest way to simulate heat conduction on a 2D plane? What I want to do is have 1024x1024 2D plane of uniform temperature 0°C and specify the temperature of a cartesian coordinate. Then, over time I want to simulate the temperature of each coordinate on the plane. So far all I found are simulations for 3d systems and programs such as FloTherm.


title: Alternatives to electromagnetism for communication?

text: It seems, from my (weak) understanding of the standard model that the only field suitable for long range communication is the one we actually use, electromagnetism. (Gravity waves could be used as well I suppose, but gravity is too weak to be practical use.) Questions:   1. Is this correct? It seems so, as you need a boson (to be able to produce 'intense' signals) and all the others are either unstable or strongly interacting (e.g. gluon)   2. Why is it like that? Is there some underlying fundamental reason that there are no other 'useful' bosons?   3. Does 'beyond the standard model' physics provide any insight? Any potential alternatives? (Graviphoton?)


title: Solving one dimensional Schrodinger equation with finite difference method

text: Consider the one-dimensional Schrodinger equation $$-\frac{1}{2}D^2 \psi(x)+V(x)\psi(x)=E\psi(x)$$ where $D^2=\dfrac{d^2}{dx^2},V(x)=-\dfrac{1}{|x|}$. I want to calculate the ground state energy(which is known to be $-0.5$) with finite difference method. Suppose $\psi(-a)=\psi(a)=0$ for sufficiently large $a$. Split $[-a,a]$ into $N$ sub-intervals with equal length $h$: $$-a=x_1<x_2<\cdots<x_{N+1}=a$$ $x_{i+1}-x_i=h.$ Now use $$D^2\psi(x_n)\to\frac{\psi(x_{n-1})-2\psi(x_{n})+\psi(x_{n+1})}{2h^2}$$ and $$-\frac{1}{2}D^2 \psi(x_n)+V(x_n)\psi(x_n)=E\psi(x_n)$$ we get $$-y_{n-1}+2(1+V_n h^2)y_n-y_{n+1}=2h^2 Ey_n$$ where $y_n=\psi(x_n)$. Recall that $y_1=y_{N+1}=0$, we get $$H \begin{pmatrix} y_2 \\\ \vdots \\\ y_N \end{pmatrix} = 2h^2E \begin{pmatrix} y_2 \\\ \vdots \\\ y_N \end{pmatrix} $$ where $$H=\begin{pmatrix} 2(1+V_2h^2) & -1 & & \\\ -1 & 2(1+V_3h^2) & \ddots & \\\ & \ddots & \ddots & -1 \\\ & & -1 & 2(1+V_Nh^2) \end{pmatrix}$$ is a $N-1$ tridiagonal matrix. If the least eigenvalue of $H$ is $\lambda$, then I expect $\dfrac{\lambda}{2h^2}$ to approximate the ground state energy $E$. So I write a MATLAB program:               function m = onedimen1(a,M)          N = 2*M+1;          h = 2*a/N;     t = -a;          for i = 1 : N-2         t = t+h;         H(i,i) = 2*(1+vp(t)*h*h);         H(i+1,i) = -1;         H(i,i+1) = -1;     end          H(N-1,N-1) = 2*(1+vp(t+h)*h*h);          m1=eig(H)/(2*h*h);     m = m1(1:5);            function y = vp(x)         y = - 1/(abs(x)) ;       end     end      but when I try               onedimen1(10,100)     onedimen1(10,1000)      the least eigenvalue of $H$ divided by $2h^2$ appears to be too small(and tends to negative infinity when $N$ increases), but the second largest divided by $2h^2$ is always near $-0.5$. Can anyone explain why? (P.S. If I change $V(x)$ to $V(x)=\dfrac{1}{|x|}$ then the least eigenvalue of $H$ is stable)


title: Can somebody provide some sort of crash course on random walk and its problems at the level of a beginning undergraduate student in physics?

text: I really need some very simple discussions of random walk (probability). Couldn't get anything from class, more so from Reif. Thanks!


title: What happens if a wind mill rotates in opposite direction?

text: Suppose if a windmill is made torotate in clockwise direction and due to air flow if it rotates in anti clockwise direction then does it produce electricity?


title: When viewed from a telescope, does an object get more magnified if its angular diameter is increased?

text: and if yes, how can the diameter be increased? Lets say I want to view saturn from a small telescope. Increasing the angular diameter will give a better magnification if the answer to the above question is yes. Then how can I increase the angular diameter?


title: Electric field from current without Maxwell's law of induction

text: > A long, straight wire carries a current that decreases linearly with time. > What is the direction of the induced electric field outside the wire? I would interpret this as follows: a current induces a magnetic field. Since the current is changing, the magnetic field is changing, which induces a changing electric field by Maxwell's law of induction. However, this question was not posed in a context where one would be assumed to know Maxwell's law of induction (a high school science contest, and the law is only covered in the most advanced high school physics courses), which makes me wonder whether this is poorly phrased and simply referring to the electric field set up by the charges in the wire. I'd greatly appreciate help.


title: Is basic quantum mechanics mathematically as robust a theory as special relativity?

text: This question is specifically about the robustness of mathematical models. Special relativity can be derived from very basic principles. Assuming that space is homogeneous and isotropic and that relative velocity is measured the same by two observers is enough to show that the ransformation from one frame to another must have the form of a standard Lorentz transformation with some universal constant c, which is infinity in Galileian relativity. Thus, the simple assumptions outlined above plus an estimate for a universal bound on speed are enough mathematically to derive all of special relativity. This shows that, in some sense, special relativity is the _only_ theory that works in flat spacetime. What about quantum mechanics? Dirac's book derives QM from general principles but throws in a few 'jumps' in logic that work great but seem to leave the door open to other mathematical theories. My question is, > Is there a small set of basic assumptions and experimental evidence which > require quantum mechanics to have the mathematical form that it does? For instance, does linear superposition of states (a simple assumption given experimental evidence) imply that observables must be linear operators (instead of other functions of states)? Is there any other inequivalent mathematical framework that satisfies linear superposition and the correspondence principle?


title: Why do objects sometimes fall on their own?

text: I have observed sometimes that when I keep some objects precariously balanced, for e.g. some objects in a cupboard, they seem quite stable at the time when I place them there but after a certain period, maybe even 8 - 12 hours suddenly you hear a noise and you can see that they have fallen for what seems like no apparent reason. What can be the reason for this? obviously, I am looking for a physics explanation and not a supernatural one.


title: Equations of motion of the double pendulum by vector methods

text: I wanted to know if anyone of you do know a book or article on the Internet in which the equations of motion of a double pendulum is deduced, only using vector methods (i.e. Newton's 2nd law as opposed to Lagrangian formalism).


title: What spin-statistics is a magnetic monopole expected to obey?

text: 1. What statistics (or spin) is a magnetic monopole expected to have?   2. Does it depend on the theory used?


title: is it possible to load a transversal wave with a longitudinal wave

text: like in communication engineering for sending information what we do, we take a high frequency carrier and modulate it with the message signal so can we do the same thing like take a high frequency carrier (transversal wave) but the message is not transversal wave rather the message is longitudinal wave say a sound wave so can we modulate or load a transversal wave with a longitudinal wave ?? also if there is any work on this regard .. kindly post the link also??


title: Negative energy solutions Dirac equation without radation field

text: In the book "Relativistic Quantum Mechanics" by Bjorken and Drell in Chapter 5.1 page 64 there is the following statement about the problem of negative solutions to the Dirac equation: > By their very existence the negative-energy solutions require a massive > reinterpretation of the Dirac theory in order to prevent atomic electrons > from making radiative transitions into negative-energy states and cascading > down to oblivion. This is no problem if we completely neglect interaction of > the electrons with the radiation field. I'm not sure of understanding this statement. Why is the issue of negative energy only related to the radiative field. Couldn't there be another mechanism by which the energy could infinitely tumble down?


title: Could this fountain, under the right conditions, technically be able to lift me up?

text: There is a big fountain in a lake in my city. ![enter image description here](http://i.stack.imgur.com/NieXt.jpg) I was talking with a friend and we were wondering whether it would be able to lift me up. I sent a few emails and obtained information about the fountain. The raw email reads in French: > Nous avons 2 pompes de 150 HP qui pousse l'eau a 195 pieds de haut et la > pression par pompe à la sortie est d'environ 380 pieds de tête. I am not knowledgable in physics in any way, but this is how I can translate it to the best of my knowledge: > We have two pumps of 150 horsepower that push water at 195 feet high and the > pressure by pump at the exit is of 165 PSI Now, could this possibly be able to lift me, a 160 pounds man? How can I find out? And under what conditions would this work?


title: Introduction to Gauge Symmetries: Good, Bad or Ugly?

text: I'm trying to come up with a good (as in intuitive and not 'too wrong') definition of a gauge symmetry. This is what I have right now: > A dynamical symmetry is a (differentiable) group of transformations that > respects system dynamics, ie maps solutions to solutions. > > A rigid symmetry is a dynamical symmetry that maps solutions to _different_ > solutions. A rigid symmetry has a Noether charge that is only conserved _on- > shell_ , ie dependent on the equations of motion. > > A gauge symmetry is a dynamical symmetry that maps solutions to _identical_ > solutions up to 'parametrization' or 'gauge'; in particular, the solutions > correspond to the same initial conditions and physics and only differ in > their mathematical description. A gauge symmetry has a Noether charge that > is conserved _off-shell_ , ie independent of the equations of motion. > > As an example, we take classical mechanics: In general, time dependence of > the solutions matter as reparametrization changes velocities. However, in > the relativistic case 4-velocities are constrained to 'length' $c$ and > dynamics need to be independent of the particular choice of the 'unphysical' > 3-velocities. First of all, is this correct? If so, is there a better choice of wording? Should anything be added?


title: Do mass and the Pauli exclusion principle conspire to make light fermions take up more space?

text: The wavelength of light fermions is longer. Wouldn't this cause them to take up 'more space' so that they didn't overlap according to the Pauli exclusion principle? Am I totally misunderstanding something? In a semi-related question... If this is true, what are the consequences for neutrinos?


title: Is the artificial gauge field a gauge field?

text: The so-called artificial gauge fields are actually the Berry connection. They could be $U(1)$ or $SU(N)$ which depends on the level degeneracy. For simplicity, let's focus on $U(1)$ artificial gauge field, or artificial electromagnetic field. One can show that it is indeed gauge invariant. My criterion for a gauge field is that it should satisfy Maxwell's equations (or Yang-Mills' equations in the non-Abelian case). Then I doubt whether the artificial gauge field is really a gauge field. How to write down the $F_{\mu\nu}^2$? What's the conserved current? And can we make gapped Goldstone modes using artificial gauge fields?


title: Point where resulting magnetic field is equal to 0

text: I am trying to solve the following task > Given 3 infinitely long current carrying wires, as shown on image ($I_1=I_2$ > , $I_3=2I_1$) and $r=5 cm$ calculate the point between them (in the same > plane; point at $I_1$ is $x_1=0cm$, point at $x_2 = 5 cm$, point at $x_3 = > 10 cm$) for which magnetic field is equal to $\vec{0}$. ![Task](http://i.stack.imgur.com/KY2ir.png) I tried doing this by calculating magnetic field produced by each current this way: $$ \frac{\mu I_1}{2\pi r} $$ but since I don't have $r$, I cannot do that. The point where magnetic field is equal to 0 is not at the same distance from each wire so i would get something like this : $\frac{\mu I_1}{2\pi r_1}$ for the first current, $\frac{\mu I_2}{2\pi r_2}$ for the second current, $\frac{\mu I_3}{2 \pi r_3}$ for the third current. If i put them in an equation to get resulting magnetic field I can go no further from that. Direction of the first two currents is, as shown on the image, into the screen and the third one out of the screen. What have I done wrong and how to fix it? Result is 3.3 cm but there is no way I can get it.


title: What would happen to wet laundry in the vacuum of space?

text: Imagine for a second that I was in a college dorm room frustrated that all the dryers were broken or in use. I'm impatient and not a jerk so I don't wait or take out someone else's laundry, I shoot it into space with the ability to get it back and land it successfully on my campus. Ignoring practicality of course: How would my clothes react? Would the water in them freeze? Would the water just dissipate into the rest of space? Would it boil if it were in the light of the sun? I'm curious about three cases: One, where the clothes are in a general vacuum regardless of the effect of the sun. Two, where my clothes are specifically behind the Earth and so they are eclipsed by the Earth. Three, where my clothes are specifically between the Earth and the sun. For those willing to make the argument that this is a silly topic: would it or would it not have implications with space suits? Thanks!


title: Properties of spilled spent nuclear fuel pellets

text: Can someone please explain to me the physics of a pile of spilled nuclear fuel pellets on the floor of a spent fuel pool? Specifically, what can we expect in terms of heat and radioactivity releases, should a damaged spent fuel assembly spill its load of fuel pellets, as it is removed from Unit 4 spent fuel pool at Fukushima Diiachi?


title: What is the difference between a skew-symmetric and an antisymmetric tensor?

text: What is the difference between a skew-symmetric and an anti-symmetric tensor? If they represent the same tensor, then why use different labeling.


title: Is this object turn alone with torque inside disk of water?

text: This is not the original question, I changed for explain details. But now it's not possible to delete the question and now the reply is not in relation with question. I post a new question, and try to modify text with original question. Sorry. ![h](http://i.stack.imgur.com/Au5Yy.jpg) ![hn](http://i.stack.imgur.com/UM7f7.jpg) An object is composed by two disks, one bigger, one smaller. There is gas in each disk. I put this object in a big disk full of water, big enough for have gravity effect. I suppose this big disk fixed. No other gravity like Earth for example (theoretical problem). Like gas object is asymmetrical it must have a torque if pressure is the same for a depth but pressure inside big disk change due to the lack of matter. So no torque. But if I add an external object of the same shape of the gas object but solid with the same density like water, it's possible to compensate near the pressure inside big disk for have a torque on gas object, it want to turn counter-clockwise. The external object want to turn counter-clockwise too ? Is it a torque on external object ? ![asy](http://i.stack.imgur.com/MTssA.jpg)


title: Why is there no gravity in the center of a celestial body, yet there is gravity on the surface?

text: I've heard few explanations, but none in good detail that make it clear to those with less mathematical knowledge on the equations of complex physics. I would like to think that since you are on the surface of the Earth, for example, everything, at least from my introspection, seems to be "pulled" towards the center of the celestial body itself. If that is true, and everything, including me, is being pulled towards the center, and our (human, for example) bodies use skeletal muscles to maintain supportive structure to fight the gravity all the time (when awake), how can there be no gravity within the very center of the Earth? It just seems hard to believe that something attracts you to it with energy, yet it has no energy in its core (gravity) to do this. Where and why does gravity stop at the center, and how does that make good, detailed sense for an average to relatively understand? Also, taking in to accounts with centers of mass, wouldn't the Earth's core be the "center of mass" in this sense? I mean if the energy or force "gluing" us to the Earth is not caused by the Earth itself, how come we are "pulled" towards Earth in particular?


I've heard that light does change our momentum a little bit which made me think: How much photons are needed to erase a complete mountain? That is to evaporate a complete mountain such as Himalaya for instance.


title: Surely force on shell can't be balanced by field momentum?

text: Imagine a particle with charge $q$ at rest at the origin. It is surrounded by a concentric spherical insulating shell, also at rest, with charge $Q$ and radius $R$. At time $t=0$ I apply a constant horizontal acceleration $\mathbf{a}$ to the particle. The electromagnetic field spreads out radially in all directions from the charge at the speed of light (Appendix 1). The momentum density in the field is directed radially and also spreads out from the charge at the speed of light (Appendix 2). As the momentum density at opposite field points cancel the total momentum in the field is always zero (Appendix 2). At time $t=R/c$ the field reaches the charged shell and imparts a horizontal force on it (Appendix 3). _What balances this force if the rate of change of horizontal momentum in the field is always zero?_ I think it has to be balanced by an opposite force on the particle that develops due to an advanced electromagnetic interaction from the spherical shell at time $t$ backwards in time to the particle at time $t_r$. **Appendix 1: EM field of an accelerated charge with velocity zero** Specializing the Lienard-Wiechert fields for an accelerating charge with zero velocity we find that the electric field is given by: $$\mathbf{E}(\mathbf{r},t)=\frac{q}{4\pi\epsilon_0}\left(\frac{\mathbf{\hat r}}{r^2}+\frac{\mathbf{\hat r}\times(\mathbf{\hat r}\times \mathbf{a})}{c^2r}\right)_{t_r}$$ and the magnetic field is given by: $$\mathbf{B}(\mathbf{r},t)=\frac{\mathbf{\hat r}(t_r)}{c}\times\mathbf{E}(\mathbf{r},t) $$ where $\mathbf{\hat r}$ is the unit vector in the direction of the field point and the retarded time $t_r$ at the charge $q$ is given by: $$t_r=t-\frac{r(t_r)}{c}.$$ **Appendix 2: Zero total momentum in EM field from an accelerated charge** The momentum density $\mathbf{g}(\mathbf{r},t)$ in the electromagnetic field is given by: $$\mathbf{g}(\mathbf{r},t)=\epsilon_0\mathbf{E}(\mathbf{r},t)\times\mathbf{B}(\mathbf{r},t).$$ The momentum density is directed radially so that: $$\mathbf{g}(\mathbf{r},t)=\frac{\epsilon_0E^2}{c}\mathbf{\hat r}$$ where the magnitude of the electric field $E$ is given by $$E = -\frac{q a \sin \theta}{4\pi\epsilon_0c^2r}$$ and $\theta$ is the angle between acceleration $\mathbf{a}$ and radial direction $\mathbf{\hat r}$. As $\sin(\pi-\theta)=\sin(\theta)$ then we have: $$\mathbf{g}(-\mathbf{r},t)=-\mathbf{g}(\mathbf{r},t)$$ so that the momentum density at opposite field points cancel. Thus the total momentum in the fields is always zero. **Appendix 3: Total force on charged spherical shell from accelerated charge** The electric field at the spherical shell consists of a static radial part and an acceleration part. The total force from the static radial part will cancel out in opposite pairs so that we only have to worry about the acceleration part. The total horizontal force on the spherical shell with charge $Q$ and radius $R$ is given by: $$F = \int_{sphere} E\sin \theta\ \sigma\ dA$$ where $$E = -\frac{q a \sin \theta}{4\pi\epsilon_0c^2R}$$ $$\sigma=\frac{Q}{4\pi R^2}$$ $$dA = 2 \pi R^2 \sin \theta\ d\theta$$ Performing this integration over the sphere using: $$\int_0^\pi \sin^3 \theta\ d\theta=\frac{4}{3}$$ We find that the total horizontal force is given by: $$F = -\frac{2}{3}\frac{qQa}{4\pi\epsilon_0c^2R}.$$


title: Reason for the Gaussian wave packet spreading

text: I have recently read how the Gaussian wave packet spreads while propagating. see: http://en.wikipedia.org/wiki/Wave_packet#Gaussian_wavepackets_in_quantum_mechanics Though I understand the mathematics I don't understand the physical explanation behind it. Can you please explain?


title: Orientation of large water drops break-up

text: I have observed that if you pour small globs of water from a considerable height (let's say more than 20 meters), the globs will break up. This is not surprising at all since the amount of water dropped was probably too much for a stable drop. In my experience (which took place many years ago while sipping water from a plastic bottle at the window of a tall building) the break-up appeared to happen at the same distance from the point of release (again, not surprisingly) and always consisted in the formation of 3 smaller drops in a (still not suprisingly) symmetric pattern. What surprised me is that the pattern had a _constant orientation_ , that is the three trajectories at 120° from each other were always oriented in the same way with respect to the building. If I remember correctly (many years have passed since 'the experiment'), one drop was always directed away perpendicular to the building wall, the other two diverged at 120 degrees towards it (eye estimate and symmetry prejudice). I have since resisted the urge of breaking in in tall buildings with different geographic orientations (N-S, sun exposition) to drop water from their highest windows - try to explain that to the police... - so... ...can someone please explain why the orientation is always the same? Is it due to the thermal gradient near the walls/windows, the earth magnetic field, the separation of charges in the drops? Or was it just sheer coincidence? (I was not alone, and three people with two different bottles repeated the experiment many times always with the same result) To cliarify: when I say "three smaller drops" I mean that from the distance the main glob appeared to 'explode' in three main globs of water. There could have been smaller drops produced, but the separation in three was clear and unmistakable.


title: What causes the lines in these photos of a flourescent tube?

text: I was taking pictures of a flourescent tube using a Moto G 2013 when I noticed these evenly spaced lines as I got closer to it. The distance between the lines reduced as I got closer to the tube. ![enter image description here](http://i.stack.imgur.com/ZAUqTm.jpg) ![enter image description here](http://i.stack.imgur.com/0p0yJm.jpg) ![enter image description here](http://i.stack.imgur.com/9YlN8m.jpg) I noticed that if I take two photographs from the same position, about 60cm from the end of the tube (bar shaking of my arms), the lines are more pronounced when focused on the tube. ![enter image description here](http://i.stack.imgur.com/hncLUm.jpg) ![enter image description here](http://i.stack.imgur.com/YvKi8m.jpg) What causes these?


title: Can you choose the variables of a state function?

text: I'm confused. I was first introduced to entropy as a state function of internal energy and volume $$S(U,V) \Rightarrow dS = C_v\frac{\mathrm{d}T}{T} - p\frac{\mathrm{d}V}{T} $$ wich is the thermodynamic identity, but now I see that it can also be written as $$S(T,p) \Rightarrow dS = C_p\frac{\mathrm{d}T}{T} + k\mathrm{d}p $$ I don't understand, if I can choose the parameters I could just decide that entropy is a function of the number of molecules of my gas so there would be no variation in entropy as long as it's in a closed container... Can someone explain to me the logic behind this ?


title: Can quanta have different energy levels?

text: The energy of a wave is h*f. How i imagine this is each cycle consisting of 1 quanta. I sometimes hear people say the energy of a photon is equal to h*f which makes it sound like each cycle has a higher energy of quanta. which way of thinking is correct?


title: Finding a angular acceleration of the rod

text: A rod of mass 6kg and length 4m is hinged at its lowest end. A disc of mass 2kg is attached to the upper end of the rod. A light string is wrapped on the disc and its free end is pulled by applying a horizontal force F=16N. initially the rod is vertical and the rod and the disc are at rest. Find the initial angular acceleration (rad/sec^2) of the the rod. Assume no friction between axle and the disc and at hinge. Also assume the light string does not slip on the disc.![enter image description here](http://i.stack.imgur.com/jjXsz.jpg)


title: Formal demonstration that minimizing the free energy equals maximizing the entropy

text: I never had great intuition when it came to thermodynamic concepts and potentials even though reading a textbook and completing the exercises has never been a huge problem. In one of them, I was asked to show that maximizing the Gibbs entropy $$-\sum_n p_n \text{ln}(p_n)$$ under the constraints of normalization $$\sum_n p_n =1$$ and the existence of a fixed average energy $\sum_np_nE_n =E^*$ gives the canonical ensemble, ie: $$p_n=\frac{e^{-\beta E_n}}{\sum_ne^{-\beta E_n}}$$. And then I am supposed to show that maximizing the entropy is equivalent to minimizing the free energy. It's quite evident looking at its definition: $$F= E -TS$$ At fixed T and E, the minimum of T matches the maximum of S (I assume S to be positive). But is there an explicit variational derivation of this result? For example, by rewriting a Lagrangian with F as a function of $p_n$ (maybe something else than replacing E and S by their definition?)


title: Imaginary angle on simple centrifugal problem

text: Say somebody is spinning a mass on a string with a period $T$ around. If $T$ is very big, the mass will describe a small circle around the person's waist. If $T$ is very small, the plane of movement should be around shoulder high, with the actual radius being the full length of the arm and string. ![](http://wstaw.org/m/2011/11/27/m35.png) The question asks for the angle of equilibrium for a given $T$ and $R$. The angle $\theta$ is measured between the person and his arm. $\theta=0$ means that the string is straight down, and $\theta = \pi/2$ means that the string is parallel to the ground. $R$ is the length of arm and string. So I thought that the two forces on the mass are the gravity, $F_g=mg$, and the centrifugal force, $F_c=m \omega^2 r=m \omega^2 \sin \theta R$. The two forces have a resulting force, which also acts in an angle $\tan \phi = F_c/F_g$ onto the mass. ![](http://wstaw.org/m/2011/11/27/m33.png) In equilibrium, both angles have to be the same. So I say: $$\tan \theta = \tan \phi = \frac{m 4 \pi^2 \sin \theta R}{m g T^2}$$ $$\frac{\tan \theta}{\sin \theta} = \frac{m 4 \pi^2 R}{m g T^2} $$ $$\cos \theta = \frac{g T^2}{4 \pi^2 R}$$ ![](http://wstaw.org/m/2011/11/27/m34.png) For the given values, $T=0.45s$ and $R=15cm$ I get an $\theta=1.23$, which seems reasonable to me. Then the problem asks for $T=0.85s$. The question is worded as asking to trouble, so I do not feel to bad about my function giving me $\theta = 0.618i$. But how is this supposed to make any sense at all? If I spin the mass slower, the angle has to be lower. If I spin with almost zero speed (infinite period $T$), I should get an angle around $\theta=0$, nothing else. But the angle is exactly zero for a given $R$ and $T$ and then goes imaginary beyond that. Can anyone enlighten me, or tell me what I did wrong in the derivation? # More thoughts based on Mark Eichenlaub's answer The forces perpendicular to the rope look like this: ![](http://wstaw.org/m/2011/11/27/m36.png) (Sorry, the restoring force has a Sine in it, no Tangent.) The net force on the mass is therefore the difference: ![](http://wstaw.org/m/2011/11/27/m37.png) (If you plot this with Sine instead of Tangent, it comes out at 1.22 like I already had before.) These are the values for the first part of the problem, and one can see the two points of equilibrium (including zero) clearly. If the frequency is lower like in the second part, there is no equilibrium other than zero. So no matter how big the angle is, the restoring force is bigger than the centrifugal force. ![](http://wstaw.org/m/2011/11/27/m38.png) (This picure is slightly different with the Sine instead of Tangent as well, but it does not change the roots.) In this light, the imaginary equilibrium point I got previously seems pretty right.


title: How do formulas get figured out?

text: Many of the formulas I learn in school are derived from more basic formulas -- as long you your math is right and assuming the more basic formulas you used are correct, you are bound to get to a correct result... But how do people 'invent' the elementary formulas, like $F=ma$; how can you be sure that that's really the right equation for force?


title: Radar Frequency Bandwidth

text: I've come across an interesting question in the course of doing some exam review in a quantum mechanics book and thought I'd share it here. "What must be the frequency bandwidth of the detecting and amplifying stages of a radar system operating at pulse widths of 0.1usec? If the radar is used for ranging, what is the uncertainty in the range?" I don't know the solution; although I'd guess that it could involve a Fourier transform (a generic first crack at anything with waves) and the uncertainty relation (as the problem does call out uncertainty in range; which could be interpreted as uncertainty in x).


title: How much would the LHC beam be attenuated by the atmosphere?

text: As I understand it, the completed Large Hadron Collider (LHC) will ultimately have a proton beam with $2,808$ bunches of $1.15 \times 10^{11}$ protons each at $7$ TeV, giving a total beam energy of $(2808)(1.15\times 10^{11})(7 \times 10^{12})(1.602\times 10^{-19}$ J$)$ $\approx 362 $ MJ. Imagine that instead of using a graphite block for the beam dump, the proton beam is aimed "straight up" at a satellite in geosynchronous orbit 42,160 km above the surface of the earth. How significantly would be the beam intensity be diminished in the atmosphere, particularly the troposphere? Could one reasonable expect to be able to blind sensors/cameras on a satellite?


title: What happens if the holonomy group lies in $SU(2)$ for a CY 3-fold?

text: I am a mathematician and reading a physics paper about the holonomy group of Calabi-Yau 3-folds. In that paper, a Calabi-Yau 3-fold $X$ is defined as a compact 3-dimensional complex manifold with Kahler metric such that the holonomy group $G \subset SU(3)$ but not contained in any $SU(2)$ subgroup of $SU(3)$. They remark "the condition that $G$ is not contained in $SU(2)$ is a really serious condition for physics since otherwise it would change the supersymmetry". Could anyone kindly explain this sentence in more detail? I think that if $G\subset SU(2)$ the physics derived from the Calabi-Yau 3-fold has more supersymmetry (because less restriction), but what is wrong about it? One possibility is that too symmetric theory is trivial. I would appreciate it if someone could kindly explain the physics behind it to a mathematician.


title: Fourier analysis in crystallography

text: What is the best reference for an introduction to the use of Fourier analysis in crystallography?


title: Resonance Raman spectroscopy vs fluorescence

text: In Resonance Raman Spectroscopy we often want to avoid the fluorescence. But what is the problem with fluorescence . What we want is a shift between exciting line and emitted radiation and both can show the vibration energy difference?? also from "What is the difference between Raman scattering and fluorescence?" I come to know the difference between Raman and fluorescence is that of lifetime of the molecule in the excited state, but how does a molecule come to know that it is studied for Raman spectroscopy so its lifetime in excited state is small or it is studied for fluorescence.


title: Why must the field equations be differential?

text: In Landau–Lifshitz's _Course of Theoretical Physics_ , Vol. 2 (‘Classical Fields Theory’), Ch. IV, § 27, there is an explanation why the field equations should be linear differential equations. It goes like this: > Every solution of the field equations gives a field that can exist in > nature. According to the principle of superposition, the sum of any such > fields must be a field that can exist in nature, that is, must satisfy the > field equations. > > As is well known, linear differential equations have just this property, > that the sum of any solutions is also a solution. Consequently, the field > equations must be linear differential equation. Actually, this reasoning is not logically valid. Not only the authors forget to explain the word ‘differential’, but they also do not actually prove that the field equations must be linear. (Just in case: this observation is not due to me.) But it seems that the last issue can be easily overcome. However, it is exactly the word ‘differential’, not ‘linear’, that is bothering me. There is a nice theorem of Peetre stating that the linear operator $D$ that acts on (the ring of) functions and does not increase supports, that is, $\mathop{\mathrm{supp}} f \supset \mathop{\mathrm{supp}} Df$, must be a differential operator. The property of preserving supports can be considered as a certain _locality_ property. Hence, the field equations must be differential because all interactions must propagate with a finite velocity. But there is another notion of ‘locality’ of an operator: the operator $D$ is called _local_ if the function $Df$ in the neighbourhood $V$ can be computed with $f$ determined only on $V$ as well, i.e., $(Df)|_V$ is completely defined by $f|_V$. The locality in this sense is not equivalent to the locality in the sense of supports' preserving. (Unfortunately, I do not have an illustrative example at hand right now, so there is a possibility of mistake $M$ hiding here.) The question is: what physical circumstances determine the (correct one) notion of locality for a given physical problem? (Assuming there is no mistake $M$.) And does my reasoning _really_ justifies the word ‘differential’ in the context of field equations? If so, are there any references containing more accurate argument than the one presented in Landau–Lifshitz's Course?


Why doesn't any object that doesn't have direct contact in a fast moving vehicle feel G-force or slam behind or ahead when braked or accelerated? For example when we are in a car (even with all windows open) and driving speed is of 100 kmph (for that matter, even plane, whose speed is over 500 kmph) and we throw up a ball (which is a non-living thing and has sufficient mass and it isn't flying on its own unlike a fly or bee etc..) in the car, and for that brief moment it is not in any direct contact with any car or any interiors, so by common physics logic it should slam into our face but it doesn't. Why?


title: Is there a general law for physics?

text: Here is why I pose this question, as I advance in physics more and more this idea haunts me and in fact helped me figure out and memorize many physical laws. It seems to me that there is a relation between all the laws of physics, something that is a general law that determines how should all other laws be _generally_ stated. I can't express this law in a complete sentence and I'm not able to verify if it has an exception or not but I will try to explain this by examples to show what I really mean: ## Mechanics: Oscillations: It seems that the whole process happens because the object is trying to get back into its **initial state** i.e the origin. ![oscillator](http://i.stack.imgur.com/I7qc4.png) ## Electricity: Lenz's law: An induced electromotive force (emf) always gives rise to a current whose magnetic field opposes the original change in magnetic flux. in other words: the current is trying to get its **initial state** by resisting the change of current. ## Atoms Radiation and Nuclear reactions happen because the atom is trying to be **stable.** Of course there are many other examples but I'm just stating few to make my point. The question is: is there something like that in physics? if not can you give me any counter example on this?


title: Would time dilation be too great for the early universe to expand?

text: I read that one second after the big bang the universe was composed of photons electrons and neutrinos. Wouldn't the density of energy/matter have caused such extreme time dilation that the universe would never expand?