title: Impossible microwave interference?

text: I was doing a microwave experiment with the following set up: there is a Gunn diode which emmits microwave radiation and a receiver (both work with polarised light). ![enter image description here](http://i.stack.imgur.com/5rLho.jpg) The strange thing is that when I put a metal plate along the line connecting the emmiter and the receiver, if the plate is perpendicular to the polarisation plane, no radiation is detected. Instead, it's deflected about 20° from both sides of the plate. However, if the plate is parallel to the polarisation plane, radiation is detected as expected. One explanation might be that polarised light will "collide" with the laterals of the plate (and that could produce interference). But when the polarisation plane is parallel to the plate, light will past without noticing. Still, my professor had no idea what was happening. **Addendum 2013/10/26** It's important to note that I doesn't matter if it a metalic or wooden plate, so it's nothing related the Fresnel equations. The effect is this: ![enter image description here](http://i.stack.imgur.com/LtfBn.png) The image is not precise at all. The black zone is where no microwaves are detected. The red line is where the maximum is detected. The incident wave is blue. Another possibility is that there is no diffraction at all, because the plate blocks the incident waves (like if you shelter on a side of a wall when someone's shooting at you).


title: Minimum energy for turning over a cube

text: This is a strange question from the physics Olympiade I wish to get some explanation for: Using a certain type of wood, cubes are sawn with different edge lengths $R$. When they are on the ground, they are turned over one of the edges of the face touching the ground, so they rotate 90 degrees in the end. What is the relationship between the minimum energy $E$ required for this action and $R$?   * A. $E$ is proportional with $R$.   * B. $E$ is proportional with $R^2$.   * C. $E$ is proportional with $R^3$.   * D. $E$ is proportional with $R^4$. The answers is D, but why?


title: Angular momentum of quantum system

text: **Problem:** A physical system is in the common eigenstate of $\hat{L^2}$ and $\hat{L_z}$. Calculate the following quantities: $\langle L_x\rangle,\langle L_y\rangle,\langle L_z\rangle,\langle L_x L_y + L_y L_x\rangle$. **Solution:** I am not sure on where to start from. Since the physical system is in the common eigenstate of $\hat{L^2}$ and $\hat{L_z}$, I presume that it is described by a wave function $\psi$ that satisfies: $L^2\psi=\hbar^2\ell(\ell+1)\psi$ and $L_z\psi=m\hbar\psi$. Another thought I made is that $\psi$ could be written as linear combination of $\psi_{\ell m}$, i.e. $\psi=\ldots + C_1 \psi_{\ell 1} + C_0 \psi_{\ell0} + C_{-1} \psi_{\ell,-1} + \ldots$, $C_j$ constants. Also, by definition it is true that: $\langle \hat{L_x}\rangle = \langle\psi, \hat{L_x}\psi\rangle$ (assuming $\psi$ is normalized). For $\langle \hat{L_z}\rangle$ specifically, it is easier because: $\langle \psi, \hat{L_z}\psi\rangle = \langle \psi, m\hbar\psi\rangle = m\hbar\langle \psi, \psi\rangle=m\hbar$. Could you people provide any directions? **EDIT:** The missing parts were that $L^2=L_x^2+L_y^2+L_z^2$ and the relationship of creation / annihilation operators $L_{\pm}$ with $L_x,L_y,L_z$.


title: How would steel degrade in space

text: If there is a steel plate floating in space, for ever, within the solar system, how is it going to degrade? Of course without oxygen it is not going to oxidize (rust), but how are cosmic rays, UV light, alpha particles and the general space environment going to affect the structure of steel on the long term? **EDIT:** Let's imagine a plane of steel on orbit around the earth. Just curious.


title: Derivation of Maxwell's equations from field tensor lagrangian

text: I've started reading Peskin and Schroeder on my own time, and I'm a bit confused about how to obtain Maxwell's equations from the (source-free) lagrangian density $L = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ (where $F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu$ is the field tensor). Substituting in for the definition of the field tensor yields $L = -\frac{1}{2}[(\partial_\mu A_\nu)(\partial^\mu A^\nu) - (\partial_\mu A_\nu)(\partial^\nu A^\mu)]$. I know I should be using $A^\mu$ as the dynamical variable in the Euler-Lagrange equations, which become $\frac{\partial L}{\partial A_\mu} - \partial_\mu\frac{\partial L}{\partial(\partial_\mu A_\nu)} = - \partial_\mu\frac{\partial L}{\partial(\partial_\mu A_\nu)}$, but I'm confused about how to proceed from here. I know I should end up with $\partial_\mu F^{\mu\nu} = 0$, but I don't quite see why. Since $\mu$ and $\nu$ are dummy indices, I should be able to change them: how do the indices in the lagrangian relate to the indices in the derivatives in the Euler-Lagrange equations?


title: How to calculate the density of relic neutrinos?

text: May be not neutrinos, but antineutrinos? Or both types? In the last case, why they didn't annihilate and what is the ratio of relic neutrinos to relic antineutrinos? Is that ratio somehow related to the barion asymmetry? For reference: Relic neutrinos or cosmic neutrino background > Like the cosmic microwave background radiation (CMB), the cosmic neutrino > background is a relic of the big bang


title: Settling of rice as it's cooked

text: I couldn't think of a better title because I'm not sure what's going on with this phenomenon--someone who has an answer, please edit to fit this better. I noticed that when I cook rice, at some point after most of the water has evaporated and the rice is settled at the bottom of the pot, there is a "lattice" of regularly-spaced gaps in how the grains at the top are arranged, with the holes extending down a grain or two. I can try to explain this better if it doesn't make sense, but for now I can't do much better because it mystifies me. What's up here? (Better tag suggestions are also welcome...)


title: General relativity theory

text: As I understand general relativity theory (please correct me if I'm wrong), time becomes dilated and space becomes compressed around mass, and this is responsible for gravity. I'm struggling with precisely how that results in gravitational force between masses. Can someone explain that to me (please keep any required math understandable to a lay person).


title: Einstein's theory tells us that gravity is a curve in space and time but how does that causes attraction in mass?

text: The sun is incredibly massive object and it causes the space around it to bend. This causes the planets to pulled to the sun or the planets move in an elliptical path around the sun. But I don't understand how the curvature in space can cause the pulling or what cause the planets to move in elliptical path?


title: Any use for $F_4$ in hep-th?

text: In high energy physics, the use of the classical Lie groups are common place, and in the Grand Unification the use of $E_{6,7,8}$ is also common place. In string theory $G_2$ is sometimes utilized, e.g. the $G_2$-holonomy manifolds are used to get 4d $\mathcal{N}=1$ susy from M-theory. That leaves $F_4$ from the list of simple Lie groups. Is there any place $F_4$ is used in any essential way? Of course there are papers where the dynamics of $d=4$ $\mathcal{N}=1$ susy gauge theory with $F_4$ are studied, as part of the study of all possible gauge groups, but I'm not asking those.


title: Scherk-Schwarz and other compactifications?

text: I have been thinking about various types of compactifications and have been wondering if I have been understanding them, and how they all fit together, correctly. From my understanding, if we want to compactify spacetime down from $D$ to $d$ dimensions by writing $M_D = \mathbb{R}^d \times K_{D-d}$. We can do this the following way: "General" compactification: > Find the universal cover of $K$, and call it $C$. $G$ is a group that acts > freely on $C$, and $K = C/G$. > > Then, the $D$-dimensional Lagrangian only depends on orbits of the group > action: $\mathcal{L}_D[\phi(x,y)] = \mathcal{L}[\phi(x,\tau_g y)]$, $\forall > g \in G$. > > A necessary and sufficient condition for this is to require that the field > transform under a global symmetry: > > $\phi(x,y) = T_g \phi(x,\tau_g y)$. "General" compactifications seem to also be called Scherk-Schwarz compactifications (or dimensional reductions if we only keep the zero modes). An "ordinary" compactification has $T_g = Id$, and an orbifold compactification has a group action with fixed points. Assuming this is correct, is this the most general definition of a compactification? Is it reasonable to introduce gauge fields by demanding that the $T_g$ action be local instead of global? I thought we should generally not expect quantum theories to have global symmetries, but any reference I've seen seems to use only global symmetries in the Lagrangian.


title: Can the concurrence be calculated in terms of the entanglement of formation?

text: If I somehow know the entanglement of formation, $E_F$ for two mixed qubits, where \begin{equation} E_F = -x \log x - (1-x) \log (1-x), \end{equation} where $x = (1+\sqrt{1-\mathcal{C}^2})/2$ and $\mathcal{C}$ is the concurrence, can I then calculate the concurrence from it? (Rather than calculating the concurrence 'normally' using $\mathcal{C} (\rho) = \max \\{ 0, \lambda_1 - \lambda_2 - \lambda_3 - \lambda_4 \\}$ where $\lambda_i$ are the square roots of the eigenvalues of $\rho S \rho^* S$ and $S = \sigma_y \otimes \sigma_y$).


title: Atomic Physics: stimulated emission

text: I'm studying a chapter about atomic physics right now but there's thing I just don't seem to understand. When stimulated emission occurs, there's an incoming photon which stimulates the atom to go an energy level lower with the result that a new photon gets emitted. Can someone explain the reason why the incoming photon stimulates the atom to emit a second one? To me it seemed more logic for the atom to absorb the photon and go an energy level up. There seems to be a phenomenon which I don't know about.


title: Technical naturalness of Yukawa couplings

text: Naturalness in the sense of 't Hooft tell us that a small parameter is a signal of a symmetry such that the parameter will be zero when the symmetry is exact. I am puzzled about how this principle is applied now that the yukawa of the top is confirmed to be -at low energy- practically $y_t=1$, while all the other yukawas are a lot smaller. It seems that this naturalness principle claims for a symmetry that does not protect the top quark, but protects all the other quarks and leptons. Is there such symmetry in the standard model, or in the most promising BSM extensions (Susy guts, etc)?


title: Is there a direct physical interpretation for the complex wavefunction?

text: The Schrodinger equation in non-relativistic quantum mechanics yields the time-evolution of the so-called wavefunction corresponding to the system concerned under the action of the associated Hamiltonian. And this wavefunction is, in general, complex, and its modulus squared yields the probabilities observed experimentally. Though, perhaps, this question has been asked many times, I am wondering if there is a direct physical interpretation - something that physically corresponds to - the wavefunction. Or is it just an intermediate calculational tool to arrive at the appropriate predictions for experimental outcomes, and nothing more? Of course, things like superposition and interference effects follow from the complex nature of the probability amplitude. So there must be something physical about it. What is it? Or are we not _supposed_ to ask that question? Is is because the probability amplitude is complex that we have difficulty in relating it to something physical? Can we do quantum mechanics without complex numbers?


title: Kerr Geometry, Separability and Twistors

text: One of the remarkable properties of the Kerr black hole geometry is that scalar field equations separate and are exactly solvable (reducible to quadrature), even though naively it does not have enough symmetries to justify this statement (there is no maximal set of Killing vectors, but apparently there is a higher rank covariantly constant tensor which does the job). I once overheard some vague statement to the effect that this simplicity is somehow related to, or more manifest in the language of twistors in this geometry, or relatedly to properties of the massless Dirac equation in this background. I am wondering if someone can help me pin down the precise statement, or provide me with entry point to the literature.


title: What does plasmon look like in 3D band structure graph?

text: Consider metal, and its reciprocal lattice representation with Fermi surface. What is the correct way to represent a plasmon in this system? M.b. rotating points on the surface? Or 3d membrane-like oscillation of surface? Or what?


title: Work done by gravity on Water

text: ![enter image description here](http://i.stack.imgur.com/qeiCo.png) Now according to me we would see change in potential energy of system and equate it to the work done by gravity. But when we see this the first column lowers by $H/2$ and right one rises by $H/2$ and thus there should be no net work done. But it's not so. Work done by gravity is +ve. Where is it wrong?


title: Alternative career paths in physics?

text: What do you guys think of alternative career paths in physics away from academia free from the usual academic shackles? Examples: Garrett Lisi who spends his time surfing and skiing while not working on E8. Or Daniel Bedingham who is able to support himself part-time as an investment banker when not working on quantum mechanics. Or Julian Barbour who supports himself by translating Russian journal articles into English when not working on quantum gravity. Is it feasible to count upon Foundational Questions Institute to support such unconventional career paths using Templeton money?


title: Lorentz engine to run a car?

text: Could a Lorentz engine (like the ones that move the write head inside of hardrives) be used to run car ? Would it be possable to get it producing 60hp, and a decent amount of tourque, if so how large would it have to be / what would its power consumption be ?


title: How to visualize a 3D scalar function?

text: In physics, we regularly deal with observables, which are functions $f(x,y,z)$ of the position vector $\vec r = (x,y,z)$. How, do you plot such a function? For functions $y=f(x)$ of 1 parameter, we use regular $x$-$y$ line or scatter plots. (These are sometimes called 2D plots.) For functions $z=f(x,y)$ of 2 parameters, we can use contour/density or surface plots (and modern plotting tools render them nicely.) The latter are often called 3D plots. For functions $f(x,y,z)$, I'm at a loss. How do I plot/visualize a three- dimensional density? The best approach I have seen so far is a set of two 2D-density plots $f(x,0,z)$ and $f(0,y,z)$ for the intersections with the $x$-$z$ and $y$-$z$ plane. The final plot shall be printable, i.e. convey the message without relying on animation or user interaction.


title: Can we impose a boundary condition on the derivative of the wavefunction through the physical assumptions?

text: Consider the Schrödinger equation for a particle in one dimension, where we have at least one boundary in the system (say the boundary is at $x=0$ and we are solving for $x>0$). Sometimes we want to impose a boundary condition in which the wavefunction vanishes (Dirichlet boundary condition). We can indirectly impose this boundary condition through the physical assumptions by using an infinite potential outside the relevant region (like in the "particle in a box" model): $$ V(x<0)=\infty ~~~~\Longrightarrow ~~~~\psi(x=0)=0 $$ What if we want to impose a boundary condition in which the _derivative_ of the wavefunction vanishes (Neumann boundary condition)? $$ ? ~~~~\Longrightarrow ~~~~ \left. \frac{\partial \psi}{\partial x} \right|_{x=0} = 0 $$ Is there a way to choose the potential, or maybe change something else in the Hamiltonian, in order to indirectly impose this boundary condition? P.S. This question is not of great practical importance, it is more of a curiosity.


title: What tree-level Feynman diagrams are added to QED if magnetic monopoles exist?

text: Are the added diagrams the same as for the $e-\gamma$ interaction, but with "$e$" replaced by "monopole"? If so, is the force between two magnetic monopoles described by the same virtual $\gamma$-exchange diagrams? I'm anticipating that the answer is 'no', because otherwise I don't see how one could tell magnetic monopoles and electrons apart (besides their mass and coupling strength to the photon). Obviously magnetic monopoles behave differently from electrons when placed in an $E$ or $B$ field.


title: Can someone explain this intuitively please?

text: In a static situation we defined voltage as energy/unit charge, or j/c. As the distance between the charged particles increased, the voltage decreased. Now why do we not apply this in a simple DC circuit? When the electrons move inside the wire, they are changing their distance from the cathode. By our definition of voltage, the voltage throughout the wire should vary along the length of the wire, resistor or not. However, it only changes when there is a resistor. Can anyone explain this? Thanks!


title: Why put something in the middle of the pan would prevent soup coming out during cooking?

text: I see a website which teach people to put something on top of the pot during cooking soup to prevent the soup coming out during cooking. What is the physical theory behind that? ![enter image description here](http://i.stack.imgur.com/YAyKl.jpg)


title: Did Penzias and Wilson need such a large horn antenna to discover the CMB?

text: Penzias and Wilson discovered the 2.7K Cosmic Microwave Background (CMB) Radiation using a 6 meter horn antenna, along with a cryogenic low-noise detector measuring at 4 GHz: http://en.wikipedia.org/wiki/Discovery_of_cosmic_microwave_background_radiation My question: Did they need such a large antenna to do this? I think that according to the Antenna Theorem, an isotropic and uniform distribution of thermal background radiation will deliver the same power to a detector regardless of the antenna's size (i.e. it's gain), as long as it is directive enough to look up in the sky and not at the hot earth background. This measured power should be kTB, where B is the bandwidth and T the 2.7K background temperature. So could the CMB have been discovered with a much smaller horn antenna, for example one maybe just ~10 inches in diameter?


title: Can we reach absolute zero one day, and if so, can we measure it?

text: I was reading this question, and I had this doubt, if we can ever reach absolute zero, how can we measure it? I ask this because as far I know, any measurement could change (increase) the temperature. **EDIT** Reach absolute zero is a hypothesis. My doubt is, if hypothetically one day we can get to absolute zero, have a way to measure this temperature? Or similarly to quantum computer, can not prove that is really quantum, can not prove that is really in -273 Celsius, or 0 K.?


title: Misconception about the expectation of a quantum system

text: For a two-level quantum system with energy eigenstates $|\phi_1\rangle$ and $|\phi_2\rangle$ at finite temperature, we can write a general state as $$|\Phi\rangle=c_1|\phi_1\rangle+c_2|\phi_2\rangle.$$ In quantum mechanics, we know that the expectation of an operator $\hat{O}$ is given by $$\langle\Phi|\hat{O}|\Phi\rangle = |c_1|^2\langle\phi_1|\hat{O}|\phi_1\rangle +|c_2|^2\langle\phi_2|\hat{O}|\phi_2\rangle +c_1^*c_2\langle\phi_1|\hat{O}|\phi_2\rangle +c_2^*c_1\langle\phi_2|\hat{O}|\phi_1\rangle.$$ However, in statistical mechanics, the expectation value is instead $$\text{probability}(\phi_1)\langle\phi_1|\hat{O}|\phi_1\rangle + \text{probability}(\phi_2)\langle\phi_2|\hat{O}|\phi_2\rangle,$$ which simplifies to only the first two terms of the expression obtained from quantum mechanics. What is wrong here?


title: Background subtraction for a signal ans Errors Analysis

text: I use a CCD to see the split of a energy level due to Zeeman effect. I have a 1 dimensional CCD of 7926 pixel of 7μm each one. My CCD analyze a region 2 dimensional, and then it steps forward 200 times. So, i have a matrix like this !["matrix"](http://i57.tinypic.com/xn5f7s.png). I select the background between 40 and 60 pixel and i calculate for each row the mean value. Then, i project these 7926 mean values over the Y axis. So, for each bins of the row of the matrix(i.e. 3040) , i subtract the value of the content of 3040 pixel of the background This is my background: !["background"](http://i.stack.imgur.com/pZUaX.png) (in x there are the pixels of my CCD and y is the intensity). I have to subtract the background from the signal, subtracting bin for bin. At each bins of the background it is associated an error $\sigma = \sqrt{N}$ where $N$ is the bin content. Each bins of the signal has an error $\sqrt{N}$, but when i subtract the backgroun how can i calculate the exact error of the pure signal? If $Z'_i$ is the bin content after the subtraction, $Z_i$ the bin content before and $Y_i$ the bin content of the background, then $Z'_i = Z_i - Y_i$ and the error (throw the propagation) $\sigma_{Y_i} = \sqrt{\sigma_{Z'_i}^2 + \sigma_{Y_i}^2 +....}$ In fact, I think, there should be a term of covariance. How can i compute it?


title: Reflection & Transmission curves of waveguides

text: I have simulated a grating-coupled waveguide in Comsol Multiphysics, the waveguide consists of 2 dielectric layers and air as the cover of waveguide. I have excited the modes of this structure by a plane wave using diode laser (1550 nm). Comsol Multiphysics calculated the reflection and transmission coefficients by scattering parameters and plotted the curves. Do the reflection and transmission curves change by increasing the air layer? I think evanescent wave could not reach the top of the layer and would not affect the curves.


title: Understanding the Charge Conjugation Operator

text: I am trying to understand the charge conjugation operator. http://en.wikipedia.org/wiki/C_parity Because the operator is Hermitian, this seems to imply that there is a (possibly spontaneous?) physical process through which a particle can (instantaneously?) change into its anti-particle. Searching for charge conjugation in general only gives me a lot of math, which I think I understand, what I don't understand is when and how charge conjugation can occur. Where can I find more information about this process? Alternatively, if charge conjugation cannot occur, why do we need a charge conjugation operator?


title: Does a reflection still transfer momentum to an mirror?

text: I have been recently wondering, if I take a powerful enough energy source (photon) and I have an **perfect** mirror exactly in front of it and assume an "emitter" shot the light towards the mirror. As perfect mirrors absorb no energy of ANY kind from photons, should this mean that the perfect mirrors would never move due to transfer of momentum of the light?


title: How to make charges of a charged conductor evenly distributed over the surface of the object?

text: Is there a way to make the charges of a charged conductor evenly distributed over the surface? I know the charges of a charged insulator are evenly distributed but I want to know that if there is a way to make the charges of a charged conductor evenly distributed too.


title: What makes a wrist-energized gyroscope rotate faster?

text: I'm considering a wrist-energized gyroscope, shown below (after my daughter let it fall and it broke open). That one was sold as Roller Ball, but variants are known as Powerball, DynaBee, Dynaball, Dyna-Flex, Wrist Ball, NSD Spinner.. ![Roller Ball](http://i.stack.imgur.com/XGJtH.jpg) There are essentially 3 relatively moving parts:   * the rotor (yellow) and its (metal) shaft;   * the washer (white);   * the casing (green); normally its two halves are glued together. There's a circular groove in the joint of the casing. The groove holds the washer and rotor, is larger than the washer's thickness, and slightly larger than the diameter of the shaft at its extremity. There are essentially 2 degrees of liberty:   * The rotor can spin w.r.t. the washer, with little friction, around the rotor's axis. It also touches the casing's groove, on which it can roll with some friction. _Update_ : the diameter of the shaft is reduced to 2.5 mm at its rolling extremity.   * The washer can rotate w.r.t. the casing, with some friction, around an axis perpendicular to the plane of the casing's groove. _Update_ : its outer diameter is 60 mm. The two axis are perpendicular and coplanar. To operate the device, the rotor is given an initial spin-up (one method is using a string wound on the rotor; that's the sole purpose of the groove on the rotor, and of the big hole in one side of the casing). Then the device's casing is held firmly in the hand, and its orientation changed in a certain manner, with the effect that **the spin rate of the rotor increases** , reportedly well over 10,000 RPM for skilled operators. I conjecture that the washer is non-essential, as the device was first described without it in US patent 3,726,146, from which the following illustration is taken. ![Mishler](http://i.stack.imgur.com/YB3Ky.png). * * * Question 1: **What causes this increase of the spin rate of the rotor**? _Update_ : In particular, if that involves friction between casing and rotor, how come that _increases_ the spin rate, even though the casing is NOT rotated faster than the rotor (in a rotation movement with the rotor's axis), giving some credibility to the argument that friction of rotor on casing can only slow the spin rate of the rotor? Question 2: Is there a convincing argument that if the rotor was practically frictionless (and not touching the casing), the spin rate of the rotor could not be increased? * * * _Update_ : I think the rotor is rolling (rather than sliding) on the casing's groove, and now realize that this friction is instrumental in spinning-up the rotor: when the casing's grove gets polished, or worse greasy or oiled (reducing friction), it becomes harder or impossible to spin-up the rotor. When the rotor is spinning steadily with the casing passively held, and I observe the washer (or equivalently the rotor's axis), there are clearly three discrete modes:   * washer does not move (relative to casing);   * washer rotates at constant speed (decreasing steadily with the rotor spin rate) in one direction;   * washer rotates at same constant speed in the other direction. This is consistent with the hypothesis of **a rolling of the rotor on casing, each extremity of the rotor rolling on a _different_ half of the casing**, with the direction of the rotation depending on which extremity of the rotor rolls on which half of the casing (depending on how the casing is held/moved, there is an occasional quick inversion). At "cruse" speed, I guesstimate the spin rate of the rotor on its axis to be perhaps 100 rotations per seconds within a factor of five, based on the sound/vibration (that's about consistent with this stating 250 Hz as a high spin rate); and the washer's spin rate to perhaps 5 rotations per seconds, based both on observing it when I stop moving the casing, and actuating the casing at a rate of 10 times per second to maintain/increase the spin rate (that casing excitation period corresponding I think to a half turn of the washer). Given how crude these spin rate estimates are, the ratio 100/5 of rotational speed is rather close to the ratio 60/2.5 of diameters.


title: Non-unique pressure for Navier-Stokes for incompressible fluid

text: The Navier-Stokes system for incompressible fluid says $$ \begin{aligned} \dot{\mathbf{u}} + (\mathbf{u}\cdot\nabla)\mathbf{u}-\nu\Delta\mathbf{u}+\nabla{p} &= 0,\\\ \nabla\cdot\mathbf{u}&=0. \end{aligned} $$ The non-slip boundary condition for the velocity $\mathbf{u}$ says $\mathbf{u}|_{\partial\Omega}=0.$ But the pressure boundary condition is unclear. Most references I found use Neumann boundary condition which is $\nabla{p}\cdot\mathbf{n}=0$ on $\partial\Omega$ ($\mathbf{n}$ is normal vector of $\partial\Omega$). As consequence, $p$ can be determined **up to an arbitrary additive constant**! Does this make any sense in physics? How can I determine the constant? There is also an initial value condition for $p$. We can add an arbitrary function $q(t)$ satisfying $q(0)=0$ to $p(\mathbf{x},t).$ That is, if $p(\mathbf{x},t)$ is a solution, so is $p(\mathbf{x},t)+q(t).$


title: How to deal with $\vec{j}\cdot\vec{A}$ or $\rho A^2$ interaction when utilizing Kubo formula? Gauge invariance?

text: If there exist electromagnetic fields in solids, electrons can feel interactions like $\vec{j} \cdot \vec{A}$ or $\rho A^2$ (these are not regarded as perturbations). But these are not gauge invariant. Then we add some other perturbation fields to study their corresponding linear response. I can't see whether brute-force Kubo formula evaluation gives gauge invariant result.


title: Interpretation of Einstein relation in kinetic theory

text: I am reviewing my (independent) study notes on diffusion and found the following comment preceding the derivation of the Einstein relation: > Now, since the particles that deliver the stochastic force, $f(t)$, to the > Brownian particle are the very same particles that supply the drag force > $=-m\zeta v$, we should, at the very least, expect a relationship between > $\lambda$ (characterizing the stochastic force) and $\zeta$ (characterizing > drag force). Here $\lambda$ appears in the autocorrelation function $\langle f(t)f(t') \rangle = \lambda \delta(t-t')$ of the stochastic force. I specifically remember writing this comment as my own interpretation, but never consulted anyone to confirm. So my question here is, is this interpretation correct?


title: In which mathematical space do magnetic fields live in?

text: A magnetic field may be described as a mapping $\mathbb{R}^3 \to \mathbb{R}^3$. Therefore, it is a function, and apparently can be described as an element in a function space. Is there some particular (perhaps Hilbert) space that magnetic fields belong to?


title: pp chain gamma decay / intermediate state

text: Based on the first equation at http://en.wikipedia.org/wiki/Proton%E2%80%93proton_chain_reaction $$_1^{1}H+_1^{1}H \to _2^{2}He+\gamma$$ Is it correct to say that in the P-P chain two hydrogen atoms fuse to form a helium-2 isotope in an excited state which decays by gamma emission to a helium-2 isotope in a non-excited state, emitting a photon in the process? $$_1^{1}H+_1^{1}H \to _2^{2}He^* \to _2^{2}He+\gamma$$ Or is it just a one-step process without excited helium-2 intermediate?


title: Art : A Particular Mirror

text: There is a particular mirror used here (1 minute 26 seconds) in this video, which reflects light in a way I've never seen. I was hoping maybe one of you might know what kind of mirror this is, so that I may use it for my own personal use.


title: Physics of donut magnets levitating vertically on a pencil?

text: When you put identical donut magnets on a pencil horizontally, the magnets all repel from each other in a way that creates equal spacing between each magnet. However, spacing is not equal when the pencil is held vertically and spacing increases from the bottom up so it looks like this: ![magnets on a pencil](http://i.stack.imgur.com/KvcRW.jpg) I think that the weight of each magnet has something to do with this, but I cannot figure out exactly why. Help?


title: what is the combined partition function of two similar but independent systems?

text: i was reading Runnels' paper on cayley tree where he has squared the partition function of a cyley tree to get that of two exactly similar trees. why square? why not add the two partition functions to get that of the whole system? im an undergraduate student, may be this is a very dumb question.


title: Real silicon atom structure?

text: Text books say that a pure silicon atom has four electrons in its outermost orbit. How do the scientists know it has four electrons?


title: Origin of Rayleigh scattering

text: Is Rayleigh scattering simply the elementary result of scattering theory, that, at low energies (long wavelengths) the scattering is dominated by $s$-wave scattering?


title: Light Waves and Light Photons gedanken Experiment

text: Suppose you have a source of light that emits light with a wavelength of 2 meters, and you set the device to be turned on and switched off alternately. You also set it so that each interval the device is turned on is only long enough for 1 meter to be emitted (1/2 a wavelength). Do you ever observe any photons?


title: Why doesn't rain fall down in streams (as opposed to drops)

text: Why is it that raindrops don't collide and 'stick together' on their descent to Earth, arriving in streams rather than separate drops?


title: When a star becomes a black hole, does its gravitational field strength become stronger?

text: I've seen in a documentary that when a star collapses and becomes a black hole, it starts to eat the planets around. **But it has the same mass, so how does its gravitational field strength increase?**


title: Black Hole mass increase from infalling body

text: When a body falls into a black hole, the black hole's mass increases. Does this mass increase depend only on the rest mass of the infalling object, or is it affected by the velocity of the infalling mass. Of course, if it is velocity dependent, then does it depend on the relative velocity between the black hole and the infaller or the velocity between the infalling mass and an external observer. If the second situation is the case, is the mass of a black hole dependent on an external frame's velocity relative to everything that has ever fallen into it (seems highly unlikely)? If it is not velocity dependant, then what happens to the excess kinetic energy (measured say, in an accelerated frame of reference at a constant distance from the black hole)?


title: How can any QM interpretations which use a linear Schrödinger Equation be used to quantitize gravity?

text: Since general relativity is nonlinear, how could we quantitize gravity with QM interpretations which use the linear Schrödinger Equation? Or is this fundamentally unworkable?


Can anyone please explain Hawking-Penrose Singularity Theorems and geodesic incompleteness? In easy to understand plain English please.


title: Dirac fermion in curved space

text: What is the connection between Dirac equation in curved space-time and effective Hamiltonian for Dirac fermion in curved space (topological insulators)? I am trying to find this connection but I am stuck a little. I tryed to do this for cylinder - just changing the metric $$ds^2 = c^2dt^2 - dx^2 - dy^2 - dz^2$$ to $$ds^2 = c^2dt^2 - dr^2 - r^2d\phi^2-dz^2$$ deriving Dirac equation (e.g. wikipedia) in this geometry and then taking $r = const.$ Unfortunatelly, there is no angle dependence in my equations, but people who do this have $sin$ and $cos$ e.g. Zhang, Rosenberg, Zhang as effective hamiltonian. Could You please help me? In deriving the Dirac equation I used the folowing tetrad: $e_{r}^{2} = 1$, $e_{r}^{3}=0$, $e_{\phi}^{2} = 0$, $e_{\phi}^{3}=r$. What about this choice: $e_{r}^{2} = \cos\phi$, $e_{r}^{3}=\sin\phi$, $e_{\phi}^{2} = -r\sin\phi$, $e_{\phi}^{3}=r\cos\phi$. I will get the same result at the end or different? Tetrads are not unigue I guess? How to choose them correctly?


title: Bosonization on the lattice fermion - a rigorous mapping

text: An inquiry: usually the bosonization is done on the field theory side. The mapping between the fermion operator to the boson operator is done for the **field theory operators**. As far as we know for the bosonization on the lattice fermion, **do we know whether there is any exact result about the rigorous mapping: from the lattice fermions to a lattice boson?** Any recommended approach or Ref on the rigorous "bosonization on the lattice fermion"? ps. I have some fair understanding of uses of field theory Bosonization. Just wonder the lattice side procedure how well is it rigorous studied. Thank you!


title: How is it that a tachyon is faster than $c$?

text: I remember reading that a tachyon is faster than light, but am still confused as to how and if they could exist in our universe?


title: How small can you be and still hear?

text: According to my understanding, at the smallest level sound is just a collection of particles colliding on a wave. So, assuming we can arbitrarily reduce the size of any human being, how small can we reduce someone while this someone can still "hear" normal size human beings talk coherently. EDIT: In other words, how small do we have to be in order for it to become impossible to differentiate random collisions with other particles from collisions that result from the wave? EDIT2: To avoid the shrinking human problem, how about thinking about small recorder? How small can the recorder be in order to record (and play back) frequencies heard by the human ear and for it to NOT play random noise generated by random collisions?


title: Videos of changing the orientation of an astronaut in space

text: Kane, Headrick and Yatteau describe in their paper "Experimental investigation of an astronaut maneuvering scheme" possible maneuvers to change the orientation in space without external torque. Is there any video (best available in the web) which shows real astronauts doing such motions?


title: Basic Question - Green's Functions in Quantum Mechanics

text: I am trying to learn about Green's functions as part of my graduate studies and have a rather basic question about them: In my maths textbooks and a lot of places online, the basic Greens function G for a linear differential operator L is stated as $$ L G = \delta (x-x') $$ which is all well and good. I am now reading Economou's text on GF in Quantum Physics where he goes to define Green's functions as solutions of inhomogenous DE of the type: $$ [z - L(r)]G(r,r';z) = \delta (r-r') $$ Where $z = \lambda + is$ and L is a time independent, linear, hermitian differential operator that has eigenfunctions $\phi_n (r)$ $$ L(r) \phi_n (r) = \lambda_n \phi_n (r)$$ Where these $\lambda_n$ are the eigenvalues of L. Where does this z come from in the second equation and what is the link between this and the first one? Edit: see my post below for a new couple of questions.


title: How can time be relative?

text: I don't understand how time can be relative to different observers, and I think my confusion is around how I understand what time is. I have always been told (and thought) that time is basically a measurement we use to keep track of long it has been since an objects inception. If that is even somewhat true, how can time be relative? If I have a rate of decay of X and you are somehow able to observe that (such as watching me age) how could I age at a different rate to 2 observers? If time slows down the faster you go, does that mean you age slower? Or do you age at the same rate, only it seems like it takes longer? If a second is currently defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom at 0K, **how** can that change at higher speeds? Assuming the temperature stays the same, shouldn't the measurement be the same? Moving faster surely can't cause cells to decay slower, or atoms to radiate slower...can it? Can someone explain, in the simplest of terms, how time can be relative?


title: Why is the time taken for something to fall proprtional to acceleration due to gravity?

text: This is a continuation of this question. I saw in a piece of writing (that I no longer seem to be able to aquire) on dimensional analysis that: $$t \propto h^\frac{1}{2} g^\frac{-1}{2}.$$ How can $t$ be proportional to $g$ when $g$ is a constant? I ask this because as $t$ rises surely $g$ can neither rise nor fall, which makes the thought that it cannot be proportional to $g$ spring to mind immediately.


title: Have I calculated the Newtonmeter ($Nm$) of torque for my corded drill correctly?

text: The corded drill is rated at: 450watts at 3000rpm the calculation I have used is: $$\frac{450}{3000} \times 2 \times \frac{3.14}{60}=0.0157 $$ I feel I have made a basic error somewhere (as the figure is very low) but dont immediately see it.


title: Why does the topological entropy scale with $\log(L)$ in 1D?

text: Why, in one dimension, does the topological entropy scale with the size of system as $S \sim \ln L$, while in a 2D system it scales with $S \sim L$? Why does dimensionality play such an important role here? I mean, is there any simple but straightforward idea to understand these results?


title: Does water in the drain follow the distance-squared rule of force?

text: When I shower I notice that the soapy water spinning around the drain often takes the form of a spinning galaxy, complete with spiral arms. Does this indicate that the water's speed around the drain is inversely proportional to the square of its distance from the drain? Actually, I have heard that spinning galaxies themselves don't follow this rule, hence the discovery of dark matter, so does the spinning water follow whatever rule that the spinning stars do? Thanks.


title: Why quantum spin liquid has negative Curie-Weiss temperature?

text: In a table in wikipedia, the Curie-Weiss temperatures of quantum spin liquids are listed. Among them, the $\Theta_{cw}(K)$ are less than zero. Why are they negative? Since temperature is defined as  $$T = \frac{dq_{rev}}{dS}$$ I would expect the additional heat let the spin liquid tend to disorder.


title: Can planes generate enough force to displace cloud, such that the displacement is visible from the ground?

text: Is it possible for planes (helicopters, jets, commercial airplanes, etc) to generate enough force to displace any amount of cloud, such that the displacement is visible from the ground? Assuming an observer is standing on the ground, and viewing clouds in the sky. Can anyone comment on whether it is possible for any planes (as above) to generate enough force to move an amount of cloud such that the movement is visible from the ground? I'm assuming clouds can be moved, as I've often seen winds that blows clouds a certain distance.


title: Magnetic moment derivation from Dirac equation

text: I am reading a text book where they show the electron has spin 1/2 using Dirac's equation. At one point in the derivation they define $\pi=P-qA/c$ where $P$ is the momentum operator and A is the vector potential. They then claim that $\pi\times \pi=iq\hbar B / c$ where B is the magnetic field. Apparently $\nabla\times A=B$ as we are assuming the scalar potential is static. My question is what happened to the $A\times P$ term in $\pi\times\pi$, why is that set to zero?


title: Large-Scale-Structure (LSS) and the Fingers-of-God

text: In the Large-Scale-Structure (LSS) artifacts named fingers-of-god are apparent in the redshift space and justified by > "The large velocities that lead to this effect are associated with the > gravity of the cluster by means of the virial theorem; they change the > observed redshifts of the galaxies in the cluster" There are few references, (Jackson 1972) (*), and Annu. Rev. Astron. Astrophys. 1988. 26: 631-86  in the WP article and I do no understand how the radial direction preference is justified and I could not find any source. I've never heard of a N-Body simulation on Large Scale able to reproduce those artifacts. Am I correct? Can anyone make this issue more clear, please ? (*) it is more a no-go than an explanation.


title: Is there any correlation between the energy density fluctuations of two separate systems in a vacuum state?

text: I think the title says it all. What I am curious to find out is if there are any observable changes in the fluctuations of zero-point energy in a vacuum state system that are the consequence of operations performed at a separate vacuum state system. I am not simply asking if there are correlations naturally, I am asking if there is any operation/setup/configuration that could be implemented to derive such a result. -Thanks-


title: why exactly is ice less dense then water?

text: The answers to this question explain that ice is less dense than water because it has a "crystal structure", but they dont explain what exactly that is and why this happens, also I saw this answer from another site stating that not all ice is less dense than water. What is the "crystal structure" that ice has? Why is ice structured that way? Can ice be more dense than water, and if yes, how and when?


title: In hydrostatics, is it possible for the bouancy to be greater than the weight of a body?

text: In the diagram we have two bodies, with density and volume $\delta$ and $V$, and a fluid $\ell$. $$\delta_1=0,7 \rm\frac{gr}{cm^3}$$ $$\delta_2=2 \rm\frac{gr}{cm^3}$$ $$V_1=100 \rm {cm^3}$$ $$V_2=10 \rm {cm^3}$$ $$\delta_\ell = 1,2 \rm\frac{gr}{cm^3}$$ The system is static. ![enter image description here](http://i.stack.imgur.com/E0db1.jpg) I need to find $(1)$ The tension of the rope $(2)$ the buoyancy $1$ receives due to the liquid. I solved it, but I'm getting strange results: Let $W$ be the weighs, $B$ be the buoyancy and $T$ the tension. Then, since the system is static: $$B_1+B_2-W_1-W_2=0$$ (the tensions cancel out) But $W_1= \delta_1 \cdot g \cdot V_1$, $W_2= \delta_2 \cdot g \cdot V_2$ and $B_2 = \delta_\ell \cdot g \cdot V_2$ (since the body is fully submerged): $$\eqalign{ 10\frac{{\text{m}}}{{{{\text{s}}^{\text{2}}}}}\left( {70{\text{gr}} + 20{\text{gr}} - 12{\text{gr}}} \right) = B_1 \cr 10\frac{\rm m}{{{\rm s^2}}}\left( {0,078\rm{kg} } \right) = B{_1} \cr 0,78{\text{N}} = B{_1} \cr} $$ It seems strange that the buoyancy is greater than the weigh of the body ($0,7 \rm N$), is that possible? Also, I can't figure out how to obtain the tension.


title: Resistance of a two-dimensional sample

text: In this review of the QHE, Steve Girvin makes the following statement (bottom of pg. 6, beginning of Sec. 1.1.1): > As one learns in the study of scaling in the localization transition, > resistivity (which is what theorists calculate) and resistance (which is > what experimental- ists measure) for classical systems (in the shape of a > [d-dimensional] hypercube) of size L are related by [12,13] > > $$ R = \rho L^{(2-d)} $$ where $R$ is the _resistance_ of the hypercube and $\rho$ its _resistivity_. For $d=2$ it is clear that the resistance and the resistivity of the sample are the same, and as Girvin goes on to say, this fact is crucial for the universality of the quantization of the resistivity plateaus in the QHE. My question is: **Is there a simple physical explanation of the above relationship?** Girvin cites two papers. Both are RMP: "Disordered Electronic Systems" by Lee and Ramakrishnan and "Continuous Quantum Phase Transitions" by Sondhi et al. I am hoping that someone can spare me the effort of going through these two papers by providing an answer.


title: Will tensile strength keep a cable from snapping indefinitely?

text: Trying to secure a wall hanging using magnets; me and a coworker came up with an interesting question: > When the hanging is hung using 1 magnet, the weight of it causes it to > quickly drag the magnet down and the hanging drops. Using _n_ magnets > retards this process; causing it to fall more slowly, but does there exist a > number of magnets _m_ such that their combined strength will prevent the > hanging from slipping, entirely and permanently? Because this doesn't make for a very good question; we worked at it and arrived at a similar one; but slightly more idealized: > A weight is suspended, perfectly still, from a wire in a frictionless > vacuum. If the mass of the weight is too great; it will gradually distend > the cable, causing it to snap and release the weight; but will a light > enough weight hang there indefinitely, or will the mass of the weight (and > indeed the cable) cause the cable to snap sooner or later?


title: Hydrostatic equilibrium of a star - simplification math error

text: I am trying to do this question and getting stuck at interpreting what the question is asking for. > $$dP/dR = -g(R)\rho(R) = -[GM(<R)/R^2]\rho(R).$$   >  Thus when we descend a distance $\Delta R$ from the surface $R_s$ where > $P=0$, the pressure will be approximately:   >  $$P = 0-\Delta P = GM(<R)/R^2\rho(R)\Delta R.$$ > > By taking $\Delta R = R = R_s$, the radius of the star, and $\rho(R) = > \bar{\rho} = 3M/(4\pi R_s^3)$, the mean density of the star, derive a rough > estimate for the central pressure of the star and show that the central > pressure is independent of mass for the scalings given above. I am substituting the mean density equation instead of $\rho(R)$ into the second equation on the image and simplifying it. I am not sure that $<R$ mathematically represents and when I simplify it I get $P = (3M)/(4\pi R_s)$. I'm not sure if this is correct because my answer still has mass.


title: Chaos is predictable?

text: I'm reading a book of computational physics [1] where the driven nonlinear pendulum is studied in deep. This is the equation used in the book: $$ \frac{d^2\theta}{dt^2} = -\frac{g}{l}\sin\theta - q\frac{d\theta}{dt}+F_d\sin(\Omega_dt) $$ Obviously the authors know when (whit which conditions) the pendulum switch to a chaotic regime. My question is the following: looking at the ODE, we can see (predict) the existence of chaotic behavior? If yes, this is valid for all the ODEs or only for a strict family of ODEs? [1] _Computational physics_ , N. J. Giordano and H. Nakanishi, Second Edition, Pearson Prentice Hall, 2006


title: For two planets in a 2:1 mean motion resonance , where will their periastron and apastron points be?

text: I want to animate the changing orbits of planets when they enter a mean motion resonance. Using a 2:1 resonance, I want to show a low-mass inner planet and a low-mass outer planet being tugged by a Jupiter-mass planet in the middle. I know the orbits of both become increasingly elliptical but I don't know in which direction the ellipse becomes extended. I want to include images of the planets' in orbit, too, so I want my points of periastron and apastron to be realistic relative to the position of the planets in their orbits. > Would the periastron point of the inner planet be near the point of 2:1 > resonance where my Jupiter-mass planet and the inner planet are nearest each > other, or on the opposite side, as I believe the case is with Neptune and > Pluto?   >  The same question for the outer planet, would it's periastron be closest to > the point where it is closest to my Jupiter mass planet, or on the other > side? Illustrating my question, I have an jpg illustration and animation in Flash at this location: Mean motion resonance animation (Flash) ![enter image description here](http://i.stack.imgur.com/8wpif.jpg)


title: Overtaking with non-constant acceleration

text: I have tried to solve this problem by adding the sum of the displacements during acceleration, constant velocity and deceleration, but it does not work out. Question: > A car accelerates from rest to $20~\text{m/s}$ in $12$ seconds ($a > =5/3~\text{ms}^{-2}$), it travels at $20~\text{m/s}$ for $40$ seconds, then > retardation occurs from $20~\text{m/s}$ to rest in $8$ seconds ($a = > -2.5~\text{ms}^{-2}$). As the car accelerates an RC car, moving parallel to > the car, is moving at $14~\text{m/s}$. When will overtaking occur and what > will the distance be? The RC car passes the car just as it starts to > accelerate. I can do this without a problem if acceleration is a constant. Is there a differential equation I can use to compute this as that is my better area or must I stick with the SUVAT equations? Again, if I could be pointed in the right direction that way I can learn.


title: Parity, how many dimensions to switch?

text: Parity is described in Wikipedia as flipping of one dimension, or - in the special case of three dimensional physics - as flipping all of them. Is there any simple rule that generalises both for any dimension? Like: "Flip an odd number of dimensions."?


title: Simple explanation of quantum mechanics

text: Can you please describe quantum mechanics in **simple** words? When ever I read this word (quantum computers, quantum mechanics, quantum physics, quantum gravity etc) I feel like fantasy, myth and something very strange that i can never understand.   So what is quantum mechanics?


title: Do perfect spheres exist in nature?

text: Often in physics, Objects are approximated as spherical. However do any perfectly spherical objects actually exist in nature?


title: Motion on a smooth surface

text: A particle of mass $m$ is moving on the inner side of smooth circular cylinder of radius $R$ whose $Oz$-axis is vertical and directed downwards. The particle started its motion from the $x$-axis with velocity $V$ which was parallel to $y$-axis. Which forces are acting on the particle during its motion?   How can I show that the Energy conservation law holds and the Angular momentum conservation law holds with respect to the $Oz$-axis?


title: Can observers be particles?

text: Generally Quantum mechanics divides a system what is to be observed and an observer. This is generally taken to be some human being. But why restrict it to such? Why not a particle? Is there a good physical reason or philosophical reason for this to dismissed as not sustainable? I'm thinking here specifically of the Copenhagen interpretation, or of its modern incarnation consistent histories. I understand that decoherence in consistent histories completely replaces the idea of the wave collapse in the Copenhagen Interpretation. Essentially, the idea of observed system and an observer is supplemented with an environment, which on the face of it seems entirely natural. The idea of decoherence comes from statistical physics. I'm suggesting that a particle that acts like an 'observer' needs to 'know' what state the observed system is in to 'know' how to react to it. Decoherence resolves the superposition of states in the observed system to a probability mixture.


title: If quantum computation is reversible, what is the point of Grover's search algorithm?

text: Wikipedia et al say the following about Grover's algorithm: > Although the purpose of Grover's algorithm is usually described as > “searching a database”, it may be more accurate to describe it as “inverting > a function”. Roughly speaking, if we have a function $y=f(x)$ that can be > evaluated on a quantum computer, Grover's algorithm allows us to calculate > $x$ when given $y$. However, since all quantum computation is reversible, can't one simply run the inverse of each step in $f$ in reverse in order to obtain $x$ from $y$?


title: Are the limits to prediction by QM provable?

text: Quantum mechanics states that only two aspects of a quantum system can be predicted with certainty: 1) the average and 2) the standard deviation of many measurements of identically prepared quantum systems. (At least, that's the message I took from my Quantum Chemistry textbook). I believe Einstein did not like the fact that you couldn't predict more than those two things -- that there's a fundamental limit to what science can help us predict about events that do occur. Any other measured information from the system is essentially "random". What I'm wondering is whether QM has definitively proven that we are unable to obtain any other information about the system? For example, assume that with each quantum system there is associated a pseudo-random number generator that we don't know about. The "random" information is actually the result of the next iteration of the generator. All statistical tests would say that the information is random, but if we knew what function the PRNG was using, we could predict this data! I remember reading that experiments have ruled out hidden variable theories. Is a PRNG function an example of a hidden variable? (By the way, I'm not trying to challenge QM; I learn best by asking contradictory questions like this).


title: What could be some applications for a highly compressed magnetic field?

text: If you had a powerful electromagnet in the shape of a hollow sphere in which the top and bottom halves are the opposing poles, what can the magnetic filed at the center of the sphere be used for?


title: Eutectic systems behavior near 100%-0% composition and low temperature

text: I can't understand why in this phase diagram ![http://upload.wikimedia.org/wikipedia/commons/thumb/5/53/Eutectic_system_phase_diagram.svg/350px- Eutectic_system_phase_diagram.svg.png](http://i.stack.imgur.com/B6FMD.png), in the area where the system is approaching the 100% $\alpha$ or 100% $\beta$ composition, the state is just a pure solid. I mean, where does the other component go? If I am interpreting it right, there's no liquid-phase mixture left, and there's neither any solid rest from the other substance.


title: Fast algorithm for maximizing the quantum fidelity

text: Consider the following optimization problem: Given a quantum state $\sigma$, a constant $b$ and a Hermitian operator $A$, find $\underset{\rho} \max F(\rho,\sigma)$ subject to $\text{Tr}(\rho A)>b$, where $F(\rho,\sigma)=||\sqrt{\rho} \sqrt{\sigma}||_1$ is the fidelity function. Does anyone know of a fast algorithm for this problem? Even better, an available package that implements such an algorithm?


title: The Momentum Operator in QM

text: I've seen the 'derivation' as to why momentum is an operator, but I still don't buy it. Momentum has always been just a product $m{\bf v}$. Why should it now be an operator. Why can't we just multiply the wave function by $\hbar{\bf k}$? Why should momentum be a derivative of a wave function?


title: The Quantization of Photon Energies

text: Despite Planck's constant being in $E=hf$, it would appear to me that energy is still not discrete, as frequency can be an fraction of a Hertz that one wants. How does this imply that electromagnetic radiation is quantized?


title: Can walking provide electrical energy?

text: This is more of a theoretical question, but is it possible to convert the energy you use to walk into electrical energy to say, power a light bulb?


title: Equilibrum position of an extensible spring

text: A stone of mass $m$ is attached at one end of a vertical spring whose another (upper) end is fixed at horizontal surface. During the motion, a spring takes vertical direction. Force in spring is proportional to extension. How can I determine equilibrium position? Will it be in equilibrium when forces that act on it are equal by intensity?


title: Is there current flowing though a power source?

text: Would there be any current flowing though an ideal voltage source? I am asking this more specifically when you are analyzing a circuit and you are doing nodal analysis. I know i=v/r, but since the source is ideal, resistance would be zero or infinity?


title: Angular momentum of anistropic harmonic oscilator

text: > _A potential given by : $$ V(x,y,z) = \frac{1}{2}m(x^2+y^2+\frac{z^2}{2}). > $$ Which component of angular momentum is conserved._ An attempt: Angular momentum along z, $ L_{z} = m(x\dot{y} - y\dot{x})$ and $ \frac{dL_{z}}{dt} = m(\dot{x}\dot{y}+x\ddot{y}-\dot{y}\dot{x}-\ddot{x}y) = m(x\ddot{y}-\ddot{x}y)$ But from equations of motion gives us $ \ddot{x} = -\frac{k}{m}x $ $ \ddot{y} = -\frac{k}{m}y $ using this, $ \frac{dL_{z}}{dt} = -kxy + kxy = 0 $ So, is $L_{z}$ conserved ? I know we can also try to work out the poisson bracket method, but this seemed slightly easier.


title: How do we distinguish between virtual particle exchange and particle decay?

text: The difference between virtual particles and unstable particles is discussed at length in this question (namely, virtual particles correspond to internal lines in Feynman diagrams and are not associated with any measurable physical state). So what is going in the case of, say, Higgs production at the LHC, where the Higgs does not live long enough to reach the detector? What are the calculational and experimental differences between producing a Higgs via quark fusion, which then decays into a pair of leptons (for example) and quark-quark to lepton-lepton scattering, proceeding via Higgs exchange? (To put it another way, we'll see an increase in lepton-lepton production from virtual Higgs exchange, but how is this distinct from seeing particles produced from the decay of an unstable Higgs?)


title: How to observe a particle with indefinite position?

text: As I understand it, when physicists talk about something behaving both like a particle and a wave, what they mean is that it has momentum like a particle, but its position is determined probabilistically by a wave function. What I, as someone ignorant of quantum mechanics, gather from experiments such as the one with the double slit is that quanta will continue to have an indeterminate position until they are disturbed ("observed"); I believe the phrase used is "collapsing the wave function". My question is the following: what does it mean to disturb something whose position is not yet well defined? What exactly is it that the "observer" interacts with?


title: After quantization of electron vibrations, do we need electrons anyway?

text: The title question is not ment in a general context, but one in which goes to the plasmon theory. In that case, how is are the statistics (boson vs. fermions) of plasmons determined? And is there an effective theory describing plasmons and their interaction with photons, such that the initial electrons are absent/integrated out/already fully represented by their oscillator excitations?


title: Does Dirac's idea of filled negative energy states make sense?

text: Please bear with me a bit on this. I know my title is controversial, but it's serious and detailed question about the _explanation_ Dirac attached to his amazing equations, not the equations themselves. Imagine for a moment that someone signed onto this group and proposed the following:   1. Empty space is a lot like a metal, or maybe a semiconductor, because like those materials its chock full of negatively charged electrons occupying different velocity states, only...   2. Unlike metals or semiconductors, the density of electrons in any one region of space is _infinite_ , because there is no limit to how fast the electrons can move. That is because these are negative energy states in which an electron can _always_ move faster simply by emitting a photon, so there's not "bottom" to how far they can drop and how dense they can become, and...   3. Unlike metals or semiconductors, there is no exactly balancing sea of positive atomic charges, well, unless maybe there are infinite numbers positively charged atoms too, and...   4. The resulting infinite negative charge density of real electrons not only doesn't matter but is in fact completely and totally invisible for some reason, and...   5. The resulting infinite mass density of electrons (recall that these are quite real electrons, only in odd negative-energy kinetic states) also doesn't matter, and...   6. Unlike the Fermi sea of a metal conduction band, removing an electron from this infinitely dense sea of electrons for some reason _doesn't_ cause other electrons to collapse into it and fill it, even though the negative kinetic velocity electrons are pushed by exactly the same Pauli exclusion forces as the ones in a Fermi band; in short, for reasons not clear, semiconductor-style hole stabilization applies while metal-style hole filling does not (is there a band gap going on here?), and...   7. Since the infinitely dense negative charges become invisible for no particular stated reason when the electrons fall into negative energy states, these unexpectedly stable open states in the negative energy sea have net positive charge, even though...   8. ... such missing states categorically should have _zero_ charge, since in sharp contrast to the positive ionic background of metallic and semiconductors, the vacuum has no background charge at all, which should leave holes in the mysteriously invisible negative kinetic energy just as uncharged and invisible as that sea for some reason is, and...   9. Even if you _do_ assume that the negative kinetic state electrons have visible charge, their infinite density would make the "comparatively" but infinitesimally smaller positive charge of such a hole invisible, and...   10. Repeat this process for every other kind of particle in existence, and...   11. If you have done all of this and done it correctly, congratulations: You now understand conceptually what anti-electrons (positrons) and other anti-particles are. First question: Have I misrepresented any of the implications of Dirac's explanation of positrons as holes in an infinite sea of negative-kinetic- energy electron states? What I have tried very hard to do is nothing more than make a list of the implications of a physics idea, just as people do all the time on this group. Who said it should not really be the issue, not if we are talking about an unelaborated _explanation_ rather than the math itself. Second question: If someone had proposed a theory in this forum like the one I just described, _and you had never heard of it before_ , what would you have thought of it? Please be honest. My point in all of this obviously is this: While Paul Dirac's amazing equations (they really are) managed to predict antimatter, his _explanation_ for why his equations require antimatter is... shall we say incompletely analyzed, to put a nice spin on it? A final thought: Has anyone ever seriously tried to make Dirac's _conceptual_ negative energy sea ideas, the ones that he espoused in his Nobel Lecture, into a real, working theory? And if so, how did they deal with the various issues I described above? (Me, I just think antiparticles are regular particles moving backward in time. Yeah, that's a pretty weird idea too, I know...)


title: Why no longitudinal electromagnetic waves?

text: According to wikipedia and other sources, there are no longitudinal electromagnetic waves in free space. I'm wondering why not. Consider an oscillating charged particle as a source of EM waves. Say its position is given by $x(t) = sin(t)$. It is clear that at any point on the $x$ axis, the magnetic field is zero. But there is still a time-varing electric field (more or less sinusoidal in intensity, with a "DC offset" from zero), whose variations propagate at the speed of light. This sounds pretty wave-like to me. Why isn't it? Is there perhaps a reason that it can't transmit energy? A very similar question has already been asked, but it used a "rope" analogy, and I feel that the answers overlooked the point that I'm making.


title: How does this thought experiment not rule out black holes?

text: How does the following brief thought experiment fail to show that general relativity (GR) has a major problem in regards to black holes? The full thought experiment is in my blog post. The post claims that GR violates its own equivalence principle at the horizon of a black hole. The principle says that the laws of physics in any sufficiently small, freely falling frame are the same as they are in an inertial frame in an idealized, gravity-free universe. Here's a condensed version of the thought experiment: > In an arbitrarily small, freely falling frame _X_ that is falling through > the horizon of a black hole, let there be a particle above the horizon that > is escaping to infinity. A free-floating rod positioned alongside the > particle and straddling the horizon couldn't be escaping to infinity as > well, or else it'd be passing outward through the horizon. However, if > instead the rod didn't extend as far down as the horizon, then in principle > it _could_ be escaping, possibly faster than the particle beside it. **In an > inertial frame, unlike in _X_ , a body's freedom of movement (in principle > and if only relative to other free objects in the frame) doesn't depend on > the body's position or extent.** Then a test of the laws of physics can > distinguish _X_ from an inertial frame. If _X_ was equivalent to an inertial > frame, I wouldn't be able to tell whether the rod could possibly be passing > the particle in the outward direction, by knowing only whether the rod > extends as far down as an imaginary boundary (the horizon) within the frame. > If _X_ was equivalent to an inertial frame, the rod could in principle be > passing the particle in the outward direction regardless of its extent > within _X_. The thought experiment above takes place completely within _X_ , which is arbitrarily small in spacetime (arbitrarily small both spatially and in duration). That is, the experiment is completely _local_. That the particle is escaping to infinity is a process occurring within _X_ ; it tells us that the particle won't cross the horizon during the lifetime of _X_. The particle needn't _reach_ infinity before the experiment concludes. It isn't necessary to be able to detect (by some experiment) that a horizon exists within _X_. It's a given (from the givens in the thought experiment) that a horizon is there. Likewise, I am free to specify the initial conditions of a particle or rod in relation to the horizon. For example, I am free to specify that the rod straddles the horizon, and draw conclusions from that. The laws of physics in _X_ are affected by the presence and properties of the horizon regardless whether an observer in that frame detects the horizon. It seems to me that the only way the equivalence principle is satisfiable in _X_ is when in principle the rod can be escaping to infinity regardless of its initial position or extent in _X_ , which would rule out black holes in a theory of gravity consistent with the principle. Otherwise, it seems the **bolded** sentence must be incorrect. If so, how? In other words, how can I _not_ tell whether the rod can possibly be passing the particle in the outward direction, by knowing only whether it extends as far down as the horizon? I'd appreciate hearing from Ted Bunn or other experts on black holes. A barrier to getting a satisfactory answer to this question is that many people believe the tidal force is so strong at the horizon that the equivalence principle can't be tested there except impossibly, within a single point in spacetime. An equation of GR (see my blog post) shows that a horizon isn't a special place in regards to the tidal force, in agreement with many texts including Ted Bunn's Black Hole FAQ. In fact the tidal force can _in principle_ be arbitrarily weak in any size _X_. To weaken the tidal force in any given size _X_ , just increase the mass of the black hole. (Or they might believe it's fine to test the principle in numerical approximation in a frame larger than a point, but not fine to test it logically in such frame anywhere. Kip Thorne disagrees, in a reference in my blog post.) Note also that the Chandra X-ray Observatory FAQ tells us that observations of black holes to date aren't confirmations of GR, rather they actually depend on the theory's validity, which is to say the existence of black holes in nature isn't proven. Edit to add: I put a simple diagram, showing GR's violation of its own EP, at the blog post. Edit to add: I'm awarding the bounty to dbrane, whose answer will likely retain the lead in votes, even though it's clearly incorrect as I see it. (In short, the correct answer cannot be that an infinitesimally small frame is required to test the EP. It is in fact tested in larger labs. The tidal force need only be small enough that it doesn't affect the outcome. Nor is the horizon a special place in regards to the tidal force, says GR.) I do appreciate the answers. Thanks! Edit to add: this question hasn't been properly answered. The #1 answer below made a false assumption about the question. I've beefed up the question to address the objections in the answers below. I added my own answer to recap the objections and reach a conclusion. Please read the whole post before answering; I may have already covered your objection. Thanks!


title: Dimensional reduction from $3+1$ to $2+1$ for $\cal{N}=2$ vector superfield

text: Let the supersymmetry transformations for the chiral multiplet $(z_k,\psi_{kL},f_k)$ be, $\delta z_k = 2i \bar{\alpha} \psi_{kL}$ $\delta \psi_{kL} = D_\mu z_k \gamma ^\mu \alpha_R + f_k \alpha_L$ $\delta f_k = 2i\bar{\alpha} \gamma^\mu D_{\mu} \psi_{kL} - 2ie\bar{\alpha}\lambda _R z^k$ Similarly let the gauge multiplet $(A_\mu,\lambda,D)$ transform as, $\delta A_\mu = i \bar{\alpha}\gamma_\mu \lambda$ $\delta \lambda = -\frac{1}{2}F^{4}_{\mu \nu}\gamma_4 ^{\mu \nu} \alpha + D \gamma_5 \alpha $ $\delta D = i \bar{\alpha}\gamma_5 \gamma ^ \mu \partial _ \mu \lambda$ Here in $F^4$ the $\mu$ and $\nu$ go over indices $0,1,2,3$ and $\gamma_4 ^{\mu \nu} = \frac{1}{2} [\gamma_4^\mu, \gamma_4^\nu]$. (these gamma matrices are defined below) One does something called "dimensional reduction" of these to $2+1$ dimensional spacetime by assuming that the fields are independent of the third spatial coordinate. In $2+1$ dimensional spacetime the gamma matrices are defined as, $\gamma ^0_3 = -i\sigma^2$, $\gamma^1_3 = \sigma ^3$ and $\gamma^2_3 = \sigma ^1$ In the so called "Majorana representation" the $4-$dimensional Gamma matrices can be written such that $\gamma_4 ^{0\1\2} = \left [ \begin{array}{cc} & \gamma_3 ^{0\1\2} \\\ \gamma_3 ^{0\1\2} & \end{array} \right ] $ and $\gamma_4 ^3 = \left [ \begin{array}{cc} I & \\\ & -I \end{array} \right ]$ One renames the third component of the gauge field $A_3$ as $C$ and splits the $4-$component fermions into $2-$component spinors in $2+1$ dimensions as, $\lambda = \left [ \begin{array}{c} \lambda_1 \\\ \lambda_2 \end{array} \right ]$ $\alpha = \left [ \begin{array}{c} \alpha_1 \\\ \alpha_2 \end{array} \right ]$ One substitutes the above in the first set of supersymmetric transformations and sets to $0$ all derivatives with respect to the $3^{rd}$ spatial direction. Then one is supposed to get, $\delta A_\mu = i \bar{\alpha}_a \gamma_\mu \lambda _a$ $\delta \lambda _a = - \frac{\epsilon ^{\mu \nu \rho}F_{\nu \rho}}{2}\gamma _ \mu \alpha_a + \partial _\mu C \gamma ^\mu \alpha ^a + D \alpha ^a$ $\delta C = i\bar{\alpha}^a \lambda _a$ $\delta D = i \bar{\alpha}^a\gamma ^\mu \partial_\mu \lambda_a$ where $a$ runs over $1,2$ and $\alpha^1 = \alpha_2$ and $\alpha^2 = -\alpha_1$ Doing the above I can get the above transformations for all the fields. (Just that I need to use a fact that is true only for these gamma matrices in the $3$-dimensions that $[\gamma_3^\mu,\gamma_3^\nu]=2\epsilon ^{\mu \nu \rho}\gamma_{3\rho}$)   * In this "dimensional reduction" the particular choice of the gamma matrices seemed crucial. Is this true?    * I don't know how but this is supposed to match the transformations of the $\cal{N}=2$ vector multiplet in $2+1$ dimensions which has componets, $A_\mu, \lambda_a,C,D$ (why?)    * I don't know how one derives the field content (as above) of the $\cal{N}=2$ vector multiplet in $2+1$ dimensions.    * From the above it is claimed that the following forms a lagrangian respecting the above symmetry and is what gets called the ``supersymmetric Chern-Simons theory" $L = \frac{\kappa}{2} (\epsilon ^{\mu \nu \rho} A_\mu \partial _\nu A_\rho - i \bar{\lambda_a}\lambda_a + 2CD)$ But to establish the above claim I need to assume two results,   * First is a property which again seems typical of $3$ dimensional gamma matrices that, $\gamma_3^{\mu \dagger} \gamma_3 ^0 = -\gamma_3^0\gamma_3^\mu$    * Secondly I need to assume the following variation for the fermion self-coupling,  $\delta (\bar{\lambda} \lambda) = 2\delta (\bar{\lambda}) \lambda$ It is not clear to me as to why should the above hold.   * A crucial difference between the above and the usual Abelian supergauge theory is that the kinetic term for the fermion is missing. How does one understand this?


title: Idea of integrable systems

text: I do not quite understand the idea an integrable dynamical system. Does it mean that the EOMs are analytically and exactly solvable? What are the necessary and sufficient conditions such that a system is integrable?


title: Quantum entanglement of spin along multiple orthogonal axes

text: Picture an entangled pair of spin 1/2-spin particles with total spin 0. In the diagram, particle 1 of the pair is moving to the left (-y), and particle 2 to the right (+y). If a z-oriented SG$^*$ is used to detect the spin direction of particle 1 on the left, then the spin direction of particle 2 can be predicted with 100% certainty by using another z-oriented SG on the right. For example, if the left SG finds particle 1 to have spin $\frac{\hbar}{2}$, there is be a 100% probability that a z-directed SG on the right will detect particle 2 as having spin $-\frac{\hbar}{2}$. Now consider leaving the left SG unchanged (pointing to +z), and rotating the right SG so it will point to +x. If particle 1 is detected on the left to have spin $\frac{\hbar}{2}$, two possibilities can be considered for what will happen when particle 2 reaches the +x-directed SG:   * Particle 2 is detected with 100% probability as having spin $-\frac{\hbar}{2}$ in the +x-direction, or   * Particle 2 is detected with a 50% probability of having spin $-\frac{\hbar}{2}$ in the +x direction, and 50% of having spin $+\frac{\hbar}{2}$ in the -x direction. In the second case, obviously, it may turn out that the total spin of the system is not equal to zero. *SG: Stern-Gerlach apparatus ![Two Stern-Gerlach apparatuses directed along z-direction](http://i.stack.imgur.com/2MPCC.jpg)


title: Positive cone of operators: if two selfadjoints $a$, $b$ obey $a^2 + b^2 =1$, must they commute?

text: The question relates to the structure of the positive cone of operators, in C*-algebra.   1. If $a$ and $b$ are selfadjoints such that $a^2 + b^2 = 1$ can one prove $a$ and $b$ commute? What one derives is $ ba = ba^3 + b^3a $ and $ ab = a^3b + ab^3 $ but I am stuck here, only higher powers seem to commute.   2. Also, if for selfadjoints $a$ and $e$, $a^2 + e^2 = e$, can one prove that $e$ is actually a square of a selfadjoint?    3. Finally, is $1-e$ actually a square of a selfadjoint? We know that $e$ is the square of a selfadjoint so that, by the textbooks, it is positive but I do not find a complete algebraic proof of this fact: a sum of squares is a square.


title: Single Photon emission from positronium in strong magnetic fields; how is momentum conserved?

text: I recently came across a paper (which I will not describe in detail, however, the authors examine the probability of positronium producing a single photon, while under the influence of a strong magnetic field). The paper insinuates that momentum is still conserved here, but it is conserved through the influence of the magnetic field. What truly astounds me is that as a magnetic field grows stronger (there is a table in the paper, which I will not post, as it could violate copyright laws), the chances of observing a single photon emission surpasses that even of a multiple photon emission. Eventually, it can become highly unlikely to witness anything other than a single photon emission. How does the strengthening of a magnetic field help conserve momentum, and how does this phenomenon eventually, hinder tremendously, the outcome of a multiple photon emission?


title: Do Physics concepts build on top of other concepts?

text: I am taking a calculus based physics class and I am struggling. I am wondering if I do not understand some concepts, will I be unable to understand future concepts introduced in the same class? In math, if you do not understand algebra you will have a hard time understanding calculus.


title: What is the motivation for the definition of a manifold?

text: In Wald's _General Relativity,_ an $n$-dimensional $C^{\infty}$ manifold $\mathit{M}$ is defined as a set, with subsets $\lbrace{O}_{\alpha}\rbrace$, which satisfies 3 properties. In particular, the third one (abbreviated) is: > _If any two sets $\mathit{O}_{\alpha}$ and $\mathit{O}_{\beta}$ overlap, we > can consider the map $\psi_{\beta}\circ \psi_{\alpha}^{-1}$, which takes > points from subset $\psi_{\alpha}[\mathit{O}_{\alpha} \cap > \mathit{O}_{\beta}]$ to points in subset $\psi_{\beta}[\mathit{O}_{\alpha} > \cap \mathit{O}_{\beta}]$. We require these subsets of $\mathbb{R}^{n}$ to > be open and this map to be $\mathit{C}^{\infty}$._ My question is: what is the purpose of this clause in the definition of a manifold? It appears to allow the existence of a transformation from different coordinate systems (from $\psi_{\alpha}$ to $\psi_{\beta}$). It requires the existence of an inverse. Other than that, is there any other motivation for defining the manifold in this way?


title: Why do we deal only with large scale magnetic fields in astrophysics, and not electric fields?

text: In astrophysics there is a lot going on about strong, large scale magnetic fields: in stars (prominences), magnetic dynamos, compact accretors collimating jets, etc. There's even a special computational formalism called magnetohydrodynamics (MHD), which allows to deal with space plasma. Yet I've never read about large scale electric fields. I know that most of the matter we model in astrophysics is plasma but, naively, one would assume that this introduces both $\mathbf{E}$ and $\mathbf{B}$ fields on an equal footing. So where does this asymmetry come from?


title: Wick Rotation in Curved space

text: So over time I have learned to do exhaustive searches before asking things here. Wick rotations are cool if you are trying to work in qft and make statements about the thermodynamics of some physical thing you are probing. You want to talk about some thermodynamics and who knows may be statistics, and like most people . . . oh Wick rotation is what you would think. Now we are in curved space, do you just throw away this tactic in the dust bin? And yes, I have read up a few things, but it seems I need a proper pointer to where to find how to attack this thing properly. Just really curious.


Is the observable universe homeomorphic to $B^3$? Where $$B^3=\\{x\in \mathbb{R}^3 : |x|\leq 1 \\}$$ Or is it even sensible to talk about space (rather than spacetime) as a 3 manifold?


title: The definition of entropy in quantum mechanics

text: I have seen entropy with several different definitions. Like Von Neumann entropy and Rényi entropy, etc. So I am curious why there are so many different definitions in quantum mechanics while only one in classical mechanics named after Boltzmann?


title: Cubic term in gauge theories

text: In ordinary classical gauge theories the term $-\frac{1}{2}\mathrm{Tr}(F_{\mu\nu}F^{\mu\nu})=-\frac{1}{4}F^a_{\mu\nu}F_a^{\mu\nu}$ in the Lagrangian is completely natural. A somehow rare term would be a "cubic" one like $$L_c=\kappa \eta_{\mu\gamma}\mathrm{Tr}(F^{\mu\nu}F_{\nu\alpha}F^{\alpha\gamma}).$$ Is there a physical reason, other than unnaturalness and Occam's razor, not to include it in a Lagrangian?


title: Normal Reaction and Punctures

text: I'm curious on how, if every object creates a normal contact force equal to the force being put on it, seemingly preventing deformation or snapping of an object, how do objects break or how are they punctured? Is there a maximum pressure (and therefore force, and therefore normal reaction) that objects have? If so, how is this calculated?


title: How to use the Coddington shape factor?

text: We are given $\sigma=\frac{r2+r1}{r2-r1}$, with $\sigma$ being the Coddington shape factor. What I am having problems with is solving for $r1$ and $r2$ when I am just given a focal length and a refractive index, along with $\sigma$. From the lensmaker's equation, we know that $\frac{1}{f}=(n-1)(\frac{1}{r1}-\frac{1}{r2})$, but I just can't seem to get it to come out where I can solve for just $r1$ or just $r2$. Any advise on how I should go about this?


title: Quantum to String: making the transition

text: We have a Hamiltonian containing a quantum simple harmonic oscillator coupled to a meter and a force. There is a term in the Hamiltonian that involves the oscillator position and the force, a term that involves the oscillator position and the meter momentum, a term containing the particle position and a final one containing the particle momentum. We want to convert this to the string theory picture: what happens when we replace the quantum point particle and the meter with strings?


title: Close-packing of equal spheres - one sphere in a pyramid with a square base and four equilateral trianglular faces

text: I'm trying to determine the "packing efficiency" of fitting on sphere into a square based pyramidal container with four equilateral triangular faces. In other words, how much space within the pyramid does the sphere occupy. The result I get is 34.91%, which surprises me (I thought it would occupy much more space). But I've done the calculation over and over again and keep getting the same result. I've also done "stacking efficiencies for 5, 14, 30 and 55 equal sized spheres in a pyramidal container, and it appears that the packing efficiency increases with the number of spheres. But in this situation I'm only interested in one sphere. Hopefully this picture will illustrate what I'm trying to describe here: ![enter image description here](http://i.stack.imgur.com/KA5oA.jpg) These are my calculations: Volume of a Square Pyramid with 4 equilateral triangles = (b^2 x h)/3 where b = length of base and h = height of pyramid For a square pyramid with a base of 1 sphere (each of the five faces in the pyramid is tangential to the sphere it contains), sphere = 2 cm in diameter (r = 1 cm),   Volume of each sphere = 4/3*π*r^3 = 4.1888 cm3 A pyramid with a base of 1 sphere contains 1 sphere,   so the volume of 1 sphere with a 1 cm radius = b = 3.4641 cm h = 3.0000 cm Volume = 4.1888 cm3 = (b^2 x h)/3 12.0000 The base of a pyramidal "box" containing 1 such sphere =   2*3^0.5 =   3.46410161513775 cm = b The height of the pyramidal box = ((2*(3^0.5))^2 x cos 30° cos 30° = 0.866025404 h = 3.000 cm Volume of a pyramidal box containing 1 such sphere =   (b^2 x h)/3   = 12.000 cm^3 "Stacking efficiency" or percent of volume of pyramidal box = volume of sphere/volume of pyramid =   **34.91%** ??? Is this correct ??? Thanks, Hope no one jumps on me for asking another stupid question. Jim.


title: How do you isolate a single photon?

text: How do scientists/researchers isolate a single photon (for single photon sources)? How do they know they have isolated it? Is it really totally "isolated"? What is the photon isolated in? Sorry if this is a basic or general question, just really interested to know what this means, considering that a "vacuum" isn't really empty space.


title: Forces in a balloon popping?

text: How much difference between internal and external pressure can a balloon withstand? how would the value be calculated? I found an equation: http://library.thinkquest.org/C003758/Function/laplacelaw.htm But, how does that relate to a material?


title: Definition of buoyancy

text: Hi I'm trying to figure out the definition of buoyancy, is this correct? So say for example I have 1 cubic meter of water which weighs 1 ton, and I placed a cubic foot of a material that at a 1 ton was 1.5 cubic meters, would it float, is this correct or am I wrong?


title: An ideal condition in Heisenbergs uncertainity principle

text: We all know that the Heisenberg uncertainity principle implies $\Delta x\, \Delta p\geq\frac{\hbar}{2}.$ But is there an ideal condition where we can measure $\Delta x$ to a particular precision and also simultaneously measure $\Delta p$ such that we can predict the position and momentum of the particle (kind of like a balance in the weighing machine) , if not with arbitary precision ,atleast to some certainity ? Consider a wave whose displacement and momentum can be altered to reach such an ideal condition that we can measure the displacement to some least accuracy but also predict where the wave is going.


title: What are the four primary fields for SU(2) level 3 WZW field

text: What are the four primary fields for SU(2) level 3 for spin $\frac{3}{2}$ WZW field in terms of operators from Potts model and chiral bosons?


title: Why the amount of entropy increase to a system is less when heat is added to a higher temperature system than to a lower one?

text: As I understand it, statistically this means that a fall from a high temperature (say $300\text{ K}$) to a middle temperature ($200\text{ K}$), and an increase from say ($100\text{ K}$) to ($200\text{ K}$) by heat exchange do not result in the same entropy changes, the first loses less entropy than the second gains. So this implies that the former loses less microstates than the latter gains. But how can this be so? Surely every increase of say $1\text{ K}$ increases the number of available microstates by the same multiplicative factor, and thus the same additive amount of entropy? Edit: I've been reading a text which states quite clearly that $\frac{\partial\log{\Omega}}{\partial U} \approx -\beta$ so the rate at which accessible microstates become available would approximately fall with increased energy. I'm trying to get a handle on why the rate of increase would typically fall with higher values of $U$, and by implication $T$. As you say the log is a concave function, which answers a lot of why this behaviour happens, but I am trying to understand it from a physical rather than a mathematical perspective. The fact that the log was used is generally presented as a way in which to keep it in check with the state variable definition of entropy - which is in fine. But I guess that then leads to a corollary question of why entropy defined in such a way that it is a concave function of the number of states rather than the number of states per see. In other words why can the increase in the number of states available for states with low initial states vs. the descrease in states for a state with high initial states be sufficient to drive spontaneous processes through a statistical process.


title: Ampere's Law and Wires?

text: For a current carrying wire, the magnetic field $B$ is given by: $$B = \frac{u_oI}{2\pi r}$$ Is this only valid for a cylindrical wire?


title: Falling metal cylinders more devastating than a nuke?

text: I just saw _G.I JOE - Retaliation_. In the movie the bad guys (spoiler alert) > destroy London by dropping a large metal cylinder from space onto the city. Could this device cause more destruction than a nuclear weapon? If so, how?


title: Obtain rate as a function of position for a force $F=-b\left( \frac{\partial x}{\partial t} \right)^{2}$

text: I have trouble getting the speed as a function of position for a force of the form: $$F=-b\left( \frac{\partial x}{\partial t} \right)^{2}$$ I had thought to use Newton's second law, so: $$ma=m\frac{\partial v}{\partial t}=-b\left( \frac{\partial x}{\partial t} \right)^{2}$$ Where $a$ is the acceleration and $v$ is the velocity. Since v (x), using the chain rule: $$ma=m\frac{\partial v}{\partial t}\frac{\partial x}{\partial t}=-b\left( \frac{\partial x}{\partial t} \right)^{2}$$ Accordingly: $$m\frac{\partial v}{\partial x}=-b\left( \frac{\partial x}{\partial t} \right)$$ However, I can not solve this equation. My questions are: Is it possible to solve this equation? I made an error in the procedure?


title: Does our current notion of velocity/acceleration (based on calculus) require absolute time?

text: When people explain special relativity, they say that a velocity in one frame leads to time dilation in that frame relative to a stationary frame. When we say something like $v=\mathrm{d}x/\mathrm{d}t$ or $a=\mathrm{d}v/\mathrm{d}t$. Isn't dt something that cannot change? A smallest infinitesimal unchanging amount. Newton described it in terms of absolute space and time. How does it make sense to define a velocity of one frame as $\mathrm{d}x/\mathrm{d}t$ yet say that the rate of time itself changes? Can $\mathrm{d}t$ change?


title: Soft brehmsstrahlung classical computation

text: On page 177 in Peskin & Schroeder there is a derivation I have a hard time with. They write the current for a charge at rest as $$j^\mu = (1,0)^\mu e \delta(x). $$ I don't understand what the four vector $(1,0)^\mu$ represents. Why is time=1?


title: Is the mechanics of the wave function in the quantum mechanics deterministic?

text: Is possible a non-deterministic propagation of the wave function in the QM?


title: Position vector, length/modulus function and differentials

text: If $\vec r$ and is the position vector of a point in motion and $r$ is its length/modulus/magnitude/size, then: Can it be true that: $$\|\mbox{d} \vec r\| \neq \mbox{d}r? $$ I think that this is true in circular motion, but if this question is generic, i.e., for almost all cases, then do you think this is correct to say? **EDIT:** Thanks for the very prompt answers and comments, but please tell me, if this question is given in an exam, which option would you mark? (As a student, I am very interested in understanding physics, but at the end of the day, I also have to think of the exam point of view...)


title: R-R fields in RNS formalism

text: In string theory I came across the fact that there are difficulties in describing the coupling of R-R fields with world sheets in RNS formalism and it can be done in GS formalism only. Can someone explain the reason(s) behind that? Also, I think if you have a space time where you can quantize strings and you have non zero R-R fields, then you need to use GS formalism to quantize strings there. It seems to me that these two are related and again my question is- why is that? Do we always need to use GS formalism there?


title: Submitting an article to another journal

text: Not long ago I submitted an article to PRE. One referee was in favor and two against (this not more than 3 months ago). Rather than pressing, I decided to submit it elsewhere. A single referee report was negative on the grounds that the article was unclear, though the details helped me revise the article. Is it all right to resubmit a substantially revised article and new supporting material to PRE under the original article ID number, or am I obligated to make a new submission? In the event of the former, should (must?) I mention the intervening submission?


title: Time Reversal Operator

text: I know that time reversal operator is an antiunitary operator. How does it work on wavefunctions? I believe in this way: $$T \psi (k,+)=e^{i\pi S_y/\hbar} K \psi (k,+) = \psi^*(-k,-),$$ but I am not sure. ("+" and "-" represent spin-up or spin-down states) Does anybody have a good explanation?


title: Significances of energy conditions

text: I know about these different energy conditions in GR, namely strong, weak and null, but never really understood the full physical significance of them or for example how to 'derive' them or how compelling is a certain condition (for that particular one to be applied). I know that they exist because you need some constraints for $T_{\mu\nu}$ to satisfy as otherwise its completely unconstrained. Can someone please elaborate?


title: Crash course in classical thermodynamics

text: I've been working on some projects lately where it would be very handy to know more about thermodynamics than I do, but sadly I never had a chance to take a proper thermodynamics course in college. Unfortunately, right now I don't have the time to work through a 500-600 page undergraduate text on the subject. Can anyone recommend a book/online resource/PDF that perhaps gives a (calculus based) broad overview of classical thermodynamics, say in something less than 200 pages? Thanks!


title: Simulating a car in an intersection

text: I'm somewhat confused. I want to simulate in real-time an intersection where cars have to turn left, right or go straight. What I have are 2 way points: One at the beginning of the intersection on the incoming street and the other at the end of the intersection on the outgoing street. As I know the next way point on the outgoing street, I know which direction the car should be pointing. How would I slow the car down to the optimal speed, calculate its steering angle and correct it in a time interval so that the car drives an optimal curve? A resource I have found, that seems quite good for this is the following paper. I just don't really understand the first part of the paper where the Circular track is calculated. At which point is the steering angle applied?


title: Water from Icebergs

text: Thousands of icebergs are calved from polar regions annually, and melt in the oceans. George Mougin, an entrepreneurial engineer, aspires to tow multi- million ton icebergs to areas of the world lacking sufficient potable water.![enter image description here](http://i.stack.imgur.com/iMUei.jpg) Simulations by Dassault Systemes suggest that a single tug is capable of towing a multi-million ton tabular iceberg from Newfoundland to the Canary Islands. Given suitable oceanic and meteorlogical conditions, the trip would take about five months with a melt loss of about 38%. _Should oceanic transportation of icebergs be succesful, how could the water be efficiently and timely harvested and transported to needy areas?_


title: Random quantum systems with asymmetric Lifshitz tails?

text: For a quantum mechanical system with a periodic Hamiltonian (Schrödinger operator) $H$, let $N(E)$ be its integrated density of states, i.e. the fraction of eigenvalues in the spectrum $\sigma(H)$ that are less than or equal to $E$. (Or see the rigorous definition in Sec. 2 of Ref. 1.) The Lifshitz exponent $\gamma_-$ is defined near the infimum $E_- = \inf \sigma(H)$ as (e.g. in Ref. 1): $$\gamma_- = \lim_{E\downarrow E_-} \frac {\ln (-\ln N(E))}{\ln(E-E_-)},$$ and similarly near the supremum $E_+ = \sup \sigma(H)$, the analogous exponent can be defined as $$\gamma_+ = \lim_{E\uparrow E_+} \frac {\ln (-\ln (1-N(E)))}{\ln(E_+-E)}.$$ These exponents measure how fast the density of eigenvalues grows or decays near the extreme ends of the spectrum. I have some empirical data for a disordered quantum mechanical system for which I appear to measure $\gamma_- \ne \gamma_+$. This behavior does not seem to be observed in the model Hamiltonians that have been summarized in the review in Ref. 1 or in its references. I haven't found any literature that describes such behavior in any theoretical studies of quantum mechanical systems or random Schrödinger operators. As I am not an expert in this field, I would appreciate any pointers to references where such asymmetric behavior in the upper and lower tails of $N(E)$ have been observed. * * * Reference   1. Kirsch and Metzger, http://arxiv.org/abs/math-ph/0608066