title: Edge states in the "half BHZ" model

text: Consider the "half BHZ" Hamiltonian $${\cal H}=\sum_{\mathbf{k}}\left(A\sin(k_{x})\sigma_{x}+A\sin(k_{y})\sigma_{y}+{\cal M}(\mathbf{k})\sigma_{z}\right)c_{\mathbf{k}}^{\dagger}c_{\mathbf{k}}$$ where $M(\mathbf{k})=M-2B\left(2-\cos(k_{x})-\cos(k_{y})\right)$ For simplicity, take A=1 and B > 0. M < 0 and M > 8B give a trivial phase, while 0 < M < 4B and 4B < M < 8B give topological phases with Chern number $\pm1$ and the corresponding edge states. The existence of the edge states can be easily verified by numerically diagonalizing the Hamiltonian for periodic boundary conditions in, say, the x direction and vanishing boundary conditions in the y direction. They can also be found analytically by expanding the Hamiltonian around the TRIM points $(k_x, k_y) = (0, 0), (0, \pi), (\pi, 0), (\pi, \pi)$ to second order and looking for solutions localized near a boundary. The trouble is, the expansion around $(0,0)$ seems to imply that the only condition for the existence of a localized state at $k_x=E=0$ is MB > 0. But we know that for M > 4B there's no such state, so what happens to it? I think it's supposed to somehow pair-annihilate with the $(0, \pi)$ edge state when M becomes bigger than 4B, but according to the expansion around $(0, \pi)$ the $(0, \pi)$ edge state vanishes on its own. Is there some other explanation for the disappearance of the $(0, 0)$ edge state? Or does it annihilate with the $(0, \pi)$ edge state after all, but there's no way to see it from the local Hamiltonians?


title: Can vorticity be destroyed?

text: I have a professor that is fond of saying that vorticity cannot be destroyed. I see how this is true for inviscid flows, but is this also true for viscous flow? The vorticity equation is shown below for reference. From this equation, it looks as if vorticity only convects and diffuses. **This would suggest that it can't be destroyed.** $$\frac{D\boldsymbol{\omega}}{Dt} = (\boldsymbol{\omega}\cdot\nabla)\boldsymbol{V} + \nu\nabla^2\boldsymbol{\omega}$$ However, consider this thought experiment: Suppose we have a closed container filled with water with initial vorticity field $\boldsymbol{\omega}_0$ at time $t_0$. If the container is allowed to sit undisturbed, as $t\to\infty$ the water will become stationary ($\boldsymbol{V}\to 0$) with zero vorticity ($\boldsymbol{\omega}\to 0$). **This suggests vorticity can be destroyed.** My professor claims the boundary-layer vorticity at the sides of the container is equal and opposite in sign to the bulk vorticity. If this is the case, the vorticity cancels out after a long time resulting in the stationary fluid and vorticity is not destroy (just cancelled out). **EDIT:** I'm looking for either a proof that the boundary-layer vorticity is equal and opposite to the bulk vorticity **or** a counter-explanation or proof. (I'm using proof in a very loose hand-wavy sense)


title: Punching - Force or Momentum?

text: If I want to punch a person inflicting maximum damage, what do I need to care about? My force of punching, i.e, do I need more acceleration? Or do I need momentum, i.e my velocity for punching?


title: How much electricity can be produced by my thermoelectric generator?

text: I am using a thermoelectric cooler from a pc's heatsink to produce electricity. Its size is 30mm by 30mm. I will cool it on one side at -10 degree Celsius and the other at 24 degree Celsius. Can anyone tell me how much electricity in watts will be produced. I'm using this heatsink cooler because it's the most easily available to me. Can anyone suggest an even better thermoelectric generator? Also how much electricity it would produce and how much it will cost?


title: Back motor effect of loaded generators?

text: The _back motor effect_ (see Counter-electromotive force) is the counter torque which opposes the rotational motion of the coils in a generator when the generator is under load. The back motor effect depends upon the load connected to the generator. The more load, the more current will be produced which will increase the counter torque. But to produce more current, the normal torque will also increase. So, shouldn't this balance the effect? If it does, then why do small generators change sound (nearly shutting down) when load in increased on them?


title: Charge in terms of wavefunctions

text: For a charged quantum particle, say, an electron or a quark, how in the particle's wavefunction is the electric charge represented? Is it truly possible to represent electric charge using the wave mechanics formulation of quantum mechanics, or is that something only matrix mechanics can satisfy? And does the same apply for other quantum properties, such as color charge or spin?


title: Optical illusion of car wheels, speeding up

text: Perhaps it is some free moving spinner attached to the wheel, but as opposed to this question: Why does the wheel of a car appear to be moving in opposite direction? I have seen car wheels that appear to spin backward both while speeding up, and moving at a constant rate. Am I just confused, or is there a reason for this?


title: Use something that slower than light in Michelson–Morley experiment

text: Follow by the Michelson–Morley experiment, What happen if we use something that has a speed lower than light instead of light in Michelson–Morley experiment? How about the result?


title: Whether Au-Ga-Bi ternary phase diagram be calculated theoretically

text: Currently, my experiment involves the three elements: `Au, Ga, Bi`, and I need `Au-Ga-Bi ternary phase diagram` to explore the underlying reason for my experimental phenomenon. Unfortunately, I can't find any information about this ternary phase diagram both in `Web of Knowledge` and `Google`. It should be impossible that nobody would think of this ternary phase diagram since the first phase diagram was published, because there are so many scientists. So I wonder whether Au-Ga-Bi ternary system has something special that impedes the calculation of its phase diagram. Otherwise, why no one does it? I'd appreciate it greatly if anyone could provide some information.


title: Einstein's "happiest thought"

text: Einstein described his discovery of the equivalence principle as the "happiest thought of my life". Why? What, in broad conceptual terms, is the logical chain of reasoning that leads from the equivalence principle to general relativity?


title: Calculate amount of energy based on height of object

text: How can you calculate the amount of energy an object produces that falls from a particular height? Or water. In an eg i would like to know how can you calculate the amount of electricity water can produce at a certain height?


title: How does one derive the equation for the speed of sound?

text: In my acoustics books I see $$c^2 = \frac{\mathrm{d}P}{\mathrm{d}\rho}$$ where $c$ is the speed of sound, $P$ is the pressure and $\rho$ is the density. Where does this equation come from? In my books it appears almost as a definition. Could you explain this, or at least point me to an article or a book that addresses this question? Thanks.


title: Why does your car lurch toward an oncoming truck as it passes you?

text: I notice that the larger the truck the greater the magnitude of the lurch. Can anyone give a physical explanation to this?


title: Can black holes form in a finite amount of time?

text: One thing I know about black holes is that an object gets closer to the event horizon, gravitation time dilation make it move more slower from an outside perspective, so that it looks like it take an infinite amount of time for the object to reach the event horizon. It seems like a similar process should slow the formation of the black hole itself: As the star collapses, its gravitational time dilation make itself collapse more slowly. This make me wonder, are what astronomers claim to be black holes really black holes, or are they stars that progressively make themselves more similar to one without actually reaching the stage of having an event horizon? EDIT: Contemplating one answer, I realize the question is ambiguous. What does finite time mean in general relativity. Here is a less ambiguous question: Is there a connected solution of 3+1 dimensional general relativity with one space-like slice not have a singularity, and another space-like slice having one.


title: Wave packets v.s. wave trains

text: Could someone please explain the difference between a wave packet and a wave train? I have rummaged around online but have not been able to find a definitive definition.


title: Specific energy and specific angular momentum of photon

text: In this PDF [1], is made reference to specific energy and angular momentum of a particle. If the particle has no mass, like a photon, how should I define these terms in the equations further down for the path of the particles? [1] Lecture XIX, Christopher M. Hirata, Caltech M/C 350-17


title: Do metals generally lose more heat by convection than non-metals?

text: I've been experimenting with radiative cooling lately and in my mind convection is always an enemy (especially forced convection since the modules are outdoors). Looking at Newton's law of cooling, it doesn't seem to me that it accounts for surfaces made of different materials losing/gaining heat (can be wrong though, not uncommon especially when working with formulas). Does it matter (much), that surfaces are made from different materials when we talk about convective heat gains/losses? I'm only saying this because metal cooled more under radiation than a type of plastic I used, and to me that doesn't make much sense as the convective heat gain should, in my mind, be greater in the metal panel and therefore it shouldn't cool more than the plastic one (both had similar emissivity because of a very thin paint coating). My experiments tell me that I'm most likely wrong in assuming the convective heat gains/losses will be greater in metals, but I thought it's most likely better to check with those who actually know what they're talking about, it may be some weird fluke in the tests.


title: Practical limits on the size of orbiting objects: could two pebbles orbit each other

text: As I understand it, gravity is inherent to mass and therefore even a small rock has its own gravitational pull. It seems entirely plausible then that a rock 1" in diameter could orbit a bigger rock, say 10" in diameter. Is this actually possible? Are there any practical limits on the size of objects that could have another object orbiting them? What kind of environment would be necessary to facilitate something like this occurring? Could it happen in a high or low earth orbit? Would it have to be far away from any stars, planets, or other large bodies? It seems to me that the speed of the objects would have to be very, very slow. I have no background in physics and don't know a ton of math so answers without a lot of complicated equations would be appreciated. Thanks.


title: Relativistic corrections to quantum mechanics of Coloumb potential

text: Systems of charged particles (such as atomic nuclei and electrons) can be described by nonrelativistic quantum mechanics with the Coloumb interaction potential. A fully relativistic description is given by quantum electrodynamics which is much more complex. Is it possible to expand various quantities in QED as power series in 1/c i.e. around the nonrelativistic approximation? Examples of relevant quantities are:   * Ground state energy of a given set of charged particles   * Excited state energies   * Scattering cross sections of charged particles & their bound states (assuming we trace over the photons in the final state)


title: Using quantum entanglement to send messages back to the past

text: > **Possible Duplicate:**   >  Entanglement in time I heard that there is an experiment that uses quantum entanglement to try to send messages back to the past. I am having a hard time understanding how such experiments would work theoretically. Can anyone offer me some insights toward these types of experiment?


title: Disregarding drag force that affects pop-up lightweight ball

text: Is it possible to disregard drag force of water with no viscosity that affects lightweight pop-up ball (its weight is assumed to be zero)? There is a discussion, on which I consider that although weight is small, the influence of drag force does not depend on ball but on water, therefore this force must be taken into account. What can you say?


title: Constructing a Hamiltonian (as a polynomial of $q_i$ and $p_i$) from its spectrum

text: For a countable sequence of positive numbers $S=\\{\lambda_i\\}_{i\in N}$ is there a construction producing a Hamiltonian with spectrum $S$ (or at least having the same eigenvalues for $i\leq s$ for some $s$)? Here by 'Hamiltonian' I understand a polynomial of $p_i$ and $q_i$ (or equivalently - $a_i$ and $a_i^\dagger$) of $k$ pairs of variables and of order $2n$. Both $k$ and $n$ can be functions of $S$ and $s$. For example, for the spectrum $$S =\\{3,5,7,9,\ldots\\}$$ one of Hamiltonians working for any $s$ is $$H = 3+a^\dagger a.$$


title: Do the energy levels of electron orbitals change relativistically?

text: When an electron emits a photon from changing energy levels, the frequency of the photon depends on the difference between the energy levels. But if someone is moving with respect to the atom, the frequency will be apparently red shifted or blue shifted. Does this mean that the energy levels of the orbits of the atom look to have different values if you are moving with respect to them? But, the apparent energy difference can be different depending on whether the photon is emitted towards you or away from you. What's going on? Aren't energy levels of orbits supposed to be a fixed value given the kind of atom?


title: Is there a Lagrangian formulation of statistical mechanics?

text: In statistical mechanics, we usually think in terms of the Hamiltonian formalism. At a particular time $t$, the system is in a particular state, where "state" means the generalised coordinates and momenta for a potentially very large number of particles. (I'm interested primarily in classical systems for the sake of this question.) Since this state cannot be known precisely, we consider an ensemble of systems. By integrating each point in this ensemble forward in time (or, more often, by considering what would happen if we were able to perform such an integral), we deduce results about the ensemble's macroscopic behaviour. Using the Hamiltonian formalism is useful in particular because it gives us the concept of phase space volume, which is conserved under time evolution for an isolated system. It seems to me that we could also consider ensembles within the Lagrangian formalism. In this case we would have a probability distribution over initial values of the coordinates (but not their velocities), and another distribution over the final values of the coordinates (but not their velocities). (Actually I guess these would need to be two jointly distributed random variables, since there could easily be correlations between the two.) This would then lead to a probability distribution over the paths the system takes to get from one to the other. I have never seen this Lagrangian approach mentioned in statistical mechanics. I'm curious about whether the idea has been pursued, and whether it leads to any useful results. In particular, I'm interested in whether the idea of phase space volume has any direct meaning in terms of such a Lagrangian ensemble.


title: Higgs boson and string theory

text: Assuming Higgs is found at 125 GeV.Is there any direct or indirect consequence on string theory ? Will it be a blow to string theory or models employing string theory ? Thanks Ps - I am just a curious pure maths student, so forgive me if my question makes no sense ! :)


title: Branching lemon drop "smoke rings"

text: This might be hard to ask, but here goes nothing. I recently poured a cup of water into a black coffee cup. There was a light source--not very bright--above the cup. Anyways, I was squeezing a lemon into my water mindlessly, and to make sure I got every last drop of lemon juice into the water, I watched the lemon juice hit the water. Then I looked closer as something pretty neat was happening. When a single lemon drop hit the water, it dissipated into a shape that many would describe as a "smoke ring." Knowing some physics/fluids, I understood what was happening here was nothing out of the ordinary. But then I kept watching. As the ring dissipated, it eventually broke off into about 5 other smaller "smoke rings." I couldn't see further down, so who knows if it continued; but I would assume that the 5 smaller rings would turn into 5 + X amount more, etc. At what point would they stop breaking up into smaller rings? Whats causing this to happen? Does this movement/shape have a scientific term? Does anything else do this that can be easily seen? I'm more interested in what the movement/pattern that is happening here, not the chemistry aspect.


title: How do I combine three resistors which are in parallel with each other?

text: For this circuit (a and b are connected by a battery), ![circuit diagram](http://i.stack.imgur.com/CVtE3.png) Will I be able to find the total resistance of the circuit by adding resistors that are in series and combining resistors that are in parallel without using the method of Kirchhoff's Voltage and Current Law? I tried adding R3+R4 and this forms one resistor, which is in parallel with R2 and R1. And R2 is in parallel with R1 and R3, and R1 is in parallel with R2 and R3. This can be verified by Kirchhoff's Voltage laws. The problem is, I get stuck here. I know that if R1,R2 and R3+R4 can be combined into one resistor, then parallel-series would solve the problem neatly. But I can't see how these resistors be combined.


title: The difference between $\mathcal{N}=2$ short multiplets and BPS states

text: I have some questions about the construction of $\mathcal{N}=2$ supermultiplets **for chiral matter**. I know that the supermultiplet should not include spin one states since they are always in the adjoint representation. So my first question is: **why are spin-one modes always in adjoint representation?** To avoid confusion, I will denote the four supersymmetry generators as $Q^A_\alpha$, where $\alpha$ is the spinor index and $A=1,2$. Firstly, massless multiplet or short multiplet. We know $Q^A_2$ generates states with zero norm. Then we can just focus on $Q^A_1$. The supermultiplet can be constructed in this way: $$|\Omega_{-\frac{1}{2}}>\\\ Q^{\dagger 1}_\dot{1} |\Omega_{-\frac{1}{2}}>, \, \, Q^{\dagger 2}_\dot{1} |\Omega_{-\frac{1}{2}}>\\\ Q^{\dagger 2}_\dot{1} Q^{\dagger 1}_\dot{1} |\Omega_{-\frac{1}{2}}>$$ The two spin-zero states in the second line form an $SU(2)$ doublet. Secondly, BPS states. In the presence of a central charge $Z$, one can write the four generators in terms of $A_\alpha$ and $B_\alpha$ after linear transformation with $\\{ B_\alpha, B^\dagger_\beta\\}=\delta_{\alpha\beta}(M-\sqrt{2}Z)$ and $\\{A_\alpha, A^\dagger_\beta\\}=\delta_{\alpha\beta}(M+\sqrt{2}Z)$. When $M=\sqrt{2}Z$, one can get $\\{ B_\alpha, B^\dagger_\beta\\}=0$. Thus, the multiplet will be: $$|\Omega_0>\\\ A^\dagger_\dot{1}|\Omega_0>, \, \, A^\dagger_\dot{2}|\Omega_0>\\\ A^\dagger_\dot{1} A^\dagger_\dot{2}|\Omega_0>$$ Again there are only four states, the same as the short multiplet. So do the two fermion states in BPS states form some kind of doublet as in the short multiplet? It's not so obvious since the relation between $A^\dagger_\dot{1}$ and $A^\dagger_\dot{2}$ are different from that of $Q^1_1$ and $Q^2_1$. From the above discussion, **is it legitimate to conclude that BPS states are different from the short multiplet in $\mathcal{N}=2$?** For instance, in the short multiplet, the two scalars form a doublet while I don't see that in BPS states. But in Alvarez-Gaume's review paper hep-th/9701069 section 2.9, he mentioned that the BPS state belongs to a $\mathcal{N}=2$ short multiplet and the two scalar modes form a doublet while the two fermions are singlet. It's like Alvarez-Gaume was saying the two states are exactly the same. So what's missing in my thinking? Thanks a lot.


title: Find initial velocity when given max height of 100m

text: A ball is thrown straight up. The height of the ball above the ground in (m) is given by the function: $$h(t)=-5t^2 + vt + c$$ What is the initial velocity if you want the ball to reach a max height of 100 m?


title: Charges lying on a Gaussian Surface

text: Let's say you have a spherical charge distribution of radius R. This distribution has some charge density as a function of radius. I know that I can determine the electric field outside of the charge density by forming a spherical gaussian surface around the charge distribution and apply gauss' law. But what if I want to find the field directly on the surface. That requires that my surface intersects some of the charge of the distribution. In that case, when I apply Guass' Law. How do I determine the charge enclosed by the sphere. Do I discount the charge which exists on the surface or is that counted as enclosed by the sphere?


title: Why the K shell only contains 2 electrons?

text: It is written in my quantum physics book that the K shell contains only 2 electrons due to the Pauli principle. I know that if $n = 1, l = 0, m = 0$, then the Hilbert space associated to the spin is of dimension $2$. I also know that Pauli principle says that if we have a vector $\lvert\psi\rangle$ which represents the state of N electrons, then $\lvert\psi\rangle$ must be antisymmetric by transposition. But how can we conclude please ?


title: Is "Witten's Dog" from a Futurama episode an actual physics concept?

text: ![Witten's Dog](http://i.stack.imgur.com/lVTfu.jpg) In an old episode ("Mars University") of Futurama which is a TV show, a character named Professor Farnsworth was trying to lecture "Superdupersymmetric String Theory" and "Witten's Dog" to some other characters. He wrote on board a Feynman diagram, the equation under the diagram looks very much like the electron capture mechanism: $p + e^- \rightarrow n + \nu_e$, except for that wavy underline. However, the Feynman-like diagram above does not look like the electron capture. Also is there any meaning to that caption:"Neutron encrusted steaming hot dark matter"? Are any of the concepts reference in this chalkboard real physical concepts? If so, what are they?


title: Convert a 200mm linear stroke into 90 degrees motion

text: Can anyone help me Convert a 200 mm linear stroke into 90 degrees motion with as much mechanical advantage as possible or into two 90 degrees motions with as much mechanical advantage as possible? Essentially I am trying to _convert the linear motion of a cylinder_ into 90 degrees to bend a wire. Preferably, I'd like this setup to bend into two sides of the wire 90 degrees at the same time. Here is a diagram of what I'm trying to do: ![enter image description here](http://i.stack.imgur.com/w9B5j.png)


title: target category of extended field theory

text: For a topological field theory to be a true “extension” of an Atiyah-Segal theory, the top two levels of its target ( _ie_ its $(n-1)^{\text{st}}$ loop space) must look like $\text{Vect}$. What other (physical) considerations constrain the choice of target category? The targets of invertible field theories are (by definition) $\infty$-groupoids and (therefore) can be represented as spectra. What constraints can we impose on target spectra of invertible theories; in particular, on their homotopy groups?


title: GR exercise: falling particles on earth's surface

text: I'm having some trouble with Exercise 5.1 in Shapiro's BH,WD&NS book, which goes as follows: > Consider two particles of mass $m$ at distance $r$ and $r+h$, such that > $h\ll r$, on the same vertical line from the center of earth. The particles > fall freely from rest a the same time $t=0$ towards the earth surface. Show > that an observer falling with one particle will see the separation between > the particles gradually increase. Translate this into a quantitative > statement about the observer's local inertial frame. This is the first exercise in this chapter, right after the first principle of GR. The author has not introduced curvature and other definitions yet. How can I understand this problem both quantitatively and intuitively? Here is how I think about it: I should solve this problem in two reference frames: the Earth, which is an inertial reference frame, with Minkowski metric, and the free falling particle frame. The two events are the other particle starting falling and after Earth time $dt$ the time and location of it. Compare in free falling frame if the particle is approaching to get the result. Am I conceptually correct? Also I am confused because even if I treat the Earth as a stationary object it has gravitational field so how is distance $r$ and $r+h$ defined? I don't think the Earth frame is inertial.


title: Why should a (physical) principle be applicable to different systems in different positions in space and time?

text: This is a question with a philosophical, as well as physical, flavor. Why should a physical principle (or a description of one), be applicable to different systems that can be in different positions in space and time? In other words why should there be such ( _modulo_ ) equivalence classes (with respect to a physical principle)? Why not a " _custom (physical) principle per case_ "? Let me give one simple example. Newton's laws are applicable to my desk, to my table, to your desk, to your table and to someone's table, say 1 year from now (or 1 year ago). Why is this? Why should the same principle be aplicable to such different systems (different in the sense stated above)? Anyone know how to approach such a question? **Note in order to avoid misunderstandings** : regardless of the tags of this question, this is **not** specifically about _covariance_ , the tag was used, by an editor, because the original tag _laws- of-physics_ was removed for other reasons


title: How obtain conserved quantities in integrable models in accordance with Liouville's theorem, via Sklyanin Poisson algebra?

text: In classical integrable models, in the discrete case we have the Sklyanin algebra, $$\lbrace T_{a}(u),T_{b}(v)\rbrace =[r_{ab}(u,v),T_{a}(u)T_{b}(v)].$$ How to prove that the conserved quantities are generated from $\ln(\tau(u))$ in the periodic case, which are the integrable models (long-range) or condition which is not this expression which provide us the local conserved quantities in the Arnold-Liouville theorem? Reference: Hamiltonian Methods in the Theory of Solitons by Ludwig Faddeev, Leon Takhtajan and A.G. Reyman. In special Part 1 - Chapter 3 and Part 2 - Chapter 3 in this book follow more references.


title: Deriving Time from Acceleration, Displacement, and Initial Velocity

text: I know the equation for displacement as a function of time is $$\vec{s} = \vec{v_i} \Delta{t}+\frac{1}{2}\vec{a}\,(\Delta t)^2$$ I need to solve for $\Delta t$ I'm having problems rearranging to do this. As of now, I created a Python program to run through the displacement formula, adding a set amount of time in each iteration, until the displacement is equal to the displacement I am looking for. I feel that this is woefully inefficient, inaccurate, and horribly CPU intensive. I am confident that this is possible to rearrange, and that I have missed something. If someone already know what this formula rearranged to solve for $\Delta t$ is, I'd much appreciate it if you would share it with me. In the meantime, I shall continue to look around for the rearranged formula and keep trying to solve for $\Delta t$. Hopefully I have not missed something painfully obvious, and that I am not wasting anyone's time.


title: Does electromagnetic field collapse the wave function of charged particles?

text: In an electron double slit experiment, let's put two charged plates behind the slits in an attempt to move the pattern up and down on the the screen. What will happen? Will it just shift the interference pattern on the screen or washes it out completely? If it washes it out, what's the minimal field that doesn't affect the pattern? Since I don't believe the electron double slit experiment was performed in an environment where all fields were exactly zero, but they still managed to get the pattern.


title: Stability of equilibrium points

text: For a spinning top, the linearised equation in the angle $\theta$ when the top is spinning about its axis of symmetry, which is vertical, is of the form $$A\ddot\theta+\left(\frac{C^2n^2}{4A}-Mgh\right)\theta=0.$$ Why should we require that the bracket coefficient be positive if we want the top to be stable, i.e. if we want small disturbances to remain small?


title: Static flow of water in the presence of a loudspeaker

text: The title, I don't know whether it's correct or not, but I came across a video in youtube, http://www.youtube.com/watch?v=_PkgQQqpH2M. The author of video used the title and hence I used the same. The video doesn't seems to be fake because of the "noise" in water flow. But what kind of freq. and why it makes the water flow in this pattern? * * * EDIT1: The title might create some confusion with the nature of the flow. The water drops **are not** stagnant. The water is moving/flowing. but the flow is sort of like standing wave pattern. There are nodes and anti-nodes in the flow. drop which seems stationary are nodes. But as such water seems to flow in and out of the nodal region.


title: A question of BRST symmetry of bosonic string theory

text: This question relates to this post I tried to verify Eq. (4.2.7) in Polchinski's string theory book vol I p. 127 but I miserably miss a sign > $$ \delta_B (b_A F^A) = i \epsilon (S_2 + S_3) \tag{4.2.7} $$ * * * Necessary equations: $$ S_2 =-iB_A F^A (\phi) \tag{4.2.4}$$ $$ S_3 = b_A c^{\alpha} \delta_{\alpha} F^A(\phi) \tag{4.2.5} $$ $$ \delta_B \phi_i = -i \epsilon c^{\alpha} \delta_{\alpha} \phi_i \tag{4.2.6a} $$ $$ \delta_B B_A=0 \tag{4.2.6b} $$ $$ \delta_B b_A = \epsilon B_A \tag{4.2.6c} $$ $$ \delta_B c^{\alpha} = \frac{i}{2} \epsilon f^{\alpha}_{\beta \gamma} c^{\beta} c^{\gamma} \tag{4.2.6d} $$ * * * My attempt to verify Eq (4.2.7), $$ \delta_B b_A F^A = (\delta_B b_A) F^A + b_A (\delta_B F^A) $$ $$ = \epsilon B_A F^A + b_A (\partial^i F^A) \delta_B \Phi_i = \epsilon B_A F^A + b_A (\partial^i F^A) (-i \epsilon c^{\alpha} \delta_{\alpha} \phi_i) $$ $$ = \epsilon B_A F^A - i \epsilon b_A c^{\alpha} \delta_\alpha F^A = i \epsilon (S_2 - S_3) $$ Why did I get $-S_3$?


title: weak interaction coupling constant

text: in wikipedia, the weak interaction coupling constant is said to be 10^-13 times weaker than that of the strong interaction but hyperphysics (http://hyperphysics.phy-astr.gsu.edu/hbase/forces/couple.html) says it's 10^-6. It seems wikipedia is right but why many other resources also provide a value similar to that of hyperphysics?


title: Determine Charge With Electroscope?

text: You have three separate glass rods, and you know one is positive, one is negative and one is neutral. You also have an electroscope that is positively charged. How can you determine which rod is positive, negative and neutral? I think you will be able to determine which rod is negatively charged, as that rod will cause the electroscope leaf to rise. I'm not sure how to differentiate between the positive+neutral though, as both will cause the leaf to collapse. Could you place it next to the negative rod and see which repels? Essentially it would be great if somebody could explain the actual physics behind the situation. Why will the negative rod cause the leaf to rise and vice versa? Thanks!


title: state on quantum statistics. 3 particles according to 3 distributions

text: consider a system of three identical particles, A B ,and C. Assume that each particle can be in one of three possible quantum states, 1,2 and 3. For the following statistics listed below, enumerate the possible states of the system and calculate the probabilities that the state 1 has zero,one,two, and three particles. a. Maxwell Boltzmann distribution b. Bose-Einstein distribution c. Fermi-dirac distribution


title: What is being deprived when a photon is being watched in double slit experiment?

text: How are photons being watched in the double slit experiment? What exactly does being observed mean, as it is obviously changes the state of the photon somehow - it must be depriving the photon of something or emitting something that interacts with the photon.


title: Measuring cross-section area of a flexible tube (5mm dia approx)

text: I want to measure the avg. inner cross-section area of a flexible tube of outer diameter 5mm. Since the cross-section isn't a circle exactly, use of vernier caliper to measure inner diameter fails. The idea that I wish to try is, to take a tube of known length, fill it with water, then pour the water on to something like a volume flask and from the volume obtained compute the cross-section area. My questions are, 1 - how valid does this method sound? 2 - Is there a better way to do this? 3 - How much error would the water stuck in the tube due to surface tension introduce? Would appreciate any help. Thanks.


title: Is there a relation between quantum theory and Fourier analysis?

text: These days I was studying the quantum theory.I found that some theories about that is similar to Fourier Transform theory.For instance, it says "A finite- time light's frequency can't be a certain value", which is similar to "A finite signal has infinite frequency spectrum" in Fourier analysis theory.I think that a continuous frequency spectrum can not be measured accurately, which is similar to Uncertainty principle by Hermann Weyl. How do you think about this?


title: is there spin degree in 1D

text: it is well known that the intrisic spin is closely related to the rotation in space. However, in 1d , it is impossible to define rotation, therefore it is meaningless to talk about spin in 1d.However, one can see many papers talking about electrons in 1d with spin Is this meaningful?


title: Lagrangian density for a Piano String

text: So I'm trying to do this problem where I'm given the Lagrangian density for a piano string which can vibrate both transversely and longitudinally. $\eta(x,t)$ is the transverse displacement and $\xi(x,t)$ is the longitudinal one. (So a point at $[x,0]$ at some later time t would be at $[x+\xi(x,t),\eta(x,t)]$). The Lagrangian density is given by $$\mathcal{L} = \frac{\rho_0}{2}[\ddot{\xi}^2+\ddot{\eta}^2]-\frac{\lambda}{2}[\frac{\tau_0}{2}+\xi'+\frac{1}{2}(\eta')^2]^2$$ where the dot is a partial time derivative and a prime is a partial x derivative. Also $\lambda$, $\tau_0$ and $\rho_0$ are parameters such as the Young's modulus, density and tension of the string respectively. So applying the action principle in the case of a continuous system I get the following PDEs for $\xi$ and $\eta$: $$\ddot{\xi} = \frac{\lambda}{\rho_0}(\xi''+\eta'\eta'')$$ and $$\ddot{\eta} = \frac{\lambda}{\rho_0}[\eta''(\frac{\tau_0}{\lambda} + \xi'+\frac{3}{2}(\eta')^2)+\eta'\xi'']$$. Linearization of the equations yields two simple wave equations, the $\eta$ wave traveling at a speed $c_T^2 = \frac{\tau_0}{\rho_0}$ and the $\xi$ wave at $c_L^2 = \frac{\lambda}{\rho_0}$. Now the part that is stumping me is that the problem asks us to show that if a given pulse of the form $\eta(x,t) = \eta(x-c_Tt)$ propagates along the string then that induces a concurrent longitudinal pulse of the form $\xi(x-c_Tt)$. Obviously I can't use the linearized equations to prove this. I would have to use some sort of iterative process by using exact PDE's to get the next step. But I tried doing this and I just get a mess. Any ideas where to go from here? (for reference this is problem 1.10 in Stone and Goldbart's book).


title: Superconductivity: why can't the resistance reach 0?

text: When we study electricity in high school we examine the resistance of conductors and its relation with temperature. Diagrams show the relationship at the beginning is pretty much a linear with temperature but at very low temperatures its starts to curve in such why it never reaches zero but it is close and the book explains it "At low temperatures it starts to curve because the conductor is not of pure material it has particles of other elements and their effect starts at low temperature and that is way it was linear beforehand" My question is how do these other elements stop it from reaching 0 and why don't they have the same effect on our conductive the whole time why does it only show at 159.7K and even less? The explanation is that as temperature is reduced, the movement of the atoms is slowed which attributes to a lower chance of the electrons' movement being slowed because of bumping into atoms increasing their speed which means it has less resistance. Why can't the resistance reach zero?


title: How to add two perpendicular 2D vectors

text: * vector D = 4 cm North      * vector J = 4.5 cm West what is D+J? In a more general sense, how can two 2D vectors that are perpendicular to each other be added?


title: Energy and time evolution of a particle in a potential well

text: Hoping this is not a silly and stupid question let me ask for help in this problem. I have a particle in an infinite square well (the box is from 0 to a), in the state described by the function $\psi (x) = Ax(a-x) \qquad \mathrm{for }\qquad 0<x<a$, $\qquad 0 \qquad otherwise$. I have to determine the most likely value of energy and the probability to obtain a value of $E = \frac{9\hbar^2 {\pi}^2}{2ma^2} $. To solve the second question I thought that $E$ it's the classic solution for energy in a potential well with $n=3$. So I calculate $\langle3| \psi\rangle$ $-$ in which $3$ is the solution wave function with $n=3$ $-$ and that's it? Right? But for the first question? Do I have to calculate $\langle H \rangle$ and compare it with a solution of the potential well? After I also have to determine the evolution of the wave function for $t>0$ when at $t=0$ we turn off the potential well, and I have no idea how to solve it. Please forgive me for my English. Hope you'll give me some hints.


title: Finding the value of velocity $u$

text: A block of mass $m$ is pushed towards a moveable wedge of mass $\eta m$ and height $h$, with a velocity $u$. All surfaces are smooth. The minimum value of $u$ for which the block will reach the top of the wedge is: ![enter image description here](http://i.stack.imgur.com/vaqFh.png)   1. $\sqrt{2gh}$   2. $\eta \sqrt{2gh}$   3. $\sqrt{2gh(1+\frac1\eta)}$   4. $\sqrt{2gh(1-\frac1\eta)}$ What's confusing me is how to proceed knowing the wedge is moveable. I tried drawing the FBD of both the objects. But I am not getting any far with it.


title: Photons in a gravitational field

text: I have been really staring for a while in a MP-Beiser book and I totally disagree with a statement he does there. On a page 85 he states that photons act as they have a mass $m$. He derives this by stating that: $$ \begin{split} p &= m v\\\ \frac{h\nu}{c} &= m c\\\ m&= \frac{h \nu}{c^2} \end{split} $$ But I totally disagree with this. We have learned and derived that momentum of a particle is: $$ \begin{split} p &= m v \gamma (v)\\\ \frac{h \nu}{c} &= m c \gamma{(c)}\\\ m &= \frac{h \nu}{c^2 \underbrace{\gamma(c)}_{=0}}\\\ m &= 0 \end{split} $$ Something here is totally wrong, but what? How can an author state what he does? I know that on Harvard they did an experiment resulting in different $\nu$ of a photons falling in a gravitational field, but they must have been wrong or something... Please someone explain.


title: How can the 5-photon absorption coefficient be estimated?

text: Imagine a large bandgap material which is irradiated by an intense laser beam. If the photon energy is only high enough for 1/5 of the bandgap, is there a way to approximate the absorption by 5-photon excitation, i.e. the ratio of transmitted to initial Intensity? All I found is related to 2 or 3 Photon absorption but not higher orders in the pertubation series ...


title: Can you use PCR to make a standard kg?

text: While reading this question: Why do we still not have an exact definition for a kilogram? , I had a crazy thought. Using PCR, you make a known number of copies of a DNA strand where the length and composition is exactly known for all the copies. The number of copies is two to the power of the number of PCR cycles, and if you start with, say, all C-G DNA, you get a known number of duplicates which are the same molecular weight. If you only provide C and G nucleotides, you don't even have to worry too much about errors in copying, because whichever way they are fixed to make pair-matching DNA, you will get the same atomic weight. You can separate out the DNA and get a known huge macroscopic batch size of macro-molecules where we know the atomic composition exactly. So we can easily make a batch of atoms where we know the number of atoms with potentially exact precision. Can you use this to make a mass standard? Just weighing the DNA won't work, because the water content will be uncertain. Drying the DNA won't work, because there will always be some completely random stuck protons or hard-to-evaporate water. But I suspect this can be done, because here is a system with an exactly known macroscopic number of atoms. Perhaps if you centrifuge the DNA and compare the density of water to the density of DNA-water? Question: Can you define the kg with PCR?


title: How can things at the event horizon slow down and appear to stop to a remote observer?

text: So they say the remote observer will never see anything fallen to the black hole, because any object will slow down as it gets closer to the event horizon and eventually stop to stay there forever. Am I getting it right so far? With this said, it turns out nothing has fallen to the black hole up till now. That is for us, as remote observers, anything that ever tried to reach the black hole got stuck at its event horizon. So we can go and see it right away - it must be still there. At the same time they say that the black hole keeps sucking matter. So question is, to a remote observer, how does all this matter fit the event horizon provided it will never fall through it?


title: Using the covariant derivative to find force between 't Hooft-Polyakov magnetic monopoles

text: I am reading this research paper authored by NS Manton on the Force between 't Hooft-Polyakov monopoles. I have a doubt in equation 3.6 and 3.7. We assume the gauge field for a slowly accelerating monopole to be $A_0 = \epsilon^2 a_i t A_1$, where $\epsilon^2$ is an infinitesimal. Also, we write $\partial_0 \phi = -\epsilon^2 a_i t \partial_i \phi$. Using this he writes $D_0\phi=-\epsilon^2 a_i t D_i \phi$, where $D_i\phi=\partial_i \phi + [A_i,\phi]$. Isnt the sign of the second term wrong? Secondly, he says differentiation wrt t gives us, $D^0 D_0 \phi = \epsilon^2 a_i D_i \phi$. Shouldnt it be $\partial^0D_0 \phi$? Cause we are taking the actual derivative wrt t rather than the covariant derivative, WE should get some extra terms, do they cancel out? How does the minus sign disappear? Does the covariant derivative behave like a normal derivative in any case?


title: Direction of rotation of a celing fan

text: From Constellation Energy > Quick energy efficiency tip: To stay cool and manage your energy at the same > time, use ceiling fans to create a “wind chill” in rooms you are using. The > wind chill will help you feel cooler than the actual temperature. **Make > sure the ceiling fan is set to turn in a counter-clockwise rotation**. I was wondering why the ceiling fan is set to turn in a counter-clockwise rotation, instead of the other way around? Thanks!


title: Majorana particles statistics

text: What is the influence of Hermitian condition ($\psi=\psi^{\dagger}$) of Majorana fermions operators in their statistical behavior? A Majorana fermion gas must obey the Fermi-Dirac statistics, or their stastistical behavior may be anyonic?


title: Questions on wave-particle duality

text: Wave-particle duality states that a particle has both wave properties and particle properties when one is _not_ observing it. 1) What is an observer? Need it be anything living or can other particles also act as observers? 2) When doing the electron double slit experiment--shooting just one electron at a time, the electron goes through both slits at the same time (if one is not observing it). Does that say that the electron is on every single geographical point at the same time?


title: Can we explain physical similarities between Black Scholes PDE and the Mass Balance PDE (e.g. Advection-Diffusion equation)?

text: Both the Black-Scholes PDE{*} and the Mass/Material Balance PDE have a similar mathematical form of the PDE which is evident from the fact that on change of variables from Black-Scholes PDE we derive the heat equation (a specific form of Mass Balance PDE) in order to find analytical solution to the Black-Scholes PDE. I feel there should be some physical similarity between the two phenomena which control these two analogous PDEs (i.e. Black-Scholes and Mass/Material Balance). My question is, can one relate these two phenomena physically through their respective PDEs? I hope my question is clear, if not please let me know. Thanks. *PDE=Partial Differential Equation


title: English translation of Helmholtz' paper: “On the Physical Significance of the Principle of Least Action”

text: I am asking about an English translation of a Helmholtz paper: > Ueber die physikalische Bedeutung des Princips der kleinsten Wirkung. > _Journal für die reine und angewandte Mathematik (Crelle's Journal), Volume > 100, Issue 2, 1887, Pages 137-166, and Volume 100, Issue 3, 1887, Pages > 213-222._ http://www.degruyter.com/view/j/crll.1887.issue-100/crll.1887.100.137/crll.1887.100.137.xml?format=INT. (Also see link and link.) The title in English: _On the Physical Significance of the Principle of Least Action._ Has it ever been translated (to English)?


title: Finding Lagrangian of a Spring Pendulum

text: I'm trying to understand Morin's example of a spring pendulum. What I don't get is his expression for $T$. I can understand the $\dot x^2$ term in the brackets. But I don't understand the $(l + x)^2\dot \theta^2$. Also, it seems rather strange to break up Kinetic Energy into tangential and radial components when it is a scalar. ![enter image description here](http://i.stack.imgur.com/apteI.jpg)


title: Why can't photons have a mass

text: Why can't photons have a mass? Could you explain this to me in a short and mathematical way?


title: Does a photon have mass?

text: I have seen questions assume photons have no mass. But I have not seen any questions that directly ask whether or not photons have mass. If photons have no mass, then how do they occupy space? How are they "there?" Do photons have mass?


title: If I shine a bright light on a neutron star, what would it look like?

text: If there are no orbitting electrons in a neutron star's makeup to interact with EM, what happens to light that strikes it?


title: For the long-term evolution of atmosphere/orbit, when is perihelion more important than mean distance?

text: When we want to figure out the long-term evolution of a planet's atmosphere/orbit, when is perihelion more important than mean distance? E.g. some processes (like Jeans Escape and escape of atmospheres) are disproportionately affected during perihelion (point of closest approach) rather than during aphelion.


Who has succeeded in demonstrating the Lense-Thirring effect? This effect is one that describes the rotational motion of the Earth from a space-time structure. This effect is the "drag" of the geometrical structure of space-time due to rotational motion of the object which causes the gravitational-inertial field.


title: Differential Forms and Densities

text: I've heard that differential forms are related to densities, however I'm still a little confused about that. I thought on the case of charge density and I came to that: let $U\subset\mathbb{R}^3$ be a region of $3$-space, and let $\rho : U \to \mathbb{R}$ give the charge density at every point of $U$. I can then create the $3$-form $\omega = \rho \ dx \wedge dy \wedge dz$, which in my understanding gives me the approximate amount of charge enclosed by a volume determined by $3$ vectors when they're given. So, if I give the vectors $v, u, w$, the value of $dx \wedge dy \wedge dz(v,u,w)$ should be the volume enclosed by those vectors, and hence $\omega(u, v, w)$ should be an approximation of the charge enclosed. Is this correct? My only problem is: in this point of view, the form isn't giving me the density, the density itself is being given by a scalar field, while the form gives me the charge instead of the density. Is this correct? The form is always meant to give the charge instead of the density? The density should always be regarded as a scalar field?


title: What really are superselection sectors and what are they used for?

text: When reading the term superselection sector, I always wrongly thought this must have something to do with supersymmetry ... DON'T laugh at me ... ;-) But now I have read in this answer, that for example for a free QFT highly excited states , that would need infinite occupation numbers to build them up, and that lie therefore outside the Fock space are said to lie in a (different?) superselection sector. If a state has finite or infinite energy depends on the Hamiltonian, and a finite energy and physically relevant Hilbert space can be obtainend from the inacessible infinite energy states of another Hamiltonian. This makes me now want to really know what a superselection sector is. What are the key ideas behind the definition of a superselection sector? Are they an underlaying concept to derive quantum field theories with a physical hilbert space that has only finite energy states, or what is their common use in physics?


title: Are the protons and neutrons in the nucleus arranged in any particular way?

text: I was wondering this: suppose you have two oxygen atoms. They will both have 8 protons and 8 neutrons in the nucleus (at least if they are the most common isotope). Now, will all those particles be arranged in the same way in both atoms? If they are, why would that be, and if not, does that affect the element's properties in any way? But then I also thought that maybe the uncertainty principle doesn't let us even ask this question. Maybe you can't tell the particles' positions so accurately, so all you can say is that you have 8 protons and 8 neutrons all together in a small space. So, which one is it? Can we even tell where all the particles are, and if we can, does it matter exactly how they are arranged?


title: Geometric structure of atomic nucleus?

text: The electrons in an atom have a certain configuration and we can predict in which space the electrons can be present of a certain energy level (s-orbitals have a sphere-shape,....). Recently I read that there are magic numbers for nuclei: special configurations of the nuclei that makes them more stable (the magic numbers are 2, 8, 20, 28, 34, 50, 82, and 126) - so there are definitely shells in a nucleus. I was wondering : do the neutrons and protons form certain geometric structures in these sub-shells of the nucleus? Or are they all stacked closely on each other?


title: The difference between the Wannier function and atomic orbit in a tight binding model

text: In a tight binding model, we usually start from the atomic orbits and linearly combine them to get the wave function of the crystal energy band. My questions are:   1. Since this kind of tight binding is an approximate method due to using atomic orbits, is it exact to use the Wannier function formalism? If so, how do I get the Wannier function systematically?   2. What is the use of maximally localized Wannier functions?   3. Why can't we get the maximum localized Wannier function when the Berry phase is not zero?   4. Also, in tight binding, formalism, taking atomic orbits or Wannier functions as the basis function, respectively, what does position operator (diagonal or not), velocity operator and angular momentum operator look like?


title: How do you fit a dipole in an electron?

text: Experiments used to observe particle spin properties (such as Stern-Gerlach) rely on a varied magnetic field and a dipole-like reaction in the particle, deflecting it in one direction or another. In the case of a point-particle such as an electron, how is it explained that a dipole with spatially separated poles exists in a single point?


title: On what planar regions are the Earth, Moon.. for that matter, all astronomical bodies, placed in the Universe?

text: When we are on Earth, we look **UP** in the Sky to see the **Moon**. How do we have to look at Earth from Moon.. Is it the way ? If so, how are these bodies actually placed in the space? Are all astronomical objects lined up on the same planar level? When we look at stars, are they actually above the Solar System planar level or below ?! Because, in general, if we have to look up to see something on a building, then from the building we have to look down to see the same object.


title: What was wrong with action a distance?

text: It is usually said that the idea of fields was introduced (electric and magnetic fields) in electricity and magnetism after Coulomb's law to cure the conceptual problems of action at a distance. Could someone explain what are the conceptual and physical difficulties or contradictions that one might have with action at a distance?


title: Find temperature of surface (Blackbody Radiation)

text: > An astronomer is trying to estimate the surface temperature of a star with a > radius of $5 \times 10^8\ m$ by modeling it as an ideal blackbody. The > astronomer has measured the intensity of radiation due to the star at a > distance of $2.5 \times 10^{13}\ m$ and found it to be equal to $0.055\ > W/m^2$. Given this information, what is the temperature of the surface of > the star? How do I do this? The hint was to use $I=\sigma T^4$ (where $\sigma = 5.67 \times 10^{-8}$).   Others: > As the star radiates energy, the total power that flows through any > spherical surface concentric with the star remains constant. The total power > flowing through such a surface can be obtained by multiplying the intensity > (power per square meter) by the surface area of the sphere. > > Using the law of conservation of energy, the surface area of the two spheres > (one at the star's surface and one with the radius equal to the observing > distance), and the intensity measured by the observer at the observing > distance, you should be able to obtain the intensity at the surface of the > star. Don't really understand the 2nd paragraph too ...


title: Does gravity affects temperature reading of a mercury thermometer?

text: I remember when I was in primary school, the science teacher put me in charge of a mercury thermometer. I do not quite understand the mechanics behind except that mercury expands when it is hot and contracts when it is cold, and that this could be read off a temperature scale along the stem of the thermometer. At the back of my mind, there is this doubt on how gravity affects the movement. Having seen most thermometers being hanged vertically on a wall, I did likewise, placing the thermometer on the floor, upright, leaning against the window sill beside my desk. This is fine until one day when a strong gust of wind blew and cause the thermometer to fall flat on the floor, smashing the glass casing and causing the mercury to leak out resulting in the evacuation of the whole class. After that incident, I learnt my lesson and place the new thermometer lying flat instead of upright. But, the lesson that I didn't learn is how gravity affects the behaviour of such thermometer. Hope to learn something here.


title: Does increasing the pressure of the gas in a container by pumping gas into it increase temperature?

text: Basis is the ideal gas law: http://en.wikipedia.org/wiki/Ideal_gas_law This law only states a mathematical relationship between many variables. It does not state what happens if one of the variables changes. So we pin down the volume and increase moles of air by pumping in more gas. n obviously changes, so does P. But what happens to T? Does the temperature change at all?


title: Solar sail area going to Proxima Centari

text: I have a physics question that I need some help with: "Proxima Centauri is a star in the Alpha Centauri solar system, it’s the nearest star to our sun (4.24light−years) http://en.wikipedia.org/wiki/Listofneareststars. How large of a solar sail would be needed to accelerate a solar sail of mass m to a velocity which will get the ship to Proxima Centauri in two lifetimes?" I'm assuming that the time is 160 years for 80 years each lifetime. Can anyone help?


title: Escape velocity for hydrogen molecules

text: The question: At what temperature is the RMS speed of Hydrogen molecules equal to the escape speed from the earth's surface? Values of radius of earth($r$) and gas constant $R$ has been supplied only. We know that the escape velocity of a body on earth is given by $(2rg)^{1/2}$. Putting in the values of $r$ and $g$ we get $11.2\ \text{km/s}$ as the answer. We also know that for a mole of ideal gas $PV=RT=\dfrac13M[c_{rms}]^2$,where $M$ is molecular weight of the gas. So substituting for $T$ we get $T=\dfrac{M[c_{rms}]^2}{3R}$ I have put $11.2\ \text{km/s}$ in place of $c_{rms}$ and $M=2$ and put the value of $R$ and I have got $904.8\ \text{K}$ as my answer, which is equal to $631.8\ ^o\text{C}$. For moon the answer came as $98\ ^o\text{C}$. My question is that moon certainly has a temperature was lower than $98\ ^o\text{C}$ so then shouldn't it have an atmosphere of hydrogen? But the moon has no atmosphere!! I cannot understand where i have gone wrong.


title: Fierz identity with Dirac spinors

text: The following Fierz relation does not seem so obvious to me : \begin{equation} \bar{\psi}_1 \gamma^\mu (1+\gamma_5)\psi_2 \bar{\psi}_3 \gamma_\mu (1-\gamma_5) \psi_4 = -2 \bar{\psi}_1 (1-\gamma_5) \psi_4 \bar{\psi}_3 (1+\gamma_5) \psi_2. \end{equation} As a first step I would have tried to do something like \begin{equation} \bar{\psi}_1 \gamma^\mu (1+\gamma_5)\psi_2 \bar{\psi}_3 \gamma_\mu (1-\gamma_5) \psi_4 = \bar{\psi}_1 (1-\gamma_5) \gamma^\mu \psi_2 \bar{\psi}_3 (1+\gamma_5)\gamma_\mu \psi_4 \end{equation} but this does not help to get rid of the $\gamma_\mu$s. Am I going the wrong way ?


title: Modeling Acoustic Lubrication

text: I thought about posting this question in the Biology StackExchange site but really it is just the application of my question that applies to biology not the core of the question itself. Can anyone help me find the math involved in modeling acoustic lubrication? My goal is to investigate whether it could be employed as a strategy to ease the passing of kidney stones which are extremely painful and can take days to pass. It would seem that if you could successfully achieve acoustic lubrication of a kidney stone that it might significantly reduce the time it takes for the kidney stone to travel down the ureter. I found this which may help: > The sound speeds were 4270 ± 80 m/s for calcium hydrogen diphosphate, 4320 ± > 40 m/s for cystine, and 4330 ± 50 m/s for calcium oxalate monohydrate. > http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2913873/ Also the size range in question would be diameters ranging from 1mm to 5mm. Stones much larger than that are not considered passable. I believe the challenge lies in the fact that the kidney stone may have a significant percentage of its surface in contact with the ureter walls so acoustic lubrication may not even be achievable, but I think it is worth investigating.


title: Remnant of a supernova

text: A supernova remnant is the structure resulting from the explosion of a giant star. The supernova remnant is surrounded by an expanding shock wave that is formed from material ejected by the explosion and interstellar material swept away during the process. When this material collides with the circumstellar or interstellar gas, it forms a shock wave that can heat the gas at high temperatures up to 10 million K, forming a plasma. The first question is: exactly what is the material ejected by the supernova? The other issue is: if we consider for a moment the idea that a "human" might be witnessing the exact moment of the explosion of a supernova, the "superhuman" would hear the explosion? That is, does the explosion produce some kind of sound?


title: Automated telescope system

text: I am living up north in Norway, 300 km above the Arctic Circle, which gives me six months per year of darkness and cold. I used to have a starter telescope when I was living in Spain, but I gave it away, and I want to build/buy a new system with the following requirements.   * operated remotely - by cables or by Wi-Fi. I want to put it outside my house - maybe even 50 meters away and stay inside (warm) and do all the stuff from my computer: moving, viewing, zooming and saving information.   * I would like the telescope to know how to track a specific star or nebula   * I would like to be able to do long exposures with an attached camera for faint objects, probably hours, so I can get nice images for you :) What gear do I need and what things may I be missing?


title: Electric Flux Density - Ring Charge

text: A ring placed along $y^{2}$ + $z^{2}$ = 4, x = 0 carries a uniform charge of 5 $\mu$C/m. Find D at P(3,0,0) Should I be using Gauss's Law to solve this problem? I was considering using a spherical Gaussian Surface, and then using the formula D = $\epsilon_0 $E to find D, but I'm not sure how to set up my integral.


title: Method of Images

text: The method of image charges is a well-known and very useful tool for solving problems in electrostatics. Unfortunately, when I was taught this method, it was presented simply as an algorithm. No real physical justification was given for its usage, and there was a complete absence of rigorous mathematics. The method has always been a little hazy to me since. If someone could give a rigorous first-principles derivation of this method from Maxwell's equations (perhaps simply Coulomb's law?), it would be most appreciated.


title: Calculating electric field

text: ![alt text](http://i.stack.imgur.com/hWGlJ.png) This problem is giving me a lot of problems. So $E=k*q/d^2$. We'd want to find the distance from q1 to P, which is .1 meters (not cm) using pythagorean thereom. So we know k, which is just $9x10^9$ times q1 which is $-2.4u$ where $u=10^{-6}$ divided by $r^2$ which is just $.1^2$. Then I get $-216,000,000$. To get x component, I take that number multiplied by $cos45$ to come up with a final answer of $-108,000,000$. And for point q2, I dont think there is an x component for the electric field since its right below the point P. What have I done wrong?


title: Still trying to understand gravitational potential and Poisson's equation?

text: A week or so back I asked a question about the gravitational potential field $$\phi=\frac{-Gm}{r}, \qquad r\neq 0, $$ and how to show the Laplacian of $\phi$ equals zero for $r\neq 0$? Eventually, (it took a while) I was able to understand that $$\nabla\cdot\nabla\phi=Gm\left(\frac{2x^{2}-y^{2}-z^{2}+2y^{2}-x^{2}-z^{2}+2z^{2}-x^{2}-y^{2}}{\left(x^{2}+y^{2}+z^{2}\right)^{5/2}}\right)~=~0, \qquad r\neq 0,$$ which was a revelation. But now I'm wondering why Poisson's equation $$\nabla\cdot\nabla\phi=\nabla^{2}\phi=4\pi G\rho$$ doesn't always equal zero as well? Obviously it doesn't, so I'm assuming that inside a mass the gravitational potential field cannot be given by $$\phi=\frac{-Gm}{r}, \qquad r\neq 0.$$ Is that correct? Also, is there a comparably easy formula for gravitational potential inside a mass or does it vary (horribly?) depending on the shape and density of the mass?


title: How was this pressure pulse propagation speed be derived?

text: Some lecture notes I was reading through claimed that a pressure pulse propagates through a liquid-filled tube (blood in a vein) with the speed $$c=\sqrt{\frac{A}{\rho}\frac{dP}{dA}},$$ where $A$ is the cross-sectional area, $\rho$ the density and $P$ the pressure (which pressure?). How can this result be derived? Why would the pressure vary with area? I suppose I'm missing some assumptions or misunderstand the expression. FYI, the expression is also written, without explanation of the variables, as $$c=\sqrt{\frac{Eh}{2\rho R}}$$


title: Can vectors in physics be represented by complex numbers and can they be divided?

text: Below is attached for reference, but the question is simply about whether vectors used in physics in a vector space can be represented by complex numbers and whether they can be divided. * * * In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms. The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, various algebraic number fields, p-adic fields, and so forth. **Any field may be used as the scalars for a vector space** , which is the standard general context for linear algebra. The theory of field extensions (including Galois theory) involves the roots of polynomials with coefficients in a field; among other results, this theory leads to impossibility proofs for the classical problems of angle trisection and squaring the circle with a compass and straightedge, as well as a proof of the Abel–Ruffini theorem on the algebraic insolubility of quintic equations. In modern mathematics, the theory of fields (or field theory) plays an essential role in number theory and algebraic geometry. In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the value of the scalar field at the same point in space (or spacetime). Examples used in physics include the temperature distribution throughout space, the pressure distribution in a fluid, and spin-zero quantum fields, such as the Higgs field. These fields are the subject of scalar field theory. In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it is an algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as an algebra, this multiplication must satisfy certain compatibility axioms with the given vector space structure, such as distributivity. In other words, an algebra over a field is a set together with operations of multiplication, addition, and scalar multiplication by elements of the field. A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied ("scaled") by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex numbers, rational numbers, or even more general fields instead. The operations of vector addition and scalar multiplication have to satisfy certain requirements, called axioms,... An example of a vector space is that of Euclidean vectors which are often used to represent physical quantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real factor is another force vector. In the same vein, but in more geometric parlance, vectors representing displacements in the plane or in three-dimensional space also form vector spaces. In classical mechanics as in physics, the field is not real, but merely a model describing the effects of gravity. The field can be determined using Newton's law of universal gravitation. Determined in this way, the gravitational field around a single particle is a vector field consisting at every point of a vector pointing directly towards the particle. The magnitude of the field at every point is calculated applying the universal law, and represents the force per unit mass on any object at that point in space. The field around multiple particles is merely the vector sum of the fields around each individual particle. An object in such a field will experience a force that equals the vector sum of the forces it would feel in these individual fields. Because the force field is conservative, there is a scalar potential energy per unit mass at each point in space associated with the force fields, this is called gravitational potential. Gauss' law for gravity is mathematically equivalent to Newton's law of universal gravitation, but is stated directly as vector calculus properties of the gravitational field.


title: Circumference of a circular path

text: Suppose a car moves with a constant speed of $20 \text{m/s}$ a quarter of a circle, and completes the quarter in $5$ seconds. One way to calculate the circumference is simply $20 \cdot 5 \cdot 4 = 400 \text{m}$. However, I know that $a=\frac{v^2}{R}$ and the circumference is $2 \pi R$. The acceleration is defined as $\frac{\Delta v}{\Delta t}$ and thus from Pythagorean theorem we get $a = \frac{\sqrt{20^2+20^2}}{5}=4 \sqrt{2} ~~ \text{m/s}^2$. Now to find the circumference we plug our result into $2 \pi \frac{v^2}{a}$, thus the circumference: $2 \pi \frac{20^2}{4 \sqrt{2}} \approx 444.29 \text{m}$ which is clearly wrong. So why the second way gave me a wrong result?


title: The definition (or perhaps, explaination) of Effective Pressure or Effectiveness in a Hydraulic System

text: I am trying to understand the effect of effective pressure or effectiveness in a hydraulic system. I have seen different units for effective pressure or effectiveness - e.g. Nm/Bar Nm/L etc. and I am a bit confused how or why this is different. Any help would be appreaciated.


title: Does the temperature of water determine how much heat will be removed from air used to evaporate it?

text: This is a question about evaporative cooling as used in residential evaporative cooling appliances. This type of cooling uses the heat in the ambient outside air to evaporate water and remove the heat from the air, then push the cooled air inside. The equation to predict the temperature of the resulting air after it's given up its heat to evaporate the water is as follows: $$T_{output} = T_{dry} - (T_{dry} - T_{wet}) * \epsilon$$ where $T_{output}$ is the output air temperature, $T_{dry}$ is the air temperature of the dry bulb, $T_{wet}$ is the air temperature of the wet bulb, and $\epsilon$ is the cooling efficiency. For example, on a very dry summer day (dry bulb 95 degrees, wet bulb 60 degrees) my evaporative cooler with 90% efficient media is capable of cooling the air to 63.5 degrees. However, this equation does not seem to take into account the temperature of the water itself. Does it matter? Intuitively, it would seem to make sense to me that hotter water would be easier to evaporate, since it's closer to its boiling point. Or maybe colder water is better because it will absorb more heat from the air? Or maybe it's a wash because the same amount of heat is required, but with hotter water, more is needed because it will evaporate faster? Help me understand this.


title: Does cosmological time expand like space does?

text: The FRW metric is given by: $$ds^2=-dt^2+a^2(t)\ dr^2$$ where $ds$ is an interval of proper length, $dt$ is an interval of cosmic time, $dr$ is an interval of co-moving co-ordinate distance and $a(t)$ is the scale factor (also $c=1$). If I take $dt=0$ then I find that an interval of proper distance $ds$ is given by: $$ds = a(t)\ dr$$ Thus the proper distance between two nearby co-moving points is proportional to the scale factor - space expands. If I take $ds=0$ then I obtain the null geodesic equation describing the path of a light ray: $$dt = a(t)\ dr$$ Thus the light travel time between two nearby co-moving points is also proportional to the scale factor. Does this imply that intervals of cosmic time expand along with intervals of space? The natural clocks in co-moving co-ordinates are expanding light-clocks. Maybe in order to derive the constant units of the "atomic" time that we experience, $d\tau$, from expanding cosmic time intervals $dt$, we need to use the conformal time equation: $$d\tau = dt/a(t)$$


title: Quantum harmonic Oscillator analytic method

text: I'm using a book from Griffiths, I got really stuck about how he arrived at the approximate solution, is it just by trying( trial solution method?), I really appreciate any help on this. $$-\frac{\hbar^2}{2m}\frac{\mathrm{d}^2\psi}{\mathrm{d}x^2} + \frac{1}{2}m\omega^2 x^2\psi~=~E\psi.$$ Change variables for convenience: $$\begin{align} \xi &\equiv \sqrt{\frac{m\omega}{\hbar}}x \\\ K &\equiv \frac{2E}{\hbar\omega} \\\ \frac{\mathrm{d}^2\psi}{\mathrm{d}\xi^2} &= (\xi^2 - K)\psi. \end{align}$$ For very large $\xi$: $$\frac{\mathrm{d}^2\psi}{\mathrm{d}\xi^2} \approx \xi^2\psi.$$ Approximate solution: $$\psi(\xi) \approx A e^{-\xi^2/2} + B e^{+\xi^2/2}.$$


title: Some ambiguous points on Spontaneous Symmetry Breaking (SSB)?

text: Almost in every textbook of condensed matter physics, the standard description of SSB could be formulated as follows: Consider the lattice Heisenberg model in an external magnetic field $H=\sum_{ij}J_{ij}\mathbf{S}_i\cdot\mathbf{S}_j+hS_z$, where $h$ is the magnitude of magnetic field and $S_z=\sum_iS_i^z$. Now the average magnetization per site is a function of both magnetic field $h$ and number of lattice sites $N$, say $m\equiv \sum_i\left \langle S_i^z \right \rangle/N=m(N,h)$, where $\left \langle S_i^z \right \rangle\equiv tr(\hat{\rho }S_i^z)$ with $\hat{\rho }=e^{-\beta H}/tr(e^{-\beta H})$ the density operator. Then if $$\lim_{h\rightarrow 0}\lim_{N\rightarrow \infty }m(N,h)\neq 0$$, we say the system has SSB at temperature $T$. Now I get some questions: (1)We know at finite $N$ and zero $h$, $m(N,h=0)=0$ due to spin-rotation symmetry. But **there is no reason** for that $$\lim_{h\rightarrow 0}m(N,h)=m(N,h=0)—[1]$$, right? Since the function $m(N,h)$ may _not be continuous_ at $h=0$, from the math viewpoint. (2)If Eq.[1] is correct, and hence $\lim_{h\rightarrow 0}m(N,h)=0$, then $\lim_{N\rightarrow \infty }\lim_{h\rightarrow 0}m(N,h)=0$, right? (3)If Eq.[1] is wrong, say $\lim_{h\rightarrow 0}m(N,h)\neq m(N,h=0)$ and hence $\lim_{h\rightarrow 0}m(N,h)\neq0$, then what about $$\lim_{N\rightarrow \infty }\lim_{h\rightarrow 0}m(N,h)?$$ And why don't we use this identity to define SSB? Thank you very much.


title: How do alpha and beta particles ionise surrounding particles?

text: I've been wondering about this question for a while. If you have alpha and beta particles released from a radioactive core, how do they ionise surrounding particles?


title: Global symmetries in quantum gravity

text: In several papers (including a recent one by Banks and Seiberg) people mention a "folk-theorem" about the impossibility to have global symmetries in a consistent theory of quantum gravity. I remember having heard one particular argument that seemed quite reasonable (and almost obvious), but I can't remember it. I have found other arguments in the literature, including (forgive my sloppiness):   * In string theory global symmetries on the world-sheet become gauge symmetries in the target space, so there is no (known) way to have global symmetries   * in AdS/CFT global symmetries on the boundary correspond to gauge symmetries in the bulk so there again there is no way to have global symmetries in the bulk   * The argument in the Banks-Seiberg paper about the formation of a black hole charged under the global symmetry I find none of these completely satisfactory. Does anybody know of better arguments?


title: Do I ever not account for temperature variation from wall temperature in pipe flow?

text: Let's say I have internal pipe flow with a constant wall temperature. If I evaluate the properties at either the mean inlet temperature of the average of the inlet and outlet, do I always have to account for property variation due to the wall temperature? For example, I would calculate the Nusselt number using a prescribed correlation, and then I would use $$Nu_D=Nu_{D,m}(\frac{\mu_m}{\mu_s})^n$$ with $n$ changing whether the problem is heating the fluid or cooling. $\mu_m$ is the viscosity at the mean temp and $\mu_s$ is the viscosity at the surface temp. Would there ever be a time in turbulent pipe flow to not do this?


title: Why consider only direction cosines?

text: Why are these called direction angles?   1. Why do we consider only direction cosines and not direction sines or tans.   2. What is its actual significance?    3. And How to use them? Why are they called direction angles?


title: How to determine the orientation of the massive Dirac Hamiltonian?

text: In the calculation of the Chern number within a 2D lattice model, let's take the Haldane model for example, the Chern number$=\pm1$ has 2 contributions coming from 2 Dirac points described by $$h_1(\mathbf{q})=q_y\sigma_x- q_x\sigma_y-\sigma_z$$ and $$h_2(\mathbf{q})=-q_y\sigma_x- q_x\sigma_y+\sigma_z$$. Both of the above 2 Hamiltonians contribute the same 1/2(or -1/2) Chern number with **the same orientation (i.e., the sign of Chern number)**. My question is: How to judge whether two massive Dirac Hamiltonians(e.g. $h_1$ and $h_2$) have the same or opposite orientations **simply from the form of the Hamiltonian**?


title: find distance using a light line

text: Let's suppose we have a particular light frequency emitter and relative sensor array, and that there is no external source of this light. This emitter has a know angle respect the receiver, and emit a point of line. Because the angle is known, we can determine the distance just by looking at which point in the sensor we read the point. This is the triangulation method, used by ![laser range finder](http://i.stack.imgur.com/mesdR.png), click here for an explanation Now, I want to understand if it possible to use a line instead of a point, and a sensor matrix instead of an array, so we can get 2D reading. This should work until the "wall" has a obliquity on the axis perpendicular to the light line, but then I can't find a way to "fix" this problem. I may use different pattern instead of a line, like point but then the problem rise if one or more of the point are out of range for the receiver. any thoughts?


title: Why is the Gupta-Bleuler gauge unfashionable?

text: In the early days of quantum electrodynamics, the most popular gauge chosen was the Gupta-Bleuler gauge stating that for physical states, $$\langle \chi | \partial^\mu A_\mu | \psi \rangle = 0.$$ However, this gauge is practically never used now. Why? Is there anything wrong or inappropriate with the Gupta-Bleuler gauge? How is the Gupta-Bleuler gauge related to the $R_\xi$ gauge?


title: What is wall crossing?

text: I know that it must have something to do with (gauge?) theories with N=2 supersymmetry, BPS states and even black holes, but most papers on the subject are too technical for me. So what is wall crossing?


title: Wikipedia's derivation of torque related to angular acceleration

text: Wikipedia derivation of the relationship between a torque and an angular acceleration is given here. Could someone help me to see how the following: $$\vec{\tau} = \left(-\sum^n_{i=1}m_i [\Delta r_i]^2\right) \vec{\alpha} + \vec{\omega} \times \left(-\sum^n_{i=1}m_i [\Delta r_i]^2\right) \vec{\omega}$$ is obtained from the following: $$\vec{\tau} = \sum^n_{i=n} (\vec{r}_i - \vec{R}) \times (m_i \vec{a}_i)$$ by using Jacobi identity, please? My attempt at the derivation is as follows: $$\vec{\tau} = \sum^n_{i=n} (\vec{r}_i - \vec{R}) \times (m_i \vec{a}_i) \\\ \hphantom{\vec{\tau}} = \sum^n_{i=n} \vec{\Delta r_i} \times (m_i \vec{a}_i) \\\ \hphantom{\vec{\tau}} = \sum^n_{i=n} m_i (\vec{\Delta r_i} \times \vec{a}_i) \\\ \hphantom{\vec{\tau}} = \sum^n_{i=n} m_i (\vec{\Delta r_i} \times (\vec{\alpha} \times \vec{\Delta r_i} + \vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i})))\;\ldots\;\vec{R}\text{ is centroid} \\\ \hphantom{\vec{\tau}} = \sum^n_{i=n} m_i (\vec{\Delta r_i} \times (\vec{\alpha} \times \vec{\Delta r_i}) + (\vec{\Delta r_i} \times (\vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i})))\;\ldots\text{ cross-product distributivity over addition} \\\ $$ Then, I try the Jacobi identity on the second term as follows: $$\vec{\Delta r_i} \times (\vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i})) + \vec{\omega} \times ((\vec{\omega} \times \vec{\Delta r_i}) \times \vec{\Delta r_i}) \+ (\vec{\omega} \times \vec{\Delta r_i}) \times (\vec{\Delta r_i} \times \vec{\omega}) = \vec{0}$$ The last term on the LHS is $\vec{0}$ because $$(\vec{\omega} \times \vec{\Delta r_i}) \times (\vec{\Delta r_i} \times \vec{\omega}) \\\ = (\vec{\omega} \times \vec{\Delta r_i}) \times -(\vec{\omega} \times \vec{\Delta r_i}) \\\ = -[(\vec{\omega} \times \vec{\Delta r_i}) \times (\vec{\omega} \times \vec{\Delta r_i})] \\\ = \vec{0}$$ So, $$\vec{\Delta r_i} \times (\vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i})) + \vec{\omega} \times ((\vec{\omega} \times \vec{\Delta r_i}) \times \vec{\Delta r_i}) = \vec{0}$$ That is different from the one stated in the Wikipedia. However, I try to continue with my own finding as follows: $$\vec{\Delta r_i} \times (\vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i})) + \vec{\omega} \times ((\vec{\omega} \times \vec{\Delta r_i}) \times \vec{\Delta r_i}) = \vec{0} \\\ \vec{\Delta r_i} \times (\vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i})) = -[\vec{\omega} \times ((\vec{\omega} \times \vec{\Delta r_i}) \times \vec{\Delta r_i})] \\\ \vec{\Delta r_i} \times (\vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i})) = \vec{\omega} \times -((\vec{\omega} \times \vec{\Delta r_i}) \times \vec{\Delta r_i}) \\\ \vec{\Delta r_i} \times (\vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i})) = \vec{\omega} \times (\vec{\Delta r_i} \times (\vec{\omega} \times \vec{\Delta r_i}))$$ to have the following: $$\vec{\tau} = \sum^n_{i=n} m_i (\vec{\Delta r_i} \times (\vec{\alpha} \times \vec{\Delta r_i}) + (\vec{\Delta r_i} \times (\vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i}))) \\\ \hphantom{\vec{\tau}} = \sum^n_{i=n} m_i (\vec{\Delta r_i} \times (\vec{\alpha} \times \vec{\Delta r_i}) + \vec{\omega} \times (\vec{\Delta r_i} \times (\vec{\omega} \times \vec{\Delta r_i}))) \\\ \hphantom{\vec{\tau}} = \sum^n_{i=n} m_i ([\Delta r_i]^2 \vec{\alpha} - (\vec{\Delta r_i} \cdot \vec{\alpha}) \vec{\Delta r_i} + \vec{\omega} \times ([\Delta r_i]^2 \vec{\omega} - (\vec{\Delta r_i} \cdot \vec{\omega}) \vec{\Delta r_i})) \\\ \hphantom{\vec{\tau}} = \sum^n_{i=n} m_i ([\Delta r_i]^2 \vec{\alpha} - (\vec{\Delta r_i} \cdot \vec{\alpha}) \vec{\Delta r_i} + \vec{\omega} \times [\Delta r_i]^2 \vec{\omega} - \vec{\omega} \times (\vec{\Delta r_i} \cdot \vec{\omega}) \vec{\Delta r_i}) \\\ \hphantom{\vec{\tau}} = \sum^n_{i=n} m_i ([\Delta r_i]^2 \vec{\alpha} - (\vec{\Delta r_i} \cdot \vec{\alpha}) \vec{\Delta r_i} + (0) - \vec{\omega} \times (\vec{\Delta r_i} \cdot \vec{\omega}) \vec{\Delta r_i}) \\\ \hphantom{\vec{\tau}} = \sum^n_{i=n} m_i ([\Delta r_i]^2 \vec{\alpha} - (\vec{\Delta r_i} \cdot \vec{\alpha}) \vec{\Delta r_i} - \vec{\omega} \times (\vec{\Delta r_i} \cdot \vec{\omega}) \vec{\Delta r_i})$$ Then, I get stuck there. If you want to help, please show the complete derivation instead of just hinting here and there unless you have the complete derivation already in your mind.


title: What is the Reduced Density Matrix?

text: The difference between pure and mixed states is the difference in their density matrix structure. For density matrix $\rho$ of mixed state the trace of $\rho^{2}$ should be less than 1. For pure state corresponding trace $Tr(\rho^{2}) = 1$. But when I tried to check the Bell two-qubit state, i got: $$ \rho = \frac{1}{2}\begin{pmatrix} 1 & 0 & 0 & 1 \\\ 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 \\\ 1 & 0 & 0 & 1\end{pmatrix}$$ $$ \rho^{2} = \frac{1}{4}\begin{pmatrix} 2 & 0 & 0 & 2 \\\ 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 \\\ 2 & 0 & 0 & 2 \end{pmatrix}$$ Trace of which is equal to 1. As I understand, reduced density matrix is the right describing of bell states. But my matrix is not reduced. Can you explain me how to find reduced matrix of bell state?


title: In space I am moving wrt to what?

text: Let us suppose I am running on a street. When my eyes are open, I can see many things moving backward, and thus it gives me an idea that I am moving wrt those things. Not even this, even if I close my eyes during the run, I can really _feel_ that I am moving. Now assume that I reach a world where there is absolutely nothing except my own body, and my eyes are closed. I am still feeling that I am running. The question is I am running wrt to what? Even if I open my eyes I would feel that I am still running, may be very fast. But visually I would not able to perceive my movement. Does it mean that my movement in space is an illusion? If not, then I am moving w.r.t what? There is absolutely nothing around me!


title: Temperature of gases

text: I can't find any law that states this (maybe the combined gas law does and I'm misinterpreting it?), but Feynman said that if you compress a gas, the temperature increases. This makes sense, for example, a diesel engine (or gas engine with insufficient octane or too high a compression ratio). Also, must thinking about a piston "hitting" particles as it is compressed makes sense that energy is imparted. But he goes on to say that when the gas expands, there is a decrease in temperature. This used to make more sense to me, but the more I think about it, it doesn't at all. Why would the particles _lose_ energy if the container expands?


title: Are Neutron stars transparent?

text: Neutrons have no charge so they would not, I think, interact with photons. Would a neutron star be transparent?


title: Applying Gauss' Law to find Electric Field

text: I'm in doubt in the application of Gauss' Law to find electric fields when the charge distribution is symmetric. Well, first of all: I know how to find the magnitude of the field - we just enclose the charge distribution with a gaussian surface on which the electric field will not change it's magnitude, and then using Gauss's Law we can write it in terms of the total charge inside and the area of the gaussian surface. My problem is: how do I find the direction of the field? I mean, in a spherical symmetric distribution it's easy, because we know what's the vector that points radially outwards (it's simply one of the unit vectors from spherical coordinates). But what about a cylindrical symmetric distribution ? Would I need to use the unit position vector of cylindric coordinates ? In the general case I would need to switch to more appropriate coordinates to write the field ? Is there a general way of treating this ? Sorry if this question is to silly or too basic. I'm just trying to understand how to use properly this law.


title: Current induced in coil when magnet falls through it?

text: So I know that magnetism can be obtained from the combination of electric fields and special relativity. I am familiar with the way one can derive the magnetic field of a current carrying wire felt by a moving charge from length contraction. Now: we all know that if a magnet falls through a coil, the magnetic flux through its cross sectional area will change with time and this gives rise to an induced current. I would like to know if we can understand it solely from relativity, without having to assume that a change in magnetic flux implies an induced electric field. Let's consider the situation in the **frame of reference of the magnet** : the magnet sees the coil moving upwards; the coil is made of a conducting material so there are lots of delocalised (free to move) electrons. There is a stationary magnetic field (produced by the magnet) and the delocalised electrons are moving upwards (with the coil) in it. The magnetic field lines are in the same direction as the upwards velocity of the electrons, so the **v** $\times$ **B** term in the Lorentz force is 0. I tried to summarise it in this diagram (it's very simplistic and not very well drawn, sorry): ![enter image description here](http://i.stack.imgur.com/wnoMw.png) How do we explain the induced current (i.e. the flow of electrons) in the coil without referring to Faraday's/Lenz's law?


title: About our voice transferred into electrical signal

text: I know when we speak to the microphone, the pitch of our voice cause the vibration of magnet in the microphone, thus causing generation of different voltages of electrical signal. But my question is: when we speak, our voice not only contains the pitchs, but also the content of our words. How are the words (such as word "hello" and "how are you") transferred into electrical signals? I am trying to figure out how our voice data delivered as packets on the Internet when we are having voice chat through Internet. What does the packets contain specifically?


title: Eigenstates of coupled Angular Momentum

text: SO I have a hamiltonian: $$H=\alpha J_1\cdot J_2$$ And I am asked to find the eigenstates and eigenvalues of this Hamiltonian in terms of products of the eigenstates of the z components of the individual spins. The wording of this question is kind of confusing me. I could solve it by rewriting it in terms of $(J_1+J_2)^2,J_1^2,J_2^2$ and then find the eigenvalues, but that is not in terms of the z components. How should I go about doing this? Thanks


title: Translation symmetry and the non-conserved momentum in Viscous fluids

text: Even though a viscous fluid has a translation symmetry (invariance) for its Lagrangian , it still 'waste' Linear momentum. How come ?, isn't the rule that every symmetry yields a conservation law ?


title: Weinberg angle measurement methods

text: I was reading up on the history of $W/Z$ bosons today and I got a little puzzled. I always assumed that people measured $M_Z$ and $M_W$ and then derived the Weinberg angle. But it appears that they knew what the Weinberg angle was by the late 70's (but not with a lot of precision). How do you measure the Weinberg angle experimentally without using $W$ and $Z$ masses?


title: Does everything with mass or energy have a gravitational pull?

text: As small as it may be, does every 'thing' have a gravitational pull? That is, something with mass or energy. No matter how obsolete or negligible it may be, is it there? If so, how is it calculated? What does 'it' affect?


title: Is there a compound denser than the densest element?

text: I'm musing about how to give students an intuitive feeling about density by letting them lift a same sized volume of different materials, e.g. 1 liter of water, a 10x10x10 cm cube of iron, lead etc. So far, the densest material accessible and affordable to a teacher would probably be mercury (I certainly remember my chemistry teacher letting us lift a small bottle, maybe 100ml, makes for an unforgettable impression). Do I have any chance of getting anywhere above the 13g/cm$^3$ of mercury? I sorted the elements in the Wikipedia Density article. Gold and Platinum require to win the lottery first. A liter of Plutonium needs connections to evil people, plus a lot of safety measures, so is right out :-) Osmium? Rhenium? Are there elements or compounds denser than mercury usable for a demonstration? Are there heavy metal compounds that can be denser than the metal itself or is there a physical constraint prohibiting such a property? (I don't want to look at the far ends of a p/T phase diagram, standard temperature and pressure is okay).


title: How do you actually define an orbit?

text: How do you actually define an orbit? I believe, Newtonian Mechanics describes an orbit as one object in free fall around another where projectile paths become elliptical. I think, Einstein describes an orbit as an object taking the shortest distance through curved space. And in Quantum Mechanics, orbits are quantized orbitals or states. Is there a definition for orbits, where all these characteristics are true?


title: Where does the "g" force that pilots experience come from?

text: I understand that it has to do with acceleration. Say a pilot does a quick maneuver and experiences a force of 5g. What exactly is happening here? And what is this force relative to? If someone can show an example with some calculations that would be really helpful. Thank you


title: Jackson, Classical Electrodynamics - page 97: how to deal with the indexes of a series to obtain Legendre polynomials

text: We want to solve Legendre's equation: $$ \frac{d}{dx}[(1-x^2)\frac{d}{dx}P(x)]+l(l+1)P(x)=0,\quad (1) $$ Jackson writes $P(x)=x^\alpha\sum_{j=0}^\infty a_j x^j$, puts this in eqn. 1 and then comes up with: $$ \sum_{j=0}^\infty\\{(\alpha+j)(\alpha+j-1)a_j x^{\alpha+j-2}-[(\alpha+j)(\alpha+j+1)-l(l+1)]a_jx^{\alpha+j}\\}=0,\quad (2) $$ and from this, he equals each term of the series to zero, and finds the recurrence relation below for the coefficients of the series: $$ \alpha_{j+2}=\frac{(\alpha+j)(\alpha+1+j)-l(l+1)}{(\alpha+j+1)(\alpha+j+2)}\cdot a_j,\quad (3) $$ Now, I don't understand how he manages to obtain eqn. 2 from eqn. 1 and eqn. 3 from eqn. 2. I rewrite eqn. 1 by doing the derivatives: $$ (1-x^2)\frac{d^2P}{dx^2}-2x\frac{dP}{dx}+l(l+1)P=0,\quad (1\text{bis}) $$ and then I evaluate the first and second derivatives of $P(x)$: $$ \frac{dP}{dx}=\frac{d}{dx}\sum_{j=0}a_jx^{\alpha+j}=\sum_{j=1}(j+\alpha)a_jx^{\alpha+j-1}=\sum_{j=0}(j+\alpha+1)a_{j+1}x^{\alpha+j} $$ e \begin{align} \frac{d^2P}{dx^2}&=\frac{d}{dx}\sum_{j=0}(j+\alpha+1)a_{j+1}x^{\alpha+j}\\\ &=\sum_{j=1}(\alpha+j+1)(\alpha+j)x^{\alpha+j-1}a_{j+1}\\\ &=\sum_{j=0}(\alpha+j+2)(\alpha+j+1)a_{j+2}x^{\alpha+j} \end{align} and put the whole in eqn. $1\text{bis}$: \begin{align} \sum_{j=0}[(j+\alpha+2)(j+\alpha+1)x^{j+\alpha}a_{j+2}&-(j+\alpha+2)(j+\alpha+1)x^{j+\alpha+2}a_{j+2}\\\ &-2(j+\alpha+1)x^{j+\alpha+1}a_{j+1}+l(l+1)a_jx^j]=0 \end{align} what's next? final edit: differentiation of the power series is wrong; see Marco81's answer for the correct development.


title: Force to use in harmonic oscillation through the inside of a planet

text: I am to find an equation for the time it takes when one falls through a planet to the other side and returns to the starting point. I have seven different sets of values - mass of object falling, mass of planet, radius of the planet, and time. I'm not including friction in the calculations. I think this qualifies as a harmonic oscillator, and thus I work with the formula $$T = 2\pi \sqrt{\frac{m}{k}}$$ To find the spring constant $k$ I need force $F$, and this is where I get uncertain. Should I work with the gravitational force between the object and the planet when the fall begins? In other words $$F = G\times\frac{m \times M}{R^2}$$ When I try this I find that $$F = kx \iff k = \frac{F}{x}$$ $$\iff k = \frac{G\times\frac{m \times M}{R^2}}{2R} = \frac{G \times m \times M}{2R^3}$$ $$\Rightarrow T = 2\pi \sqrt{\frac{m}{\frac{G \times m \times M}{2R^3}}} \iff T = 2\pi \sqrt{\frac{2R^3}{G \times M}}$$ Using this equation for the values I have, however, I get the wrong results - $T = 7148$ instead of $T = 5055$. What am I doing wrong?


title: Is the solar energy Infinite?

text: Is the solar energy coming from the sun infinite and will continue to be radiated to our earth forever? (discarding any outer factors) what's the sun's fuel?


title: Find the elongation in the wire

text: ![enter image description here](http://i.stack.imgur.com/lRJAa.jpg) > Two wires of diameter 0.25 cm, one made of steel and the other made of brass > are loaded as shown in fig. The unloaded length of steel wire is 1.5 m and > that of brass wire is 1.0 m. Compute the elongation of the steel and the > brass wires. I'm having problem with brass that since it is not connected to a rigid support so then how to calculate the elongation in the brass wire? I think their will be less elongation in brass wire in this situation as compared to a situation where the brass wire is connected to a rigid support. Am I right? Why?


title: How do you calculate angle of projection?

text: At what angle the projectile should throw with initial velocity v in order to reach distance d? discard the air resistance, only gravitation acts. So far I got the equations for horizontal and vertical velocity. Can someone point me in right direction?


title: How does time dilation work in this situation?

text: It is my understanding that when moving near the speed of light, time slows down relative to other things not moving so fast. Based on this principle, would it be theoretically possible to travel a thousand light years in a year, with the thousand years only having passed on the place you're moving relative to? If something is wrong with my premises or question, that knowledge is welcomed, too. (I am still a bit shaky on the whole subject of time-dilation because I don't fully understand how it's possible without a privileged reference frame.) Understanding is the prime goal, here.


title: Why would spacetime curvature cause gravity?

text: It is fine to say that for an object flying past a massive object, the spacetime is curved by the massive object, and so the object flying past follows the curved path of the geodesic, so it "appears" to be experiencing gravitational acceleration. Do we also say along with it, that the object flying past in reality exeriences NO attraction force towards the massive object? Is it just following the spacetime geodesic curve while experiencing NO attractive force? Now come to the other issue: Supposing two objects are at rest relative to each other, ie they are not following any spacetime geodesic. Then why will they experience gravitational attraction towards each other? E.g. why will an apple fall to earth? Why won't it sit there in its original position high above the earth? How does the curvature of spacetime cause it to experience an attraction force towards the earth, and why would we need to exert a force in reverse direction to prevent it from falling? How does the curvature of spacetime cause this? When the apple was detatched from the branch of the tree, it was stationary, so it did not have to follow any geodesic curve. So we cannot just say that it fell to earth because its geodesic curve passed through the earth. Why did the spacetime curvature cause it to start moving in the first place?


title: Why does Se-82 undergo double beta decay?

text: Looking at the decay chain, I saw it undergoes double beta decay. How is it feasible for something to undergo a simultaneous double decay?