title: Extending a solution for $\left(H_\text{osc} + \delta^{(3)}(\vec r) \frac{\partial}{\partial r} r \right) \Psi(\vec r)$ to Gaussian potentials

text: I am a undergrad and currently trying to understand and use the following paper: > Th. Busch u. a. Two Cold Atoms in a Harmonic Trap. 1997. URL: > http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.54.9047. There, the authors derive an analytical solution to $$ \left(H_\text{osc} - E + \sqrt 2 \pi a_0 \delta^{(3)}(\vec r) \frac{\partial}{\partial r} r \right) \Psi(\vec r) = 0, $$ which models the center-of-mass part of two interacting atoms in a harmonic trapping potential. Their interaction is the regularized $\delta$-function $$ \delta^{(3)}(\vec r) \frac{\partial}{\partial r} r. $$ The solution they give is $$ \Psi(\vec r) = \frac 12 \pi^{-3/2} A \exp\left(-\frac{r^2}2\right) \Gamma(-\nu) U\left(-\nu, \frac 32, r^2 \right),$$ where $A$ is for normalization and $U$ is the confluent hypergeometric function. With _Mathematica_ , I have verified that this $\Psi(\vec r)$ is indeed a solution of the Schrödinger equation. In section 3 of the paper, they sketch how this can be done in one dimension as well. I have not succeeded in deriving $\Psi(x)$ for that case yet, but I hope that this will come eventually. My goal is to find an analytical solution to a one dimensional anharmonic oscillator that is pertubed with a gaussian potential. Since I am currently lacking $\Psi(x)$ for the one dimensional $\delta$-pertubed oscillator, I wanted to try the three dimensional case first. In electrodynamics we had solved potential problems with a Green's function and used convolution to extend it. I think that the same might be possible here. Convolving the Gauss potential with the unregularized $\delta^{(3)}$-function gave me $\sqrt{2} \pi \mathrm e^{-r^2} (2 r^2-1)$. Then I did a convolution of the 3D-$\Psi(\vec r)$, but that does not solve the adapted Schrödinger equation, as far as I can tell. Is that even a legitimate way to extend the solution in the paper to other potentials? If so, what would be the right way to do this?


title: Stop Motion 3D model of airflow over an airfoil

text: I am looking for a 3D stop motion image of air represented as molecules over a wing. Does anyone know where I can find an image that I can build a model from?


title: Can the wave function of the hydrogen atom be written as a superposition of plane waves with a single frequency?

text: The solutions of the Schrödinger equation for a hydrogen atom have definite energy. Does this mean that they could be written as a superposition of plane waves of a single frequency - corresponding to that energy - with only the phases and directions differing?


title: Why are higher generation of matter unstable?

text: My secondary school physics textbook has mentioned that protons and neutrons are made up of down and up quarks in different amounts. It has also mentioned that other quarks exist. It states that particles from these quarks are unstable. It also lists the charges of these quarks. It doesn't provide much more detail beyond this. My question is this: Why is (for example) a particle made up of ccs (Wikipedia tells me this is called a Double charmed Omega baryon) any less stable than a proton which has the same charge?


title: Should I begin to study quantum mechanics or wait until I have a stronger base on easier topics?

text: I apologize if this is off topic since it does not relate directly to the study of physics, but I could not think of a better place to ask. I am a high school student graduating in a few days and have recently developed an interest in quantum mechanics. I began to study physics about one and a half years ago and calculus two years ago, and I feel confident about my abilities in integral and vector calculus (at a typical college Calc I–III level), differential equations, mathematical problem solving, and some set theory. I have been introduced to quantum behavior, particle physics, and nuclear physics, but none of these were delivered with a heavy focus on mathematics. Quantum physics, however, caught my attention as a future field of study due to its bizarre nature and inquisitive mathematical models. Without prior knowledge of more difficult topics, I am concerned that attempting to study quantum mechanics will only provide entertainment rather than benefit my college career. I have selected a few books (Griffiths, Sakurai, Feynman, etc.) of which I will choose one to read over the summer of this year and onto college if I determine to do this. Should I follow these plans without first strengthening my knowledge in other fields of physics and mathematics, as I dive directly into the topic? Or should I wait until I am more experienced in other fields before discouraging myself with the difficulty of quantum mechanics? If so, what areas of physics or mathematics would you recommend as prerequisites to the topic?


title: Is work is equal to $mv^2$ (without $\frac{1}{2}$)?

text: I was trying to come up with an equation for work that doesn't include time, because I don't know time. Here's what I did: $$ work = Fd = mad = m{v\over t}d = m{v\over\left({d\over v}\right)}d = mv^2 $$ Now, $mv^2$ is familiar, from kinetic energy equation I think, but I believe that was $\frac{1}{2} [* mv^2]$. Where did I go wrong?


title: Are all metals good conductor of electricity?

text: I am writing an article for kids, which is on conductors and insulators of electricity. If I make a statement that "All metals are electrical conductors and all non- metals are electrical insulators" Am I right? Are there any metals which are bad conductor of electricity? Are there any non-metals which are good conductor of electricity?


title: Oil drop experiment--How was the result so accurate?

text: I remember while learning about Millikan's oil drop experiment and being pretty skeptical about the setup. I know that there is a lot of controversy regarding manipulation of data, but the fact is; he still got a pretty close answer. I mean, we can get the correct answer by manipulation of data if we know the answer already. If we don't all we can do is 'reduce' the error box (which Millikan purportedly did). But I'm surprised that he still got an answer close to 1% of the actual value. I personally would have expected an error of maybe even an order of magnitude for these reasons:   * Viscosity of air: The viscosity is dependent upon humidity and other factors. I know he _did_ use the wrong value here.   * $g$: How was this known to the accuracy required for this experiment? It varies across the Earth, and besides, could it be measured to whatever accuracy using the equipment available at that time?   * Velocity of the oil drop: I don't know how fast the oil drops fell, but to me, terminal velocity looks pretty fast. How did he accurately measure the velocity? A micrometer eyepiece takes time to focus, and more time to take a reading. And anyways micrometer scales aren't that precise, are they?   * Potential difference: How did he accurately know this?   * "Selecting one drop": How was this done? If all drops had a charge as a small integral multiple of q, there should be more drops with the same charge which behave the same. How can you get rid of those?   * Densities of air/oil: Air density varies. Oil density may not be known accurately.   * Electric field: Between plates, E is not exactly constant or equal to $V/d$ There are fringing effects which may alter the value of E. IMO since the distance between the plates is comparable to the area (it look like it), fringing effects are pretty significant.   * Stokes law is only applicable for perfectly spherical bodies. Is a falling oil drop perfectly spherical? Basically I'm skeptical of how one can get such a tiny number with 1% accuracy with all those uncertainities. Could someone convince me how he _did_ get it so accurately? Were there any measures he took to minimise these errors (I can't find any)?


title: Correlation Functions, Symmetries and Measurements

text: Is there a book that goes deep into correlation functions? What I'm interested in a book/article that explains in the detail the relation of the correlation functions with symmetries and how one can understand the link between the use of correlation functions and measurements in the lab. I am interested in experimental examples/theory not restricted to quantum, but all kind of physics examples that use correlation function.


title: What was the apparent magnitude of the June 15th 2011 lunar eclipse?

text: My plan was to observe and estimate the apparent magnitude of the Moon during totality of the June 2011 lunar eclipse, but the clouds rolled in at the exact moment, so I couldn't make any useful estimates. What was the apparent magnitude?


title: Particular targets of high angular resolution infrared telescopes

text: I work in the field of infrared interferometry, specifically instrumentation. As such, I need to be aware of the science goals of such an instrument. Is there a quick list of important and contemporary targets that such an instrument of high angular resolution would be pointed at? (By "high", I mean objects that subtend a small angle on the sky.)


title: What objects/states of objects with absolute magnitude do we know of?

text: For measuring distances the knowledge of absolute magnitude or luminosity is often crucial, especially for very big distances. Unfortunately we can't measure the diameter of far distant objects and calculate and derive absolute magnitude due to resolution limits. That's why objects or better named states in the life cycle of specific objects, like Type Ia supernovas, are so important. What additional objects do we know of sharing this property? Are there objects theoretically predicted to have a absolute magnitude but until now not yet discovered? Name the object and the spectral range of emitted light or particles.


title: New theories and publications

text: When someone develops a new theory on physics, which is barely on schetch (so there are no measurements, nor simulations) with just a mathematical and conceptual description, in which scientific journals people use to publish these theories?


title: Please recommend a good book about physics for young child (elementary school aged)

text: I'm looking for a book that would be appropriate for an advanced elementary school aged kids (say, 6-11 YO) **describing the basics of physics (or sciences in general) in entertaining way**.   * The structure of the book should preferably (though not necessarily) be "Curious question" followed by entertaining-but-advanced for their-age physics answer. As a good example of a chapter/story I had in mind: "Why can the soap wash off dirt/grease?" "Why can the water be poured into containers?" "How does a plane fly?" "How does a rocket fly?"   * The topics that must be covered are atoms/molecules, how they go together to make various matter etc...   * Doesn't have to be 100% limited to physics - could include chemistry, biology, other natural sciences.  P.S. For those fluent in Russian, i'm looking for an English equivalent of the soviet popular kid's physics/science book "почему вода мокрая").


title: Simplifying some math for an ant-on-rubber-band problem

text: OK, I've been doing this problem for fun (it's a great problem, BTW!): http://www.physics.harvard.edu/academics/undergrad/probweek/prob76.pdf Here is the solution: http://www.physics.harvard.edu/academics/undergrad/probweek/sol76.pdf However, I'm having a problem understanding the solution. How does the first line lead to the second line in the simplification of $F(t + dt)$? It looks like the author has taken $Vdt$ to be zero to get $x/\ell$ but has left in $Vdt$ for the second term in line two. Is that legal?


title: Scale invariance and self organized criticality

text: On wikipedia i have found this statement: > In physics, self-organized criticality (SOC) is a property of (classes of) > dynamical systems which have a critical point as an attractor. Their > macroscopic behaviour thus displays the spatial and/or temporal scale- > invariance characteristic of the critical point of a phase transition, but > without the need to tune control parameters to precise values. I have two questions on respect of this:   1. What is a "critical point" on a general statistical system?   2. What is the relation between critical behaviour and the scale invariance?


title: Why water inside a vessel placed in a body of boiling water does not boil?

text: Imagine a plastic sac filled with water placed in a pan of boiling water. I wonder why the water inside the sac does not boil. The temperature of the water inside the sac should be the same as that of outside water, i.e., 100 deg. celsius. Similarly if to place a glass tumbler of water inside a pan placed on fire, the water will boil only inside the pan, but not in the tumbler.


title: How to explain relativistic mass with 2 moving systems, but not 3?

text: All the visual explanations I know work in some kind of "If you are moving relative to something A, while inside A something is moving, the stuff in A has to move slower due time dilation and therefore the mass has to increase so the impact(and momentum) stays the same." Since all moving objects and not only the ones inside of moving objects have their mass increased I have big trouble understanding why these kind of pictures should lead to any understanding. Could I argue that if A was moving his whole point-of-view would be moving?   As if everything already would be inside of something moving?   This sounds very wrong and just raises more questions in my head.   You could simply determine who was moving if that would be true, so it isn't.   Or is it? I don't know where my thoughts got messed up. Hopefully you do.


title: Question about indutive loads in power system

text: Our home appliances are mostly resistive loads and the bill we pay for consuming power is actually real power. If we use more inductive loads at our home, will it just cause problems in power factor or does it affect our bill as well (in the form of consuming more units despite it consume reactive power)?


title: Galaxy rotation equation problem

text: I would like to know if in this equation for the rotation of galaxies: $$v^2=\frac{GM}r.$$ Is $M$ the mass of the black hole at the center of our galaxy? Someone said to me that > No, $M$ is the mass of all matter that is enclosed in the spherical shell of > radius $r$.This includes the mass of the black-hole,mass of the galactic > disk up to radius $r$ and also the mass in the halo. However, the quantity $GMm/r^2$ indicates the opposite because we derived $v^2=GM/r$ by equating $v^2 m/r=GMm/r^2$, where $M$ is the mass of the object at the centre (e.g. the Sun in our solar system) and the velocity is a function of the distance. What is the mass of that black hole?


title: Why are neutrinos more weakly interacting than light?

text: When people describe neutrino interactions they describe them as rare/infrequent due to the fact that the neutrinos are electrically neutral and have little mass, if any. Well why then is the photon much more strongly interactive considering that it is neutral and massless? What is the discrepancy between the two? Why is the neutrino so elusive, whereas light is so prevalent? Does this perhaps have anything to do with the fact that a neutrino is a fermion whereas a photon is a boson?


title: A question about the Bosonization of the Thirring model

text: Is there a way or sense in which one can Bosonize this kind of a Lagrangian, $L = \bar{\psi}\gamma^\mu \partial _\mu \psi + f(x) \bar{\psi}\psi$ for $f(x)$ being some function on space-time.   * What is the most pedagogic reference which explains the Thirring Model/Sine-Gordon bosonization from scratch?


title: Energy of one-electron state $E(k) \sim k^2$ "$\implies$" occupied region in $k$-space indistinguishable from sphere

text: When we consider a system of $N\gg 1$ one-electrons the occupied region in $k$-space is stated to be indistinguishable from a sphere, since the energy of a one-electron level is directly proportional to $k^2$. I would rather expect this argument when the energy is proportional to the cube $k^3$ of its wave vector, since a volume in $k$-space goes as $k^3$. How does $E(k) \sim k^2$ imply that the volume is indistinghuishable from a sphere? > **Note**. Ashcroft and Mermin say: If it were not spherical it would not be > the ground state, for we could then construct a state of lower energy by > moving the electrons in levels farthest away from $\mathbf k = 0$ into > unoccupied levels closer to the origin.


title: Does the frequency of a vibrating substance like a string depend on its properties or simply depend on the agent which vibrates the string?

text: We know that if someone vibrates a substance like a string, the string oscillates with its natural frequency. But I think the frequency depends on the rate the agent like a man's hand does vibration on the string. But my book says natural frequency only depends on the properties of the string. Please explain.


title: Working of dry cell battery!

text: please help me out in this query!! actually i need to know why when we use a battery or dry cell, As we know it consists of :- 1)electrolyte 2)cathode and anode. but i need to know as electrolyte is consisting of a salt (we take it as NaCl salt for it ) and the ions is formed from the salt due to potential we make on both ends of the battery, but why it cant happen as if the ions can directly form without connecting the battery into a wire , because the two terminals consists of positive and negative ions which can directly put force on NaCl compound and can break it ,but why their is a need to create a potential by attaching the battery with a wire externally? In other words:- Why molten salt as an electrolyte cant ionise normally in a cell without connecting the cell with a wire outside? regards :- Harjot singh


title: What are the chances that a deadly asteroid will hit Earth in the next decade?

text: What are the chances that an asteroid that will kill multiple people will hit Earth in the next decade?


title: Graph Theory and Feynman Integrals

text: In Vladimir A. Smirnov's book _Analytic Tools for Feynman Integrals_ , Section 2.3, the alpha representation of general Feynman integral takes the form $$ F_{\Gamma}(q_1,\ldots,q_n;d) = \frac{i^{-a-h}\pi^{2h}}{\prod_l\Gamma(a_l)} \int_0^{\infty}\mathrm{d}\alpha_1 \ldots \int_0^{\infty}\mathrm{d}\alpha_L \prod_l\alpha_l^{a_l-1} \mathcal{U}^{-2} Z e^{i\mathcal{V}/\mathcal{U} - i\sum m_l^2\alpha_l} $$ where $\mathcal{U}$ and $\mathcal{V}$ are defined as sums running over trees and 2-trees of the given Feynman graph. I know that $\mathcal{U}$ is equivalent to $\det{A}$ in the $4h$-dimensional Gauss integrals, but I can't figure out how it can be expressed in the language of graph theory. Could anyone provide some help? References on the topic of graph theory and Feynman integrals are also desired. Thanks a lot!


title: Mass gap for photons

text: I am puzzled by the answers to the question: What is a mass gap? There, Ron Maimon's answer gives a clear-cut definition, which I suppose applies to any quantum field theory with Hamiltonian $H$, that the theory has a mass gap if there is a positive constant $A$ such that $$\langle \psi| H |\psi \rangle\geq \langle 0 |H | 0 \rangle +A$$ for all nonzero (normalized) $\psi$. But then, Arnold Neumaier says _QED has no mass gap, as observable photons are massless states._ I would quite appreciate a brief explanation of this statement. The definition is concerned with the minimum possible _energy_ for non-zero states. So I don't see why the photons having zero _mass_ would imply the absence of a mass gap.


title: Can a current carrying loop or wire produces no magnetic field?

text: A current carrying wire produces magnetic field around it. We can find the direction by Fleming's Right hand rule. We know change in electric filed produces magnetic field and change on magnetic field produces electric field. It is mutual relationship. **My question is that is there any condition such that current carrying loop or wire produces no magnetic field**


title: How does the central peak in moon craters form?

text: The central peak in many of the moon's large craters are visible with a telescope and they seem a little odd to me. Can someone explain how they form. ![enter image description here](http://i.stack.imgur.com/hi25I.jpg)


title: Finding Electric Field outside a Charged Cylinder

text: I'm trying to solve a problem that involves finding the electric field due to a uniformly cylinder of radius $r$, length $L$ and total charge $Q$. Well, my thought was: if I am to use Gauss' Law, I'll have to use a gaussian surface enclosing the cylinder. Then here arises my doubt: My try was to enclose the cylinder with another cylinder with radius $R > r$, parametrize it and find it's normal $\hat{n}$ at each point. Then, I would need to calculate $\mathbf{E}\cdot \hat{n}$ and integrate. This would only be good to calculate $\mathbf{E}$ if it's magnitude is constant on the gaussian cylinder so that the magnitude would drop out of the integral. Also, I would need $\mathbf{E}$ parallel to the normal. It seems that both those things are true, but how can I see it and justify/prove it? I just want suggestions, hints, in how can I see/prove that $\mathbf{E}$ has constant magnitude on the gaussian cylinder and how can I see/prove that $\mathbf{E}$ is parallel to the normal at each point. I also thought on some argument arround symmetry but I didn't find any consistent one. Thanks in advance for your help!


title: Why is a critical system equal to a gapless system?

text: In condensed matter physics, people often say that a system without energy gap is a critical system. What does it mean? Any help is appreciated!


title: Some basic questions related to electricity

text: _Note_ These questions may be incorrect. 1) Why do _positive charges_ flow from high potential to lower, but the electrons do not? Am I wrong? 2) Does battery supply _charge_ to a conductor? I read it from a Youtube video. 3) If I increase the _potential_ or charge of a conductor, will the _capacitance_ of the conductor change?


title: Intuition on topologically nontrivial 2D-band structures?

text: I want to get more intuition on topologically nontrivial band structures. There's this popular 2D two-band model for a topological insulator where $H=\sum_{k}h(\boldsymbol{k})$ (see Qi, Hughes, and Zhang, PRB 78, 195424 (2008)), \begin{eqnarray*} h(\boldsymbol{k})=\left(\begin{array}{cc} m+\mbox{cos}k_{x}+\mbox{cos}k_{y} & \mbox{sin}k_{x}-i\mbox{sin}k_{y}\\\ \mbox{sin}k_{x}+i\mbox{sin}k_{y} & -m-\mbox{cos}k_{x}-\mbox{cos}k_{y} \end{array}\right) \end{eqnarray*} Let me write its real space representation by Fourier transforming $h(\boldsymbol{k})$. This gives two sets of non-identical orbitals $a$, $b$ per site $n$ in a square lattice: \begin{eqnarray*} H & = & \sum_{mn}t_{mn}^{(a)}a_{m}^{\dagger}a_{n}+t_{mn}^{(b)}b_{m}^{\dagger}b_{n}+t_{mn}^{a\leftarrow b}a_{m}^{\dagger}b_{n}+t_{mn}^{b\leftarrow a}b_{m}^{\dagger}a_{n}\\\ & = & \sum_{n}\frac{1}{2}\Big[ma_{n}^{\dagger}a_{n}+a_{n}^{\dagger}a_{n+\hat{x}}+a_{n}^{\dagger}a_{n+\hat{y}}\\\ & & -mb_{n}^{\dagger}b_{n}-b_{n}^{\dagger}b_{n+\hat{x}}-b_{n}^{\dagger}b_{n+\hat{y}}\\\ & & -ia_{n+\hat{x}}^{\dagger}b_{n}+ia_{n-\hat{x}}^{\dagger}b_{n}\\\ & & -a_{n+\hat{y}}^{\dagger}b_{n}+a_{n-\hat{y}}^{\dagger}b_{n}^{\dagger}\Big]+\mbox{h.c.} \end{eqnarray*} A rudimentary code in Mathematica allows me to numerically calculate the Chern number and get what the authors say (that for $-2<m<2$, I get a topollogically nontrivial band structure, and a trivial otherwise). However, I would like some further intuition on how this works. People normally talk about calculating a Berry-like phase from the hopping amplitudes $t_{mn}$ of the tight-binding model and seeing that for a closed circuit they get a phase that is not 0, $\pi$, or $-\pi$, indicating breaking of time- reversal symmetry. I don't understand how that works in the context of this simple model: Suppose I go through all the hopping in a square plaquette, completing the closed circuit $b_{(1,1)}\rightarrow a_{(1,2)}\rightarrow b_{(2,2)}\rightarrow a_{(2,1)}\rightarrow b_{(1,1)}$ where $n=(1,1)$ indicates the $x,y$ coordinates of the site in the lattice, then $t_{(1,1)(2,1)}^{b\leftarrow a}t_{(2,1)(2,2)}^{a\leftarrow b}t_{(2,2)(1,2)}^{b\leftarrow a}t_{(1,2)(1,1)}^{a\leftarrow b}=\left(\frac{1}{2}\right)^{4}(i)(+1)(-i)(-1)=-\frac{1}{16}$ which has an argument of $\pi$. What's going on? Do I also need to consider the loops of the hopping amplitudes between identical sites? If so, why? Any intuitions will be appreciated. (I'm not a string theorist, so the least abstract, the simplest qualitative explanation, the better). Thanks a lot.


title: A question about Hamiltonian phase flow

text: > Show that if a one-parameter group of difeomorphisms of a symplectic > manifold preserves the symplectic structure then it is a locally hamiltonian > phase flow. Note that > A locally hamiltonian vector field on a symplectic manifold $(M^{2n}, > \omega^2)$ is the vector field $I \omega^1$, where $\omega^1$ is a closed > 1-form on $M^{2n}$ and $\omega^1(\eta) = \omega^2(\eta, I\omega^1)$.


title: Quantum Mechanics - Rectangular Potential Barrier - Normalisation

text: I have a quick question regarding the normalisation of the wave function of a particle incident on a potential barrier specifically regarding the normalisation of the wave functions. The problem is set up as on this webpage: http://www.ntmdt.com/spm-basics/view/tunneling-effect And through consideration of the boundary conditions (continuity of wave function and derivative of wave function) the final wave function looks like this: ![Diagram 1](http://www.ntmdt.com/data/media/images/spm_basics/scanning_tunnel_microscopy_stm/stm_physical_backgrounds/tunneling_effect/img04.gif) My question is: are the wave functions either side of the barrier normalisable? And if not does this mean the situation is not physical?


title: If two ultra-relativistic billiard balls just miss, will they still form a black hole?

text: This forum seems to agree that a billiard ball accellerated to ultra- relativistic speeds does not turn into a black hole. (See recent question "If a 1kg mass was accelerated close to the speed of light would it turn into a black hole?") On the other hand, LHC scientists take seriously the formation of black holes by colliding protons. Therefore I presume it will be agreed that two ultra-relativistic billiard balls colliding head on with the requisite energy will form a black hole. The question is, suppose the two relativistic billiard balls are on antiparallel tracks that just miss each other, but pass within the radius of the putative black hole formed by their combined mass-energy. Will the black hole still form? If not, why not?


title: Removing any obstacles, if I were to make a long plank of wood equal to the Earth's diameter, would it appear curved from space?

text: Let's say I was able to make a really long plank of wood equal to the Earth's diameter and it was able to go through all obstacles- i.e oceans, mountains, etc. My first question:   1. Once the plank of wood reaches the Earth's diameter in length, would both ends appear to touch although still look straight from the point of view of someone on Earth   2. Would the wood appear curved from space?  Note: This wood stays on the Earth's surface and doesn't go up.


title: Could one measure a stick to an arbitrary precision by having its length estimated by enough people?

text: I remember reading somewhere that the problem of exact time-keeping on ships could have been solved a lot earlier than it was if somebody would have had the idea of keeping time with a whole array of imprecise clocks - taking the average of clock-times would have given a precise time. Using the same reasoning, could I measure the length of a stick to atomic precision by showing it to enough people and having them guess at its length?


title: Why does decay of correlations imply absence of order?

text: In a few articles I have read, a two-point correlation function $\langle g(x)g(y) \rangle$ is shown to decay with increasing distance of $x$ and $y$, and this is then taken to imply an absence of the corresponding order. For example $\langle g(x)\rangle$=0 in the thermodynamic limit. A concrete example: lets consider a heisenberg model on a square lattice with N sites: $$ H=\sum_{x,y}J_{x,y}\ \vec{S_x}\cdot\vec{S_y} + h\sum_x S^3_x $$ why does $$ \lim_{N \to \infty}|\langle S_x^3 S_y^3\rangle|≤e^{-|x-y|} $$ imply that $\lim_{N \to\infty} \frac{1}{N}\sum_x \langle S_x^3\rangle = 0$? Is it important how fast the correlation function decays? Would the mean magnetisation also disappear in the case of a polynomial decay of the correlation function? Does the result depend on the dimension of the lattice?


title: Vibration in a broken cylinder

text: The physics S.E. may be wrong for this question, but I wonder what happens to a vibration in broken, cracked, or otherwise damaged cylinder. Say you have a 1 meter long steel cylinder, you break it in two, and then put the pieces back together (without welding or anything). You then put in a sinusoidal vibration in one end, and measure the frequency and amplitude in the other end. As far as I can tell, the frequency would not change, but only the amplitude, as energy of the frequency wave is lost in the translation from the complete steel cylinder to the damaged cylinder and back to the complete cylinder again. Is this interpretation correct? If so, how would a cylinder that is only cracked or bent behave? I know they use vibration to, for example, diagnose motors and propellers.


title: How exactly does gravity work?

text: The electromagnetic force and strong and weak forces require particles like photons and gluons. But in case of gravity there is no such particle found. Every _mass_ bearing object creates a gravitational field around it, and whenever another mass bearing object enters its field the gravitational force comes into operation. If all other forces of nature have some particles associated with them why should gravity be an exception? And if there is no such particle, what exactly is the gravitational field and how does it spread over an infinite distance and cause the gravitational force to operate? * * * **Note:** I am a high school student and have not studied quantum mechanics.


title: Colliding particles at speeds aproaching c

text: (In natural units where $\hbar=c=1$.) Two particles are to be collided. Each of these particles has a rest mass of 0.9 GeV and they will be collided at equal but opposite speeds. What is the minimum speed required for the collision to produce a single particle of rest mass 5.6 GeV? I've written down the relativistic equations for momentum and energy, but haven't had much luck. I got to $$\frac{\gamma_1}{\gamma_2} = \frac{28v}{9u}$$ and attempted to calculate the ratio $v/u$, but that didn't really go anywhere helpful. Any hints would be much appreciated. Thanks.


title: Advantage of taking qutrits in place of qubits

text: In general, all the quantum algorithms which I have read so far use qubits (so the space is $\mathbb{C}^2$) and the tensor products of the qubit spaces (space is ${\mathbb{C}^2}^{\otimes n}$). So my question is, are we going to get any advantage if we take space of other dimensions. For instance, what if we do the similar works with qutrits (and things in higher dimensions as basic units). My guess is that, it may have some advantages for error correcting codes. Can somebody point out some reference in this direction (coding) or in something else, of which I am not sure or. Further, are we going to get any advantage from the point of experimental realization? I am form Maths and I do not have much idea regarding this. Advanced thanks for any help/ suggestion/references.


title: How could the multiverse theory be disproven?

text: Theorists (physicists) suggest that there is the term/entity, the Multiverse that contains a huge number of universes not necessarily like our own. I personally find this theory very elegant because its explains the probabilistic outcomes of the experiments with fixed conditions on a quantum level. It's obviously very hard to prove the validity of the theory of multiverse, but   * How can we disprove the existence of multiverse?


title: Schrödinger's Equation and its complex conjugate

text: I would like to know why there is a minus sign on the right-hand side of the Schrödinger's complex conjugate equation, whereas in the Schrödinger's equation there isn't. I know it is a simple question, but I don't know where this comes from. $$ -\frac{\hbar^2 }{2m}\frac{\partial^2\psi}{\partial x^2} + V(x)\psi = i \hbar \frac{\partial \psi}{\partial t} $$ $$ -\frac{\hbar^2 }{2m}\frac{\partial^2\psi^*}{\partial x^2} + V(x)\psi^* = -i \hbar \frac{\partial \psi^*}{\partial t} $$


title: Is preheating plates in microwave dangerous? If so, why?

text: My understanding of microwaves was that due to the high frequency they are easily absorbed into any material. The more "loose" the material, the easier the absorption. I know it's dangerous to let microwave on when empty, mostly because there is nothing that will absorb the microwaves and that may cause magnetic "storm" inside :) But I heard today that it's equally dangerous to use microwave to preheat plates. I would like to know if and why it's true. I think that the plate is a material like any else. But due to the high density it absorb the microwaves slower than say a glass of milk. And also the vibrations may cause it to break after extended period of time.


title: Picture changing operators

text: What is a picture-changing operator in string theory? How do they work?


title: The wave equation in general relativity, special relativity, and Cartesian coordinates

text: The relativistic wave equation is $$\square\varphi=\rho$$ where $\varphi$ is the field, $\rho$ is the source, and $\square$ is the D'Alembert operator, defined by $$\square=\nabla_\mu\nabla^\mu=g^{\mu\nu}\nabla_\mu\nabla_\nu$$ where $g^{\mu\nu}$ is the inverse metric tensor and $\nabla$ is the covariant derivative operator. In Minkowski spacetime, this can be reduced to $$\square=\partial_\mu\partial^\mu=\eta^{\mu\nu}\partial_\mu\partial_\nu$$ where $\partial$ is the partial derivative operator. Using Cartesian coordinates $(t,x,y,z)$, this can be further reduced to $$\square=\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\frac{\partial^2}{\partial x^2}-\frac{\partial^2}{\partial y^2}-\frac{\partial^2}{\partial z^2}$$ Is this characterization of the wave equation correct? In particular, what is the correct distinction between using $\partial$ and using $\nabla$, and the correct distinction between curved _spacetime_ and curvilinear _coordinates_ , when manipulating these expressions?


title: Creating a 3-D object from 2-D objects

text: I'm reading a book, 'Warped Passges' by Lisa Randall. In it she mentions how we can envision higher dimensions by explaning how higher dimension ojects can be created from lower dimension objects. So she says > "This book is three-dimensional. However, its pages have only two > dimensions. The union of the two-dimensional pages comprises the book." (p > 17). So here's the question: can we really create a 3-D book from 'true' 2-D pages? Because 'true' 2-D pages will have no 'thickness' (the 3rd dimension) and hence no matter how many pages we stack on each other, that construct will strictly remain 2-D. Generalizing this, I think we can not create higher dimension objects from lower dimension objects unless those lower dimension objects have non-zero extent into the higher dimension. And if they do have this extent, they're already a higher dimension object. So isn't this example paradoxical?


title: Should all theories of gravity have Schwarzschild solution?

text: A consistent theory of gravity must include the Newton's classical theory of gravity as a weak field approximation. Moreover, to satisfy the experiments in the solar system, the Schwarzschild solution is necessary. Thus, I wonder whether all reasonable theories of gravity should have the Schildschwarz solution. Certainly, I know in some modified gravity no Schwarzschild solution exist. What does this mean?


title: How to easily calculate lengths (or relative lengths) of paths between symmetry points in BZ

text: I am trying to easily calculate the length between special kpoints within the BZ of the 32 point groups in a crystal system. I am calculating the lengths in order to scale k point sampling along these paths for electronic structure calculations. I could use the reciprocal lattice vectors to calculate each and every special symmetry point for every Bravais Lattice to get the lengths but I am wondering if there is an easy way to at least find the ratio between the lengths that I am just missing. I for example want to find the lengths of the paths for Rhombohedral Bravais Lattice crystal so I use the paths ![enter image description here](http://i.stack.imgur.com/G5Bhw.png) but for shorter parts of the paths I want to sample less k points and for paths that are longer I sample more (so that they're for the most part evenly spread out) I can calculate these points and path lengths by hand but my intent is to automate the process so I can have the reciprocal lattice vectors and the type of lattice be inputs to a function that gives me the lengths between each of the points automatically. I am under the impression that since the fractional coordinates of some these points are dependent to the cell's reciprocal lattice vectors, that some of the path lengths change when the lattice vectors change. So I am wanting to have an easy way to get these lengths without hardcoding all of the equations with if/else statements. I fear I might just have to do that. Hopefully you can tell me otherwise!


title: Canonical Stress Tensor for the Free Electromagnetic Field

text: I have the followwing Lagrangian for the free electromagnetic field, $$\mathcal{L} = -\frac{1}{4} F^{\mu \nu}F_{\mu \nu},$$ and the canonical stress tensor is, $$T^{\alpha \beta}=\frac{\partial \mathcal{L}}{\partial \left(\partial_{\alpha} A^{\lambda}\right)}\partial^{\beta} A^{\lambda}-g^{\alpha \beta}\mathcal{L}.$$ My $T^{0i}$ is, $T^{0i} = (\vec{E}\times \vec{B})_{i}$ but it's the symmetric stress tensor, and I need to find it: $$T^{0i} = (\vec{E}\times \vec{B})_{i} + \nabla \cdot (A_{i}\vec{E})$$ I don't understand where this divergence term came from. How can I do this asymmetric $T^{0i}$ tensor? (Just for reference, see Jackson - Classical electrodynamics, Third ed.- page 606)


title: I think only x and t need obey Lorentz transform rule, other 4 vectors not necessarily true

text: Lorentz transformation aims at an invariant speed of light for all reference frames. Speed only involves x and t. Therefore it's reasonable to understand the Lorentz transform can act on spacetime four vector (ct,x1,x2,x3) But speed does not directly involves other physical quantities such as E field, B field, current density and so on. My questions is why do we need these four vector covariant under Lorentz transform? For example: $x_{\mu }x^{\mu}=const$ directly involves space time $\partial_{\mu }{\color{Red} j^{\mu }}=0$ indirectly involves space time $\partial_{\mu }\partial ^{\mu }{\color{Red}A^{\nu} }= \frac{4\pi}{c}{\color{Red} j^{\nu }}$ does not involve space time (if only judge from this equation, we can conclude that A and j obeys Lorentz transform rule)


title: Inertial frame: Sun earth and moon

text: From the perspective of sun, is the moon orbiting the earth considered a single inertial frame or are moon and earth separate inertial frames? Or am I missing something completely fundamental?


title: Choosing proper centripedal force in Newtonian gravitational forces

text: I am trying to do an orbit simulation, I took Newton's law of Newtonian gravitational forces. It seems to work correct, but I can't really determine the velocity of an object needed to stay in free fall. I didn't want to use the masses in our planet systems, as the values are to high (in kg) So when my earth's mass is 1, what should be the masses be of the orbiting satellites and what should the initial velocity be to stay in freefall, when I want to altitude be 50 units (meters). It would be nice, if you could give me some hints!


title: Understanding how mass spectroscopy works

text: Let me start by saying that I've posted this question here as well. I've posted it here because I think the questions I've asked involve the physics of molecules. So I’m trying to get a deeper understanding of how mass spectroscopy works. Most tutorials and textbooks I’ve encountered omit certain details about the process and I’m hoping someone out there who understands the process can fill in the gaps. I’ll start by explaining the way I currently understand the process and then list where I start to get confused. 1 First a particular molecule is bombarded with a beam of electrons (ionization step). This step works to free an electron from the molecule under analysis which generates a cation. **Note** : It’s important to remember that the only reason why this molecule was ionized is to make use of a magnetic field to bend its direction. Only ions will be affected by this magnetic field which allows us to pinpoint the molecule in question. 2 Next, these ions are accelerated by an electric field toward a magnetic field that bends the moving ions by a certain amount. > Question 1: What’s the difference between an electric field and a magnetic > field? And why does the magnetic field bend the ions while the electron > field does not. Is this just a consequence of the shape of the apparatus? > Could you use a magnetic field to accelerate a particle and a electric field > to do the bending? I’m just really confused as to the difference between > electric fields and magnetic fields. 3 The ions that bend will travel around a tube by a certain degree and hit a detector. The amount of bending will tell us about the mass of a particular molecule. Heavier molecules will bend less and lighter molecules will bend more. > Question 2: I’ve come across a tutorial that said that what this detector is > actually measuring is the mass to charge ratio. How is this calibrated and > how does this step allow us to measure the mass of a single molecule. Won’t > there be multiple particles hitting the detector at the same time, will this > not affect the mass reading of a particular molecule? I'm just confused > about this final step and how we can get an accurate mass reading for a > particular molecule. Any help understanding this concept would be appreciated.


title: What causes the orbital eccentricity?

text: I found some websites trying to attribute this to the external other forces affecting the 2 body problem to follow an elliptical rather than a circular orbit, but I am not convinced though. I think any 2 body problem can have an elliptical orbit (isolated from other external forces) as this is the general solution of Kepler's first law if derived from Newton's law of universal gravitation. Can we for example say that initial formation conditions are the main cause?


title: What happens when relativistic effects stop?

text: I'm currently learning special relativity in high school and we only primarily deal with what happens when an object is moving at constant relativistic speeds. But what if the object slowed back down to a stop? I observe length contraction for an object at relativistic speeds, when it returns to be stationary, does it go back to its original length?


title: Is there a graphical representation of the Lorentz transformation equations?

text: I always loved theoretical physics as a kid and when I came upon this site while seeking computer advice via superuser I had to stick my silly little head into an oasis of intelligence. I have often wondered what it would look like to view an object moving via the Lorentz transformation equations from a nominal speed of less than $c$ towards $c$ at an acceleration that a human could survive, let's say $3g$. And the space-time distance could be 100 light years. Would anyone be willing to produce a graph of this 1 dimension? I have tried to visualize it since I read "A Sophisticate's Primer of Relativity" 2nd Ed PW Bridgman when I was 19. As a novice I always wanted an answer from a higher power and now I think I can get a response.


title: Voyager Golden Record

text: Has there been any subsequent researches or any respond from deep outer space in any type of form with these images and sounds?


title: Partition Function - TOTAL internal energy vs Average Energy

text: Given a partition function Z, the books sometimes uses the terms 'total internal energy' and 'average energy' exchangeably. It confuses me to no end. On one hand they say that Internal energy is average energy: $$\bar U = -\frac{\partial ln Z}{\partial \beta}$$ Then they use this to calculate specific heat: $C_v = (\frac{\partial U}{\partial t})_V$. On the other hand they say internal energy is $$U = \sum \epsilon_p n_p = \int \epsilon \space n_p g_{(\epsilon)} d\epsilon$$ where $n_p$ is mean number of particles with energy p, $g_{(E)}$ is density of states. Then they use this to calculate heat capacity. Which is which?!


title: How is force exerted on a wall equal to derivative of hamiltonian with respect to wall position?

text: I'm trying to understand a solution of a problem in Landau, Lifshitz "Quantum mechanis. Non-relativistic theory" in $\S22$ "The potential well": > Determine the pressure exerted on the walls of a rectangular "potential box" > by a particle inside it. The first sentence in the solution makes me wonder how what it says is true: > The force on the wall perpendicular to the $x$-axis is the mean value of the > derivative $-\partial H/\partial a$ of the Hamilton's function of the > particle with respect to the length of the box in the direction of the > $x$-axis. Here $a$ is length of the box in $x$ direction. As I understand, this is a result from classical mechanics. But having re-read the chapters on Hamilton's function in "Mechanics" by Landau and Lifshitz, I still don't quite get how the force on the wall appears to be the derivative above. So, the question is: how to derive this result?


title: How do I enforce the no-slip boundary condition in time dependent incompressible pipe flow?

text: This is a technical problem which must have been solved already. It won't be in beginners textbooks but there should be a solution somewhere. I welcome reading suggestions. Maybe someone with experience in solving Navier-Stokes equations numerically can help me. Here goes: An incompressible fluid flowing down a pipe obeys the Navier-Stokes equations $$\partial_t \mathbf{v} + (\mathbf{v} \cdot \nabla) \mathbf{v} = \nu \Delta \mathbf{v} - \nabla P \, ,$$ and the pressure is related to the velocity field by the incompressibility condition, $\nabla \cdot \mathbf{v} = 0$. When the flow is time dependent one usually specifies the velocity field at $t=t_0$, $\mathbf{v}(t_0,\mathbf{x}) = \mathbf{v}_0(\mathbf{x})$. The no-slip boundary condition requires the velocity to vanish on the edges of the pipe for all time $\mathbf{v}(t,r=R,\theta,z) = 0$. I use cylindrical coordinates $\mathbf{x}=(r \cos{\theta},r \sin{\theta},z)$. $R$ is the radius of the pipe. In practice however the pressure must be expressed in terms of the velocity field at all times. To the best of my knowledge this is achieved by taking the divergence of Navier-Stokes equations. Using the incompressibility condition one can write this as \begin{align} \nabla \cdot \left[(\mathbf{v} \cdot \nabla) \mathbf{v}\right] = (\partial_i v_j) (\partial_j v_i) = -\nabla^2 P \, . \end{align} Then the time evolution is computed as follows: Assume that instead of the no- slip boundary condition we specifiy the pressure on the boundary for all times, $P(t,R,\theta,z) = P_R(t,\theta,z)$. Then the initial velocity field $\mathbf{v}_0(\mathbf{x})$ is changed by a small time increment from $t=t_0$ in the following way, $$\mathbf{v}(t_0+dt,\mathbf{x}) = \mathbf{v}_0(\mathbf{x}) + dt \left[ - (\mathbf{v}_0 \cdot \nabla) \mathbf{v}_0 + \nu \Delta \mathbf{v}_0 - \nabla P(t_0) \right]\, .$$ The pressure at time $t_0$ is then the solution of an inhomogenous Laplace problem with the specified boundary condition given by $P_R(t_0,\theta,z)$. Specifying the pressure on the boundary $P_R(\theta,z)$ is mathematically fine. We can use this to integrate the flow equations for all times by iterating the procedure that I just outlined. It is however not very physical. We usually control the velocity field on the boundary. One then goes back to the no-slip boundary condition by tuning $P_R(t_0,\theta,z)$ in such a way that $\mathbf{v}_0(t_0+dt,R,\theta,z)=0$. If we choose initial conditions such that $v_0(R,\theta,z)=0$ we extract the pressure through $$ \nu \Delta \mathbf{v}_0(R,\theta,z) = \nabla P(t_0,R,\theta,z) \, .$$ My problem is the following: How can we be sure that this last step is possible? On the left-hand side we have an arbitrary (?) vector. It may not be possible to write it as the gradient of a scalar field. It looks like the no- slip boundary condition contains more information than the specification of $P(t,R,\theta,z) = P_R(t,\theta,z)$.


title: Stress energy tensor and the covariant derivative of the 4-momentum

text: Another basic question. I have usually seen the stress energy tensor $T^{ij}$ described as the flow of the 4-momentum field $p^i$ along direction $x^j$ in spacetime with $p^0$ as energy and $x^0$ as time as is standard in relativity. At the level I have been reading this is usually not defined further, but it sounds to me somewhat like $$T^i\,_j = \nabla_j p^i$$ or equivalently $$T^{ij} = g^{jk} \nabla_k p^i$$ but I have never seen it written that way. Is it correct? If not what is the relationship between $T^{ij}$ and $\nabla_j p^i$? If it is correct what properties of the 4-momentum field are needed to show the symmetry and continuity of the stress-energy tensor if it is defined this way? **EDIT** After some thought $T^{ij} = g^{jk} \nabla_k p^i$ does not seem right since $$ T^{0i} = p^i = T^{i0} $$ I am still curious if there is some other relationship.


title: Is $H_0^1$ something reasonable for the electric field for a perfect conductor?

text: I'm trying to pull over some concepts that were derived for Navier-Stokes like equations to Maxwell's equations for the perfect conductor. At a certain point, I am about to assume that the electric field $\mathbf E$ is in the Sobolev space $H_0^1(\Omega)$, i.e. $E$ is differentiable wrt. the space variables and $E=0$ at the boundary. This is different from the common assumptions that locate $\mathbf E$ in $H_0(curl;\Omega)$, i.e. $\mathbf E$ is a bit less smooth and only its tangential component, namely $n\times E=0$, is zero at the boundary. Since $H_0^1(\Omega) \subset H_0(curl;\Omega)$, my assumption is more restrictive. **My question** : Does the assumption that $E=0$ at the boundary exclude common types of electrical fields? I am concerned about the condition that $E=0$ on the boundary, rather than about the additional smoothness. In other words: Is it reasonable to assume that a magnetic field is zero at a boundary.


title: Problem in progresive wave

text: In this progressive wave  we need to find out the phase difference. If we define it by $\phi$ in the x distance, we get . $$\phi = \frac{2 \pi}{\lambda}x$$ But when we write the prograsive wave equation we , write as, $$y= A \sin (\omega t - \phi)$$ A author says that this phase difference is less than the phase difference in the left side. My problem is, we can insert also + sign. But what is the physics for inserting the plus or minus sign. Thanks


title: Fields with SO(3) diagonal subgroup symmetry

text: I read about a Higgs field $\vec{\phi}=\frac{1}{2}a\hat{r}\cdot \vec{\sigma}$ (in the context of 't Hooft-Polyakov monopole) with SO(3) diagonal subgroup symmetry consisting of simultaneous and equal rotations in real and isotopic space, wherein $\sigma^i$ is the Pauli matrix, $\hat{r}$ is the spatial unit vector. I only know a little bit of introductory group theory for physics student. Unfortunately, I cannot see how this form of field $\phi$ comes out. Does it mean that we are **using Pauli matrices as kinda bases of the isotopic space**? If so, why? I've learned nothing about such thing. And by the way, what does _isotopic space_ mean?


title: QM: How to compute position/momentum relation in polar coordinates

text: So if we are working in one dimensional space, we have the formula: $$\langle x|p\rangle = \frac{1}{\sqrt{2\pi\hbar}} e^{ipx/\hbar}$$ Suppose instead we are confined to a circle of radius $R$ so that the position is given by $\theta$ and the generalized momentum is $$p_\theta=-i\hbar\frac{1}{R}\frac{d}{d \theta}$$ Then what is the value of $\langle \theta|p_\theta\rangle$? I know we can compute $\langle x|p\rangle$ by using the fact that $p$ is the generator of spatial translations. But the generator of rotations is $L_z$, and the analogous derivation becomes messty


title: Effective Spin Interaction in Helium

text: My question is: what is the point of forming an effective spin-spin interaction for the exchange term in the variational analysis of Helium? Is there any significance to this term other than on aesthetic grounds, e.g. does this spin-spin term arise in similar problems? In the first round of variational analysis of the Helium atom, one typically neglects the electron-electron interaction and solves the Schrodinger equation to get states that are products of hydrogen wavefunctions. After this, we add back in the electron-electron interaction and calculate the value of the energy shift and get something like $\left< V_{ee} \right> = J_{n,l}\pm K_{n,l}$ where $J_{n,l}$ is the contribution due to the coulomb interaction of the charge densities of the electrons, and $K_{n,l}$ is the exchange term. Then you typically see the energy from the electron-electron interaction written as: $\left< V_{ee} \right> = J_{n,l}-\frac{1}{2}(1+\frac{4}{\hbar^{2}}\vec{S}_{1}\cdot\vec{S}_{2})K_{n,l}$ To me, this just seems like an a clever way to explicitly write the triplet/singlet distinction into the plus/minus formula. Is that it or am I missing a broader concept?


title: Weak measurement formalism: Weak value can be too big?

text: A question to experts in the weak measurement formalism: In the famous paper How the Result of a Measurement of a Component of the Spin of a Spin-1/2 Particle Can Turn Out to be 100 by Aharonov, Albert, and Vaidman they explain in equation (4) under what condition the weak measurement formalism holds: $$ \Delta \ll \underset{n}{\text{max}} \frac{\langle \psi_{f}|\psi_i\rangle}{\langle \psi_{f}|A^n|\psi_i\rangle^{1/n}} $$ where $\Delta$ is the spread of the pointer function. As customary $\psi_f$, $\psi_i$ are the initial and post-selected state and $A$ is the observable to be measured. My problem is that the pointer function is a gaussian function. Does someone know if there are generalisations of this statement in the sense that the pointer function is more general and not a gaussian?


title: Time dilation all messed up!

text: Okay! There is a problem with my logic and i cannot seem to point out where. There's a rocket ship travelling at close-to-c speed **_v_** without any acceleration (hypothetically), and there is an observer AA with a clock A on earth, and there's another observer on the rocket BB with a clock B and these two clocks were initially in sync when the rocket was at rest with a FoR(frame of reference) attached to the earth. Now, this rocket's moving and AA tells b is running slower than A, and is running slower by a factor of $ \gamma $ where $$ \gamma = 1 /(1-v^2/c^2)^{1/2} $$ $$ t_a/t_b = \gamma $$ where v is the relative velocity between the two i.e the earth and the rocket! That would mean the time elapsed on A is greater than that on B but this will happen only in the FoR of AA? So $t_b$ in this equation must be the time on B as observed by AA? Is this correct? What do the terms mean in the equations? If the symmetry holds and BB doesn't accelerate, then BB could say that $$ t_b/t_a = \gamma $$ right? where $t_b$ and $t_a$ are the times on B and A with respect to FoR of BB? but I was solving this problem and I took the earth FoR of A, but the prof took the rocket FoR of B? Like how will I know which FoR to solve the problem from? It'd greatly help if the terms in all the above equations were laid down neatly! DO we even need these FoRs?? Because in all the solved problems the prof is't specifying any and is using random ones! Please help!!!! This is the question where i messed up. The first rocket bound for Alpha Centauri leaves Earth at a velocity (3/5)c. To commemorate the ten year anniversary of the launch, the nations of Earth hold a grand celebration in which they shoot a powerful laser, shaped like a peace sign, toward the ship. 1.According to Earth clocks, how long after the launch(of the rocket) does the rocket crew first see the celebratory laser light? This must be 25 years. My reasoning is: If v = 3c/5 10v+ vt= ct where t is time taken by the light to reach the rocket from earth as calculated from earth.. and I solved that for t. and added 10 years to that because the time starts at the launch of the rocket! 2.According to clocks on the rocket, how long after the launch does the rocket crew first see the celebratory laser light? This is 20 years. Here, I say: If it takes 25 years as observed by clocks on earth for the laser to reach the rocket, what should be the corresponding time as seen on a clock on the rocket? Using the formula: 25 = $\gamma$t where $\gamma$= 5/4 solved for t! 3.According to the rocket crew, how many years had elapsed on the rocket's clocks when the nations of Earth held the celebration? That is, based on the rocket crews' post-processing to determine when the events responsible for their observations took place, how many years have passed on the rocket's clocks when the nations of Earth hold the celebration? For this, I did the following: 10 years on earth = T years on rocket ship where T must be lesser than 10 as observed from Earth FoR! Therefore, T= 4(10)/5 years = 8 years! But, prof says, 10 years in earth = T years on rocket ship where T must be GREATER than 10 as observed from the Rocket FoR??? Therefore, T = 10(5/4) years = 12.5 years!! What does this question actually want?


title: Intersecting Wilson loops in 2D Yang-Mills

text: I am currently trying to understand 2D Yang-Mills theory, and I cannot seem to find an explanation for calculation of the expectation value of intersecting Wilson loops. In his _On Quantum Gauge Theories in two dimensions_, Witten carries out a curious calculation: For three reps $ \alpha,\beta,\gamma $, we fix a basis of the tensor product space belonging to $\alpha \otimes \beta \otimes \gamma$ called $\epsilon_\mu(\alpha\beta\gamma)^{ijk}$ ($\mu$ indexes the $\mu$-th basis vector, the $i,j,k$ are the indices from the original reps) with the property that $$ \int \mathrm{d}U {\alpha(U)^i}_i' {\beta(U)^j}_j' {\gamma(U)^k}_k' = \epsilon_\mu(\alpha\beta\gamma)^{ijk}\bar{\epsilon}_\mu(\alpha\beta\gamma)_{i'j'k'}$$ A minor question is why this is possible - I would be fine with accepting that I can always find some vectors that fulfill that relation, but why are they a basis? The real part I do not understand comes now: By the above, each edge of a plaquette carries some $\epsilon_\mu$, and at a crossing of two lines, we have thus four of these coming from the edges, and four other reps $\delta^{j}_c$ (j runs from 1 to 4) belonging to the plaquettes themselves. Without any explicit computation, Witten now simply says that after summing the $\epsilon$ over all their indices (as required by the decomposition of a trace beforehand), we get a local factor associated to this vertex $G(\alpha_i,\delta^{(j)}_c,\epsilon)$, which is the 6j Wigner symbol (but he won't pause to show why). I cannot find any source that would spell that relation out, i.e. show why we get precisely the 6j symbols in this computation (though their connection to the associator of the tensor product makes it plausible that we do). The real question is - the 6j symbol of what associator is this, and how would one go about and prove this? I would be very grateful to anyone who can either explain this to me or direct me to a reference where this is discussed in more detail.


title: Strong Newton's third law of action and reaction: Mathematical Interpretation

text: According to the strong law of action and reaction for internal forces(Goldstein): "$F_{ij}=-F_{ji}$ and the forces lie along the direction joining the particles." Now consider the statement > _If those internal forces are conservative we can associate the internal > forces with a potential of the form $V(|\vec{r_i}-\vec{r_j}|)$_. How can one justify this statement **mathematically** though it seems intuitively obvious?


title: Adjoint representation of the Lorentz group

text: Is it possible to construct an adjoint representation for the Lorentz group?


title: Angular momenta of photon

text: $A^\mu$ can have multipole expansions in classical electrodynamics. This gives rise to dipole photon, quadrupole photon etc. For dipole photon $j=1$ (In electrodynamics books they write it as $l=1$). Since, $\vec J=\vec L+\vec S$ and $j=|l-s|$ to $|l+s|$, in steps of unity.   1. Can we apply this formula because I think S is not a good quantum number to specify photons but helicity is. If it propagates in z-direction then $S_z$ is good quantum number. Right? Then, can I directly use this formula or should I instead use $m_j=m_l+m_s$? Does it mean whenever I have gamma-transitions between two nuclear levels I should always use $m_j=m_l+m_s$ and not use $\vec J=\vec L+\vec S$ and $j=|l-s|$ to $|l+s|$?   2. We know that, the projection $S_z=0$ is not allowed for a photon propagating along z-direction. But is it true that $L_z=0$ projection is also not allowed for photons?


title: Why doesn't the number of space dimensions equal the number of time dimensions?

text: It seems as though symmetry is a driving force behind theoretical physics. With symmetry in mind, should we expect that the number of time dimensions should be the same as the number of spatial dimensions?


title: Significant digits of time divided by 10

text: We're testing the period of a pendulum in physics class by measuring the time it takes to complete 10 periods then dividing that by 10. Our timing equipment measures to the nearest 100th of a second. There's a big debate at school over whether or not we can go into precision of 1000ths if, for example, we time 10.00 seconds for 10 periods then divide by 10 to get 1.000s per period (preserving 4 sig digs). Yes, the timer cannot measure beyond 100ths, but the rules of sig digs dictate that division does not change the number of sig digs. It also makes sense anyway because a difference of .001 seconds per period should be detectable if aggregated across 10 periods and measured with precision to 100ths of a second. But a lot of students and teachers do not accept that we can have more precision using "math tricks". Who is right?


title: Examples to illustrate temperature dependant radiation by examples of temperature to color relation

text: I'm looking for a set of examples to **illustrate** the relation of temperature and color of "glowing" bodies.   It should allow to build an intuitive understanding of this relation, so it's not about spectra, but just color in terms of visual appearance. The examples should cover infrared to somewhere above visual light if possible (or reasons why it's not possible). * * * Some good temparature/color-relation-examples froom comments/answers: An example for about **2500K** could be a **tungsten wire** of an incandescent lamp. It is not a good example because the wire in gas is not a good model for a black-body, as C. Towne Springer explains in his answer.   He provides an an **electric kiln** as an example for about **1400K** giving very good black-body approximation. As said, it's about the illustration, not about perfectly matching the spectrum of a black body - "similar enough to look the same" would qualify. From Robin's comment, the **sun** could be good as example for about **5800K**.


title: How many gas centrifuges are needed to build two small nuke within a year?

text: imagine this is 1960-70.....a country has nuclear plants already (for peaceful purposes) now if they get the technology of gas centrifuges what things they are going to need for building two small nukes within a year? and if they want to detonate this like a time bomb, is it possible?


title: A question about canonical transformation

text: I have posted this question in math.stackexchange before with no answer till now. It may be more suitable to post here. There is a problem in Arnold's Mathematical Methods of Classical Mechanics which says that: > Show that the map $A: \mathbb{R}^{2n} \rightarrow \mathbb{R}^{2n}$ sending > $(p, q) \rightarrow (P(p,q), Q(p,q))$ is canonical(p206) if and only if the > Poisson brakets of any two functions in the variables $(p,q)$ and $(P,Q)$ > coincide: $$ (F,H)_{p,q} = \frac{\partial H}{\partial p} \frac{\partial > F}{\partial q} - \frac{\partial H}{\partial q} \frac{\partial F}{\partial p} > = \frac{\partial H}{\partial P} \frac{\partial F}{\partial Q} - > \frac{\partial H}{\partial Q} \frac{\partial F}{\partial P} = (F,H)_{P,Q}. > $$ I cannot solve this problem and think about it as following: From $(F,H)_{p,q} = (F,H)_{P,Q}$ I can induce that $$ \sum_i \det\left( \frac{\partial(P_j, P_k)}{\partial(p_i, q_i)} \right) = \sum_i \det\left( \frac{\partial(Q_j, Q_k)}{\partial(p_i, q_i)} \right) = 0, \sum_i \det\left( \frac{\partial(P_j, Q_k)}{\partial(p_i, q_i)} \right) = \delta_{j,k}, $$ and $$ \sum_i \det\left( \frac{\partial(p_j, p_k)}{\partial(P_i, Q_i)} \right) = \sum_i \det\left( \frac{\partial(q_j, q_k)}{\partial(P_i, Q_i)} \right) = 0, \sum_i \det\left( \frac{\partial(p_j, q_k)}{\partial(P_i, Q_i)} \right) = \delta_{j,k}. $$ But in the other hand, to induce $dP\wedge dQ = dp \wedge dq$ I need that $$ \sum_i \det\left( \frac{\partial(p_i, q_i)}{\partial(P_j, P_k)} \right) = \sum_i \det\left( \frac{\partial(p_i, q_i)} {\partial(Q_j, Q_k)} \right) = 0, \sum_i \det\left( \frac{\partial(p_i, p_i)}{\partial(P_j, Q_k)} \right) = \delta_{j,k}, $$ or $$ \sum_i \det\left( \frac{\partial(P_i, Q_i)}{\partial(p_j, p_k)} \right) = \sum_i \det\left( \frac{\partial(P_i, Q_i)}{\partial(q_j, q_k)} \right) = 0, \sum_i \det\left( \frac{\partial(P_i, Q_i)}{\partial(p_j, q_k)} \right) = \delta_{j,k}. $$ Is there something wrong in the above reasoning? Can you show me how to solve this problem?


title: Feynman Toy Model Constraints on Number of Each Kind of External Lines

text: (From my homework) In our toy model we have three kinds of spinless particles: $A$, $B$, and $C$. The primitive vertex of decay/interaction is shown below: ![enter image description here](http://i.stack.imgur.com/lVPMN.png) My actual problem statement says: "Suppose a diagram in our toy model for the Feynman Calculus has $N_A$ external $A$ lines, $N_B$ external $B$ lines, and $N_C$ external $C$ lines. Develop a simple rule for determining whether it is an allowed reaction." If all the initial particles are $A$ then I was able to determine that $2N_A\geq N_B + N_C$ is a necessary restriction but I am not sure if this basic toy model restricts all initial particles to be $A$. Additionally, if the model restricts all final particles to be $B$ or $C$ then the inequality in my restriction becomes an equality. Does anyone have enough experience with this toy model to know if the only initial particles are $A$? If not, can you offer some suggestions to help me figure out an appropriate restriction for the general case, where $A$ $B$ and $C$ particles can all be initial or final particles? EDIT: After talking with my professor he told me that the masses of each particle obey $m_A>m_B+m_C$, so that $A$ particles are able to decay into $B$ and $C$ but that reactions that produced $A$ particles as real (instead of virtual) particles are not allowed.


title: Quantum Decoherence and Macroscopic Phenomena

text: I don't really know what quantum decoherence is, but I have heard that it is used to explain why macroscopic objects do not seemingly follow quantum dynamics. Could someone please give a simple explanation of what decoherence is and why it follows from it that macroscopic objects do not seemingly follow quantum dynamics.


title: Classical Limit of Commutator

text: In Dirac's book _Principles of quantum mechanics_ (4th ed., pgs 87-88), he seems to give a very elementary argument as to how the commutator $[X,P]$ reduces to the Poisson brackets ${x,p}$ in the limit $\hbar\to 0$. However, I don't understand the argument that he is making. Could someone please explain this? Dirac P.A.M. Principles of quantum mechanics (4ed., Oxford, 1958)


title: Differentiating the ideal gas law

text: In reading Fermi's Thermodynamics, to show that $C_p = C_v + R$, the author differentiates the ideal gas law for a mole of gas ($PV = RT$) to obtain: $PdV + VdP = RdT$. Now, the only way I am able to interpret this, is by assuming that all three variables depend on time, so that one could differentiate both sides, yielding $$\frac{d}{dt}(PV)=\frac{d}{dt}RT \equiv P\frac{dV}{dt} + V\frac{dP}{dt} = R\frac{dT}{dt}$$ You could then "multiply both sides with $dt$" to get the desired result. (I apologize if this last phrase is non-sense, but I am still very bad working with infinitesimals). Is this the correct reasoning, or is there some completely different path to the same result?


title: What's the isospin of photons?

text: What's the isospin of photons? Why PDG book says $0, 1(1^{--})$ for its $I(J^{PC})$? What does $0, 1$ mean here? Is it that the isospins aren't determined and can be 0 or 1?


title: Calculating collisions in realtime - dealing with a delay in time

text: _I wasn't sure in which stackexchange to post this, since the questions deals with a lot of different topics. I've posted it first at stackoverflow, but I've been redirected here..._ Suppose:   * You're coding a program in which 10 circles move around on the screen in realtime, all with a different velocity, that can change at certain times due to physics-calculations.   * The calculations can only be calculated on every frame   * Each frame, you have to make sure the circles who "collide"/"have collided during the time between this frame and the last" with eachother will 'bounce off' by using physics-calculations   * During the time between frame x and frame x+1, three circles will collide with eachother. However, during frame x none of the circles touches the other. In frame x+1, the same thing applies (none collides) I will try to better illustrate this with an image with my epic paint skills: http://imageshack.us/photo/my-images/854/collisionproblem.jpg/ **The question:** What are good ways to keep track of collision like this, so that a collision wouldn't get skipped due to some (unexpected) large delay in time between two frames? _This question has been spooking around in my head for way too long..._


title: Is it possible to create a new element that doesn't exist in the universe?

text: When I say something new I do not refer to something already made like H,O etc and when I mean something new I do not refer to a transformation like tritium to helium and gold. If so how ?(I mean is there a specific way to do that ?)


title: Assumptions in Heisenberg's 1925 paper

text: I am not entirely clear as to what were the bases for Heisenberg's assumptions in his 1925 paper. He claims that one cannot consider relations between quantities that are unobservable "in principle", like the position and period of revolution of an electron. To quote some text : "These rules (the abovementioned relations) lack an evident physical foundation, unless one still wants to retain the hope that the hitherto unobservable quantities may later come within the realm of experimental determination. This hope might be justified if such rules were internally consistent and applicable to a clearly defined range of quantum mechanical problems." My first query is why does he claim the position and period of an electron to be unobservable "in principle"? There was theoretically no reason (at THAT time) to doubt that these quantities could be measured, though certainly they were indeterminate practically. Secondly, just because a theory dealing with those quantities is inconsistent, or not general enough, why does it imply that we cannot define or measure quantities that that theory deals with? We may be able to measure some quantities perfectly, but still formulate an incorrect theory around them. Finally, is there any ad-hoc basis to decide what these "uncertain" quantities are? More specifically, how could Heisenberg pinpoint position of an electron as an uncertain parameter and not any other quantity (like some electric field, etc.)? Thanks in advance. (by the way I'm studying the original paper solely to look more closely at the motivation for assumptions underlying the theory)


title: Why does light always travel in a straight line?

text: No matter the frame light is in, it always moves in a straight line in that frame. Why is that? It doesn't seem like something to me that should necessarily be true. If some one runs forward and sends something perpendicular to their motion, why would it continue to move forward with them-it doesn't carry the source's velocity.


title: Why Lorentz Transformation in Special Relativity has to be like this?

text: Basically I think Albert Einstein (A.E.) was trying to find a transformation that: >   1. Always transform a constant-velocity movement into a constant-velocity > movement. >   2. Always transform a light-speed movement into a light-speed movement. >   3. If an object with a speed $v$ in frame $A$ is rest in frame $B$, then > any rest object in $A$ has a speed $-v$ in $B$. > A.E. gave: $$x'=\frac{x-ut}{\sqrt{1-u^2/c^2}},$$ $$t'=\frac{t-(u/c^2)x}{\sqrt{1-u^2/c^2}}.$$ But there are more than one transformation that can do this. Multiply a factor, we can get another one: $$x'=\frac{x-ut}{\sqrt{1-u^2/c^2}}(1+u^2),$$ $$t'=\frac{t-(u/c^2)x}{\sqrt{1-u^2/c^2}}(1+u^2).$$ It satisfies all three postulations given. So why the latter can't be the Lorentz Transformation?


title: how to reduce pressure drop in a 5/16 ss tube

text: I have a fluid line which passes high density fluid at $60 \text{psi}$. At the end there is a valve. I have a $30 \text{psi}$ dynamic pressure drop in the line valve itself creates a back pressure of $10 \text{psi}$. How do I reduce my pressure drop without changing tube parameters?


title: How is the distance between Geneva detector and the Gran Sasso one measured?

text: Does anyone know how is the distance between detectors in the now famous neutrino experiment measured? Also, how was the time of flight measured?


title: Current status of Penrose-Hawking "thunderbolts?"

text: I recently came across the interesting statement that most definitions of cosmic censorship ( _CC_ ), and most work on proving it, don't exclude something that Hawking calls a _"thunderbolt"_ (Penrose 1978, Hawking 1993, Penrose 1999). Penrose gives a verbal description as follows. > ... censorship proposals [of a certain] nature ... do not ... eliminate the > possibility of what Hawking (1993) refers to as thunderbolts, first > considered in Penrose (1978). This is the hypothetical situation according > to which a gravitational collapse results in a 'wave of singularity' coming > out from the collapse region, which destroys the universe as it goes! On > this picture, the entire space-time could remain globally hyperbolic since > everything beyond the domain of dependence of some initial hypersurface is > cut off (‘destroyed’) by the singular wave. An observer, whether at infinity > or in some finite location in the space-time, is destroyed just at the > moment that the singularity would have become visible[...] For convenience, I'm going to state the relevant technical definitions here:   * $S$ - past set, a set whose causal past $J^-(S)$ is the same as itself;   * $IP$ - indecomposable past set, a past set that is not the union of two past sets;   * $PIP$ - proper $IP$, an $IP$ that equals $J^-(x)$ for a single point $x$;   * $TIP$ - $IP$ that isn't a $PIP$, also known as an "ideal point";   * $TIP_\infty$ - $TIP$, generated by a timelike curve, that extends infinitely far into the future;   * $TIP_S$ - singular $TIP$, that is not an $TIP_\infty$. One possible statement of cosmic censorship is that no $TIP_\infty$ contains a $TIP_S$. This particular definition does rule out thunderbolts. Penrose 1999 discusses the interpretation a little, and gives the impression that this statement of _CC_ may be strong and therefore too hard for people to make progress on. **Has any progress been made since 1999 on proving any version of _CC_ that rules out thunderbolts?** Please accept my apologies in advance if a Penrose thunderbolt wipes us out before I can upvote your helpful answer. **Sources:**   * Penrose (1978). Theoretical principles in astrophysics and relativity, eds. Liebowitz et al.   * Hawking (1993).The rennaissance of General relativity" (in honour of D. W. Sciama), eds. Ellis et al.   * Penrose (1999).The Question of Cosmic Censorship, _J. Astrophys. Astr._ **20** (1999), 233–248


title: Water under high pressure

text: If you were to sink a container to the bottom of a deep ocean and seal it there, then bring it up to the surface, would it retain its pressure? The answer for a gas is obviously yes, but what about for a liquid like water which is incompressible? Once the crushing weight of the water column above is removed, does the water retain it's quality of "pressurizedness" or return to normal water? I guess a clear way to test this would be to bottle up a deep water fish and bring it up to the surface and see if it explodes. While we're at it, what about a solid? Barring any elasticity and incidental temperature change, will a solid object break a non-sealed glass container which is exactly fitted to it and then placed in vacuum?


title: Bubble in a pipeline

text: I am just thinking about this phenomenon: We have a horizontal pipeline with a flowing liquid, which contains a small bubble of gas. How do the dimensions of this bubble change when it reaches a narrower point of the pipeline? Are there practical applications that track bubble sizes to estimate the properties of the flow?


title: How long does it take for the speed of the train to decrease to $1~\text{m/s}$?

text: Here's the full problem: > _The speed of a train coasting on a level track satisfies the differential > equation $\mathrm{d}V/\mathrm{d}t = kv^2$. The initial speed of the train is > $v(0)=10~\text{m/s}$ and the speed is decreasing at the rate of > $-1~\text{m/s}^2$ when $v=5~\text{m/s}$. How long does it take for the speed > of the train to decrease to $1~\text{m/s}$? Under this model, when does the > train stop?_ I'm very bad with word problems and have been struggling quite a bit. I'd go to my school's tutoring center but currently we are on labor day vacation. -_- EDIT 1: I've attempted it a few times and here's my answer: $vknot=10~\text{m/s}, v=5~\text{m/s}, a=-1~\text{m/s}^2$ I found $k = -1/25$ and the proceeded to integrate the $\mathrm{d}V/\mathrm{d}t = kv^2$. The equation I then got was $v=kv^2t + vknot$ and plugged all my conditions in. My answer was $t=5~\text{s}$. I have no idea if I am correct as word problems always throw me off. EDIT 2: I found k=-1/25 by plugging into the dV/dt equation. dV/dt is the same as acceleration and it's stated that our acceleration is -1m/s^2, leading to this: $-1 = k(5)^2$ ===> $-1 = 25k$ ===> $-1/25 = k$. I don't know if this was valid to do, but I wasn't sure how to approach so made the assumption that it was valid. EDIT 3: After realizing the error with my integration, after integrating I got the formula $v=v(vknot)kt + vknot$. Time was then t=2.5 after plugging in my information. Now I just don't know how to tell when the train stops...


title: OAM states for wireless communications

text: Can someone give me an overview of how OAM states are used in communications? Using Orbital Angular Momentum States seems like a hot topic for communications. I read a few articles about the basic principles and experiments done in Italy that use OAM states to double the capacity of a wireless link. Also if anybody knows where to find a good tutorial on OAM states or any other good place to look for more information, experiments, data, etc. related to this topic that would help me out a lot :).


title: Geodesic Equation from energy-momentum conservation

text: I've been reading the excelent review from Eric Poisson found here. While studying it I stumbled in a proof that I can't make... I can't find a way to go from Eq.(19.3) to the one before Eq.(19.4) (which is unnumbered). I've been able to do some progress (which I present below), but can't get the right answer... Please somebody help me. It's getting really frustating... Thank you Given the energy-momentum tensor$$T^{\alpha\beta}\left(x\right)=m{\displaystyle \int_{\gamma}\frac{g^{\alpha}{}_{\mu}\left(x,z\right)g^{\beta}{}_{\nu}\left(x,z\right)\dot{z}^{\mu}\dot{z}^{\nu}}{\sqrt{-g_{\alpha\beta}\dot{z}^{\alpha}\dot{z}^{\nu}}}\delta_{4}\left(x,z\right)d\lambda,}$$ one can take it's divergence\begin{alignedat}{1}\nabla_{\beta}T^{\alpha\beta} & =m{\displaystyle \int_{\gamma}\nabla_{\beta}\left[\frac{g^{\alpha}{}_{\mu}g^{\beta}{}_{\nu}\dot{z}^{\mu}\dot{z}^{\nu}}{\sqrt{-g_{\alpha\beta}\dot{z}^{\alpha}\dot{z}^{\nu}}}\delta_{4}\left(x,z\right)\right]d\lambda}=\\\ & =m{\displaystyle \int_{\gamma}\nabla_{\beta}\left[\frac{g^{\alpha}{}_{\mu}g^{\beta}{}_{\nu}\dot{z}^{\mu}\dot{z}^{\nu}}{\sqrt{-g_{\alpha\beta}\dot{z}^{\alpha}\dot{z}^{\nu}}}\right]\delta_{4}\left(x,z\right)d\lambda+m{\displaystyle \int_{\gamma}\frac{g^{\alpha}{}_{\mu}g^{\beta}{}_{\nu}\dot{z}^{\mu}\dot{z}^{\nu}}{\sqrt{-g_{\alpha\beta}\dot{z}^{\alpha}\dot{z}^{\nu}}}\nabla_{\beta}\left[\delta_{4}\left(x,z\right)\right]d\lambda.}} \end{alignedat} But using Eq.13.3 of the reference one finds that the divergence of the energy-momentum tensor is also given by\begin{alignedat}{1}\nabla_{\beta}T^{\alpha\beta} & =m{\displaystyle \int_{\gamma}\nabla_{\beta}\left[\frac{g^{\alpha}{}_{\mu}g^{\beta}{}_{\nu}\dot{z}^{\mu}\dot{z}^{\nu}}{\sqrt{-g_{\alpha\beta}\dot{z}^{\alpha}\dot{z}^{\nu}}}\delta_{4}\left(x,z\right)\right]d\lambda}=\\\ & =m{\displaystyle \int_{\gamma}\nabla_{\beta}\left[\frac{g^{\alpha}{}_{\mu}\dot{z}^{\mu}\dot{z}^{\nu}}{\sqrt{-g_{\alpha\beta}\dot{z}^{\alpha}\dot{z}^{\nu}}}\right]g^{\beta}{}_{\nu}\delta_{4}\left(x,z\right)d\lambda-m{\displaystyle \int_{\gamma}\frac{g^{\alpha}{}_{\mu}\dot{z}^{\mu}\dot{z}^{\nu}}{\sqrt{-g_{\alpha\beta}\dot{z}^{\alpha}\dot{z}^{\nu}}}\nabla_{\nu}\left[\delta_{4}\left(x,z\right)\right]d\lambda,}} \end{alignedat} Which means that$$m{\displaystyle \int_{\gamma}\frac{g^{\alpha}{}_{\mu}g^{\beta}{}_{\nu}\dot{z}^{\mu}\dot{z}^{\nu}}{\sqrt{-g_{\alpha\beta}\dot{z}^{\alpha}\dot{z}^{\nu}}}\nabla_{\beta}\left[\delta_{4}\left(x,z\right)\right]d\lambda=-m{\displaystyle \int_{\gamma}\frac{g^{\alpha}{}_{\mu}\dot{z}^{\mu}\dot{z}^{\nu}}{\sqrt{-g_{\alpha\beta}\dot{z}^{\alpha}\dot{z}^{\nu}}}\nabla_{\nu}\left[\delta_{4}\left(x,z\right)\right]d\lambda,}}$$ so it must be zero (please correct me if I'm wrong). Then, using Eqs.(5.14) and (13.3), the divergence of the energy-momentum tensor is simply\begin{alignedat}{1}\nabla_{\beta}T^{\alpha\beta} & =m{\displaystyle \int_{\gamma}\nabla_{\beta}\left[\frac{g^{\alpha}{}_{\mu}g^{\beta}{}_{\nu}\dot{z}^{\mu}\dot{z}^{\nu}}{\sqrt{-g_{\alpha\beta}\dot{z}^{\alpha}\dot{z}^{\nu}}}\right]\delta_{4}\left(x,z\right)d\lambda}=\\\ & =m{\displaystyle \int_{\gamma}\frac{D}{d\lambda}\left[\frac{g^{\alpha}{}_{\mu}\dot{z}^{\mu}}{\sqrt{-g_{\alpha\beta}\dot{z}^{\alpha}\dot{z}^{\nu}}}\right]\delta_{4}\left(x,z\right)d\lambda}+\\\ & \,\,\,+m{\displaystyle \int_{\gamma}\frac{g^{\alpha}{}_{\mu}\dot{z}^{\mu}}{\sqrt{-g_{\alpha\beta}\dot{z}^{\alpha}\dot{z}^{\nu}}}\nabla_{\beta}\left[g^{\beta}{}_{\nu}\dot{z}^{\nu}\right]\delta_{4}\left(x,z\right)d\lambda.} \end{alignedat} If what I've done is correct then comparing with the reference's result the last term must be zero. Can anyone think why? I thought that, since the covariant derivative is taken in the point $x$ then $g^{\beta}{}_{\nu}\nabla_{\beta}\dot{z}^{\nu}$ is zero but then, what forbids $g^{\alpha}{}_{\mu}g^{\beta}{}_{\nu}\dot{z}^{\nu}\nabla_{\beta}\dot{z}^{\mu}$ to be equally zero?


title: Can a pool ball change direction mid-roll?

text: In this youtube video, a pool shark consistently gets the cue ball to drastically change direction mid-roll (i.e. while after he's hit it). Is this theoretically possible without using trick balls? If so, how? If not, why not? **Note: I am not asking whether the particular video is fake, I'm asking whether such shots are _possible_.**


title: Experimental realization of Quantum Teleportation of Spin, not polarization, not ions or atoms

text: I've looked everywhere in databases my school provides, to google searches, to the questions asked in physics forums, and here. As I understand, the original QT (quantum teleportation) protocol thought of by Bennett has been "realized" with polarization states of photons which is awesome proof. It validates QT because the entangled polarization state of light is a perfect analog. However, it hasn't been exactly realized because it seems to have not been done with spins! Has there really not been any papers on teleporting the spin state of a particle?


title: Why are black hole singularities stable?

text: The Friedmann equations says that huge matter densities lead to huge expansion rates. In Newtonian gravity, two massive point particles separated by an infinitesimal distance will experience an enormous force. So what is different about a black hole?


title: Apparent velocity of an approaching object

text: What would be the apparent velocity of an object approaching us at an actual velocity of $v$? I know $cv/c-v$ is a possible answer but here's another argument - shouldn't the velocity be the Doppler shift $$v\left(\frac{c+v}{c-v} \right)^{1/2}$$ times the actual velocity?


title: Uniform Acceleration

text: > _A steel ball is allowed to roll from rest down a 20.0 m long slop and over > a 25 m long level surface. It takes 4.0 seconds to roll down the slope. > Calculate the speed at the bottom of the slope._ I found the average speed on the slope to be 50 m/s. But, I'm not sure what formula to use to find the speed at the bottom of the slope.


title: Paper plane between two fans - is this possible?

text: The setup: two fans facing each other, distance around 1m. Both are turned on. In between them, place a simple paper plane and according to this video, it will fly. http://www.bieraffe.com/1184/videos/papierflieger-schwebt-trick/ Is this really true? If yes, what is the physics behind it? The fan that is blowing from behind seems to run at less RPM.


title: Voros onde dimensional zeta function

text: in the paper http://arxiv.org/pdf/math-ph/0005029v2.pdf formula 23 Voros evaluate and get the following spectral theta /(semiclassical) function $ \sum (E_{n}+\lambda )^{-s}= \frac{\Gamma (s-1/2)}{\Gamma(s)\sqrt \pi}\int_{0}^{\infty}\frac{dx}{(V(x)+\lambda)^{s-1/2})}= \zeta (s, \lambda)$ $ H=p^{2}+V(x). $ i believe he has used the identity for the Gamma function $ \Gamma (s-1/2)a^{1/2-s} = \int_{0}^{\infty} dt t^{s-1/2}e^{-at} $ However how could i compute the derivative $ \frac{ \partial }{\partial s}\zeta (s, \lambda)$ at $ s=0 $ should i ignore all the divergent termes and simply take the finite part ?? thanks.


title: Conversion from barn unit to square meter?

text: I read Wiki: http://en.wikipedia.org/wiki/Barn_(unit) on this and calculations under the 'Conversion from squared prefixed SI units' do not look like correct. Is 1fm^2 = 10 mb ? (or 1b = 100m^2 ) . Which one is correct and how do you calculate it? Thanks.


title: Standard notation reference

text: I'm searching for a compresensive and somewhat complete list of suggested standard notation (the symbols one ought to use in (theoretical) physics and also mathematics). Is there such a collection, has someone thought about that, or should I do it myself?


title: Why is $\phi$ constant in the hyperbolic Lorentz transforms?

text: If two spacetime events are separated by $(dx,dy,dz,dt),(dx',dy',dz',dt)$ in two reference frames $K,K'$ respectively, whose relative motion is purely along the $x$-axis, then, purely due to the fact that $$dx^2+dy^2+dz^2-c^2dt^2=dx'^2+dy'^2+dz'^2-c^2dt'^2.....(*)$$ it follows that $$dx'=dx\cosh\phi+cdt\sinh\phi.....(1)$$ $$cdt'=cdt\cosh\phi+dx\sinh\phi.....(2)$$ for $\phi \in \mathbb{R}.$ Mathematically, any $\phi\in \mathbb{R}$ will satisfy $(*)$, though if $K,K'$ and the two spacetime events are specified, $\phi$ should be fixed. **Question:** In Landau and Lifshitz, the authors seem to implicitly assume that $\phi$ is solely a function of $K,K'$, and not of the two spacetime events being studied. They make this assumption thusly: they find $\phi$ for the special case of the two events satisfying $dx=0$ (i.e. a particle travelling between the two events travels with speed $0$ relative to $K$), but then insert this value for $\phi$ into $(1),(2)$, not justifying this. What is the justification for this? _Note_ : In Landau and Lifshitz, they analyse the $x',x,t',t$ equations, not the equations involving their differentials, but they later take differentials of $(1),(2)$, so eventually do what I described above.


title: Frequency of Vortex Shedding

text: Suppose we have a long cylinder with diameter $D$, then we move it parallel to its axis with a velocity $V$ in a medium such that the Reynolds number of the flow is in the range $250<R_e<2\times10^5$. As a result, vortex shedding will takes place with a frequency $f=\frac{S_tV}{D} $ source: http://en.wikipedia.org/wiki/Vortex_shedding Where $S_t$ is the _Strouhal number_ This formula looks so simple, except for the Strouhal number. So I think maybe there are some simple explanations for it. Can anybody provide a simple model to explain at least the proportionality of the vortex shedding frequency with $V/D$? (Without using dimensional analysis of course)


title: Effect of boundary conditions on partition functions

text: While computing partition functions in statistical mechanics models (say) on a 2d lattice one usually makes use of "circular boundary conditions" which thus gives the lattice topology of a torus. It makes the expressions, and calculations simpler; and one usually assumes that in the infinite volume limit boundary effects will be negligible. However I have never come through any general proof of this physical argument. What if we choose more complicated boundary identifications so that lattice has topology of some higher genus surface ? How will it affect the final answer for partition function or other correlation functions ?


title: Minimizing the energy of a Slater determinant: why are the Lagrange multiplier elements of a Hermitian matrix?

text: If I want to minimize the energy of a Slater determinant subject to the constraint that the spin orbitals are orthonormal (as in the Hartree-Fock approximation), I can use Lagrange's method of undetermined multiplier, i.e. $$L[\\{\chi_{a}\\}] = E_{0}[\\{\chi_{a}\\}]-\sum_{a=1}^{N}\sum_{b=1}^{N}\varepsilon_{ba}([a|b]-\delta_{ab})$$ where $\\{\chi_{a}\\}$ are the spin orbitals, $E_{0}$ is the ground state energy, $[a|b]$ is the overlap integral between spin-orbitals $\chi_{a}$ and $\chi_{b}$ and $\varepsilon_{ba}$ is a Langrange multiplier. By setting the first variation of $L$ to zero I can proceed to minimize the energy of a single determinant $|\Psi_0\rangle=|\chi_{1}\chi_{2}\cdots\chi_{a}\chi_{b}\cdots\chi_{N}\rangle$. A problem in Szabo and Ostlund's "Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory" asks me to prove that the Lagrange multipliers are elements of a Hermitian matrix, i.e. $\varepsilon_{ba} = \varepsilon_{ab}^{*}$, given that $L$ is real and $[a|b] = [b|a]^{*}$. How would one go about proving this? Where I am starting at the moment is that since $L$ is real, then $L = L^{*}$. If I find $L^{*}$ then making this equal to $L$ will show that it is necessary for the Lagrange multipliers to be elements of a Hermitian matrix. However, following through with this doesn't seem to get me to a point where I can say that $\varepsilon_{ba} = \varepsilon_{ab}^{*}$.


title: Uniform Circular Motion with Banked Road and Car

text: In Uniform Circular Motion, if a car is rounding a curve at a certain speed, and the angle of the road allows the car to drive around at that speed, that speed is called the "design speed." If the measure of the angle is θ, the radius is r, the design speed is d, and the coefficient of static friction on the tires is μs, how does changing the speed of the car affect the minimum coefficient of static friction needed to keep the car along the same path?


title: When can we add a total time derivative of $f(q, \dot{q}, t)$ to a Lagrangian?

text: The other day, I was listening to this lecture on the Lagrangian for a charged particle in an electromagnetic field, and at one point in the video, the lecturer mentions that we can add any total time derivative of a function $f(q, t)$ to the Lagrangian without altering its equations of motion. This is nothing new to me, and I understand it fully, but shortly afterwards (approximately two minutes after the linked starting point), he goes on to say that you can, in fact, add a total time derivative of a function $f(q, \dot{q}, t)$, given certain conditions. This definitely surprised me, and I would love to know more about it, but the lecturer quickly moves on, so my question is as follows: under what conditions can one add the total time derivative of a function which depends on the particle's generalized velocities in addition to its generalized coordinates and time without affecting the particle's equations of motion?


title: Can someone explain to me the Rocksar-Kivelson Hamiltonian?

text: The following paper shows the hamiltonian of the 2D quantum dimer gas (page 2) http://www- thphys.physics.ox.ac.uk/people/ClaudioCastelnovo/Talks/050209_MIT.pdf Here are some questions I have.   * Why is one term kinetic and the other potenial? Since it is all about singlet states, wouldn't they all be potential?   * What is the significance of the t and v variables?   * what are the eignenstates of said hamiltonian? I'm also not sure how the 2d dimer gas comes as a result of the fractional quantum hall effect. Can anyone explain that to me ?


title: Is energy simple or composite?

text: From $E=mc^2$ it seems mass and energy are interchangeable. I can understand that mass is distinguishable, one type from another, such as protons and electrons. But in the reverse process, how is it possible that the energy that became one particle is completely indistinguishable from the energy that became a different type of particle. Shouldn't there be only one type of particle? does this imply that energy is some kind of composite thing with the potential to be this or that, like a stem cell in biology. or is it a simple thing? Take an electromagnetic wave (photon) as an example (which is massless). it is completely indistinguishable from another photon of the same frequency. yet one can "condense" into one kind of particle and the other into a different kind. how is this possible if both photons were completely indistinguishable.


title: Inelastic Scattering and coherent scatterng

text: Another Scattering Question So I have this Bravais Lattice of sites R vibrating with some normal mode with a small displacement amplitude $u_o$, some wave vector k and some frequency $\omega$. We can clearly describe $u(R)=u_o \cos(k R - \omega t)$. If we scatter a beam of radiation from the lattice with a change of wave vector q, the amplitude at the detector is $$\psi(t)=\Lambda e^{-i \Omega t} \sum_{\vec{R}} e^{i\vec{q}\vec{r}(\vec{R},t)}$$ Where $\Omega$ is the frequency of this incident wave, $\vec{r} (\vec{R},t)$ is the position of atom R and time t. So I need to find the wave vectors q that are coherently scattered as a result of vibrations. $u_o$ is assumed small, so we can say, for all q of interest, $q u_o << 1$. So we are interested in Thompson Scattering here. Can some one give me a hint on how to set my parameters and start on this? What confuses me is how to operationalize finding all the wave vectors q and making sure that they are only the q that have been coherently scattered; I think this means I will have conditions on q I must set, then evaluate for my wave vectors. If they are coherently scattered, I think this means I want the set of wave vectors that contribute to the peaks constructive interference. Thanks!


title: How does static friction increase with increase in the applied force?

text: My book says that 'Between surface of contact between two bodies there are contact forces applied between each other (they are equal in magnitude but opposite in direction,obeying Newton's 3rd law) due to electromagnetic interaction. The perpendicular component of the contact force is the Normal force while the parallel component is the frictional force. Friction is caused due to the coldwelding ie. formation of bond between the micro contact areas. When they distort due to external force,they create vibration waves which become damped releasing heat.' But it didn't tell about the mechanism of static friction, how it happens in the microscopic level. As the external force on the body increases, static frictional force also increases its value (upto a level). How does the force of static friction increase ? What is the cause of it at the microscopic level? Does the bond become more strong? How does this increase in static friction happen?