title: Converting angular velocity to linear velocity through friction

text: A very basic question here; it's related to this one, but not quite the same. If a rotating rigid body (a sphere for the sake of discussion) with mass $m$, radius $r$ and inertial tensor $I$ has initial linear velocity $v_0=0$ and non-zero, initial angular velocity $\omega_0$, what are the idealized final angular and linear velocities if the body finds itself instantaneously on a surface with infinite friction and mass, assuming $\omega_0$ is parallel to the surface (and the surface doesn't absorb any of the energy, nothing is lost to heat)? The body should end up rolling along the surface at some constant velocity. So I think we know that $v_f = r\times \omega_f$ with $r$ anti-parallel to the surface normal. Clearly the inertial tensor must come into play since if all of the mass is concentrated on the outside of the body, it has a lot more angular momentum than if the mass is mostly at the center of the body. I tried using conservation of momentum here, but the units don't match up between angular momentum and linear momentum.


title: If it is given which intervals are spacelike, can be determined which intervals are lightlike?

text: Provided that the notion of "$\mbox{spacelike}$"-ness (of an interval) is symmetric: $$\text{spacelike}( \, x - y \, ) \Longleftrightarrow \text{spacelike}( \, y - x \, ),$$ then for any set $X$ (of sufficiently many elements) the set $X^2$ (of all pairs, regardless of order, of not necessarily distinct elements of $X$) may be partitioned into three disjoint and generally non-empty subsets   * of pairs containing the same element twice: $I_X := \\{ x \in X: (x x) \\}$,    * of pairs of distinct elements whose interval is (called) "$\text{spacelike}$": $S_X := \\{ x \in X \, \& \, y \in X \, \& \, x \ne y \, \& \, \text{spacelike}( x - y ): (x y) \\}$, and   * of all remaining pairs $K_X := X^2 \backslash (I_X \cup S_X)$. Provided further that the notion of "$\mbox{lightlike}$"-ness (of an interval) is symmetric as well: $$\text{lightlike}( \, x - y \, ) \Longleftrightarrow \text{lightlike}( \, y - x \, ),$$ and given a suitable set $X$ (of sufficiently many elements) and a suitable set $S_X$ satisfying $X^2 \cap S_X = S_X$, how would the corresponding set $K_X$ be partitioned further into two disjoint and generally non-empty subsets   * of pairs of distinct elements whose interval is (called) "$\text{lightlike}$": $L_X := \\{ x \in X \, \& \, y \in X \, \& \, x \ne y \, \& \, \text{lightlike}( x - y ): (x y) \\}$, and   * of all remaining pairs $T_X := K_X \backslash L_X$ ?  **Edit** The wording of this question (apart from formatting issues) as it presently stands appears not adequate to the title. (Helpful responses to it have been received nevertheless, which I try to incorporate in going forward.) In trying to improve the detailed wording, what would need to be considered first is (put roughly, for the time being, as I've come to consider it only recently): (1) Whether and how the topological notion of " _boundary_ " can be suitably generalized to the context of sets of pairs such as $X^2$ and $S_X$, and (2) Whether, given a particular set $X$, the predicate "$spacelike()$" in the definition of set $S_X$ implies certain relations to a corresponding set $K_X$ which I did not state explicitly above; such as the absence of " _impossible figures_ " wrt. membership of certain pairs in $S_X$ or $K_X$. I plan to defer editing (apart from possible formatting) until these preliminary questions have been expressed more adequatly elsewhere.


title: Why is static friction omitted here? And comparison with this problem on Newton's second Law. Solutions are all provided

text: The problem itself is short, but my concept question is...long. So be prepared. Note that problem 1 and problem 2 are related to my concept questions, so I am not taking advantage of asking two in one. **Problem #1** ![](http://img815.imageshack.us/img815/7890/problem1.jpg)! Note it should read "to the right with an acceleration of $3.00m/s^2$ **Solutions #1** ![](http://img233.imageshack.us/img233/8065/sol1s.jpg)! **Concept Questions #1** a) In the free-body diagram for the 5kg mass, I am guessing $n_1$ is the gravitational force from the 2kg b) Now if you look carefully in the solutions, you noticed that the static friction between the 2kg and 5kg mass is ignored. $\sum F_x = F - \mu_k n_2 = ma$ But shouldn't it actually be? $\sum F_x = F - (f_s + f_k) = ma$ If not, why was it ignored? **Problem #2** ![](http://img215.imageshack.us/img215/6696/problem2f.jpg)! **Solution #2** ![](http://img59.imageshack.us/img59/4281/sol2pp.jpg)! **Concept Question #2** a) For the cart, what happened to the force exerted by the hanging mass on the cart in the free-body diagram? b) How could the mass on the top have an acceleration and the hanging mass NOT accelerate? I ask this because they had the condition $T = m_1 a$ $T - m_2 g = 0$ **General Concept Question (both problems)** This is mostly concerned with Newton's second law for both the masses in the bottom cart/box. How come on the "ma" side of the equation, we set it equal to $(\sum m)a$. I thought in Newton's second law, we set it equal to the single mass alone? I understand that in all the problems, they treated all the mass stacked **as a single mass** , but how do you do it systematically like I do? The solutions given kinda skipped a lot of trivial things, and I am not used to it. For instance, for **Problem #1** , for the mass on the bottom. Shouldn't my system of equations be $\begin{cases} \sum F_{2x} = f_s = 2a \\\ \sum F_{5x} = F - (f_s + f_k) = 5a \end{cases}.$ Compared with **Problem 2** , if I had added the force exerted by $m_2$ on M, then it would be (and i had set the Ma side to only a single mass) **For M** $\begin{cases} \sum F_{x} = F - {F_{2M}} = Ma \\\ \sum F_{y} = n = (M + m_1 + m_2)g \end{cases}$ **For $m_1$** $\begin{cases} \sum F_{x} = T = m_1a \\\ \sum F_{y} = n_1 = m_1 g \end{cases}$ **For $m_2$** $\begin{cases} \sum F_{x} = {F_{M2}} = m_2a \\\ \sum F_{y} = T = m_2 g \end{cases}$ **Note that $F_{2M}$ means the normal/contact force exerted by $m_2$ on to mass M and $F_{M2}$ is the force exerted by mass M on $m_2$ I am so sorry for making it this long and I realize normally I can only ask quick concept checks, but I feel like if I overcome this concept I can handle **any** Newtonian problems with forces. This is not homework, this is just self-study (it is the holidays afterall!). Thank you so much for reading


title: Equivalence of classical and quantized equation of motion for a free field

text: Suppose a classical free field $\phi$ has a dynamic given in Poisson bracket form by $\partial_o\phi=\\{H, \phi\\}$. If we promote this field to an operator field, the dynamic after canonical quantization is given by $\partial_o\phi=i[H, \phi]$. How do we prove the equivalence of these two equation of motions? Does the procedure break down for interacting fields? Edit: The following is my understanding of how the question is answered. I consider a scalar field for simplicity. 1/ Because the field is **free** we can write the operator $\phi$ as a superposition of plane-wave operators: $$ \phi(x)=K\int (d^3p) a_p e^{-ipx}+a_p^{\dagger}e^{ipx}$$ Where K is a normalisation constant. The problem is now to find what equation governs the time evolution of $a_p$ 2/As an operator, $a_p$ evolves according to $\frac{d}{dt}a_p=i[H, a_p]$ 3/ Because the field is **free** we can write $H$ in the form: $$ H=\int(d^3p)\omega a_p^{\dagger}a_p$$ 4/Inserting into the commutator and using commutation relations this gives $$\frac{d}{dt}a_p=i\omega a_p$$ $$\frac{d^2}{dt^2}a_p=-\omega^2 a_p$$ 5/These equations for operators are the same as the classical equations, which justifies the use of the classical Euler-Lagrange equations for the operators. Is this correct?


title: Can we solve California's drought crisis by pointing parabolic mirrors at the ocean?

text: If you concentrated sunlight to points in the ocean, could you make the water evaporate significantly faster to induce a substantial amount of rain? Assuming you could do that, could you use silver iodide to bias rain toward certain areas of land?


title: Motivation for "discretized quantum state spaces"

text: I know that the title of the question is rather vague, so let me clarify what I mean. For a quantum system, the set of states has infinitely (even continuously) many extreme points, i.e. there are infinitely many pure states. By the set of states, I mean the convex set of _density operators_ $\mathcal{S} \subseteq \text{Herm}(\mathcal{H})$ _on_ a Hilbert space $\mathcal{H}$ rather than the set of unit vectors _in_ the Hilbert space $\mathcal{H}$. So I am talking about a convex subset of the vector space of Hermitian operators on $\mathcal{H}$ (for my puroposes, we can restrict to finite dimensions). For example, the set of states for a spin-1/2 particle is given by the Bloch ball. Every point on the boundary of the Bloch ball is an extreme point of the Bloch ball and therefore a pure state. This boundary is a sphere, containing continuously many states. I am wondering whether there are physicists who have come up with reasons why the set of states of a (quantum) system should be discretized instead of continuous in the sense that it should only have finitely many extreme points / pure states. To see geometrically what I mean, see the following figure: ![http://i46.tinypic.com/2chmkc6.png](http://i.stack.imgur.com/gcNMM.png) More precisely, my question is: **What are possible reasons or motivations why the set of states of a (quantum) system should be the convex hull of finitely many points (i.e. a polytope), instead of having infinitely many pure states?** Looking for such reasons, I have come across papers like this one: Discreteness and the origin of probability in quantum mechanics They say that quantum-gravitational considerations might suggest some kind of discretization of the states of a system. First of all, I don't really understand their argumentation yet why there should be discreteness. Moreover, I'm not sure whether the kind of discreteness they propose is the same as the one I'm asking for (e.g. I'm not sure whether their state spaces are convex). If anyone can help me with this kind of argumentation (by explaining it or pointing to additional sources) or if anyone has come across other arguments for discreteness, I would be glad to see them. I'm aware of the fact classical systems can be described by a simplex, therefore bearing this kind of discreteness. However, I'm interested in reasons why _quantum_ state spaces should be discretized. **EDIT:** Thanks for the answers so far. Perhaps I should be clear about a few things: I do _not_ think that quantum state spaces are discrete, and I do _not_ really try to justify discreteness of quantum state spaces. I also see reasons why a quantum state space cannot be discrete. I am just _curious_ whether other physicists have come up with justifications or reasons for discreteness. Why? To see what ideas are ruled out by arguments against discreteness. The answers so far are not uninteresting, but they provide reasons why a quantum state space should _not_ be discrete. I see that my question somehow asks for justification of something that cannot be, but I am still interested in what possible motivations for discreteness there could be.


title: Infinity Corrected Microscope - Building from Scratch

text: I took an optics course a few years back, and am trying to figure out how to build an infinity-corrected microscope from discrete optical components which are listed in references [2] (lenses) and [3] (mounting equipment). ![Diagram of Infinity Corrected Objective, \(c\) Olympus Micro 2014](http://i.stack.imgur.com/OttSp.jpg) I have gone through a number of books trying to figure out how to actually assemble this with some simple lenses, tubes, and lens mounts. The problem is that all of them have an example like that above, but don't spell out what I need to do to determine:   1. Total magnification (I'm assuming it is the ratio of the focal length of the eyepiece lens over the focal length of the objective lens).   2. How close the lenses should be (I think the objective and tube lenses should be close, just not closer than the focal length of either, and I don't know what the focal length should be for the tube lens either). What about the distance between the tube lens and eyepiece lens? I thought the gap between the tube lens and eyepiece lense could be arbitrary in length, but the diagrams I find seem to indicate otherwise.   3. What kind of lenses to actually use (Do all of them have to be convex lenses? Some examples should the use of a "plano convex" lens instead of a "convex convex" lens for the eyepiece). If anyone could point out how I would design this or steer me towards some resources that can provide a definitive answer on how to design an infinity corrected microscope, eyepiece and all included, it would be greatly appreciated. Thanks! **References**   1. _Microscope Optical Components Introduction_ , Accessed 2014-03-04, `<http://www.olympusmicro.com/primer/anatomy/components.html>`   2. _Experimental Grade Optics_ , Accessed 2014-03-04, `<http://www.anchoroptics.com/pages/category/Experimental%20Grade%20Optics.cfm>`   3. _Modular Mounting Components_ , Accessed 2014-03-04, `<http://www.edmundoptics.com/optomechanics/modular-mounting-components/>`


title: Does angular momentum conservation imply that angular momentum $J$ is parallel to angular velocity $\omega$?

text: In other words, does $\frac{dJ}{dt} =0$ imply $J \times \omega =0$? Counterexamples or proofs would be helpful! EDIT: This question originally asked if $\frac{dJ}{dt} =0 \Leftrightarrow J \times \omega =0$, but clearly the $\Leftarrow$ direction is not true.


title: conservation of momentum when a bullet hits a block

text: why momentum is conserved when a bullet hit a block horizontally even when force of bullet is acting on it and net external force is not zero ?


title: Mathematics of AdS/CFT

text: To date, what is the most mathematically precise formulation of the AdS/CFT correspondence, and what are the most robust tests of the conjecture?


title: How many atoms per light year does light encounter when traversing interstellar space?

text: Interstellar space is pretty empty but there a small number of of atoms (mostly hydrogen?) floating around. How many atoms per light year would a photon encounter while traversing interstellar space?


title: Why does this cowbell appear to vibrate at the wrong frequencies?

text: The cowbell in question: http://www.youtube.com/watch?v=OuU0BzXm-qk In most cowbells, the larger the metal dome, the lower the frequency produced. What would cause a phenomenon where the opposite is true: the larger dome creates a higher-pitched sound, as the one in the video does.


title: Why is free neutron unstable but free proton is stable?

text: Aren't they basically made up of the same stuffs (quarks) in almost the same configuration?


title: What is the simplest possible topological Bloch function?

text: Kohmoto (1985) pointed out in _Topological Invariant and the Quantization of the Hall Conductance_ how TKNN's calcuation of Hall conducance is related to topology, in which topologically nontriviality is said to be equivalent to impossiblility choosing a global phase of Bloch function $u_k (r)$ in Brillouin zone. As shown in the Figure, we can choose two distinct gauges in sector I and II, and the curvature is the loop integral of phase mismatch on boundary $\partial H$. ![enter image description here](http://i.stack.imgur.com/POOL2.png) What is the simplest possible Bloch function that is   * topologically nontrivial, and   * an eigenstate of Bloch Hamiltonian?  Bloch Hamiltonian: $H(k_x,k_y) = \frac{1}{2m}(-i\partial + {\bf k}+e{\bf A}(x,y))^2 + U(x,y)$ where $U$ is lattice periodic.


title: Linear dependence of magnetic potential on current density

text: I'm a mathematician learning physics to provide some background for my mathematical work (especially pde's!). I have been reading through Jackson's Classical Electrodynamics (3rd edition), and I was puzzled by an assumption he makes. On page 214, he derives the equation $W=\frac12 \int_{V_1} \mathbf{J} \cdot \mathbf{A}$ under the assumption that the magnetic potential $\mathbf{A}$ and the charge density $\mathbf{J}$ are related linearly. However, this seems to be a very strict condition, since they are related in each coordinate by the Poisson equation. It seems like only eigenvalues of the Poisson equation could satisfy the linearity condition (after diagonalizing the linear relationship). And yet on the very next page, he uses the above formula for work in the very general setting of a system of $N$ arbitrary circuits. So, my question is, how common is a linear relationship between the vector magnetic potential and the current density? And do Jackson's results hold in the settings he uses them in?


title: Do stars appear to move with uniform motion?

text: The Ancient Greek astronomers had quite an obsession over uniform circular motion; I was wondering if there was a logical reason for this. Did it develop through actual observations of the stars? Do they appear to move uniformly? If so, why? I've read a few things online that attributes this belief mainly to "Pythagorean mysticism", saying that the Ancient Greeks simply looked for an idealistically satisfying description of celestial motions. This too makes me wonder though: 1.) Why uniform? 2.) Why circular?


title: Why is the thomson cross section constant over cosmological time?

text: In all astrophysical calculations I'm aware of, the thomson cross-section (for electron scattering) is taken as a constant in time, why is this the case? I have only weak experience in cosmology and GR, but naively I would have guessed that if the wavelength of photons is red-shifted from expansion, then the debroglie wavelength of an electron should increase* and thus the thomson cross-section.   * I guess you could include, 'why isn't matter-wavelength similarly redshifted' in my question.


title: Calculate speed of proton at infinity

text: I have this problem to solve: Two protons are on the x axis at x=-1 and x=0. An alpha particle is placed at x=2, and the proton on the left is released, find its speed at infinity. I know that this involves the change in kinetic and potential energies of the particles, but I dont understand which particles are included in that calculation. Is the proton on the left neglected because it is the one released? An explanation in general would be appreciated. For the purposes of following the rules in asking a homework question properly, the specific physics topics I was asking about were: which particles are included in the kinetic and potential energy calculation for determining the speed at infinity of a proton, and why. My effort in solving this question was finding out that work is the change in kinetic energy + change in potential energy, but not knowing which particles to include in the calculation. This should clear up any confusion that this question is off topic.


title: Three-plate capacitor

text: ![enter image description here](http://i.stack.imgur.com/d7Ggl.png) What is the total capacitance of this capacitor? Is it analogous to the case of inserting a plate into a capacitor (essentially creating two capacitors in series)?


title: How to obtain the field equations in Brans-Dicke theory from the action?

text: The action for the Brans-Dicke-Jordan theory of gravity is $$ \\\S =\int d^4x\sqrt{-g} \; \left(\frac{\phi R - \omega\frac{\partial_a\phi\partial^a\phi}{\phi}}{16\pi} + \mathcal{L}_\mathrm{M}\right). $$ And the field equations of the gravitation field are $$ G_{ab} = \frac{8\pi}{\phi}T_{ab}+\frac{\omega}{\phi^2} (\partial_a\phi\partial_b\phi-\frac{1}{2}g_{ab}\partial_c\phi\partial^c\phi) +\frac{1}{\phi}(\nabla_a\nabla_b\phi-g_{ab}\Box\phi). $$ I tried to vary this action w.r.t $g_{ab}$ but failed. How can I obtain these from the action? Where can I get the detailed derivation? Thx!


title: Is the wave function of a particle re-created after a measurement stops?

text: Yeah, I haven't quite understood, or been told, what happens to, for example an electron and it's wavefunction, when you stop to measure it? I mean, an electron has a wave function describing it's position and so on in the region of space it is confined. When I measure it, I get an output, and the wavefunction collapses - as far as I understand. But then, when I stop measuring, the electron has not been "destroyed" has it? So does it resume it's original wavefunction, and continue as nothing has happened, or have I totally misunderstood something? :)


title: What about the other side of the Black Hole

text: I was wondering if on the recieving end of a black hole can you see the photons being sucked in


title: How to derive the schwarzchild metric?

text: I'm having trouble differentiating the following when making a change of co- ordinates to determine the Schwarzchild metric. $$r'^{2}=r^{2}C(r)$$ Then taking the total derivative of both sides, the answer says; $$dr^{2}=\frac{4r'^{2}dr'^{2}}{2C(r)r+r^{2}C'(r)}$$ I cannot seem to get this answer??


title: What effects would we see on Earth if one star of a distant binary system went supernova?

text: Assume that astronomers on Earth have been monitoring a distant binary star system for a long time (relative to the orbital periods of the binary system) and then they see one star go supernova. Let $c_g$ = "speed of gravity" and $c$ = speed of light. What differences would we expect between observations in the following hypothetical cases: \begin{align} \mathrm{(i)} \quad c_g & = 0.5 c \\\ \mathrm{(ii)} \quad c_g & = 1c \quad \text{(the generally accepted value of $c_g$)} \\\ \mathrm{(iii)} \quad c_g & = 2c \end{align} Focus on differences which would allow us to infer a value for $c_g$. This question is prompted by an earlier Q&A (What is the status of existing measurements of the speed of gravity?) which I do not have enough points to comment on.


title: Two-oscillator system coupled to reservoir

text: I am implementing the Monte Carlo wave-function approach to dissipation problems. So far, I have simulated the quantum harmonic oscillator coupled to a finite temperature reservoir given in section 5A of the above paper. I must now implement the two-oscillator system, whose Hamiltonian I am told is of the form $$ H=\hbar\omega_aa^\dagger a+\hbar\omega_bb^\dagger b-\hbar ga^\dagger a(b^\dagger+b). $$ My question is this:   1. Can I form my wave-function vector in such a way that, for example, every odd entry represents the first oscillator in the corresponding state and every even entry represents the second? As an example, if both oscillators were in the second state, the wave-function would be $$ \left|\psi\right\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\0\\\1\\\1\\\0\\\0\\\\\vdots\end{pmatrix} $$   2. If this is possible, what sort of form would the ladder operators $a,a^\dagger,b,b^\dagger$ take and would the corresponding number operators for each oscillator still be $a^\dagger a$ and $b^\dagger b$? Sorry if this is a dumb question, I have not yet taken a real QM course. Thanks for the help.


title: How are low energy effective actions derived in string theory?

text: For example the eq 2.1 here with regards to Type IIB. Unless I am terribly missing/misreading something Polchinski doesn't ever seem to derive these low energy supergravity actions. I would like to see a beginner's explanation (maybe together with review paper for further information) to getting these actions from string theory (and hopefully also something about deriving the black-hole and the brane metrics from them)


title: Compton scattering multiple wavelengths?

text: The formula given for compton scattering shows that when x-ray of one specific wavelength hits carbon or some materials, emitted x-ray will be of one new specific wavelength. However, according to scattered x-ray intensity (y-axis) vs wavelength (x-axis) graph, it shows that there are multiple wavelengths for each scattering angle. So, what is the wrong part of my thoughts? How should compton scattering formula be interpreted? And how can one explain that scattered x-ray frequency can be higher than original x-ray frequency?


title: scattering singularity

text: In QFT when one works out the cross section between two colliding electrons one gets a formula which is proportional to $\theta^{-4}$ where $\theta$ is the scattering angle which is due to a nearly on-shell photon with momentum $\approx 0$. See for example Peskin and Schroeder pg 155. This is also seen in ordinary Rutherford scattering and apparently is not a problem experimentally as one can't detect particles along the collision beam. Also, in Rutherford scattering the singularity gets removed if one takes into account the screening of the nucleus by the electron cloud (see QM by Messiah (1961), pg 422) which modifies the Coulomb nature of the potential. I was wondering if this singularity is indicating some problem with the calculation? Is there some effect in the calculation which has been neglected which when included would remove the singularity? Could renormalization effects maybe have a similar effect to the electron cloud in the Rutherford case?


title: NED velocity to redshift conversion?

text: I've done some search with the Nasa Extragalactic Database (NED) and I have a very basic question about the velocity/redshift conversion. For example, for the first object of this page, we have $v=19791km/s$ and $z=0.066016$. If we use the simple formula $z=v/c$, we find the result of the database : $19791/299792=0.0660157709345$. But now if we use the formula : $z = \sqrt{\frac{c+v}{c-v}}-1$ we find $0.0683461749892$ Which is the correct one and why ? Thank you very much.


title: Statistical analysis of data in Physics

text: Analysis of data is integral in bridging the gap between theory and experiment. My question is about the two major methods of approaching the analysis - Bayesian and Frequentist. How much do the results of the analysis depend upon the choice of the method? For instance, consider experiments in Particle Physics. This paper by Louis Lyons states that Particle Physicists use a hybrid approach to analysis. (Frequentist hypothesis testing and Bayesian parameter estimation.) If a purely Bayesian/Frequentist approach was applied, would there be a difference in the results of the analysis? What implication would any difference have on the interpretation of data?


title: What are "cycles of anomaly" and "cycles of longitude"?

text: In several early (pre-1600) astronomical texts I read about "cycles of anomaly" and "cycles of longitude", but it us unclear to me what these terms mean. They were clearly familiar to authors at the time and are undefined in any of the texts I've looked through. How were these terms employed in pre-1600s solar system models? What modern astronomical terms or observable events do they correspond to?


title: Using Moments to influence the thickness of beams in structure?

text: Are moments relevant to the thicknesses of beams in a structure (ie. the larger the moment the thicker the beam)?


title: A reading list to build up to the spin statistics theorem

text: Wikipedia's article on the spin-statistics theorem sums it up thusly: > In quantum mechanics, the spin-statistics theorem relates the spin of a > particle to the particle statistics it obeys. The spin of a particle is its > intrinsic angular momentum (that is, the contribution to the total angular > momentum which is not due to the orbital motion of the particle). All > particles have either integer spin or half-integer spin (in units of the > reduced Planck constant ħ). > > The theorem states that: > >   * the wave function of a system of identical integer-spin particles has > the same value when the positions of any two particles are swapped. > Particles with wavefunctions symmetric under exchange are called bosons; >   * the wave function of a system of identical half-integer spin particles > changes sign when two particles are swapped. Particles with wavefunctions > anti-symmetric under exchange are called fermions. > > > In other words, the spin-statistics theorem states that integer spin > particles are bosons, while half-integer spin particles are fermions. **Question.** What would you recommend as the best means to understanding the proof of the spin statistics theorem? To put it another way: if you took (for instance) the proof-sketch for the spin-statistics theorem given later in that same Wikipedia page, and wanted to add enough material to it to create a textbook whose entire purpose was to take someone from a third- or fourth-year undergrad level of physics up to understanding the spin-statistics theorem inside-out, what references would you use to flesh out the material of that book? **Context.** I'm a researcher in quantum computation, coming from the computer science end of things: I have a very solid grasp on the basic mathematical framework of _non-relativistic_ QM, and some special relativity, if not all applicable techniques. I have very basic familiarity with fermionic and bosonic operator algebras, as algebras generated by creation/annihilation operators satisfying certain axioms, though I have not had occasion to use them much myself. I take for granted that understanding the proof of the spin-statistics theorem will involve learning a non-trivial amount of physics (and also likely mathematical) background. Any recommendations? **Edited to add:** If it is sufficient to learn quantum field theory, please recommend a suitable text as an answer. For instance, if you know of a good book on QFT that does not assume much of a background in particular topics such as E&M, and which definitely covers all of the concepts pertinent to the proof-sketch given on the Wikipedia page, and/or itself has a good self- contained proof of the spin-statistics theorem -- in short, a book which can take me from Schrodinger and Einstein all the way to the spin-statistics theorem -- then please recommend the book in question as an answer.


title: Adding rotation to internal coordinates

text: I'm optimizing the geometry of a system composed of several interacting masses (a molecule). The energy of the system depends on the relative position of the masses, and all velocities are zero. In a simple case, the energy is invariant with translation and rotation of the whole system, so I can optimized using internal coordinates, defining these internal coordinates as the different distances, angles and dihedral angles between the masses. For converting coordinates to and from Cartesian I have the matrix of coordinate derivatives, etc. J. Chem. Phys. 117, 9160 (there are small mistakes in eqs. 34 and 35). Now the question is how to proceed when the energy is not invariant with translation and rotation. I have to include translation and rotation of the whole system to the degrees of freedom of the optimization. Translation is easy, because I can add the Cartesian coordinates of the center of mass, and the derivatives with respect to the coordinates of the masses are easy. But how can I add rotation? I guess it will be related to the orientation of the principal axes of inertia, but I can't find a clear expression that will give me the three rotational coordinates I need, and their derivatives with respect to all the masses' coordinates. Can anyone give me some help or pointers?


title: Examples of nearly perfect fluids and gases

text: When learning for physics (hydrodynamics and gases) I wanted to know what would be examples of nearly perfect fluids (no surface effects and no friction) and perfect gases (only elastic collisions and a simplified interaction potential) that are used in experiments etc. When searching for that in the Internet I only found things like _ultra-hot quark-gluon plasma_ and _optically trapped 6Li atoms_ for perfect fluids. I rather thought of more practicable examples like fluid helium, is it near to a perfect fluid? For gases I think that would be hydrogen and light halogens (because the volume of the molecules is neglected), but at which temperatures?


title: Do plastic shoes electrically insulate people from the ground?

text: I've read on the Internet that wearing plastic shoes doesn't allow electric charges we collect from our environment (PC, TV, appliances) to be earthed. Is this reasoning correct?


title: Intuition behind Fourier transformed spaces

text: Intuitively I've been able to understand a Fourier transform a change-of-basis formula - you're basically moving from position to momentum basis or from time to frequency - but what does it mean that these spaces are 'conjugate' to each other? Does this have to do with them being complete bases? A related question comes from considering the electric field generated by a travelling electron, $\textbf{E}(\textbf{r},t)$. If we consider sending $\textbf{E}$ to position-frequency space $\tilde{\textbf{E}}(\textbf{r},\omega)$, I find it weird that there is no longer a time dependency. Have we 'smeared' the electron across its trajectory and computed some quasi-average electric field? Is there some other interpretation that might make more sense?


title: Can an orbit be calculated using two points and transit time?

text: Working in only two dimensions and assuming that the central body is at the origin of the coordinate system, given two points in space and knowing the transit time between those points, as well as the direction of motion, is it possible to calculate a body's orbit? It seems to me that there should be enough information - at $T=0$, the object was at $(θ_0, r_0)$ and at $T = T_1$, $(θ_1, r_1)$. Two points + knowing one focus gives a set of ellipses - but the additional information of transit time seems to me like it should be enough to narrow it down to 1-2 orbits in the general case, travelling in opposite directions if there are two. Is it possible? If so, how would I go about doing so (numeric approximations are fine)?


title: Could a spacecraft dock with the voyager 1?

text: Is the Voyager 1 (or any similar exploration spacecraft) equipped with the ability for space to dock with it (attach to it)? Hypothetically, if a voyager or similar spacecraft were in Earth's orbit, could a NASA spacecraft dock with it in a similar fashion as it would the ISS?


Will a machine or a technique ever be possible that allows artificial gravity in interplanetary space? And I ask out of pure curiosity. I know there is the "Vomit Comet" that allows for a non-gravity 'simulation' (for lack of a better word) on Earth, and I was thinking about if there would be a way to reverse that and rather have gravity in space. edit: Say I was building a space shuttle because I'm dying to go to the moon (true story). Unlimited budget, the smartest people you can fathom helping me build my space shuttle. I get motion sickness and the whole "anti-gravity" thing wouldn't really work for me. Would there be a way to have gravity on my space shuttle? Perhaps by using a type of machine or magnets or something of the like?


title: Intuitive meaning of Newton (units)

text: Is there any intuitive reasoning behind why the units of a "Newton" is $kg \frac{m}{s^{2}}$ and how it represents force? I always wanted to understand why objects of different mass fall at the same rate on earth and I found an explanation that uses Newtons Second Law of Motion and it made sense, but I'm confused as to why the units of force have this weird formula. Did Newton just use those units to make his second law work out mathematically via cancelling? Or is there some way to interpret $kg \frac{m}{s^{2}}$ as meaning force? Any insight would be appreciated.


title: How is an arbitrary operator usually denoted in quantum mechanics?

text: Which symbols are usually used to denote an arbitrary operator in quantum mechanics, such as _O_ in the following example? $O \mbox{ is Hermitian} \Leftrightarrow \Im{\left< O \right>} = 0$


title: Invariance of Lagrangian in Noether's theorem

text: Often in textbooks Noether's theorem is stated with the assumption that the Lagrangian needs to be invariant $\delta L=0$. However, given a lagrangian $L$, we know that the Lagrangians $\alpha L$ (where $\alpha$ is any constant) and $L + \frac{df}{dt}$ (where $f$ is any function) lead to the same equations of motion. Can we then consider that the Lagrangian is invariant under a transformation if we find $\delta L=\alpha L$ or $\delta L=\frac{df}{dt}$ instead of $\delta L=0$?


title: No Voltage in a Circuit but still a Current, How?

text: Hi I've done an experiment in physics using a basic circuit. Can anybody explain how come the data logging equipment I was using continued to register a current when there isn't a positvie voltage in the circuit.


title: Using the Scalar Electrostatic Potential to Calculate Transition Probabilities

text: **transition probabilites** of atomic systems prone to some time-varying electromagnetic field are very often calculated using **perturbation theory** leading to expressions including the vector potential $\mathbf{A}$. This approach seems to be very complicated and I am wondering: > ### Is using the electrostatic potential $U$ to calculate "electric > transitions" (approximately) sufficient? This was the short version of my question; below you will find some further notes that (hopefully) contain all information needed to clearify it. In **relativistic quantum mechanics** , the action of an an **electrodynamic field** on an electron can be realized by minimal coupling using the replacement $\partial_\mu \rightarrow \partial_\mu - \mathrm{i}eA_\mu$. Thus, in a certain sense, (quantum) electrodynamics can be derived from the Dirac equation using the invariance of the free-electron action with respect to a phase change. ### Electromagnetic Fields and the Schrödinger Equation Nevertheless, often one must ignore the beauty of gauge theory to be able to **practically calculate** something under certain simplifications using the non-relativistic Schrödinger-equation. There, one can incorporate an external electromagnetic field applying the replacements $$H\left(\mathbf{p},\mathbf{x}\right) \rightarrow H\left(\mathbf{p}-e\mathbf{A}\left(t,\mathbf{x}\right),\mathbf{x}\right)+eU\left(t,\mathbf{x}\right)$$ to the Hamiltonian of the system where $\mathbf{A}$ is the known vector potential and $U$ is the potential corresponding to the electric field. In the Coulomb gauge $\mathrm{div}\mathbf{A}=0$ one can always achieve $U\equiv 0$ if no sources are present and the overall Hamiltonian will look like $$H = H_0 -\frac{e}{m}\mathbf{A}\left(t,\mathbf{x}\right)\mathbf{p}$$ where $H_0 = \frac{\mathbf{p}^2}{2m} + V(\mathbf{x})$ and the term in $\mathbf{A}^2$, the ponderomotive potential has been neglected. ### Time-Dependent Perturbation Theory The last expression for $H$ has the form $$H = H_0(\mathbf{p},\mathbf{x}) + H_t(\mathbf{p},\mathbf{x},t)\ .$$ Using the Dyson series, one can formally write down the time-dependent solution in the interaction picture as $$|\psi_I(t)> = T\exp\left[\frac{1}{\mathrm{i}\hbar}\int_{t_{0}}^{t}H_{t,I}\left(t^{\prime}\right)dt^{\prime}\right]|\psi_I(t_0)> .$$ Then, one takes only the first two terms into account to find that **transition amplitudes** are approximately given by $$A_{n\rightarrow m}\approx\frac{1}{\mathrm{i}\hbar}\int_{t_{0}}^{t}\left\langle m\right|H_{t,I}\left(t^{\prime}\right)\left|n\right\rangle dt^{\prime}\ \mathrm{with}$$ $$H_{t,I}\left(t\right) = e^{\mathrm{i}\left(t-t_{0}\right)H_{0}/\hbar}H_t\left(t\right)e^{-\mathrm{i}\left(t-t_{0}\right)H_{0}/\hbar}\ .$$ ### Relation to the Question In the given case we have $$H_t = -\frac{e}{m}\mathbf{A}\left(t,\mathbf{x}\right)\mathbf{p}$$ and assuming that $$\mathbf{A}\left(\mathbf{x},t\right) = \mathbf{A}\left(\mathbf{x}\right) T(t)\ ,$$ and further using $$\mathbf{p}=\frac{\mathrm{i}m}{\hbar}\left[H_{0},\mathbf{x}\right]\ ,$$ one will have to solve $$A_{n\rightarrow m}\propto <m|\mathbf{A}\left(\mathbf{x}\right)\cdot \mathbf{x}|n>\int_{t_0}^t e^{-\mathrm{i}\omega_{mn}\tau}T(\tau) d\tau\ +\ \mathrm{h.c.}$$ with $\hbar \omega_{mn} = E_m - E_n$, where the $E_i$ are energy levels of the unperturbed $H_0$. Using here the vector potential $\mathbf{A}$ is the correct way to incorporate the field, of course. For me, it seems **extremely complicated** to do so since to calculate the transition probabilities for an **arbitrary field distribution** one will have to expand $\mathbf{A}$ into vector spherical harmonics. So, the use of a scalar potential like $U$ would greatly simplify computations and my question can now be re-asked as > ### Why can't we just simply use $U(\mathbf{x})\cdot T(t)$ to calculate the > transition amplitudes $A_{n\rightarrow m}$ for "electric excitations"? **Thank you very much** for insights, comments and corrections.


title: Why electrons get excited?

text: Why and how are electrons get excited and what happen inside an atom when electrons get excited?


title: Time Evolution Operator in Interaction Picture (Harmonic Oscillator with Time Dependent Perturbation)

text: **1\. The problem statement, all variables and given/known data** _Consider a time-dependent harmonic oscillator with Hamiltonian_ $$\hat{H}(t)=\hat{H}_0+\hat{V}(t)$$ $$\hat{H}_0=\hbar \omega \left( \hat{a}^{\dagger}\hat{a}+\frac{1}{2} \right)$$ $$\hat{V}(t)=\lambda \left( e^{i\Omega t}\hat{a}^{\dagger}+e^{-i\Omega t}\hat{a} \right)$$ *(i) Compute $\hat{U}_S(t,0)$ using the interaction representation formula (Equation 1 in next section) to second order perturbation theory. (ii) Compute $\hat{U}_S(t,0)$ using (Equation 2 in next section) to second order perturbation theory. **2\. Relevant equations** EQUATION 1: $$U_I(t,0)=1-\frac{i}{\hbar}\int_0^t dt' V_I(t')+\left( \frac{-i}{\hbar} \right)^2 \int_0^t dt' \int_0^{t'} V_I(t')V_I(t'') + \dots$$ EQUATION 2: $$U(t,0)=1+\sum_{n=1}^{∞}\left( \frac{-i}{\hbar} \right)^n\int_0^t dt_1 \int_0^{t_1} dt_2 \dots \int_0^{t_{n-1}}dt_n H(t_1)H(t_2)\dots H(t_n)$$ **3\. The attempt at a solution** So I know that for the interaction picture the transformation of the operator $\hat{V}_I$ is.. $$\hat{V}_I=e^{\frac{i}{\hbar}\hat{H}_0 t} \hat{V} e^{\frac{-i}{\hbar}\hat{H}_0 t}$$ I also know that both operators and kets evolve in time. So I use the interaction picture equation of motion on the ladder operators so I can obtain an expression for them as a function of time. $$\frac{d\hat{a}}{dt}=\frac{1}{i\hbar}\left[ \hat{a},\hbar \omega \left(\hat{a}^{\dagger}\hat{a} + \frac{1}{2} \right) \right]$$ $$\frac{d\hat{a}^{\dagger}}{dt}=\frac{1}{i\hbar}\left[ \hat{a}^{\dagger},\hbar \omega \left( \hat{a}^{\dagger}\hat{a} + \frac{1}{2} \right) \right]$$ I then got.. $$\hat{a}(t)=\hat{a}(0)e^{-i\omega t}$$ $$\hat{a}^{\dagger}(t)=\hat{a}^{\dagger}(0)e^{i\omega t}$$ I plugged these into the expression for V to get, $$\hat{V}=\lambda \left[ \hat{a}^{\dagger}(0)e^{i(\Omega + \omega)t} + \hat{a}(0)e^{-i(\Omega + \omega)t} \right]$$ So now what needs to be done, is to transform this into the interaction picture and then plug it into Equation 1 from above and integrate. But this seems very messy and I am having doubts if this is the correct way to I also know that both operators and kets evolve in time. So I use the interaction picture equation of motion on the ladder operators so I can obtain an expression for them as a function of time. go about this problem. If anyone can shed some light onto this I would really appreciate it!


title: Do batteries lose charge?

text: Say I have a charge of +q and -q on the positive and negative terminals of a battery. If I connect wires to each terminal, but don't connect the wires (essentially creating an open circuit), the charges move around to neutralize the field in the wire. During this transient state, charges move around. Would the charge on each terminal of the battery decrease a little during the transient state, since a field exists for a short period of time? In other words, does the voltage and charge of the battery slightly decrease during the transient state?


title: Why is the independence of orthogonal vector-quantities always implicit in books/lectures?

text: The "theorem" that I can "just" separately deal with orthogonal quantities (like horizontal and vertical force or velocity, etc), I never found explicitly mentioned, but just implicitly in the solution of exercises. Why? I do not think that this is obvious. For example projectile motion exercises are always solved such that vertical and horizontal motion is dealt with separately. One might think > when projectile has larger horiz velocity, then it hits more air particles > per unit time, and since it moves vertically upwards, it hits them more > often from below then from above and so with larger horiz velocity the > vertical deceleration will be larger Why is it always implicit in books, lectures, exercises, that the orthogonal (eg: verical and horizontal) components (of velocity/force/etc) are independent, and never explicitly mentioned as a theorem and explained why we can do that? Or is it in some?


title: Which symmetry is associated with conservation of flux?

text: Which symmetry is associated with conservation of flux (e.g., in electromagnetism)? For example, when working with Gauss's law in electromagnetism, net flux through an arbitrary volume element remains unchanged when the net charge is unchanged.


title: Can a system entirely of photons be a Bose-Einsten condensate?

text: Background: In Bose-Einstein stats the quantum concentration $N_q$ (particles per volume) is proportional to the total mass M of the system: $$ N_q = (M k T/2 \pi \hbar^2)^{3/2} $$ where k Boltzmann constant, T temperature Questions: A) For a B-E system "entirely of Photons" - what is the total mass of the system? (answered, see below) B) Does an ensemble of photons have a temperature? (answered, see below) C) Is this a Bose-Einstein condensate? I've found a paper _here_ (like the paper put forward by Chris Gerig below) which finds a BEC, but it is within a chamber filled with dye, and the interaction of the photons with the dye molecules makes it a dual system, as to one purely of photons. I think in this case there is a coupling between the dye molecules and the photons that is responsible for the chemical potential in the partition equation   $$N_q = \frac{g_i}{e^{\left.\left(\epsilon _i- \mu \right)\right/\text{kT} - 1}}$$ where $g_i$ is the degeneracy of state i, $\mu$ is the chemical potential, $\epsilon_i$ is the energy of the ith state. I suspect an Ansatz along the lines of $\mu$ = 0, and $\epsilon_i$ = $\hbar \nu_i$, where $\nu_i$ is the frequency of the i photon. _another edit_ : After going for a walk, I've realized the Ansatz is almost identical to Planck's Radiation Law but the degeneracy = 1 and chemical potential = 0. So, in answer to my own questions: A) is a nonsensical question, as photons have no mass, noting from wiki on Quantum Concentration: "Quantum effects become appreciable when the particle concentration is greater than or equal to the quantum concentration", but this shouldn't apply to non-coupling bosons. B) yes the ensemble has a temperature, but I was too stupid to remember photons are subject to Planck's Law. C) _Is this a Bose-Einstein condensate?_ No, as photons have no coupling or chemical potential required for a BEC. So, for an exotic star composed entirely photons, all the photons should sit in their lowest energy levels and the star will do nothing more than disperse. **Is this right?**


title: Physical intuition for independence of components of velocity in derivation of Maxwell–Boltzmann distribution

text: Maxwell derived the shape of the probability distribution of velocity of gas particles by starting with just two assumptions. These are:   1. The probability distribution is rotation invariant.   2. The components (of velocity of a gas particle) in the direction of the coordinate axes are statistically independent. And the rest is lovely deduction, but I found that as a layman I don't have any physical intuition as to why the second assumption is plausible. Is there an intuitive explanation behind the second assumption? If not, is there a way to derive the second assumption from a set of more plausible-looking assumptions?


title: Super-dense coding protocol with a key

text: I have this assignment: > Show that super-dense coding protocol with the key in the state > $\frac{|00⟩⟨00|+|11⟩⟨11|}{2}$ is equivalent (in a sense of transmission rate > and security) with classical one-time pad. I understand that super-dense coding protocol is a _coding protocol_ , not an _encryption protocol_ , so what exactly is the phrase "...super-dense coding protocol with the **key**..." supposed to mean? How is a key, usually used in encryption protocols, related to this coding protocol? Or has the word key a different meaning here? Thank you for your answer.


title: Why do some of the iron fillings arrange in a particular pattern when sprinkled around a magnet, instead of getting attached to the magnet?

text: My sir started a new chapter "magnetism and matter". Sir said that iron fillings form a particular arrangement when sprinkled around a magnet. I saw the figure in my book where it was shown the arrangement of some iron fillings at a distance from the magnet. I didn't get much convinced. If a magnet, which can pull iron fillings lies in front of them, why do they remain at a distance from the it? Why can't they directly attach to the magnet instead of forming those arrangements?


title: Third-order topological quantum phase transition in p+ip superfluid

text: A two-dimensional spinless non-relativistic p+ip superfluid undergoes a quantum phase transition between the BCS (weakly-coupled) and BEC (strongly- coupled) regimes. This transition is driven by massless fermions and believed to be of the third order. Most easily it can be shown, but relating the problem to the Dirac equation in 2+1 dimensions (Read/Green 2000) and calculating its energy $$ E \sim \int d^2 p \sqrt{m^2+p^2} $$ which contains the non-analytic piece of the form $\sim |m|^3$. This is not a rigorous argument since the non-relativistic superfluid is not completely equivalent to the Dirac problem, but the third order phase transition is also found rigorously in the mean-field treatment of the spinless p+ip superfluid, e.g. http://arxiv.org/abs/1008.3406 Since third order phase transitions are rare and usually require fine-tuning, I am wondering now if the order of the phase transition is preserved if one goes beyond mean-field, i.e., can the order of the phase transition change if we include bosonic fluctuations of Cooper pairs? Is there a general argument which forbids or allows that? In summary, is there a simple general argument that protects the order of the phase transition in this model?


title: Definition of "Quantizing"

text: Could anyone explain to me what "quantize" means in the following context? > Quantize the 1-D harmonic oscillator for which $$H~=~{p^2\over 2m}+{1\over > 2} m\omega^2 x^2.$$ I understand that the energy levels of a Q.H.O. are quantized. But here it is used as a verb... Aren't the energy levels naturally quantized?


title: How do you determine the torque caused by the mass of a lever?

text: Suppose we have two objects sitting on two side of a lever, and the lever also has a mass, and those objects have masses. Then how we can balance $\sum τ$? This is what I have done: $$\begin{multline}M_\text{heavier}\times g\times D_\text{from center to heavier} - M_\text{lighter}\times g\times D_\text{from center to lighter} \\\ - \text{torque caused by mass of lever}\end{multline}$$ where $D= \text{distance from center}$ I don't know to to include torque cased by mass of lever


title: Causality in a gedanken experiment on the hydrogen atom

text: Consider a gedanken(=thought) experiment where I am tracking the motion of the electron in a hydrogen atom with a time resolution of (say) $\Delta t = 10^{-20}$ seconds. Further assume (for simplicity) that the proton is stationary. In quantum field theory we talk about forces carried by virtual particles. Therefore, the Coulomb interaction between the proton and electron in this hydrogen atom is a result of the exchange of virtual photons. However, we do not need to pull out our QED machinery because the electron on average is not energetic (or fast) enough. Also, the number of virtual photons exchange between a _single_ pair of proton-electron that the quantum-ness of EM field is washed out and we can treat it classically. In other words, in this limit we do not need to think in terms of virtual photons. Now, in this thought experiment the proton vanishes at $t = 0$. The absence of the proton will not be felt by the electron until $t = a_0/c \,(\approx 17\Delta t$); $a_0$ and $c$ are the Bohr radius and speed of light. Assuming causality I would expect the electron to fly off (because it is no longer bound) at this point in time ($17\Delta t$). But virtual photons can exceed $c$. Then won't the electron know that proton is disappeared before $17\Delta t$?


title: exponential potential quantization

text: What are the energies $E_{n}$ of the Schroedinger operator $$ -\frac{d^{2}}{dx^{2}}y(x)+ae^{bx}y(x)=E_{n}y(x) $$ for some real and positive 'a' and 'b' with the Boundary conditions $ y(0)=0=y(\infty) $? Working a bit I get the quantization condition $$ J_{E_{n}/4}(\sqrt{-a})=0,$$ ($ J_{a}(x) $ is a Bessel function) but unfortunately this equation yields only imaginary roots.


title: Walking and Friction

text: If walking is a result of the reaction to a kick backwards to the ground (reaction being friction), it appears that it should be true that the kick will have to be less than the kinetic friction (which is less than the static friction threshold = $\mu$mg) or the acceleration will never be more than $\mu$g. What happens if the kick exceeds this force, there still has to be a reaction? Is that reaction distinguished from the kinetic friction?


title: If it turns out that neutrinos do travel at faster than lightspeed, how will the success of special relativity be explained?

text: As per in the title. If it turns out that neutrinos do travel faster than the speed of light, how will the success of special relativity be explained? My apologies if this has been asked before; I've just browsed through few questions, and I couldn't find one that matches (although it was slightly touched on in one answer). Thanks.


title: Energy in an EM wave should depend on frequency

text: I just finished reading Feynman's _Lectures on Physics_ vol.I, §34-9: "The momentum of light". The author explains that there is a relation between the wave 4-vector $k^{\mu}$ and the energy-momentum 4-vector $p^{\mu}$ of an EM wave, namely $$p^{\mu}=\hbar k^{\mu}, $$ or equivalently $$\tag{deB}W=\hbar \omega, \mathbf{p}=\hbar \mathbf{k},$$ and those equations are called _de Broglie relations_. However, as I learned in my classical electromagnetism course, flux of energy in such a wave is quantified by Poynting's vector, yielding formulas such as the following: $$\tag{1} I=\frac{1}{2 \mu_0 c} E_0^2, $$ where $I$ stands for "average intensity" of the wave and $E_0$ for "maximum amplitude of electric field". > **Question** Where is $\omega$? It does not appear in formula (1) nor in any > other formula based on Poynting's vector. But as of equations (deB) it > should do so. Am I wrong? Thank you.


title: String matrix models with c>1

text: Question 1: What is the status of string random matrix models (~ triangulated random surfaces) with c>1? In my limited search, I have just come across a few papers by Frank Ferrari (in 2000-2002) on this. But, they applied, if I understand correctly, to a few cases with some specific kind of singularities in their moduli space or some higgs field. See his lecture at http://arxiv.org/abs/hep-th/0205171 and references therein. * * * On a slightly different note...   Question 2: How is the BFSS matrix model (and its derivatives, like the ones mentioned at Good introductory text for matrix string theory) related to the random matrix models of triangulated worldsheets? My understanding is that the former is a nonperturbative formulation of some critical string theories, while the latter is a perturbative formulation of (noncritical only?) string theories. But, what are the similarities and connections between these two applications of random matrix theories? PS: some references (especially some pedagogical ones) would also be of immense help!


title: Quantum operator catastrophe

text: Assume we look at an interaction between 2 fermions $V \sum_{k_i,k_j,k_m,k_n} c_{k_i}^\dagger c_{k_j}^\dagger c_{k_m} c_{k_n} \delta_k $ where $\delta_k$ conserves momentum. We can directly write down a few terms from the sum $ ... + \underbrace{c_{k_1}^\dagger c_{k_2}^\dagger c_{k_3} c_{k_4}}_{i=1,j=2,m=3,n=4} + \underbrace{c_{k_2}^\dagger c_{k_1}^\dagger c_{k_3} c_{k_4}}_{i=2,j=1,m=3,n=4} + ... \ . $ and then use the anticommutator relations $ [c_i,c_j]_+ = [c_i^\dagger,c_j^\dagger]_+ = 0 $ $ [c_i,c_j^\dagger]_+ = c_i c_j^\dagger + c_j^\dagger c_i = \delta_{ij} $ to exchange the first two operators in the second summand ($ c_{k_2}^\dagger c_{k_1}^\dagger = -c_{k_1}^\dagger c_{k_2}^\dagger $) such that $ ... + c_{k_1}^\dagger c_{k_2}^\dagger c_{k_3} c_{k_4} - c_{k_1}^\dagger c_{k_2}^\dagger c_{k_3} c_{k_4} + ... $ the summation vanishes. For terms that do not vanish as pairs such as $ c_{k_1}^\dagger c_{k_1}^\dagger c_{k_3} c_{k_4} $ we can see that due to the anticommutator, these terms vanish individually. Now where is the mistake?


title: How can the surface of a liquid exist if there are no attracting molecules above it?

text: Okay this is probably a stupid question but here goes: We know that molecules in a liquid are attracted from all the molecules around it so there is no net attraction. Well, then how do the molecules at the surface exist because there are only molecules below it and not above it? Won't they accelerate downwards? Also, does this have something to do with surface tension? If it does, then isn't surface tension a force that acts along the surface and not into or out of it?


title: Physics tension question

text: Two blocks are pulled across a frictionless surface by a $240\ N$ force, as shown in the diagram below: $60\ kg$ - - - cord - - - $20\ kg$ → $240\ N$. If the blocks are accelerating at $3\ m/s^2$ what is the tension of the cord in between the two blocks? So I don't really have a strategic approach to this problem. I Isolate $20\ kg$ and see that $f = (20\ kg)(3\ m/s^2)$. Therefor it takes 60 newtons for that block to accelerate at $3\ m/s^2$ and the other 180 newtons are used to pull the second block and therefor $180\ N$ is the tension.   1. Is that correct, my teacher never gives me the answer sheet :(   2. Is there a step by step approach I can apply to tension questions?  Thanks a lot!


title: Radial Schrödinger equation

text: I found a problem that says: > Show by direct substitution that $R_{10}$ is a solution of Schrödinger's > radial equation. AFAIK Schrödinger's radial equation is $\frac{-\hbar^2}{2m}[\frac{d^2}{dr^2}+\frac{2}{r}\frac{d}{dr}-\frac{l(l+1)}{r^2}]R+E_p(r)R=ER$ and $R_{10}$ in this case is: $2(\frac{Z}{a_0})^{3/2}e^{-\rho/2}$ where $\rho$ is $\rho = \frac{2Zr}{na_0}$ and $E_p(r)$ is: $E_p(r) = -\frac{Ze^2}{4\pi\epsilon_0r}$ The thing is I don't understand very well what $E$ is in this case and (I think) I need it to solve the problem. It's the total energy but in the book (Alonso y Finn) they never talk about the kinetic energy. I suppose it depends on $r$, $n$ and $l$ because once I do the substitution on the left side of the equation and I factorize it I get $R_{10}$ multiplied by a term that's full of those variables.


title: Formula for molar specific heat capacity in polytropic process

text: I found this formula for a polytropic process, defined by $PV^n = {\rm constant}$, in a book: $$C = \frac R{\gamma-1} + \frac R{1-n} $$ where $C$ is molar specific heat and $\gamma$ is adiabatic exponent. I do not know how it was derived, can someone guide me?


title: What is the status of existing measurements of the speed of gravity?

text: In replying to a recent question I stated: > Gravitational waves have not been yet experimentally observed so as to have > their velocity measured. Which after the fact prompted me to try and verify it. I found a "speed of gravity" claimed measurement, which has been in a controversial discussion. In the last paragraph of this wiki article I found: > In September 2002, Sergei Kopeikin and Edward Fomalont announced that they > had made an indirect measurement of the speed of gravity, using their data > from VLBI measurement of the retarded position of Jupiter on its orbit > during Jupiter's transit across the line-of-sight of the bright radio source > quasar QSO J0842+1835. Kopeikin and Fomalont concluded that the speed of > gravity is between 0.8 and 1.2 times the speed of light, which would be > fully consistent with the theoretical prediction of general relativity that > the speed of gravity is exactly the same as the speed of light.[18] There has been criticism, described in the paragraph. Has this matter been resolved ? is the measurement valid? The speed of gravity has also been calculated from observations of the orbital decay rate of binary pulsars and found to be consistent to the speed of light within 1% ( same link). This is the speed that gravitational fields transfer information, and though the number itself depends on the theoretical framework solving for the system parameters it is true that just the existence of gravitational damping in the binary system would imply that the speed cannot be infinite. Along these lines I was wondering whether studying the second star in a binary system where the first has gone nova would not give a cleaner and model independent measurement of the speed of transfer of information gravitationally. In this day and age where everything is digital one should be able to have data on this from now on.


title: Math formula for crystal glass resonance

text: For a music therapy experiment, I have to use several crystal glasses with water and different cylindrical shapes, and make them vibrate and produce different sounds. Is there a math formula that binds together the radius of the glass, the level of the water (or the volume occupied), and the frequency that produces the sound? Thanks in advance.


title: Creation of particle anti-particle pairs

text: I was reading some QFT notes and there is one point that I don't understand, they are justifying why we need QFT saying that the number of particles is not preserved once we consider special relativity. For example, if a particle is in a box of side $L$, then by Heisenberg's principle: $\Delta p \geq h/L$ and then $\Delta E \geq hc/L$. But when $\Delta E > 2mc^2$ creation of particle anti-particle out of vacuum may happen. I don't understand why the uncertainty of the knowledge of energy of the particle inside the box implies the creation of particles by the vacuum. I mean why particle and vacuum are related somehow?


title: Aharonov-Bohm Effect electricity generation

text: This question is based on highly intuitive picture of the Aharonov-Bohm effect (perhaps a naive one). From what I have read, the current explanation of the AB effect is that although the electron passing through the two slits does not experience any magnetic field, the phase difference that leads to interference results from the interaction with the potential. My question is "If the electron is not really at one single position in the area between the slits and the detection screen, why can it not be inside the solenoid, and thus, experience a magnetic field?". My question is based on my understanding that the electron acts like a wave between the slits and the detection screen. Also, if the answer to my question is "yes, it can be inside the solenoid.", can we generate electricity by cutting power off the solenoid and letting it act like a passive wire through which some of the electron "wave" passes (Induction) ?


title: Notation: What is $\delta_{mn}$?

text: In a textbook, I found this relation for eigenvalues of total angular momentum: $$(L^2)_{mn} = \langle l,m \rvert L^2 \lvert l,n\rangle = \hbar^2l(l+1)\delta_{mn}$$ What is the $\delta_{mn}$ refer to? The book does not explain. source


title: Hilbert space and Lie algebra in quantum mechanics

text: We are looking for a publication or website that explains the Standard Model in terms of Hilbert space and Lie algebra. We are reading Debnath's _Introduction to Hilbert Spaces and Applications_ and Iachello's _Lie Algebras and Applications_. Is there a book or website that combines the two approaches (Hilbert and Lie)? If they can't be combined, can you provide a link to articles that compare/contrast them?


title: How can you test to see if a dice is weighted?

text: I was browsing Etsy today and came across this. What tests are there to see if the dice are usable, ie, if one side isn't favored over another, and if all sides are balanced? Would this just be to roll the dice a large nubmer of times and collect the data? Or are there other tests that could be done without doing that?


title: Ground state energies with fermions of same spin?

text: > Consider two non-interacting Fermions (half-integer spin) confined in a > 'box'. Construct the anti-symmetric wavefunctions and compare the > corresponding ground-state energies of the two systems; one with particles > of identical spin and the other with particles of opposite spin. Now, the wavefunction with opposite spin is incredibly easy, One can just duplicate the spin wavefunction for helium, right? What confuses me is the "ground state" energy of a wavefunction with two fermions of identical spin. Isn't that impossible? Wouldn't this look like non-excited helium with the same spin electrons in the 1S orbital (which can't be done)?


title: What software programs are used to draw Mechanics, Electricity & Magnetism, Waves, Thermodynamics, Optics & Modern Physics diagrams?

text: For training students in **competitive exams & physics olympiads**, I want to illustrate deeper meanings in problem solving through diagrams. I have my teaching material hand drawn in OneNote. I want to create some professional quality **colorful diagrams** that we find in **Halliday, Resnick & Walker - Fundamentals of Physics**. To manually place Inductors or wedges or springs or lens or to create field diagrams on Adobe Illustrator is hard. You can imagine all the alignment troubles... Any software with scripting/programming is also welcome. So I decided to look out for help from the experienced ones in these regard. Help very much appreciated. Thanks.


title: The amount of energy required to break the Coulomb Barrier

text: This web page says that only **3-10 keV of energy** is required to break the Coulomb barrier for colliding hydrogen nuclei. Pretty far from what originally said. Is this true? Does this have something to do at all with cold fusion? P.S: I'm just a Physics enthusiast, very lay.


title: Atmospheric Circulation

text: What is the simplest simulatable model giving our rotating earth its 3 circulation cells (Hadley, Ferrel, Polar)? The model should also show 1 circulation cell if the earth's rotation were stopped (or in a case like Venus), and no circulation at all if radiation were somehow uniform to/from the earth (instead of true equator-heating due to the sun's point source). By the way, it would be nice if the same model could show 8 circulation cells in a case like Jupiter. I expect the earth's $23^{\circ}$ tilt can be ignored. The important parameters seem to be the planet's radius, the planet's rotation speed, the planet's mass, the atmosphere's mass, the atmosphere's molar density (treated as an ideal gas with energy absorption from the earth's surface, but not directly from the sun's rays), and the atmosphere's viscosity. I hope the earth's surface could simply be modelled with a constant temperature profile, increasing from the pole to the equator (or, I wonder if daily thermal cycling of this surface temperature profile might be needed to generate multiple circulation cells). Anyway, using parameters like these, I'm hoping to see a justifiable formula giving 3 (I can't yet accept the qualitative "Hadley cell air falls with inertia and friction" explanations on the internet).


title: What lepton neutrino would be in this reaction?

text: If I have a reaction (shown below), and I'm supposed to work out what the products will be, what would the lepton neutrino $\nu$ be? e.g electron neutrino: $$ \nu + p \to $$


title: What is the meaning of colors in deep space pictures?

text: Are the telescopes really capturing these colors in deep space or are they just "artist's impression". What could make such vast space "red" or "green"? ![deep space photo](http://i.stack.imgur.com/qWc1Y.png)


title: Spontaneous breaking of a discrete non-Abelian symmetry

text: Can someone give an example of an one dimensional local gapped quantum lattice model with a discrete non-Abelian global internal symmetry that is spontaneously broken in the ground state? In particular, where ground states transform as an irreducible representation with dimension > 1?


title: Converting Angular Momentum?

text: I am aware of three apparently different forms of angular momentum: spin, orbital and macroscopic, like a spinning crystal or baseball. Can angular momentum be transferred or converted from one of these forms to another? If so, how?


title: Are the rest masses of fundamental particles certainly constants?

text: In particular I am curious if the values of the rest masses of the electron, up/down quark, neutrino and the corresponding particles over the next two generations can be defined as constant or if there is an intrinsic uncertainty in the numbers. I suppose that since there is no (as of yet) mathematical theory that produces these masses we must instead rely on experimental results that will always be plagued by margins of error. But I wonder does it go deeper than this? In nature do such constant values exist or can they be "smeared" over a distribution like so many other observables prior to measurement? (Are we sampling a distribution albeit a very narrow one?) Does current theory say anything about this? (i/e they must be constant with no wiggle room vs. no comment) I am somewhat familiar with on-shell and off-shell particles- but I must confess I'm not sure if this figures into my question. I'd like to say that as an example I'm talking about the rest mass of the electron as could be determined by charge and charge/mass ratios. But perhaps this very number is itself influenced by off-shell electron masses? Perhaps this makes no sense. Needless to say I'd appreciate any clarification.


title: Are Lorentz transformations linear transformations?

text: My textbook says that Lorentz transformations are linear transformations and present them as matrices. Lorentz transformations relate different coordinate systems with each other. It seems that coordinate systems are linear spaces, but coordinates are just labels for spacetime points, with no other structures attached. So what is the linearity all about?


title: How do heated showers/faucets work?

text: I'm not sure if this question is for Physics SE, but I'd go ahead and ask. I was wondering how the heated showers work (the ones with the hot and cold knobs). I have always been curious and this may clear my head on the questions such as :   1. Why is the temperature so volatile? (A slight turn to Hot makes it REALLY hot, and the inverse as well)   2. How is the temperature regulated?


title: High-energy scattering matrix a become wilson line

text: I would like to ask why the scattering matrix $S^{ab}(x)=\left\langle gluon\: b|gluon\: a\right\rangle$ can be approximated for fast particle interacting with a traget field $A^{-}$ by wilson line $S^{ab}(x)=\left[P\exp\left\\{ ig\int dx^{+}T_{c}A_{c}^{-}(x^{+},x_{\perp})\right\\} \right]^{ab}$


title: Why the uncertainty principle can be used for estimation?

text: It is usually said/done in textbooks and classes that if $\Delta x$ is known then $\Delta p_x$ can be estimated using the uncertainty principle as $\Delta p_x \sim \hbar/\Delta x$. But the uncertainty principle does not say that, it says $\Delta p_x\ge\hbar/\Delta x$. That means we can only set a lower bound on $\Delta p_x$, i.e. $\Delta p_x$ could be anything between $\hbar/\Delta x$ and $\infty$ Why then the lower bound is chosen for estimation? Are there certain situations where the products in uncertainties is of the same order as $\hbar$?


title: Query in Carroll Section 5.2 Birkhoff's Theorem

text: This is regarding the proof of Birkhoff's theorem. A part of the proof requires one to show that the most general spherically symmetric metric can be written in the form $$ ds^2 = -e^{2\alpha(t,r)} dt^2 + e^{2\beta(t,r)}dr^2 + r^2 d\Omega_2^2 $$ In order to show this, Carroll briefly describes how one would go about setting the coordinate system in which the metric takes the form above. For this, he begins by stating that "spherical symmetry implies existence of 3 Killing vectors $K_i$ satisfying $[K_i, K_j] = \epsilon_{ijk} K_k$" (this is the Lie algebra of $SO(3)$ which is the symmetry group of $S^2$). He then invokes Frobenius's theorem to say that starting from any point $q \in M$, we can construct a two-dimensional submanifold $S_q$ (he then uses the word "sphere" instead of submanifold due to the obvious relation of $S_q$ to our ordinary understanding of spheres). From here, I present the following quote from the book (p. 199-200 below eq. (5.28)) > ...Begin by considering a single point $q$ lying on a sphere $S_q$ (note > that $q$ must not be a degenerate point at which all of the Killing vectors > vanish). Put coordinates $(\theta, \phi)$ on this particular sphere only, > not yet through the manifold. At each point $p$ on $S_q$, there will be a > two-dimensional orthogonal subspace $O_p$, consisting of points along > geodesics emanating from $p$ whose tangent vectors at $p$ are orthogonal to > $S_q$. Note that there will be a one-dimensional subgroup $R_p$ of rotations > that leave $p$ fixed; indeed, these rotations keep fixed any direction > perpendicular to $S_q$ at $p$, and hence the entire two-surface $O_p$ is > left invariant by $R_p$. Consider a point $r$ that is not on $S_q$, but on > some other sphere $S_r$ in the foliation, and that lies in the two-surface > $O_p$ orthogonal to $S_q$ at $p$. Since $p$ is arbitrary, this includes any > possible point $r$ in a neighborhood of $S_q$. Note that $O_p$ will be > orthogonal to $S_r$ as well as to $S_q$. To see this, consider the two- > dimensional plane $V_r$ of vectors in the tangent space $T_rM$ that are > orthogonal to the two-surface $O_p$. Since $O_p$ is left invariant by the > rotations $R_p$, these rotations must take $V_r$ into itself, because they > are an isometry, and hence preserve orthogonality. But $R_p$ also takes the > set of vectors tangent to $S_q$ into itself, since these rotations leave the > spheres invariant. In four dimensions, two planes that are both orthogonal > to a given plane at the same point must be the same plane; hence, the > vectors tangent to $S_r$ must be orthogonal to $O_p$. I follow every statement up to the last few lines. The points of confusion are > But $R_p$ also takes the set of vectors tangent to $S_q$ into itself, since > these rotations leave the spheres invariant. In four dimensions, two planes > that are both orthogonal to a given plane at the same point must be the same > plane; hence, the vectors tangent to $S_r$ must be orthogonal to $O_p$. While I can still follow the above lines, I do not see how he reaches the final conclusion and I feel like the crux of the argument is there. I also feel like the following line is important, but I do not quite see how > Since $p$ is arbitrary, this includes any possible point $r$ in a > neighborhood of $S_q$ **Does anyone have a clearer explanation for why $S_r$ is also orthogonal to $O_p$?**


title: Source of potential energy in a capacitor being pulled apart

text: When the plate spacing is large the capacitor system is storing more energy than when it is small. If one manually changes the spacing, where does the additional energy "come from" as the plate spacing increases?


title: Setting $\delta R =0$ on boundary of hypersurface

text: Does requiring $\delta R=0$ on the boundary of hyper-surface create any restrictions or problems in deriving the field equations from Einstein-Hilbert Action?


title: Can we see light as it 'interferes' with itself and produces the characteristic double-slit pattern?

text: This TED talk suggests that we can now watch as a beam of light propagates through a bottle filled with water. My question is: can we use this new technology to perhaps 'see' the photon as it makes its way to the detector screen in the double slit experiment? What implications can femto-photography have on the single/double-slit experiment and physics in general?


title: Is a singularity a real thing?

text: I've heard the work a few times now, the most recent in the star trek film. Is a singularity a real thing? If so what is it?


title: Dispersion in a rectangular prism

text: Why are the colours visible only at the fringes of the light refracted through a rectangular prism?


title: Energy comparison, evaporate water vs compress air, same mass

text: I would like to compare the energy requirements of the 2 following tasks:   1. Total energy to evaporate 1 litre of water at normal atmospheric pressure at normal temperature?   2. Total energy to compress air into a 1 litre container so that the compressed air had a mass of 1kg? What air pressure would atain this mass? If its too hard to calculate could you give an estimate?


title: Generating unlimited electricity through magnets

text: Well i had seen a toy, which i till date don't know was it battery powered or not? and also could it be used to actually create energy (I know that it is not possible but whats the problem?) **Structure** of a toy: Imagine a toy man having magnets at his feet, he is fixed to an axle and can just perform a motion in forward or backward direction, and also note that at the base of the man's feet (attached to that axle) is another magnet with same pole as that of the magnet of feet. **What was happening** : What was happening was that the man was continuously performing the motion and not stopping at all. The thing is that i saw it from a distance and so i could not know that wheter or not it used battery **Questions** Will it stop? Will it generate electricity forever if i put an turbine to at the axle 1\. On Earth 2\. In outer space 3\. Under gravity + No Friction Well i know the simple answers in of most of them in form of yes/no but Why? **Front View** : ![enter image description here](http://i.stack.imgur.com/eSC4E.png)


title: Formalism Of Quantum Field Theory vs Quantum Mechanics

text: How far can we extend the formalisms on quantum mechanics (QM) to quantum field theory (QFT)? In particular,   1. How is a Fock space $\mathcal{F}$ different from a Hilbert space $\mathcal{H}$? Can a general state in Fock space $\mathcal{F}$ be written as the superposition of number operator eigenstates? If yes, are the number operator eigenstates the only basis states for the Fock space states?   2. Are all the postulates of quantum mechanics hold in QFT as well? What is the interpretation of norm of one or many particle states in QFT? Is there a concept of position basis or momentum basis in Fock states?


title: Creating matrix Hamiltonian for Feynman's CCNOT

text: I'm trying to read Quantum Mechanical Computer and to implement the CCNOT logical gate with Mathematica. Since i wish to use the SWITCH implementation of the CNOT [Fig.8] i've realized that i need to use 7 cubits (there's no difference with registry cubits and command line cubits). Also, i know that every operation is described by unitary matrix (we can only rotate ours cubit) i was thinking to use Pauli matrices, composed by tensorial product. As Feynman said, I'll obtain as results a $M=2^n * 2^n$. The vector will rapresent every possibile configuration of my system (it's composed by $2^n$) Also, using the observation that the upsite in the command line are invariant for my system (the expected falue of the number of the spinup commute with the hamiltonian of the spin) i can readuce the dimension of my matrix to $s*2^n$, and so the vector will be a unitary vector with only one $1$ in the corrispondence of the atom with the spin up. I have to create an algebra of operation that preserve the invariant of the clock, so not every possible operation is allowed on my chain (ex: NOT is not allowed because i will have 2 or 0 spin up that is an unacceptalbe substate). An example of a matrix that change my vector state from state $|1>$ to $|2>$ is the matrix $|2><1|$


title: Identity in CFT

text: I heard and read couple of times reference to a certain identity in conformal field theory (maybe specific to two dimensions). The identity relates the trace of stress-energy tensor to the beta functions of all the operators in the simplest way: $$T^{\mu}_{\mu} = \sum_{i}\beta({\cal{O_i}}){\cal{O_i}}$$ It's an operator identity and the sum runs over all operators (primary and descendants). It makes senses since vanishing beta functions and "tracelessness" are two marks of conformal symmetry. If it exists, I would find it really elegant (who cares?) and I would appreciate to read the demonstration, it strangely isn't part of the textbook material on the subject.


title: Is there a Lagrangian whose Euler-Lagrange equation is the gradient?

text: I am trying to recast a problem I am working on in terms of Lagrangian mechanics. I am in the following situation. Suppose I have a function $f:X \rightarrow \mathbb{R}$ (a field). In the its simplest case, I am studying constant functions. Since $X$ is an open ball, this is the same as studying: $$ \nabla f = 0. $$ Embarrassing enough, I don't know how recast this simple differential equation in terms of Lagrangian mechanics. Namely, I don't know which function $L(x,f, \nabla f)$ has the above as its Euler-Lagrangien equations. More precisely, I think I cannot recast the problem as a Lagrangian over several variables but only one field. Is there a standard way of doing this? Perhaps more importantly: Else, is there a standard $L(x,f,\nabla f)$ whose solution is the constant functions?


title: What is the potential field of an ion near the Bohr radius?

text: I figure that at large enough distances, the potential field of an ion is just the Coulomb potential for its net charge. But what happens at scales comparable to the ion's Bohr radius? Could there be, for example, some sort of screening effect from the electron shell that changes the potential? (depending on what the test charge is, like if you dropped a single electron near an ion) I'm a bit rusty on quantum mechanics, but I do remember that the math for atoms that aren't hydrogen gets complicated. Is there a known good way to approximate this potential? Or is my best bet to go download some quantum chemistry software?


title: Amount of heat required at for unit rise in temperature at different temperatures of water

text: In my text book, it is given : > One calorie is defined as the amount of heat required to raise the > temperature of water from 14.5 °C to 15.5 °C. I found out in wikipedia that this is actually the definition of 15 °C calorie. I want to know that as temperature increases, will the amount of energy for unit increase in temperature decrease? I find 15 °C calorie > 20 °C calorie. Is this relation uniform till 100 °C or is there any anomaly? And does the increased kinetic energy of the molecules of water, at an elevated temperature cause the amount of heat - required for unit rise in temperature, to be greater than the amount of heat required for the same unit rise, but at a lower temperature? For example, the heat energy required to produce unit rise in temperature of water at 14.5 °C is 4.1855 joules and the amount of heat energy required to produce unit rise in temperature of water at 19.5 °C is 4.182 joules. So does the increased K.E of water molecules at 19.5 °C cause the required heat energy to be lower than the heat energy required at 14.4 °C?


title: Other frequencies in a cavity

text: This is a fairly basic question but is something that I've never properly understood. If you have a cavity with perfectly reflecting walls, I understand that there are obviously frequencies which generate standing waves but I'm not sure I understand why they are the only frequencies which need to considered. What happens to waves of other frequencies as they propagate in the cavity? The energy of the waves cannot just disappear, so by what mechanism are they suppressed?


title: role of pure mathematics in fusion?

text: Is there any way a pure math guy can get his rigorous math fix while simultaneously moving fusion research forward? Is fusion simply too complicated to take a pure math approach? What open math questions are there pertaining to fusion?


title: Visualizing Ricci Tensor

text: By definition Ricci Tensor is a Tensor formed by contracting two indices of Riemann Tensor. Riemann Tensor can be visualized in terms of a curve, a vector is moving and orientation of the initial and final vector. However, the operation of contraction of indices which give rise to Ricci Tensor doesn't give any visual sense. I would like you guys to help me visualize Ricci Tensor.


title: Does inducing a current in a wire have any effect on the strength of a permanent magnet?

text: Does inducing a current in a wire result in any changes in the strength of a permanent magnet? Specifically, what would the results of the following controlled experiment be? You set up two alternators that each simply consist of a permanent magnet located inside a stationary loop of wire as depicted here. With the first (control) alternator, you simply rotate the magnet at a certain velocity for a short time, once per day, and measure the amount of induced current on a daily basis. With the second alternator, you hook the magnet up to your favorite (external) source of rotational motion and spin it continuously and measure the amount of induced current at the same intervals as with the control alternator. Would the amount of current change over time (in either one)? If it would, in which direction and why? And if this is still not well-specified enough for an definite answer, what would it depend on? (how fast the magnets are rotating? what kind of circuit the wire is hooked up to? what the magnet is made out of? etc.)


title: Is the $mL_c$ value for triangular and rectangular fins the same value?

text: I am looking at the solutions that my professor put up and I feel that he did something wrong. Here is the question and I will give my stab at the solution so you can see why I think that it is wrong. A straight fin is fabricated of aluminum alloy with conductivity of 185 $\frac{W}{mK}$ with a base thickness of t = 3mm and L = 15mm. Its base temperature is $T_{b}=100 C$ and it is exposed to a fluid for which $T_{\infty}=20C$ and $h=50 \frac{W}{m^{2}k}$. Compare the efficiencies of rectangular, triangular and parabolic profiles assuming a fin of unit width. Solution for **rectangular** : $$m=\sqrt{\frac{hP}{kA_{c}}}=\sqrt{\frac{2h(w+t)}{kwt}}=13.4432m^{-1}$$ $$mL=(13.4432)(.015)=.201649$$ $$mL_{c}=13.4432(.015+\frac{.003}{2})=.221813$$ This is where the problem lies. My professor then uses the $mL_{c}$ value in the efficiency for the rectangular fin which is fine. But he then goes on to use the above mL value for the triangular and parabolic fins. I would think that the mL values for these shapes would be different than for a rectangular fin. I assumed that for fins of non-uniform surface area like the triangular and parabolic fins then you would use $$mL_{c}=\sqrt{\frac{2h}{kA_{p}}}L_{c}^{\frac{3}{2}}$$ $$m=\sqrt{\frac{2hL_{c}}{kA_{p}}}$$ to find the m value for non-uniform surface areas. Then you would multiply that by L to get the mL values used in the efficiency equations for the triangular and parabolic shapes. So my question is, **Should the mL values for the rectangular and triangular fins be the same value or should they be different? And when should I use the equation that uses the area profile if it cannot be used here?** Clarification of variables: P: Perimeter w: fin width (unit width) $A_{c}$: cross-sectional area t: fin thickness $A_{p}$: profile area L: Fin length $L_{c}$: characteristic length


title: Riemann Tensor Calculation trick(number of element)

text: When we calculate Riemann Tensor for different curvature we have lots of components. However, there are many components that are zero. How can we argue, based on the symmetry of connection , that those elements are zero? For example if I am calculating the Riemann Tensor of $S^2$ sphere, I get only one non zero component i.e. $R_{\phi,r,\phi}^{\theta}$ = $sin^2 {\theta}$ and other components are zero. So, How can I argue, without calculating that all other components are zero. Edit: (Dimension, No. of independent Riemann Components) = (2,1; 3,6 ; 4,20)


title: Rough diffusion rate of sugar water through a larger volume of water

text: I hope this a question that can be answered and isn't too vague. I'm also only after a very rough answer. I'm adding a small amount (~500mL) of sugar solution into a carboy with a much larger volume of water (~20L).   I don't particularly want to stir it in very much because there is a lot of sediment that has already settled out. I'm wondering roughly how long it would take for the sugar water to diffuse roughly evenly through the larger volume?


title: Reducing the evaporation of water at room temperature

text: Water drops kept on different surface evaporate after a few minutes. How can I reduce the rate of evaporation of these drops? Can I add some additives which suppress evaporation?


title: CARS microscopy

text: I have probably very naive question. In case of CARS microscopy http://en.wikipedia.org/wiki/Coherent_anti- Stokes_Raman_spectroscopy we have $ω_{pr}+ω_{p}-ω_{S}$, how can we have minus $ω_S$? In fluorescent microscopy we always add energy after hitting/absorption. I can think that, from mechanical point of view, in case of three photons ($pr$, $p$, and $S$), $S$ should hit the electron (atom) in opposite direction in order to decrease kinetic energy; is it right? It is interesting to see why we have minus?


title: How is the time independent potential term a solution of Schrodinger equation

text: Consider a time-independent potential: $V(x)$. Then, it is usually stated that $$ \Psi(x,t)=\rho(x)\exp{\left(-\frac{i}{\hbar}Et\right)} $$ is the general form of a solution of the Schrodinger equation $$ i\hbar \partial_t\Psi=H\Psi $$. Why is this the case?


title: Normalizing Propagators (Path Integrals)

text: In the context of quantum mechanics via path integrals the normalization of the propagator as $$\left | \int d x K(x,t;x_0,t_0) \right |^2 ~=~ 1$$ is incorrect. But why? It gives the correct pre-exponential factor for the free particle and the incorrect one for the harmonic oscillator. It seems to me that the propagator should describe the probability for the particle beginning at $(x_0,t_0)$ to arrive at $(x,t)$. So, the probability for the particle starting at some particular space-point to go somewhere (any x) must be 100%. The result that $$\left | \int d x K(x,t;x_0,t_0) \right |^2 ~=~ 1$$ isn't satisfied seems to attack this ideas.. What is wrong with my though? Other thing... There is some normalization procedure to obtain the pre- exponential term using the phase? (avoiding functional determinants) **Comment 1 : ** Now I found the question Normalization of the path integral which is similar. But seems to say that the normalization procedure is correct! I am not finding this to the harmonic oscillator. So, I don't have an answer yet.


title: What is gravity?

text: I am not asking why it occurs, but what it is. Everything understood is something. Something would entail the existence of a property, whether or not tangible. We know photons exist, but we know they are subatomic particles. Speaking on states and matter, is gravity matter? Does it have states? Is it solid, liquid, gas, plasma? Is it atomic, subatomic, etc.? Natural phenomenom doesn't tell me absolutely anything about _what gravity is_. So I ask again in pure simplicity ... what is gravity? Is gravity magnetism? As such it would be a force. But I've been told that's incorrect. Also, Eintein's "theory" doesn't answer any of these questions: 1.What causes gravity? 2.What _is_ gravity? 3.Is gravity a force? 4.Can gravity be controlled?


title: The magnitude of spacecraft anomalous velocities?

text: That Voyager and several other spacecraft have had anomalous changes in speed is known. According to the news reports, the differences are unsettling. But just how large are these anomalies really? Are they 'way down there' at the levels of computational precession? Or, if I understood how to calculate them, could I do so with my home computer in a reasonable period of time? (Say a few days calculation.)


title: Why is the Legendre transformation an application of the duality relationship between points and lines?

text: When I read the Wiki about Legendre transformation, there is a statement > The Legendre transformation is an application of the duality relationship > between points and lines. What's the meaning of this statement?


title: Nuclear decay information

text: I'm looking at some gamma spectroscopy calibration data, produced by a known source (semi known... the source was last calibrated 2006...) My question is, looking at for example http://www.nucleide.org/DDEP_WG/Nuclides/Co-57_tables.pdf or http://ie.lbl.gov/toi/nuclide.asp?iZA=270057 In the table for the gamma from Co-57, the KeV and #photons values have a number in a bracket beside their main number. **What is the meaning of those numbers?** My guess is it's to make sure we only have whole numbers of photons (As required).


title: Nobel Prize 2013: What is it about?

text: I would really like to understand Higgs-Englert’s discovery that earned them the 2013 physics Nobel prize. I tried reading their work, but understood nothing of it unfortunately. The reason why I’m asking here is because, unfortunately, it is very hard to find physics principles explained in such a way that a non-physicist can understand them. Either the explanations are far too simplified and/or incomplete, or they require a lot of prior knowledge and/or ask you to accept certain concepts that most of us would find unusual or unnatural. I’m a mechanical engineer. As such, I have a basic understanding of physics, but no way near enough to understand Higgs-Englert’s work. I have many friends and colleges in the same situation as I am who would also like to understand what has been discovered, how, and why it’s important. Therefore, could someone provide a **simple** , but **complete and thorough explanation** of this work? Preferably including **explanations of the basic concepts/principles it makes use of**.


title: Lethality of sounds and extreme "loudness"

text: In theory, could pure sound be lethal? How loud would it have to be? Also, which events are the loudest in the universe, and how loud are they? I'm confining attention to events which occur regularly, e.g. starquakes.


title: Age of the universe in years

text: It seems to be commonly accepted that the Big Bang occurred roughly 13.7 billion years ago. My question is what is the meaning of the year in this context? ## When I type year definition into google, it returns this: ![Definition of a year according to google](http://i.stack.imgur.com/yxvxh.png) The age of the earth likewise is accepted 4.54 billion years. I'm assuming that by years old, we mean Earth years. **Now, if the earth only existed for 4.54 billion years**   1. What is the meaning of the year before the Earth existed?   2. Why is the age of the universe even expressed in years as opposed to something like distance? It appears that 13.7 billion (Earth) years is completely meaningless since a year depends on the existence of the Earth.


title: Linear Operators and their representations

text: I am currently learning Quantum mechanics on a slightly advanced level. I am curious in knowing if there are Linear Operators (Linear Maps) in the Hilbert Space (finite dimensional ones) that don't have isomorphic with Matrices? In those case are there any other representations that we can choose?


title: (Co)homology of the universe

text: In this post let $U$ be the universe considered as a manifold. From what I gather we don't really have any firm evidence whether the universe is closed or open. The evidence seems to point towards it being open but closedness is within the error range of present data. However, this seems to be an awfully crude distinction, basically of whether $H_1(U)$ is trivial or not. For argument's sake, let's consider the latter case. Do present physical theories place any restrictions on $H_1(U)$? For example, is $H_1(U)=\mathbb{Z}_2$ allowed, or $H_1(U)=\mathbb{Z}\times \mathbb{Z}$? What about $H_1(U)=A_4$, the alternating group of order 12? I am especially interested in higher (co)homology/homotopy groups. For example, the topological censorship theorem states that the universe is simply connected, so $H_1(U)=0$. However, there are spacetimes where space is a 3-sphere. More generally, we can postulate what would happen if $U$ was a homology sphere of a certain dimension.


title: Is there a mathematical derivation of potential energy that is *not* rooted in the conservation of energy?

text: For simplicity I'll consider only gravity, but in general this question only applies to conservative forces. As per my understanding, the way one gets to the equation for gravitational potential energy is this: $(1)$ $\ddot{r}=-\frac{MG}{r^2}$ is given experimentally $(2)$ $KE=\frac{1}{2}m\dot{r}^2$ is derived theoretically and (thought) experimentally $(3)$ Energy is conserved $(4)$ For a set of generic initial conditions,   * Using $(1)$, find $r(t)$ and then derive $\dot{r}(r)$   * Using $(2)$ find $KE(r)$   * Using $(3)$, $PE(r)=E_0-KE(r)$ **Is this a correct method?** If the answer is 'no' my question ends here, if not: surely $(3)$ is _by definition_ correct (circularly so), and so the equation for potential energy is arbitrary? I would be interested in seeing a method that does not do this (or is correct at all, if the answer _was_ 'no').


title: Wire morphing shape?

text: Can one morph a wire by electrical current into a shape different than static shape at beginning? As in the beginning shape being in a spiral, then when electrical energy is passed through the wire it becomes straight. When the current is shut off it would return to a spiral shape as in the beginning form. OR vise versa?