title: Understanding the "$\pi$" of a rotating disk

text: Let us say you are in an inertial reference frame with a circular planar disk. If you take your meter measuring rods (or perhaps tape measure) you can find the diameter and circumference of the disk. If you divide the circumference by the diameter, you will get exactly $\pi$. Now you start rotating the disk. A book I have claims that now the ratio will be different (lets call it $\pi_\circ$ to avoid ambiguity.) This has caused to reevaluate my understanding of special relativity (this is a set up to general relativity, but the problem itself only requires the special theory.) My problem is that, sure, the measuring rods (or tape) would contract on the circumference, but wouldn't the circumference contract a similar amount, therefore giving a measurement of $\pi$ in every reference frame according to your own measuring rods. Of course, this reasoning goes on to make me question how I understood all of special relativity up this point.


title: Why doesn't a stationary electron lose energy by radiating electric field (as per coulomb's law)?

text: If an electron in a universe constantly generates an electric field why does it not get annihilated ? I am confused because I read that an accelerating charge radiates and loses energy. So, why won't a stationary electron radiate and lose energy?


title: Lie algebra of lorentz group

text: I'm stuck in following calcualtion from sredniki's QFT book.(Its actually in the solution manual) How can i get from $$\delta\omega_{\rho\sigma}(g^{\sigma\mu}M^{\rho\nu} - g^{\rho\nu}M^{\mu\sigma}) $$ to $$ \frac{1}{2}\delta\omega_{\rho\sigma}(g^{\sigma\mu}M^{\rho\nu} - g^{\rho\nu}M^{\mu\sigma} -g^{\rho\mu}M^{\sigma\nu} + g^{\sigma\nu}M^{\mu\rho}) $$


title: The electric field of a conductive sphere containing a charge - grounded vs not grounded

text: Let's suppose we have a sphere but unlike theoretical ones it'll have some thickness say $\Delta r$ and inner radius $R$. What I was wondering about is how will it behave if we place some charge $q$ in the center? What would the field look like and what will be the charge density on either side. When it's grounded and when it's not. My attempt: When it's not grounded I though that outside if we create a spherical surface Gaussian with $r>R+\Delta r$ then there's only the charge $q$ to consider so it'll behave like a exactly like $q$ on it's own meaning $E=k \frac{q}{r^2} \hat{r}$ , and the same goes for $r<R$. inside the shell it'll be zero as it's a conductive surface so $\phi = const$. On the inner surface and the outer surface the sum of all charge will have to be $-q , q $ accordingly as they have to cancel each other so the density will be $\sigma = \frac{-q}{4\pi R^{2}} , \frac{q}{4\pi\left(R+\Delta r\right)^{2}} $ . When it comes to the grounded version I'm a little confused as I don't really understand what's the difference apart from the fact that the initial $\phi =0 $ but electrical charge will still be pulled to the sphere from $\infty$ so it seems like there's no difference but I'm a beginner so I'm not sure whether my deduction are valid or not and it seems kind of fishy to me but I can't really point to what is essentially wrong.


title: I need to calculate the temperature of some flat copper wire as it loses contact with a chilled capstan roller

text: I need some help in designing a system for cooling wire as it comes off a resistance annealer. I need to calculate the temperature of that flat copper wire as it comes off a chilled roller (call it a capstan) on this annealer. The capstan, which is copper, is 6.00 inches in diameter, but the wire will only contact this chilled capstan for 180 degrees of rotation. With the capstan rotating at 1018 rpm, the flat wire will be in contact with this capstan for a distance of about 9.42 inches, and a length of time of approximately .029 seconds. This capstan is at a temperature of about 286K, being chilled with internal coolant. Its containment wall is .312 inches thick. The copper wire being annealed is .0007 inches thick and .150 inches wide. It will be at a temperature of about 672K when it initially comes in contact with this capstan. What I need to know is the temperature of the wire when it loses contact with the capstan 180 degrees away from initial contact. If I can get the wire adequately cooled when running over that chilled capstan, I do away with a coolant bath, and hence eliminate a lot of attendant problems with the coolant. Thanks for helping me out with this problem.


title: In a 2-body problem, when is the moving path closed?

text: In a 2-body problem is it true that, in all situations, the moving path is closed? In which cases are the paths closed? Fixing the coordinate system or fixing one of the bodies gives us different paths; when are these paths closed?


title: Has the Reissner-Nordstrom metric ever been experimentally verified?

text: In contrast to the solution of the conventional Reissner-Nordstrom problem, where the Schwarzschild metric takes on an additional $1/r^2$ term due to the added electric charge, P. Mannheim has in _Solution to the Reissner-Nordstrom Problem in Fourth Order Conformal Weyl Geometry_ found instead the same $1/r$ proportionality for both the mass and charge induced terms in the metric. This raises the question.


title: Electron motion in a wire

text: For my introductory course to electromagnetism (I'm an undergraduate student, so ELI5), I'm trying to get the right conceptual model of electron movement in a thin wire (with constant but non-zero cross section, like a cylinder) due to a constant current. It seems to me, that there are several contributing factors, and I wonder which dominate, and if some kind of order exists.   1. Electrostatic repulsion between electrons. If this was the only force at play, electrons should all travel on the surface of the wire.   2. Magnetostatic attraction caused by the magnetic field from charges in motion. If this was the only force then the electrons would all travel in the center of the cylinder.   3. Collisions with matter and thermal scattering, which would make the motion chaotic.   4. Material properties of the cylinder, and if in particular the cylinder is a metal, then I suspect that the lattice planes of the metal and orientation of these planes is somehow relevant.  Basically 1 and 2 create order, 3 and 4 seem to destroy it. I can't seem to find anything relevant in my courseware or online. Specifically I'm interested in common DC household wires.


title: Testing covariance of an expression?

text: This is something I've been unsure of for a while but still don't quite get. How does one tell whether an expression (e.g. the Dirac equation) is covariant or not? I get it for a single tensor, but how is it defined when there is no overall up/down index to base it on? Any advice?


title: What is the highest possible frequency for an EM wave?

text: What is the highest possible frequency, shortest wavelength, for an electromagnetic wave in free space, and what limits it? Is the answer different for EM waves in other materials or circumstances? How could waves of this frequency be generated and transmitted, again if that is possible?


I am an absolute neophyte regarding physics. What are the challenges to achieving cold fusion? I'm not sure this is a duplicate of Why is cold fusion considered bogus?, because that question is talking largely about the validity of some claims of cold fusion being realized. This question is specifically asking about the challenges of cold fusion and how they may be addressed in the emerging study of "low energy nuclear reactions"


title: Astrophysical Data

text: I am working on a project that requires some amount of number crunching. I require some basic data about the galaxy M82 and our own Milky Way (particularly luminosities). My advisor is a nut and he blurts out numbers from his head that seem to be right. All the more, I would like to be sure. So is there any place where I can find reliable data? (Google isn't helping much here and I also tried http://nedwww.ipac.caltech.edu/). Thanks in advance! EDIT: I am working on gas outflows from galaxies. I am focusing on galaxies which have luminosities beyond their Eddington limit. Thus to compare some calculated results, it would be useful to have the bolometric luminosities of M82 and Milky Way. Hope this clarifies.


title: Generator using Earths's Magnetic field?

text: In the past I've been shown Internet sources claiming a specific generator that uses nothing but the earth's magnetic field to generate power, I'm looking for the name of the generator? I realize other discussions on this site have claimed (using math etc) that the energy is so minimal it isn't worth the effort, but that's not my question. I've seen the evidence for the device on the Internet in the past, and I wish to examine the evidence of the device in detail again, yes it's not possible I'm sure by showing me all the equations, but if there are practical implementations of this technique I'd like to take a look.


title: Capillary action and conservation of energy

text: When I dip a paper towel in a cup of water the water gets drawn up due to capillary action. How is this reconciled with conservation of energy, as it would seem on the surface that the potential energy of the system has gone up with no work being done?


title: where does the energy for capillary action come from?

text: I think capillary action occurs because water molecules have a little magnetic field with poles so they can stack up on each other somewhat. When it happens though, where is the energy derived from to create this kinetic movement?


title: Is the neutral pi meson a relativistic electron positron pair?

text: An electron position pair orbiting at relativistic speeds is proposed as a model of a neutral pi meson by Ernest Sternglass. However, his mathematics is semi-classical. Has anybody done a QED approximation?


title: Killing vector field in terms of the tetrad basis

text: I have come across the following equations in Wald. For a static spherically symmetry metric $$ds^2 = -f(r)dt^2 + h(r)dr^2 + r^2 ( d{\theta^2} \sin^2{\theta}d{\phi^2})$$. If $(e_{\mu})_a$ are the basis vectors of the orthogonal tetrad system, and $\frac{\partial}{\partial t}, \frac{\partial}{\partial r}$ are the killing vectors corresponding to transformations of $t$ and $r$ then $$1. \,\, (e_0)_a=f^{\frac{1}{2}}(dt)_a=f^{-\frac{1}{2}} (\frac{\partial}{\partial t})_a$$ $$2. \,\, (e_1)_a=h^{\frac{1}{2}}(dr)_a=h^{-\frac{1}{2}} (\frac{\partial}{\partial r})_a $$ The first equality of each of the equation is clear to me, but how do you related th killing vector to the tetrad basis vector? (i.e. how do I get the second equality of each of the equations)


title: Difficulties with bra-ket notation

text: I have started to study quantum mechanics. I know linear algebra,functional analysis, calculus, and so on, but at this moment I have a problem in Dirac bra-ket formalism. Namely, I have problem with "translation" from ordinary linear algebra "language" to this formalism. For better understanding of the problem, I'll give some definitions which I use: 1) Assume, that $\mid u\rangle$ is a vector in Hilbert vector space $V$. Bra $\langle v \mid$ is a vector of dual vector space $V^{*}$ $\left( \langle v \mid: V \longrightarrow C \right)$, defined by $\langle v \mid u \rangle=g\left(\mid v\rangle, \mid u\rangle \right) $, where $g: V \times V \longrightarrow C$ is metric on $V$. 2)$A$ is linear operator on $V$. Consider bilinear form $(\quad)$: $\left(f,x\right )=f(x)$. In this notation we can define _adjoint_ operator $A^{*}$ on $V^{*}$:$(f,Ax)=(A^*f,x)$. I tried to understand two following equations:   * There is expression $\langle v \mid A \mid u \rangle$. In my text book was written the following phrase: "Operator $A$ acts on ket from the left and on bra on the right". But according to the difenitions that i use, adjoint operator $A^{*}$ acts on $V^{*}$.But in this case operator $A$ acts on $V^{*}$. I dont absolutely get it.  One possible solution which i see is that this is just a notation of the next thing:$\langle v \mid A \mid u \rangle=(v,Au)=(A^{*}v,u)= \langle A^{*}v \mid u \rangle; \quad A^*\langle v \mid :=\langle v \mid A$ The second way is to use isomorphism between $V$ and $V^{*}$, and then operator $A$ is able to act on $V$ and $V^{*}$ (Dual Correspondence). The third way is that we use a matrix representation everywhere;and in expression $\langle v \mid A \mid u \rangle$ we multiply a row $v$ on matrix of operator $A$ on column $u$. Then this expression absolutely clear because the multiplication of matrix's is associative.   * The same difficulties i have with expression $(A \mid v \rangle)^{*}=\langle v \mid A^ \dagger$. Could you explain it to? I would be happy, if you will say which way is right and if all of my suggestions are wrong, please, tell me the right one.


title: Neutrino Oscillations and Conservation of Momentum

text: I would like to better understand how neutrino oscillations are consistent with conservation of momentum because I'm encountering some conceptual difficulties when thinking about it. I do have a background in standard QM but only rudimentary knowledge of particle physics. If the velocity expectation value of a neutrino in transit is constant, then it would appear to me that conservation of momentum could be violated when the flavor eigenstate at the location of the neutrino source is different from that at the location of the interaction, since they are associated with different masses. For this reason I would think that the velocity expectation value changes in transit (for instance, in such a way to keep the momentum expectation value constant as the neutrino oscillates), but then it seems to me that the neutrino is in effect "accelerating" without a "force" acting on it (of course, since the momentum expectation value is presumed constant, there may not be a real problem here, but it still seems strange). Any comments?


title: Why no basis vector in Newtonian gravitational vector field?

text: In my textbook, the gravitational field is given by$$\mathbf{g}\left(\mathbf{r}\right)=-G\frac{M}{\left|\mathbf{r}\right|^{2}}e_{r}$$ which is a vector field. On the same page, it is also given as a three dimensional gradient$$\mathbf{g}=-\mathbf{\nabla\phi}=-\left(\frac{\partial\phi}{\partial x},\frac{\partial\phi}{\partial y},\frac{\partial\phi}{\partial z}\right)$$ As this second equation is also a vector field, why doesn't it contain a basis vector of some sort and why isn't $\mathbf{g}$ given as a function of something or other? Also, how do you actually get from the gravitational potential field $$\phi=\frac{-Gm}{r}$$ to the second equation? I can see that you apply the operator $\nabla$ but how does that give you $\mathbf{g}$? Thank you


title: Quantum physics before the observer and relic interferences

text: It is known that the concept of an observer plays important role in quantum mechanics. In interpretations with collapse only the observer can trigger the wavefunction collapse. From this naturally arises a question, whether the laws of nature were different before the observer appeared. It seems there can be at least two theories on this matter.   * First theory is that before the observer appeared the universe was in a mixed state and underwent unitary evolution following the universal wave function. With the first observation (when he experiences the first ever qualia) the universal wave function collapsed to a random (but probable) state consistent with the observer's existence. Since before the first observation the universe was undergoing unitary evolution, it acted as a well-isolated from the external world quantum computer. This means that the observer in principle may be able to spot relic interferences that distort the statistical behavior of the past, such as quantum jumps in biological evolutionary progress, similar to how quantum annealing allows jumps from one local minimum to another in an adiabatic quantum computer. Unfortunately such differences can be spotted only by statistical methods, but this idea possibly completely compromised by the much greater influence of the anthropic principle on the observed statistics of the past. It is not evident whether the anthropic principle itself can be explained in terms of relic interferences. Another major problem with detection of relic interferences is that with the increase of the complexity of a quantum system, the chances of interferences between different evolutionary paths and their eventual reconciliation quickly decreases. This fact has been proven by Thomas Breuer who showed that the increase of the complexity of a system (i.e. its entropy), its evolution under objective decoherence becomes virtually indistinguishable form its evolution under both objective and subjective decoherences (see below) with the contribution of the latter exponentially decreases (indeed it is quite more probable to see an interference of two states of an electron than two states of a Schroedinger's cat). This suggests that relic interferences may be only observable at the very first moments after Big Bang when the entropy of the Universe was still very low.   * The second theory is that the laws of nature never changed, and wavefunction collapse happened always. This means that the observer effectively existed always and also always existed subjective decoherence. So the differentiation between these two theories lies in the existence of subjective decoherence before the birth of the observer.  A problem with this theory that postulating existence of the observer forever (even if unconscious) allows us to trace him (by the parental lineage of his cells for example) only up to 4 billion years back from now and then we only can refuge to the first theory. That said I should note that Breuer published an article titled "Ignorance of the Own Past" that seemingly supports the second theory, but the result was obtained only with an assertion of determinism and the author notes that he does not know whether the result still holds in an indeterministic world. So my question is whether this result was since improved to include indeterministic case so we could decide which of the two theories to prefer?


title: How to check the unitarity of the theory by having field equation?

text: Let's have some field equation of some field corresponding to particles with mass $m$ and spin $s$. How to check the unitarity of the theory? May I do it without getting $S$-matrix? May the scalar product in a field space help?


title: Factor 2 and equations for the weak gravitational field

text: By using some axioms people derives equation for Lorentz force and, then, Maxwell's equations from the Coulomb's law and Lorentz transformations. When I used analogical methodology for Newton's law of universal gravitation, I derived some equations like gravitomagnetic equations deriving from the GR equations in the weak field limit. But I lost factor 2, which multiplies "magnetic" component. It's binded with representation of gravitation field as the tensor. Can you explain me, how to, using methodology described above and representative of gravitation field by tensor, get factor 2?


title: Why aren't stratospheric balloons compressing their helium to descent?

text: I can't find a reference of a working stratospheric balloon mounted with a compressor to control its altitude. Is there a big physical difficulty (such as compressor weight/power ratio, or power source availability) which prevents this, or just no one as an interest in it ?


title: What happens to the wavelength/frequency of a photon as it passes through an event horizon?

text: I've asked a similar question about photons and black holes but wanted to rephrase it more specifically, so here goes... Ever since I learned how a photon's wavelength and frequency are indivisibly linked with the speed of light, I've wondered what this tells us about the speed of light itself. Bring red / blue shifting into the picture and I began to think about the edge cases, like could a photon ever be shifted to infinity (an infinitely long wavelength or infinitely large frequency). One of the answers to my original question mentions that a photon released at exactly the event horizon in a radial direction directly away from the center of the black hole would be stuck there. Using the terminology of limits we would say that 'as the distance from the photon's release point and the event horizon goes to zero', we seem to imply that at the event horizon the photon will not move, this of course violates other laws of physics which is why I'm asking the question. This seems to be a scenario that should illuminate the darkness inside the black hole. I'm not being flippant, I truly believe that there is something about our approaches which we've been missing, for example the study of physical limits like absolute-zero. Is there a relationship between why an atom cannot be cooled to 0K just like a photon can never travel at less than c, through whatever medium it finds itself. Forgive me and please correct me if I describe this poorly, I don't have the proper vocabulary; my goal is to understand whether a connection exists not write something for a peer-reviewed journal. If atoms cooled to temperatures approaching 0 Kelvin start to form a Bose-Einstein condensate due to the overlapping of their quantum wave functions, would there be a similar effect as photons approach an event horizon (EH)? If photons cannot actually pass the EH, then is the EH more like a physical surface in that since photons cannot pass through the EH are the equations attempting to describe the interior actually nonsensical in the same way it would be to try to divide by zero or find the square root of -1? This leads to other questions like if photons (and I'm assuming atoms too) can not pass the EH then any matter falling into the black hole after the formation of the EH would never reach the singularity and thus the distribution of this new mass would form a sphere near the EH? Would it be correct so say that this new matter 'appears' to add mass to the singularity due to its distribution while never actually being in the singularity? I know this is more than one question but I'm not sure how to break it up into better chunks, suggestions are appreciated.


title: CGS Units for Magnetism

text: Why does the formula for magnetic field force include the speed of light in the denominator in cgs units? Where does the extra $c$ go in SI units?


title: Why aren't the graphs for black body radiation straight lines?

text: We know that a wave which has greater frequency will have low wavelength and high energy. So, by decreasing the wavelength, the frequency and consequently energy (intensity) of that wave will increase or vice versa. Now, i want to ask a question in black body radiation by looking at the following graphs ![enter image description here](http://i.stack.imgur.com/bod9S.jpg) My question is that if by decreasing the wavelength of a wave, the energy of that wave increases then why do these graphs fall down after reaching a maximum value. Shouldn't the graph just move straight on?


title: Why can't a single photon produce an electron-positron pair?

text: In reading through old course material, I found the assignment (my translation): > Show that a single photon cannot produce an electron-positron pair, but > needs additional matter or light quanta. My idea was to calculate the wavelength required to contain the required energy ($1.02$ MeV), which turned out to be $1.2\times 10^{-3}$ nm, but I don't know about any minimum wavelength of electromagnetic waves. I can't motivate it with the conservation laws for momentum or energy either. How to solve this task?


title: Any case of a particle seemingly decaying into copies of itself?

text: Is there any case reported that seems to resemble the following: there is a particle and at some moment, the particle seems to break down into two or more particles that are all identical to the original particle?   * additional information: I am not talking of antimatter/matter issues.


title: Does spin alone have any effect on the physical interactions of particles?

text: In Hartree-Fock theory the time-independent electronic energy of a single (restricted) determinant electronic wavefunction consists of one electron terms, $h_{ii}$, Coulomb interaction energies, $J_{ij}$, and exchange interaction energies $K_{ij}$. Exchange interaction energy terms result from integration over the 2-electron repulsion operator $r_{12}^{-1}$ where the electrons in the 2-electron integral have parallel spin. I have read that the exchange interaction is not a true physical interaction, but arises because of the single determinant description of the system. It is a partial description of the correlated motion of electrons, which is better described in many-determinant methods. My question is this: does spin alone have any effect on the physical interactions of particles? For example, consider 2 uncharged particles with spin 1/2. They may either have parallel or antiparallel spins. In physical reality, would these 2 particles repel one another if they have parallel spin? If I was to model this system with a Hartree-Fock calculation, I believe there would be no exchange interaction, because the Hamiltonian would feature no two particle operator, so no Coulomb or exchange integrals would arise. However, perhaps if I used a higher level of theory, I might see an interaction?


title: What is known about some massive Gaussian models on a lattice?

text: Recently I started to play with some massive Gaussian models on a lattice. Motivation being that I work on massless models and want to understand the massive case because it seems easier to handle (e.g. cluster expansion makes sense thanks to contours carrying mass). Consider a Hamiltonian $$ H(x_{\Lambda}) = \sum_{<i,j> \in E(\Lambda)} (x_i - x_j)^2 +\sum_{i \in \Lambda} U(x_i) $$ with $\Lambda$ a lattice (think mainly of ${\mathbb Z}^d$ with $d = 2$), $E(\Lambda)$ the set of its edges, $x_{\Lambda} \in (\Lambda \to {\mathbb R})$ and $U(y)$ a potential. I would like to know what is known about this class of models in general. E.g. for a related class of models without mass but with arbitrary two-point function Funaki and Spohn showed that there is no phase transition if that function is convex. > 1. I wonder whether similar result is known for convex (replace with any > other reasonable condition) $U$. Similarly, > 2. Is there a necessary condition on $U$ for there to be a phase transition? > Try to give some examples. E.g. it seems natural that there will be a transition for a symmetric double- well model (with spontaneous symmetry breaking at low temperatures) and one could also investigate the case with one of the wells being more favorable. I think I should be able to prove these things with some work but I suppose it's already been done by someone. > 3. Could you point me to a reference about some double-well models? Other kind of model I've been thinking about is (setting $\beta = 1$ and letting $p$ play the role of temperature) $$\exp(-U(y)) = p \exp (-a y^2) + (1-p) \exp (-b y^2)$$ with $a$ suitably small and $b$ suitably large. This is a big well with a smaller well inside. Intuitively the system should sit in the small well at low temperatures ($p = 0$) and jump out and behave freely (with small mass $a$) at higher temperatures ($p = 1$), so this is a toy model of melting. Trouble is, I have no idea whether this really works and I can't decide whether there will or won't be a phase transition. > 4. Any ideas about this model? Pointing me to a reference would be wonderful > but I am not sure this has been studied before.


title: How to correctly show units with a vector

text: Supposed I have an position vector $$\vec{r}=\begin{pmatrix} 10.0 & -30.0 & 25.0\end{pmatrix}$$ expressed in $\mathrm{millimeters}$. What is the correct notation to display $\vec{r}$   1. $\begin{pmatrix} 10.0 & -30.0 & 25.0\end{pmatrix}\text{ mm}$   2. $\begin{pmatrix} 10.0\text{ mm} & -30.0\text{ mm} & 25.0\text{ mm}\end{pmatrix}$   3. $\begin{pmatrix} 10.0 & -30.0 & 25.0\end{pmatrix}\text{ in }\mathrm{mm}$ If the answer is **2.** then why add all those redundunt units when all elements of a vector have to be of the same unit. If you have a long list of values then usualy you present this a table with the units in the header (and not on each element). What if the units are complex (with powers and fraction), do we really have to write them out for each element? How would you consicely write out a vector while describing the units those values are in also at the same time? PS. I did not post this in the `Math SE` because they have never heard of units :-) and only physics deals with real situations.


title: Visualization of proton's wavefunction

text: Visualizations of hydrogen's wavefunctions / electron orbitals are abound. I could not however locate a visualization of the wavefunction of a proton. The reason I was looking for one is to see whether the three quarks that make it up "occupy" disjoint regions of space (i.e. the maximum probability locations are separated). Can someone please point to one such visualization, or explain why they can't be created (or why my question is nonsensical, which is always a possibility...)


title: Symmetries in Wilsonian RG (2)

text: This question is related to the paper http://arxiv.org/abs/1204.5221 and is a continuation of the previous question Symmetries in Wilsonian RG   * In the liked paper why do the equalities in equation 2.7 and 2.11 hold? (the LHS of both the equations is the same and hence the two equations are 2 different ways of writing the full connected functional W)  I guess one reads 2.7 to say that when one is flowing down to the IR from UV one develops only "relevant" (dim <4) operators and one I guess reads 2.11 to mean that one develops only irrelevant (dim >4) operators when one flows up to the UV from IR. Why?   * In the linked paper just below equation 2.2 the authors comment that if there is a CFT in the UV then this UV behaviour can change if irrelevant operators are added. why? I would think that (dim>4)/irrelevant operators would come suppressed with positive powers of the cut-off and hence if one pushes the cut-off to infinity then they would vanish and hence the UV is not affected by them. But the authors don't seem to think so...


title: Symmetry, Transformations and non-linear transformations

text: I am a physics student. My mathematical background is quite weak. I just want to know the similarities (if there are any) or differences between coordinate transformation of two kinds :   1. Rotation of coordinate (and hence new transformed coordinate system) or translation and so on. (Most of which are symmetry associated)   2. Transformation to a different system of coordinates - like Cartesian to cylindrical or spherical to parabolic and so on. In this regard I would like to know the difference between these two in the linear algebra language. I can see in both cases the inner product is preserved. Secondly, I am also interested in knowing if we can associate symmetry to the second kind of transformation and hence some generator of transformation, if not what are the cases in which we can't associate such generators to transformation. I have already posted this question here. However I have not obtained satisfactory responses there.


title: Moose Models (Purpose, Examples)

text: A problem set for my QFT class is titled "Moose Models" and deals with the moose model for a gauge symmetry of $U(1)\times U(1)$. I was wondering if I could get an explanation of what a Moose Model is, and what it's useful for. Googling things such as "Moose Model Particle Physics" led me to a thesis from Harvard on Minimal Moose Models in $SU(5)$ but it's frankly too advanced for me to do much with. Any help would be appreciated.


title: Using $\sin()$ or $\cos()$ for computing SHM?

text: In simple harmonic motion, you can use either the sin or cos form of the equation but my question is which one do you use when and why? I am having a tough time understanding this, so any help would be thoroughly appreciated


title: Ground and first excited state of non interacting spin system Hamiltonian

text: For a non interacting spin system containing two $\frac{1}{2}$ spin particles I am trying to determine its Hamiltonian. If the energy of a up spin is $+\mu {\bf B}$ and a down spin is $-\mu {\bf B}$, the possible energies are $+2\mu {\bf B}$ for $|UU\rangle$, $-2\mu {\bf B}$ for $|DD\rangle$ and 0 for both $|UD\rangle$ and $|DU\rangle$. I am confused what will be the Hamiltonian which will allow me to determine the energy of this system. If $|DD\rangle = \begin{pmatrix} 1 \\\ 0 \\\ 0 \\\ 0 \end{pmatrix} $, $|DU\rangle = \begin{pmatrix} 0 \\\ 1 \\\ 0 \\\ 0 \end{pmatrix} $, $|UD\rangle = \begin{pmatrix} 0 \\\ 0 \\\ 1 \\\ 0 \end{pmatrix} $ and $|UU\rangle = \begin{pmatrix} 0 \\\ 0 \\\ 0 \\\ 1 \end{pmatrix} $, I am not sure how to determine the matrix representation of the Hamiltonian. Although I suspect that the Hamiltonian matrix will remain same if I use an alternative set of bases i.e. $\begin{pmatrix} 0 \\\ {1 / \sqrt{2} }\\\ {1 / \sqrt{2} } \\\ 0 \end{pmatrix}$, $\begin{pmatrix} 0 \\\ {1 / \sqrt{2} }\\\ {-1 / \sqrt{2} } \\\ 0 \end{pmatrix}$, $\begin{pmatrix} {1 / \sqrt{2} } \\\ 0\\\ 0 \\\ {1 / \sqrt{2} } \end{pmatrix}$ and $\begin{pmatrix} {1 / \sqrt{2} } \\\ 0\\\ 0 \\\ {-1 / \sqrt{2} } \end{pmatrix}$.


title: Trying to speak correctly of spacetime intervals and how to compare them

text: Is it correct to speak of "magnitude of a spacetime interval"? For instance, considering a pair of (distinct) events, $A$ and $B$, which are lightlike separated, is it correct to say that "the magnitude of the spacetime interval of event pair $A$ and $B$ is equal to 0"? Further, considering two pairs of (distinct) events, both of which are spacelike separated, is it correct to say that "the ratio between the magnitudes of the spacetime intervals of these two pairs of events is a real number"? Likewise, considering two pairs of (distinct) events, both of which are timelike separated, is it correct to say that "the ratio between the magnitudes of the spacetime intervals of these two pairs of events is a real number"? Finally: Is "$s[ P, Q ]$" an appropriate notation to abbreviate "the magnitude of the spacetime interval of event pair $P$ and $Q$"?


title: When is the "minus sign problem" in quantum simulations an obstacle?

text: The "minus sign problem" in quantum simulation refers to the fact that the probability amplitudes are not positive-definite, and it is my understanding that this leads to numerical instability when for example summing over paths, histories or configurations since large amplitudes of one phase can completely negate other large contributions but with the reverse phase. Without controlling this somehow, the sampling of alternative configurations has to be extremely dense, at least for situations where interference is expected to be important. Assuming I haven't misunderstood it so far, my main question is when this is a show-stopper and when it can be ignored, and what the best workarounds are. As an example, let's say that I want to simulate Compton scattering or something similar, numerically, to second order. I could evaluate the Feynman diagrams numerically to a certain resolution and sum them. I assume this won't work well. In Lattice QCD, complete field configurations are randomly generated and the action calculated and amplitude summed I guess (I have only superficial knowledge of Lattice QCD unfortunately).


title: Phase of Elements

text: There are 11 gaseous elements and two liquid elements at standard temperature and pressure. The rest are solid. Can phase be predicted from quantum mechanical principles?


title: Church–Turing Thesis

text: Can the Church–Turing Thesis be proved assuming classical mechanics, how is the proof or disproof? Edited: I was looking for a proof of "everything computable by a device obeying CM is computable by a Turing machine", but it seems obvious now. My followup question is why does it hold for QM?


title: Why so many arguments for the transformation equations of generalized coordinates?

text: For a system of $N$ particles with $k$ holonomic constraints, their Cartesian coordinates are expressed in terms of generalized coordinates as $$\mathbf{r}_1 = \mathbf{r}_1(q_1, q_2,..., q_{3N-k}, t)$$ $$...$$ $$\mathbf{r}_N = \mathbf{r}_N(q_1, q_2,..., q_{3N-k}, t)$$ Each particle in space can be uniquely identified by 3 independent variables, so why aren't the above of the form $$\mathbf{r}_i = \mathbf{r}(q_{i1}, q_{i2}, q_{i3})?$$ Note there is only one transformation $\mathbf{r}$ for all $\mathbf{r}_i$, a function of only three generalised coordinates and independent of $t$.


title: Why is heavier object more reluctant to get falling down?

text: Is it because of the upward force that stops the object? for example- ![http://i.stack.imgur.com/gqXyY.png](http://i.stack.imgur.com/80t15.png)


title: Can negative energy explain blackholes?

text: I think black holes are peculiar and I want to know if there is a relation to negative energy. Does negative energy have positive mass and therefore can explain the nature of black holes? IS there any relation between negative energy and blackholes or is negative energy just a way of saying that is takes energy?


title: Using A.C or D.C to create an electromagnet

text: Does it matter if I use A.C or D.C as a source to create an electromagnet? Or Does it depend on the voltage of current? **For example:** I have a coil which has about 50 turns. If I use same amount of A.C or D.C voltage, which one of these will create a stronger magnetic field?


title: Gravity anomaly in Lacnov, Slovakia: Gravity ashamed

text: http://www.youtube.com/watch?v=4qxwqNqL-Nc http://tvnoviny.sk/sekcia/spravy/technologie/v-lacnove-sa-predmety-kotulaju- do-kopca.html   (Sorry: above links in Slovak.) Can someone explain this to me? Is it a hoax? (In case of link rot try this or this Youtube search.)


title: What should a physics undergrad aspiring to be a string theorist learn before grad school?

text: The question I guess is pretty clear. I am a physics undergrad wishing to pursue research in quantum gravity(string theory?). What are the subjects I should learn other than the usual compulsory undergrad courses such as quantum mechanics, and GR, etc? Things that come to my mind are QFT, lie algebras, differential geometry, algebraic geometry etc? Other than this, could you suggest any topic that are not entirely essential but interesting in mathematical physics, gauge theory etc (e.g. solitons, monopoles etc). Actually I am looking for a mini project of sorts which would give me an idea of what these fields really are. I have done some lie algebras, classical field theory and am currently studying QM.


title: Road to String Theory

text: I have a question for our theoretic SuperUsers. How much knowledge and which fields of physics you have to know to start studying string theory? I am now on QFT, after I think I'll start study Supersymmetry or maybe special cases of QED. How much more should I know to start Strings? :)


title: Should it be obvious that independent quantum states are composed by taking the tensor product?

text: My text introduces multi-quibt quantum states with the example of a state that can be "factored" into two (non-entangled) substates. It then goes on to suggest that it should be obvious1 that the joint state of two (non-entangled) substates should be the tensor product of the substates: that is, for example, that given a first qubit $$|a\rangle = \alpha_1|0\rangle+\alpha_2|1\rangle$$ and a second qubit $$|b\rangle = \beta_1|0\rangle+\beta_2|1\rangle$$ any non-entangled joint two-qubit state of $|a\rangle$ and $|b\rangle$ will be $$|a\rangle\otimes|b\rangle = \alpha_1 \beta_1|00\rangle+\alpha_1\beta_2|01\rangle+\alpha_2\beta_1|10\rangle+\alpha_2\beta_2|11\rangle$$ but it isn't clear to me why this should be the case. It seems to me there is some implicit understanding or interpretation of the coefficients $\alpha_i$ and $\beta_i$ that is used to arrive at this conclusion. It's clear enough why this should be true an a classical case, where the coefficients represent (where normalized, relative) abundance, so that the result follows from simple combinatorics. But what accounts for the assertion that this is true for a quantum system, in which (at least in my text, up to this point) coefficients only have this correspondence by analogy (and a perplexing analogy at that, since they can be complex and negative)? Should it be obvious that independent quantum states are composed by taking the tensor product, or is some additional observation or definition (e.g. of the nature of the coefficients of quantum states) required? * * * 1: See (bottom of p. 18) "so the state of the two qubits _must be the product_ " (emphasis added).


title: Nanotube chiral angle as a function of $n$ and $m$

text: I'm looking into nanotubes and I thought I'd assure myself that the basic geometry equations are indeed correct. No problems for the radius, I quickly found the known formula: $$R = \sqrt{3(n^2+m^2+nm)}\frac{d_{CC}}{2\pi}$$ if $d_{CC}$ is the carbon bond length. For the chiral angle, however, the equation should be $$\tan{\theta} = \frac{m\sqrt{3}}{2n+m}$$ but I got $$\tan{\theta} = \frac{m\sqrt{3}}{2n-m},$$ a sign difference. I thought I'd look for the mistake and quickly correct it, but I cannot for the life of me find it. I'll show my derivation below. The starting point is of course the nanotube structure: ![Nanotube structure](http://i.stack.imgur.com/Q6DRD.jpg) I'll simplify this picture to make it more clear what I did: ![Simplified picture](http://i.stack.imgur.com/0LHIH.png) From the above picture, it's easy to see that $$\sin{\theta} = \frac{h}{na_1}$$ and $$\sin{\left(120^{\circ} - \theta\right)} = \frac{h}{ma_2}.$$ This can be rewritten using the trigonometric identity $$\sin{\left(\alpha-\beta\right)} = \sin{\alpha}\cos{\beta} - \cos{\alpha}\sin{\beta}$$ where $\alpha = 120^{\circ}$ and $\beta = \theta$, giving rise to $$\frac{\sqrt{3}}{2}\cos{\theta} + \frac{1}{2}\sin{\theta} = \frac{h}{ma_2}.$$ Now, using the fact that $a_1 = a_2 (=\sqrt{3}d_{CC})$, we find the following equations: $$\left\\{\begin{array}{rcl}n\sin{\theta} & = & \frac{h}{a_1}\\\ m\left[\frac{\sqrt{3}}{2}\cos{\theta} + \frac{1}{2}\sin{\theta}\right] & = & \frac{h}{a_1}\end{array}\right.$$ which obviously allow for combination: $$\begin{eqnarray} \frac{n}{m}\sin{\theta} & = & \frac{\sqrt{3}}{2}\cos{\theta} + \frac{1}{2}\sin{\theta} \\\ \left(\frac{n}{m}-\frac{1}{2}\right)\sin{\theta} & = & \frac{\sqrt{3}}{2}\cos{\theta} \\\ \frac{2n-m}{2m}\sin{\theta} & = & \frac{\sqrt{3}}{2}\cos{\theta} \\\ \tan{\theta} & = & \frac{m\sqrt{3}}{2n-m}. \end{eqnarray}$$ This differs from the known expression in that it has a minus sign in front of $m$ in the denominator, but I fail to see my mistake. It could be really silly and I'm just being blind... Thanks for taking a look at it.


title: What's the distinctions between Yang-Mills theory and QCD?

text: So Yang-Mills theory is a non-abelian gauge theory, and we used a lot in QCD calculation. But what are the distinctions between Yang-Mills theory and QCD? And distinctions between supersymmetric Yang-Mill theories and SUSY QCD?


title: Cusp anomalous dimension in QCD

text: What is the "cusp anomalous dimension", starting from the basics? I came across this term while reading some QCD papers.


title: Is the number of independent constants of a system equal to the number of degree of freedom of it?

text: Maybe the question is not very clear myself since I am not a physics major.But can you help me make this question clearer and then give me some comments on it? I got that this holds in gravitional field.Since in a gravitional field, we need the position and velocity of a particle to determine the trajectory of it.Yet from Newton's law $m\frac{d^2x}{dt^2}=F=m\frac{d^2x}{d(t-t_0)^2}$, we see that we actually need $6-1=5$ variables to determine the motion of a particle.On the other hand, we know that the total energy of the particle $E$, the angluar momentumn $L$ and the Runge-Lens vector $A$ are constants of the motion.Yet we also have that $L$ is perpendicular to $A$ and $||A||$ is a function of $L$, so all in all we have $7-1-1=5$ constants of motion. Then does this hold in general?I learned from someone that it is difficult to determine the number of independent constants of a motion?Is it right?Will someone explain it to me? Will someone be kind enough to give me some hints or comments on this problem? Thank you very much!


title: how does phase-shifting principle in interferometry work?

text: I have been using the coherence scanning interferometry (or also known as white light scanner), where it collects images containing the fringes. The system is able to produce the depth profile for one image stacks with the fringes. If we plot the intensity of one pixel of a particular point in which the fringe will passing by on a 2d-plot (stack vs intensity), we can see some sinusoidal wave profile. I like to understand how the phase shifting method works on the fringe images to derive the depth information? Is it based on the phase difference on one point ? Can the sinusoidal profile be used in the method at all ?


title: About defining "baryons" and "mesons"

text: I want to understand the proof of the claims (of the construction as well as of its uniqueness) of gauge singlet states given around equation 2.13 (page 10) of this paper.   * Also does the listing of gauge singlet states there depend on the fact that these are superconformal primaries? (Are they claiming that any gauge singlet state is a primary?) What exactly is the connection between the construction of the gauge singlets and them being superconformal primaries? Let me repeat the claims here again,   * If you have $N_f$ fields in the fundamental representation of $U(N_c)$ then apparently these can't be combined (tensored?) into an $U(N_c)$ invariant (gauge singlet).    * But $N_f$ in the fundamental of $SU(N_c)$ can be combined into "baryons" - gauge singlets of $SU(N_c)$ as, $\epsilon_{i_1\dots i_{N_f-N_c}j_1\dots j_{N_c}}\epsilon^{a_1\dots a_{N_c}}$ $\prod_{k=1}^{N_c} \phi^{j_k}_{a_k}$    * If with the same $SU(N_c)$ the $N_f$ fields happen to be in the adjoint of $SU(N_c)$ then there exists forms invariant under $SU(N_c)$ given as $Tr[\prod_{k=1}^n \phi_{i_k}]$ (for any $n$ of these $N_f$ fields)    * If one has a pair of fields in the fundamental and the anti-fundamental of $SU(N_c)$ then the gauge invariant operators under $SU(N_c)$ are given as the "mesons" - $\phi^i_a \bar{\phi}^a_j$ (where $a$ is the $N_c$ index and $i,j$ is the $N_f$ index) - I wonder if some Hilbert's invariant theory has gone into these claims. If yes, how? I guess somewhere it is being used that the gauge invariant states are finitely generated since these are invariant under the action of these reductive gauge groups.


title: Why Does Light Not Become Polarized In A Magnetic and/or Electric Field?

text: I am familiar with the Faraday Effect, but I remain confused as to why the electric and/or magnetic components of light do not naturally align themselves with a magnetic or electric field (in a vacuum) and thereby become polarized.


title: observation and implied time since creation

text: I read on a post Big Bang and Cosmic microwave background radiation? We detect light from another 13 billion years ago does this mean that one billion years ago we could only detect light from about 12 billion years ago. If so why and is the same true for the inverse


title: Interaction of matter with EM fields

text: For the interaction between electromagnetic fields and matter,   1. when do we have to include quantization of the EM field and when we can ignore it?    2. when do we have to include quantization of atomic energy levels and when we can ignore it? Update: I am aware that part of the answer might depend on the accuracy we are looking for. Part of the problem here is that I do not know how to estimate such things, or what quantity will quantify the accuracy we are looking for which can tell us if we can ignore quantization of either energy levels or the fields or not.


title: Would the universe get consumed by blackholes because of entropy?

text: Since the total entropy of the universe is increasing because of spontaneous processes, black holes form because of entropy (correct me if I'm wrong), and the universe is always expanding, would the universe be consumed by black holes or would its area increase too quickly for that to happen?


title: What are the local covariant tensors one can form from the metric?

text: Normally in differential geometry, we assume that the only way to produce a tensorial quantity by differentiation is to (1) start with a tensor, and then (2) apply a covariant derivative (not a plain old partial derivative). Applying this to GR, I think one way of stating the equivalence principle is that the only tensorial object that we expect to be "built in" to the vacuum is the metric. Since the covariant derivative is basically defined as a derivative that produces zero when you apply it to the metric, this means that you can't get anything of interest (i.e., local and tensorial) by appying the process described by #1 and #2 to the vacuum. This can be used as a fancy way of arguing that the Newtonian gravitational field $\mathbf{g}$ isn't a tensor, since in the Newtonian limit, it's essentially the gradient of the metric. However, the process described by #1 and #2 is sufficient but not necessary. In fact, one way of defining curvature is by taking non-covariant derivatives on the metric to form the Christoffel symbols, and then doing further operations involving non-covariant derivatives to get the Riemann curvature tensor -- which surprisingly ends up being a valid tensor. It seems, then, that the Riemann tensor is a special case. I originally thought that there might be a uniqueness theorem that proves that if we want to produce a local, tensorial quantity from the metric, the only possibilities are the Riemann tensor or curvature polynomials formed from the Riemann tensor and its covariant derivatives. [EDITS] A comment by joshphysics and the answer by BebopButUnsteady helped me to refine this conjecure as follows. Joshphysics pointed out that things like $g_{ab}g_{cd}$ might be considered trivial counterexamples. I can think of two possible ways of dealing with this: (1) BebopButUnsteady's answer shows that this is in some sense not a counterexample at all, since the metric itself can be expressed as a Taylor series in terms of the Riemann tensor and its derivatives. If the metric is analytic, and if we're willing to accept infinite series, then this means that there is no information in the metric that isn't recoverable from the Riemann tensor. (2) What _doesn't_ seem to exist, apart from curvature polynomials formed from the Riemann tensor and its covariant derivatives, is (a) any varying scalar field, or (b) any vector field. (Part b is basically the equivalence principle.)


title: Is the redshift of ancient stars less than constant linear expansion predicts?

text: When accelerated expansion of universe was discovered, the astronomers used redshift of ancient stars. Say if star is 10 billion years in the past, then it should be moving faster than star from 5 billion years ago because of distance. But not 100% faster, only close to 100% faster. So red shift for 10 bln star is slightly less than two redshifts of 5 bln year star. Is it what was measured in the discovery of accelerated cosmological expansion ?


title: Is there a way to obtain the classical partition function from the quantum partition function in the limit $h \rightarrow 0$?

text: One would like to motivate the classical partition function in the following way: in the limit that the spacing between the energies (generally on the order of $h$) becomes small relative to the energies themselves, one might write $$Z_{quantum}=\sum_i e^{-\beta E_i} \stackrel{?}{\rightarrow} \int e^{-\beta E}dE$$ (which is clearly wrong on dimensional grounds, but illustrates the idea). Yet this is not of the same form as the classical partition function, namely $$Z_{classical}=\frac{1}{h^{3N}}\int e^{-\beta H({\bf p},{\bf q})}d{\bf p}d{\bf q}. $$ So one might look for a measure $f(\lambda)$ to append to the quantum partition function such that $$\lim_{\lambda \rightarrow 0} \sum_i e^{-\beta E_i} f(\lambda) = \frac{1}{h^{3N}}\int e^{-\beta H({\bf p},{\bf q})}d{\bf p}d{\bf q}.$$ Does such a measure exist, or is there an alternate procedure for deriving the classical partition function from the quantum one?


title: Capacitors' working in a circuit

text: Does charge in a capacitor containing circuit stop flowing when the potential of the capacitor becomes equal to the potential of the battery??


title: Charge On parallel plate capacitor?

text: Suppose if we supply two plates of parallel plate capacitor with charges $Q_1$ and $Q_2$.What would be the charge of the capacitor? As the field inside plate should be zero and hence using gauss law I can understand that charges on inner surface should be equal in magnitude and of opposite nature.Further charges on outer surface would be $\frac {Q_1+Q_2}2$ but what do we call the charge of capacitor in these cases?


title: Can a rotating-ring space-station design be applied to a rotating sphere?

text: Suppose engineers built a rotating space station similar to Space Station V from the film 2001: A Space Odyssey (circa 1968), but with a large sphere, instead of a ring? Could this be rotated or pushed through space in such a way as to give it mostly consistent gravity on all parts of the surface within the sphere?


title: Perpetual motion

text: Does this video have a physics behind or it is fake? Where the energy comes from? It looks like a perpetual motion!


title: Capillary action in a Capillary tube

text: Consider a capillary tube or radius r .And it is placed in a beaker containing water.what proportion exist between the radius of the capillary tube and the height to which it came transport the water through the capillary action. Is there any formula relating these two factors???


title: Will this type of engine produce thrust?

text: I was wondering that if we replace the combustion chambers in a jet engine with heaters heating the compressed air up to temperature near to plasma then what will be the case, will it produce any kind of thrust or it will be just useless


title: When will PLANCK release all-sky CMB anisotropy measurements?

text: After hearing so much about PLANCK, people these days are still constraining models with WMAP results. When will PLANCK finally release its much more superior data?


title: Help getting the distance an object has moved

text: _What is a distance an object went through at the speed of 14hm/min for 6 hours?_ I am having problems with conversions. First, I tried passing 14hm/min to m/s. I did it alright. But now I am unsure. Do I multiply it by 6 hours and I'm done?


title: Dirac Field Ehrenfest Proper Time derivative?

text: The Dirac Equation has a corresponding Ehrenfest time derivative for the evolution of operators relativistic quantum mechanics. Is there a similar theorem for the evolution of relativistic operators, but the derivative is taken with respect to proper time?


title: What is the most natural definition of the weak hypercharge coupling constant if grand unification is wrong?

text: A tricky question. Here is the famous graph of the running of the three coupling constants in the standard model: http://www-ekp.physik.uni- karlsruhe.de/~deboer/html/Forschung/unification_eng.eps . The graph shows, in its top curve, the running of the coupling constant alpha_1. This is the coupling of the weak hypercharge coupling constant for the weak hypercharge group U(1)_Y, which is one of the three gauge groups of the standard model of particle physics. But there is a tricky detail. In that curve, alpha_1 is multiplied with 5/3. This factor 5/3 comes from the assumption that GUTs are valid. The factor ensures that the various group traces of U(1)_Y, SU(2) and SU(3) are normalized in the correct way when they form the SU(5), SO(10) or any other grand unification gauge group. In the case that grand unification is wrong, the factor 5/3 cannot be deduced. Which factor would be natural in this case? Clarification added, after remarks by Lubos Motl: it is assumed in the question that the usual definition of the weak hypercharge is used, Y_W=2(Q-T_3), in which left-handed quarks have hypercharge 1/3.


title: Do and can phone signals come inside AC car which is glass-packed

text: Can phone signals penetrate glass, so can I expect phone signals to come if I am sitting inside a closed AC car. Thanks,


title: Infinite chain of alternating charge

text: ## The problem There is an infinite chain (1d) of alternating charges that are in a distance a from each other. That is there is one charge +q and next to it -q and so on. Now calculate how much work it would take to insert one charge somewhere in the chain. ## What I've tried The work required has to be the potential after the insertion minus the potential energy before the insertion. The potential energy of the chain before the insertion is: $E_{pot} = \frac{1}{4\pi\epsilon_0}\sum_{k=1}^{\infty}\left(\frac{q^2}{2ka}-\frac{q^2}{(2k-1)a}\right)$ This simplifies to: $E_{pot} =\frac{q^2}{4\pi\epsilon_0 a}*(-log(2))$ ## Where I'm stuck The problem is though that that is "from the view of one individual charge" in the sense that I took the potential energy between one specific charge and all other charges in the chain. The chain as a whole has of course a potential energy of infinity because it would take infinitely much work to seperate all those charges. Because there is no special point somewhere in the chain I felt that it was ok to just pick some arbitrary charge. When you insert another charge though, there is one very special point in the chain: The point where those two same charges are next to each other. Therefore I feel like I cannot do the same with the chain after the insertion. I have no idea how to calculate the potential energy difference then.


title: Basic Thermodynamics: Quasistatic Adiabatic Process

text: I'm going through the exercises in a Thermodynamics book, just to revise and build my intuition. Right now, I'm working on: > Show that for a quasistatic adiabatic process in a perfect gas, with > constant specific heats: > > $PV^\gamma = constant$ > > with $\gamma = \frac{C_p}{C_V}$ where P is pressure, V is Volume, and C is heat capacity. **I'm not looking for the answer, just for a hint (I'm stuck and want to find the solution myself).** So those are my thoughts:   * perfect gas means: $PV = RT$ , (R is the universal gas constant)   * adiabatic means: $dQ = 0$, (Q for heat)   * since there is no heat exchange, the process is reversible   * reversible means: $dW = -PdV$, (W is for work)   * heat capacity is defined as $C_V = \left( \frac{dQ}{dT} \right)_V$, respectively $C_P = \left( \frac{dQ}{dT} \right)_P$ If I draw a PV diagram for this situation, it looks like this: ![enter image description here](http://i.stack.imgur.com/8LAQf.png) Now I want to show that $PV^\gamma=const$ by going from state 1 to state 2 in the PV diagram. I've started like this: $W = -\int_{V_1}^{V_2}PdV = -\int_{V_1}^{V_2} \frac{RT}{V}dV = RT ln (V_2/V2)$ This leads me into the wrong direction though. I thought about using $R = C_p - C_V$ here, but it doesn't seem to work. Any suggestions? **Please just give me a hint, not the solution.**


title: What happens to atoms inside a black hole?

text: Black holes have very high gravitational force that tends to crush everything. So as we know atoms in a molecule have inter-atomic spacing between them and further electrons also revolve at a certain distance to the nucleus and inside the nucleus there is also some empty space. So does the strong force of a black hole crush them all the way, leaving no space in between them? And, if that happens, then it should give rise to a completely new species of atoms and molecules, right?


title: Does the ice melt faster when breaks free from larger sheet and drops into Antarctic Ocean waters?

text: My question is, when Ice is intact with the large sheet and exposed only to air, it melts at some pace. But does it melt faster or slower when the big piece breaks free from the sheet and drops into the water ? Why?


title: Electric field energy

text: Is it true that $\int_V \rho \phi $ is the equation for the energy of a charge distribution in an external potential and $\frac{1}{2} \int_V \rho \phi$ is the equation for the energy of a charge distribution in its own potential? I never noticed this difference before and there are not many reference that explicitely touch this topic, so I thought that I could ask this.


title: Can a black hole collapse in itself?

text: As we know that the black hole is a lump of highly dense matter, and that's the reason for it's so strong gravitational force. Bat let us assume that it has sucked up a huge amount of mass and it's gravitational force has become of very high strength. Than what will happen. Will the black hole collapse in itself like a dying sun an give birth to another worst nightmare for the cosmologist or anything else will happen.


title: Long Range Spin-Spin Interactions

text: A recent article on probing Earth's interior mentioned the potential use of a "fifth force", long range electron spin-spin interactions, as a tool in the endeavor. Has anybody published any experimental results confirming detection of this "force'? Is existence of long range spin-spin interaction compatible with current quantum mechanics theory?


title: Model the gyroscopic effect for 3axes stabilisation

text: **Context** I am trying to stabilise a platform on a guimbal using 3 axes gyroscopes. I have made tons of research (same problem for weeks even months now) without being able to properly model the 3 orthogonal gyroscope actuators (high inertia flywheels). **Question** What is the dynamic model of a 1-axis gyroscope actuator? Can it be used 3 times on different directions (I have a Simulink model) to get a 3-axes one or is it more complex? From what I have seen, $\boldsymbol{w_{precession}} =\boldsymbol{h_{gyro}}\times \boldsymbol{\tau_{ext}}$ which may or may not be what I'm looking for (experts?) but I don't see the rotation being instantaneously triggered: what's the corresponding torque? **Additional info** My Simulink plant implements $\boldsymbol{\theta}=\int_0^t\int_0^t\frac{\boldsymbol{\tau_{ext}}}{\boldsymbol{J}}dt²$; and the gravity torque, to counter, is given by $\boldsymbol{\tau_{grav}}=\boldsymbol{r}\times\boldsymbol{W}$ (weight with moment arm centre of guimbal-CG). If this is the right way to go, I could inject the $\omega_{p}$ after the first integration, but what's the associated torque that counters the gravity's?


title: Does a Photon leave trace in a silicon tracker?

text: I am having this image from ATLAS Detector. ![](http://i.imgur.com/2dnmlHK.jpg) In gray you can see the ATLAS's Si Tracker.In Green you see the Electromagnetic Callorimeter. In red there is the Hadron Callorimeter and in Blue there are the muon Telescope. Those two traces are suppossed to be elecrtrons. My question is, why cannot they be photons? Nuclear Physisists are using Si Detectors for gamma ray spectroscoopy, which means that they can actually detect photons, with them. Just for the record, there is a cut in transversal momentum, of 10GeV... Any ideas on why to reject those particles from being photons? ![](http://i.imgur.com/zM2DY8g.jpg)


title: Pair correlation function for a inhomogeneous Laughlin droplet

text: Pair correlation function for the usual Laughlin droplet is defined as $g(\vec{r})$: $$\rho_0 g(\vec{r})=\frac{1}{N}\langle\sum_i^N \sum_{j \neq i}^N \delta(\vec{r}-\vec{r_i}+\vec{r_j})\rangle$$, where $\rho_0=\frac{\nu}{2\pi l_0^2}$. So if now the Laughlin droplet is inhomogeneous, meaning that the density function $\rho(\vec{r})$ is direction dependent, then should the pair correlation function still be defined the same as above, or do we need to change $\rho_0$ to something else?


title: Superconductors on each side of a flexible membrane what happens?

text: I have a sheet of flexible Kevlar 16th of an inch thick coated on each side with a high temperature superconductor, what happens when you put hundred amp of current into each side, and then 1000 and so on. And I'm assuming that the superconductor is bonded very well to the Kevlar, and that it is at a temperature cohesive to it superconducting. Will the 2 sides repel each other? Turning the Kevlar into a rigid structure? How rigid? Please put in the layman's terms!


title: Is a world with constant/decreasing entropy theoretically impossible?

text: I'm not 110% sure exactly what I mean by this question. It was sparked by a friend who said he wished the law of entropy were reversed, so he wouldn't have to worry about cleaning the bathroom. Basically, we can imagine many changes to the laws of physics - you could scrap all of electromagnetism, gravity could be an inverse cubed law, even the first law of thermodynamics could hypothetically be broken - we've all imagined perpetual motion machines at one time or another. However, the second law of thermodynamics seems somehow more 'emergent'. It just springs out of the nature of our universe - the effectively random movement of physical objects over time. Provided you have a Universe whose state is changing over time according to _some_ set of laws, it seems like the second law must be upheld, things must gradually settle down into the state of greatest disorder. What I'm particularly wondering is if you can prove in any sense (perhaps using methods from statistical mechanics)? Or is it possible to construct a set of laws (preferably similar to our own) which would give us a universe which could break the second law. Thanks


title: What are the practical applications of decoherence?

text: Let me clarify this question somewhat. I know decoherence is ubiquitous in nature and explains the emergence of a classical world from quantum physics. My question is really about how a knowledge of how decoherence actually works can be put to use in a practical application. An application we can't design in the absence of such a knowledge, even though decoherence is still happening all the time. Thanks


title: Scattering Processes in Scalar Yukawa Theory

text: I'm trying to compute nucleon-nucleon scattering in scalar Yukawa theory. Here we view a nucleon as a complex scalar field $\psi$ and a meson as a real scalar field $\phi$. They interact through $H_I=g\int d^3x\psi^{\dagger}\psi\phi$. Suppose we want to compute the amplitude for scattering from an initial state $|p_1,p_2\rangle$ to a final state $|p_1',p_2'\rangle$, which we assume are **eigenstates of the free theory**. At second order in perturbation theory we have the term $$\frac{(-ig)^2}{2}\int d^4xd^4y\ T\left[\psi^{\dagger}(x)\psi(x)\phi(x)\psi^{\dagger}(y)\psi(y)\phi(y)\right]$$ using Dyson's formula, where all fields are in the interaction picture, so the Heisenberg picture of the free theory. Using Wick's theorem we may compute this amplitude explicity. My notes claim that the only term giving a nonzero contribution is $$:\psi^{\dagger}(x)\psi(x)\psi^{\dagger}(y)\psi(y):\overbrace{\phi(x)\phi(y)}$$ I disagree with this however. Surely also we get contributions from the disconnected and unamputated diagrams? For example $$:\psi^{\dagger}(x)\psi(x):\overbrace{\psi^{\dagger}(y)\psi(y)}\ \overbrace{\phi(x)\phi(y)}$$ and the various other permutations of such terms will produce extra terms (delta function divergences) won't they? I know that if we are considering true scattering where our initial and final states are eigenstates of the interacting theory then there's a theorem which says we may ignore the disconnected and unamputated diagrams. Perhaps I'm meant to implicitly use this here, and the assumption in bold above is an error in the notes. To summarise - am I right in thinking that disconnected and unamputated diagrams give a nonzero contribution to scattering for eigenstates of the free theory? And is the correct way to deal with this to assume we're working with eigenstates of the interacting theory? Are all my arguments and intuitions above correct? Many thanks in advance!


title: Quantum mechanical angular momentum and spin formalism/notation

text: I am currently stuck on the following notation: $\frac{1}{2}\otimes\frac{1}{2} = 0 \text{ (antisym) } \oplus 1 \text{ (sym) }$ No matter what I tried, I couldn't derive the identity. I am sure that it is trivial, but I can't figure how to treat the notation. It would be great, if somebody could write this in matrix notation or in bra/ket notation with explicit spins ($\uparrow\downarrow$).


title: Could the attractive force of gravity be modeled as a repulsive force?

text: If space intrinsically contained gravitons of many different wavelengths then could gravity be model as a repulsive force simular to how casimir attraction works? or any other repulsive mechanism you can think of.


title: Coronal Mass Ejection and its effect on Earth's Orbit

text: Do CMEs (Coronal Mass Ejections) have an effect on Earth's Orbit around the sun? Given the the Earth and Sun are essentially two large magnets, one in orbit around the other. If so, how much of an impact is there?


title: Are objects in a gravitational well shortened?

text: Bob is in a gravitational potential well, he moves a long vertical stick up and down a distance of 1 meters. Alice observes the upper end of the stick, at upper location. There is the phenomenon of redshift of energy, so either Alice measures a shorter motion than 1 meters, or alternatively Alice measures a weakened force. So I'm asking: is Bob weak, or is Bob short? ( EDIT: It interests me what REALLY happens. It seems that when you fall into a black hole, you maybe become contracted: your lower part is already slowed down, while your upper part is still moving a little bit faster. I'm certain that when you are being lowered into a gravity well, you REALLY lose energy. Therefore there REALLY is no redshift of energy when energy travels back up from a gravity well. Now I try to make a thought experiment out of these two things: a shortened person in a gravity well, sends his reduced energy up, by poking with a stick ) EDIT 2: Question 2: Bob moves the vertical stick horizontally a 10 meters distance, what distance and what force does Alice observe at the upper end of the stick?


title: Finding the moment of inertia through superposition?

text: Let's say I have a body consisting of two homogenous spheres/balls that touch each other. I also have a body fixed coordinate system which consists of that body's principal axes. I know the the moment of inertia for each of the spheres alone, $I=\frac{2}{5}mr^2$. Can I use that to find the moment of inertia of the entire body? I need this because for such a body, with such a coordinate system, I don't think integration would be very simple... I'm looking for all three moment of inertias, however two of them should be the same because of rotational symmetry.


title: A (smart) way to map out the Brillouin zone of a 2-D material

text: I am currently writing a tight-binding calculation model for various 2-D cells as part of a homework assignment. Whilst solving the problem set was quite easy I struggle at a smart way to plot the band diagram in the usual "lets visit all high symmetry points in the first BZ"-fashion. ![http://img46.imageshack.us/img46/7431/graphene.png](http://i.stack.imgur.com/rURGx.png) right now I am manually 'sticking together' the vectors from point to point but that is rather tedious and doesn't seem smart to me. Unfortunately I am lacking a good idea so I am hoping there are some people here who do band calculations and can give me a hint!


title: Would a collapsing Universe have the density of water?

text: I understand that the typical density of the super massive black hole is close that of the water. It is also my understanding that this density is not true matter density because the volume used to calculate the density is measured from event horizon (so I suppose this is rather potential gravitational energy density?). If it was the case that the universe were to collapse into a super massive black hole, can we still assume that it would be close to that of the water? If so, why, and if not, how would you calculate the density? Note that the equations using the Shwarzschild Radius don't work in this case.


title: Angular momentum in a rod rotating around one end?

text: Sorry if I can't get straight to the point, I have to give a lot of details before I actually state the question. The formula for angular momentum is $L=I \omega$. If we look up $I$ for a thin rod pivoted around one end we get $I=\frac 13 ML^2$ so $L=\frac 13 ML^2 \omega$. However, $L$ is also equal to $p \cdot d$, where $p$ is the linear momentum and $d$ is some distance, so $I \omega=pd$ My question is as to why in this case $d$ comes out to be what it is $($it is $\frac 23 L)$ We can work out the linear momentum of the rod: Assume the rod, which has a mass $M$, length $L$, and a constant angular speed of $\omega$ is split into $n$ equal pieces. Then each piece has mass $\frac Mn$ and a linear momentum of $\frac Mn \cdot \omega r_i$, where $r_i$ is the distance from the pivot to the end of the $i^{th}$ segment. The linear momentum is approximately $\sum_{i=1}^{n} \frac Mn r_i \omega$. We can make this exact by taking the limit as $n$ approaches infinity. We have $r_1=\frac Ln$, $r_2=2\frac Ln$, $\cdot \cdot \cdot \cdot \cdot r_i=i \cdot \frac Ln$, so we can substitute this in the sum: $$ \lim_{n \to \infty} \sum_{i=1}^{n} \left(\frac Mn i \cdot \dfrac Rn \omega\right)$$ Since $M$, $\omega$, $R$, and $n$ are not relevant to the summation, we can factor them out: $$ p=MR \omega \lim_{n \to \infty} \dfrac {1}{n^2} \sum_{i=1}^{n} i$$ $$p= MR \omega \lim_{n \to \infty} \dfrac {1}{n^2} \dfrac {n(n+1)}{2}$$ $$p=\dfrac 12 ML \omega$$ Again, we have $$L=\dfrac 13 ML^2 \omega=pd=\dfrac 12 ML \omega d$$ If we solve for $d$, we get $d=\frac 23 L$. Why is this? I was expecting $d$ to be a more relevant point, such as $L$ or $\frac 12 L$ (because that's where the center of mass is. But why $\frac 23$? It seems pretty random.


title: How do speakers play sounds with multiple sources?

text: I know that speakers produce sound by creating a pressure wave, but I don't understand how when they play music or games you can hear the different sources. I guess what I'm asking is how do they play (or seem to play) multiple frequencies and amplitudes simultaneously?


title: Visualisation of vortex flows

text: Are there examples of vortex flows (eg: von Karman vortex street or large eddies) which are visualized in a way apart from displaying where the mass is displayed? I can imagine there to be a display based on velocity. But is there a display based on pressure? From what I have seen, in a display of the vortex street, you can see how a mass of fluid is swirled around and eventually detached from a body, but that information isn't really useful quantitatively. Is it possible to find previous work where someone has measured velocity/forces/pressure around the vortex, is it possible and will a quantitative measurement help in any way?


title: Alcubierre drive and inertia.

text: What is the inertia or velocity of a vehicle upon exiting or shutting down an Alcubierre bubble? Would the vehicle maintain the velocity it had in the bubble? I'm not sure I asked the question in a meaningful way so I'll try another question that targets what I'm after. If a vehicle was traveling in one of these bubbles how hard would it be to come to a full stop?


title: Can a local magnetic field when changed introduce an electric field till infinity?

text: Here is a **question** from my school textbook: ![the question](http://i.stack.imgur.com/1AgIx.png) Here is the **solution which they expect** : Change in magnetic flux is $\pi a^2 B $, and since $\oint\vec{E}.\vec{dl} = -\frac{d\phi}{dt}$, the total tangential electric force acting on the rim is $(\pi a^2 B \lambda)/(\delta t)$, and multiplied by torque and divided by moment of inertia gives the angular acceleration which multiplied by $\delta t$ gives the angular velocity as $\frac{\lambda B \pi a^2}{MR}$. **Doubt :** If the magnetic field is limited to radius $a$ from the centre of the wheel, is it right that the electric field induced by it when it is diminishing reaches out till the rim of the wheel?


title: How do you know which atoms are in a specific piece of matter?

text: If you have a piece of matter in front of you, how can you know of which atoms it consists? And starting from nothing, how can I discover all the elements (or just a lot of them) of the periodic table? I can imagine some unefficient methods based on atom's characteristics (e.g. heat the matter till it becomes a gas, then calculate it's **spectrum** ; weigh and compare pieces of the matter very accurately to "discover" the quanta of **mass** ; ...), but all those methods seem quite theoretical/impossible. Moreover, when people started tabling the atoms (end of the 19th century), things like the spectrum of an atom weren't even known yet. Or is it all done by **chemical reactions** , and then comparing the ratios of the compounds and their masses? This seems for me the most obvious way to tackle the problem, although it still looks like a very hard job to determine all possible elements like that experimentally (>100 years ago, or even now if I had to do it myself in some kind of way). I cannot imagine having all the elements in my daily neighbourhood right know, to start experimenting with them and "discovering" them. Measuring characteristics like the **boiling point** etc. doesn't seem to be a good idea either, because you don't know whether you're dealing a mono-atomic specimen or a compound. So my question is actually twofold: how did people manage to discover them more than 100 years ago (historically), and how can I do it myself now (practically)?


title: Does QED provide a closed form for Coulomb logarithms?

text: The classical models for the integrand as well as the cut-offs in computing the Coulomb logarithm are pretty rough. > Does quantum electrodynamics have definite expressions for the quantity > $\mathrm{ln}(\Lambda)$? Specifically, I'm motivated by the applications for electron diffusion coefficients. And I could imagine the answer also answers this: Is the kind of Colombo logarithm for other charges (color charges, etc.)?


title: Reference frame and centrifugal force

text: Imagine you are on a theme park ride in which you sit in a car and are spun around in a circle, basically like a giant centrifuge. An observer from the outside would say that there is no centrifugal force acting on the person in the car, only centripetal. However, for the person in the car, there is the force pulling them inwards (centripetal), and its reaction force pushing outwards against the side of the car (centrifugal). However, from this point of observation, the person in the car is stationary and everything else is moving around him. Does this not mean that there is no circular motion in this frame of reference, and therefore the reaction force is not centrifugal, as there is no circle centre? Basically, I am confused as to why a fictitious centrifugal force will show up in a frame of reference where circular motion is not occurring.


title: Sum of torque from a sphere

text: A sphere (grey color) turn in rotation at $\omega$ rd/s. There are 2 walls that prevent sphere to escape. Walls can only turn around center of rotation. The sphere turn only at $\omega$ rd/s too. **The center of gravity of sphere is the center of circle. The center of gravity of {system less sphere} is red point.** Two forces on walls give a torque. Magenta force is centrifugal force. Red forces = magenta force on walls. F1 and F2 are forces perpendicular on radial axis, there are projection, there are not friction. ![enter image description here](http://i.stack.imgur.com/kkk9E.png) * * * I done the resolution with the projection in each axis. There is a difference on $d$ but very little difference, sure forces F1 and F2 change in the same time. I drawed new projection: ![enter image description here](http://i.stack.imgur.com/WXiRa.png) If I add friction, this change the equilibrium, this don't change $d$ nor $R$ so like friction can be different on interface sphere/wall1 and sphere/wall2, how can I have a sum of torque at zero ? I drawn all forces with friction: ![enter image description here](http://i.stack.imgur.com/R36r6.png) I give same friction like that sphere don't turn. I hope I can do that, I'm not sure. I exagerate the size of friction for look with good precision. Forces F1 and F2 changed. I suppose sum of torque is at 0 before friction. I suppose the center of gravity of {system except sphere is red point} and I suppose the center of gravity of sphere is in the center of circle (black point). Friction give torque to {system - sphere} and give the same torque to sphere (in other direction). But forces apply to the center of gravity, no ? So for me the torque on sphere is lower than the torque give to {system - sphere}. What's wrong ? If I give different friction on wall1/wall2. This would say the sphere turn ? If the sphere turn it's on its center of gravity. So this rotation transmit/cancel energy to the rest of the system ? This transmission of energy is from center of gravity of ball to external wall1/2. How torquescan cancel themselves ? ![enter image description here](http://i.stack.imgur.com/f3o5S.png) With friction I done simulation for look at sum of forces. Green color: sum of forces are not collinear, the software shows a global torque. Maybe it done the same error like me with center of gravity. If you could explain where forces act: on which center of gravity ? ![enter image description here](http://i.stack.imgur.com/wkt1f.png) Pieces 1 are link together (it's one piece). Piece 2 is alone. The system turn clockwise. Friction is at 0.5. If I added weight to (A) this move the center of gravity of (1), and the torque increase more. The weight of pieces are: Big circle: 15 kg Small circle: 0.6 kg   Big stems: 0.7 kg   Small stem: 0.2 kg Maybe the software is not precise enough. But when I look at forces I can understand the torque increase more and more. So what force is false ? * * * I simulated with a lot of parameters. Frequency change results until 4800 Hz. From 4800 Hz to 12000 Hz results don't change (below 3%). From 9600 to 12000 Hz I can't measure the difference. Big circle turn around its center (center of the circle). Small circle turn around its center (center of the circle) too. I have isolated 5 cases: **First:** The most common : here the system win energy ![enter image description here](http://i.stack.imgur.com/5EmCf.png) **Second:** Sometimes: I can't say if system win or lost energy because it's too short ![enter image description here](http://i.stack.imgur.com/W7sCx.png) **Third:** It depend, this is the most difficult to say, with x rd/s it's sometimes (less than first case), with y rd/s it's often (more than first case). The energy is lost or win, it depend. But that I find strange, it's the energy. Angular energy is lost (the torque must increase angular energy in this case) but linear velocity increase sometime more sometime less than angular velocity. ![enter image description here](http://i.stack.imgur.com/b3qEX.png) **Forth:** Very rarely (very less often than second case), I estimated this case like 20 cases on 12000 cases in a turn: Here the system lost energy. The energy lost is important because I can see it even the forces appears very quickly. ![enter image description here](http://i.stack.imgur.com/JSf5q.png) **Fifth:** Sometimes: I can't say if the system win or lost energy. No forces N. * * * If you can explain each case ? I would like to understand forces and why energy move up/down. The case 4 is a bug I think, but it lost energy don't win. Maybe in the first case the software add energy but in fact the system lost. * * * I think I understood something but I'm not sure: In the following case (it's the same case than before but with more straight surfaces for show if it was not the mesh of Algodoo, but not and I added an axis for show if there is the same problem with energy and it is even energy increase slowly): ![enter image description here](http://i.stack.imgur.com/lIIX2.png) With forces like that: ![enter image description here](http://i.stack.imgur.com/7R9kh.png) In Algodoo the free big circle turn around itself. It's logical that big circle in free rotation receive a torque on it. This torque act in its center of gravity. Forces act on another object from axis, like radius is not the same the torque is not the same and energy too I think. I resume forces like that: ![enter image description here](http://i.stack.imgur.com/xxvVQ.png) Energy can be like that: ![enter image description here](http://i.stack.imgur.com/2EZkg.png) Some datas: ![enter image description here](http://i.stack.imgur.com/Xxsoh.png) What do you think about that ? Free big circle can turn around itself with a particular forces on it, yes/no ? If yes, these forces are apply in its center of gravity, yes/no ? And these forces don't work in the same value to other object due to the difference of radius, yes/no ? I added angular velocity of 2 objects, there are not the same (never), there is always a difference of 0.1 rd/s. It's possible to see the object 2 turn around itself with software. I tested with Comsol and with Algodoo, the angular velocity is not the same. And even I force the angular velocity of object 2 at the same value than object 1, some seconds after angular velocity return to the previous values. If there is a difference for angular velocity this could say torque from friction don't work the same value for 2 objects and maybe it's the reason software can give more and more energy even all the system turn alone. ![enter image description here](http://i.stack.imgur.com/xAtxC.png)


title: Understanding the displacement of a particle in a wave

text: I have a question regarding waves and the equation we use to describe their motion. My understanding feels shaky, so i'd like to see if I can get a good explanation/way of thinking about it. The displacement of a particle in a $+x$ moving wave is given by; $y = A \sin(2 \pi ft - \frac{2\pi x}{\lambda})$ But when this is contrasted to a specific example, is it correct to equate parts of the equation to one another? For example, if I have $y = 0.021\sin(25t - 2x)$ does this imply that $2\pi ft = 25t$ and $\frac{2\pi x}{\lambda} = 2x$ And you can solve for frequency and wavelength respectively? I hope my question makes sense.


title: Does the law of Universal Gravitation apply to every matter?

text: If I have a big ball of 20,000,000kg and another of 100g, does it mean that the big ball will pull the small ball towards it?


title: Quantum Logic and Quantum Field Theory

text: Quantum Logic is a very interesting and powerful answer to the problem of Quantum Mechanics foundations. Nevertheless this approach is usually developed in a non-relativistic framework. Is it still valid in a relativistic one? Can one talk about Yes/No experiments, elementary propositions lattice and so on also in the context of Quantum Field Theory?


title: Definition for Chiral Spin Liquid

text: What is the definition of chiral spin liquid? Especially what does chiral mean here? I encounter a lot of terminologies with chiral. It seems they mean differently in different contexts. If you could generally answer what chiral means, it would be great.


title: Is it possible to find the hydrogen atom's radial wavefunctions?

text: Is there a way to actually find the equation of $R(r)$ without looking at a table with these equations already given? I'm given $n$, $\ell$, and $m$.


title: Faraday's Law - recursive?

text: So we know that the EMF is induced by change of flux. The thing that was always confusing me is the following:   * we start changing the magnetic field   * which in turn induces electric field which makes charge carriers move   * this e-field also, in turn, makes another magnetic field that changes   * and the whole process seems to go ad infinitum from there! As far as I understand, this is the basis for the electromagnetic radiation. But Faraday's equation takes into account only the "first" field that is changing, or so I was led to believe, e.g. when you're calculating the self- impedance of the solenoid, you will only look for the first derivative of the magnetic field caused by the current going through it, not all the subsequent magnetic fields. Since it's also natural to assume that the Law is valid and my reasoning is wrong, where am I wrong?


title: Can quantum weirdness be explained by waves in space?

text: Looking at the reflections of the Sun off a wavy lake the Sun appears located in many places at once and jumps positions instantly and randomly. Might quantum weirdness simply be a particle's reflection off waves in the fabric of space? ![reflections](http://www.vibrakeys.com/wordpress/wp- content/uploads/2011/03/WaterLight.jpg)


title: Hamiltonian function for classical hard-sphere elastic collision

text: I'm trying to find the Hamiltonian function for a system consisting of a single particle in one dimension colliding elastically with a wall at x = 0. Everything I've read on the topic (e.g. this question Why can't collisions be elastic?) says that the wall can be represented by a step function potential barrier $V = K\theta(x)$ with K any number higher than the maximum momentum of the particle -- often taken to be infinite for simplicity (e.g. V=0 if x < 0, V = $\infty$ if x > 0, so the Hamiltonian would be $H = p^2/2m + K\theta(x)$, where $\theta(x)$ is the step function, $\theta(x) = 0$ for x < 0, $\theta(x) = 1$ if x > 0, and K is a constant that we eventually take to infinity)...but I cannot get that to work. When I apply the Hamiltonian equations of motion to that Hamiltonian, I end up finding that the collision is only elastic and energy-conserving if K is not only finite, but also dependent on the particle's momentum. I'm fairly sure that there's just something wrong with my derivation, because using an infinite potential barriers to exclude particles from certain positions is used everywhere in physics (e.g. hard-sphere dynamics in molecular dynamics or billiard dynamics, the Van Der Walls forces, etc)...but I just can't see where I'm going wrong. So, below is my full derivation (I start with two particles of diameter D, and then take the mass of the second particle to be infinity at the end; however, I get the same results if I start with just one particle of diameter D/2). Note: the problem is specifically getting the Hamiltonian; the equations of motion themselves are trivial to get via e.g. conservation of energy/momentum arguments; x is proportional to |t|, and p to sign(t)). **Variables** D := diameter of particle K := height of "wall" (nonzero; assumed by most references to be a constant, often infinite, but always greater than the maximum kinetic energy; this derivation only assumes that it is position-independent) $m_i$ := mass of particle i (for a wall, $m_2 \to \infty$) $ x_1, x_2, p_1, p_2 $ := coordinates and momenta of particles $r = |x_1 - x_2|$ := absolute distance between centers of particles (so $(D - r) >= 0$ iff they are touching) $H = T + U$ $U(r) = K\theta(D - r)$ $T = p_1^2/2m + p_2^2/2m$ **Definitions / abbreviations** $\epsilon(x) := sign(x) $ $\theta(x) := step(x)$ $\partial r/\partial x_1 = \epsilon(x_1 - x_2)$ $\partial r/\partial x_2 = -\epsilon(x_1 - x_2)$ $\partial\theta/\partial x = \delta(x)$ **Derivation** (0) Hamilton's equations: $\partial H/\partial x_1 = -dp_1/dt = \partial U/\partial x_1$ $\partial H/\partial x_2 = -dp_2/dt = \partial U/\partial x_2$ (1) Chain rule (prime means differentiation wrt appropriate x): $\partial U/\partial x_1 = K\theta'(D - r)(-r') = -K\delta(D - |x_1 - x_2|)\epsilon(x_1 - x_2)$ $\partial U/\partial x_2 = K\theta'(D - r)(-r') = K\delta(D - |x1_ - x_2|)\epsilon(x_1 - x_2)$ (2) note that the derivatives are equal/opposite $ -\partial U/\partial x_1 = dp_1/dt = -dp_2/dt = \partial U/\partial x_2$ (3) find total change in momenta $\Delta P_i $ (and final momenta $p_i(t_2)$) Let $t_1$ := the time where $x_1 - x_2 = +D$; let $t_0 := t_1 - dt, t_2 := t_1 + dt$ Let $p_1(t_1) = P_1, p_2(t_1) = P_2$ Let $\Delta P_1 := \int_{t_0}^{t_2} (dp_1/dt)~dt = K = -\Delta P_2$ Then $p_1(t_2) = P_1 + K; p_2(t_2) = P_2 - K$ (4) But, energy must be conserved: $T(p_i) = T(p_i + \Delta P_i)$ $(P_1 + K)^2/2m_1 + (P_2 - K)^2/2m_2 = P_1^2/2m_1 + P_2^2/2m_2$ $(P_1^2 + 2KP_1 + K^2)/2m_1 + (P_2^2 - 2KP_2 + K^2)/2m_2 = P_1^2/2m_1 + P_2^2/2m_2$ $(2KP_1 + K^2)/2m_1 + (-2KP_2 + K^2)/2m_2 = 0$ Multiply through by $2m_1m_2$ $(2KP_1 + K^2)m_2 + (-2KP_2 + K^2)m_1 = 0$ $2K(m_2 P_1 + 1/2 K m_2 - m_1 P_2 + 1/2K m_1) = 0$ (5) By definition, |K| > 0. So, $m_2 P_1 - m_1 P_2 + 1/2K(m_1 + m_2) = 0$ **Conclusion** : $K = 2(m_1 P_2 - m_2 P_1)/(m_1 + m_2) $ So to conserve kinetic energy, K (and hence U) must depend on the momentum. (6) In the special case where $m_2 \to \infty$: $ 1/2K = (m_1 P_2)/(m_1 + m_2) - (m_2 P_1)/(m_1 + m_2)$ $ 1/2K = (m_1 P_2)/(m_2) - (m_2 P_1)/(m_2)$ $ 1/2K = 0 - P_1, K = -2P_1$, as expected. (7) However, since the potential now depends on the momenta, $ dx_i/dt$ is no longer simply $p_i/2m_i$; we have $ H = p_1^2/2m_1 + p_2^2/2m_2 + 2(m_1 P_2 - m_2 P_1)\theta(D - r)/(m_1 + m_2)$ $\partial H/\partial p_1 = dx_1/dt = p_1/m_1 - 2m_2\theta(D - r)/(m_1 + m_2)$ $\partial H/\partial p_2 = dx_2/dt = p_2/m_2 + 2m_1\theta(D - r)/(m_1 + m_2)$ (8) Or in the special case of the wall, $ H = p_1^2/2m_1 + p_2^2/2m_2 -2 P_1\theta(D - r)$ $\partial H/\partial p_1 = dx_1/dt = p_1/m_1 - 2P_1 \theta(D - r)$ $\partial H/\partial p_2 = dx_2/dt = p_2/m_2 + 2P_1 \theta(D - r)$ ...which is zero outside of the barrier, but there is ambiguity in the definition of the step function at zero; we can retain the normal equations of motion only if $\theta(0) = 0$. Any thoughts would be greatly appreciated.


title: What sets the length scale of stratified rock formations?

text: Driving through areas of the American Southwest such as Utah and Arizona, you notice stratified rock formations everywhere. The grand canyon is the best known example. One striking thing is that the thickness (or height) of individual layers of rock seems to be roughly constant, within an order of magnitude. (Perhaps 20 feet tall? Hard to tell from the road.) This is true even for very tall cliffs which have dozens of layers. Does this imply some kind of very long-term periodicity in climate over geologic time scales? If so, what is it? If not, why is there a dominant length scale in these features? What sets this scale?


title: How to prove Wick's Theorem (Zee's eq. I.2 (16))  via Gaussian integration?

text: I'm working through Zee's QFT in a Nutshell but there's an integral [I.2 (16)] I couldn't quite derive. The problem is to find $$\langle x_i x_j ... x_k x_l\rangle=\frac{\int ... \int dx_1 ... dx_n x_i x_j ... x_k x_l\exp(-\frac{1}{2}\vec{x}\cdot A\cdot\vec{x})}{\int ... \int dx_1 ... dx_n \exp(-\frac{1}{2}\vec{x}\cdot A\cdot\vec{x})} $$ using $$ \int ... \int dx_1 ... dx_n \exp(-\frac{1}{2}\vec{x}\cdot A\cdot\vec{x}+\vec{J}\cdot\vec{x})=\sqrt{\left(\frac{(2\pi)^n}{\det[A]}\right)}\exp(\frac{1}{2}\vec{J}\cdot A^{-1} \cdot \vec{J}) $$ with $A$ a real symmetric matrix. Zee suggests taking repeated derivatives with respect to $\vec{J}$ and then letting $\vec{J}\rightarrow0$. Here's my attempt at a solution: The top of the numerator in the first equation can be obtained by differentiating the the above equation by $\frac{\partial}{\partial J_a}$ for $a=i,j,...,k,l.$ Dropping the constant for now, we find that: $$ \int ... \int dx_1 ... dx_n x_i \exp(-\frac{1}{2}\vec{x}\cdot A\cdot\vec{x}+\vec{J}\cdot\vec{x})=\left(\sum_m^n A^{-1}_{im}J_m\right)\exp(\frac{1}{2}\vec{J}\cdot A^{-1} \cdot \vec{J}). $$ This process of taking derivatives can be repeated via the product rule to get messy stuff, but what worries me is that when one takes $\vec{J}\rightarrow0$ the integral will vanish, will it not? Because the sums over $m$ that pop out will all vanish. The answer, Zee says, is not zero, but instead $$ \langle x_i x_j ... x_k x_l\rangle = \sum_{{\rm Wick}}(A^{-1}_{ab})...(A^{-1}_{cd}) $$ "where the set of indices {$a,b,...,c,d$} represent a permutation of the set of indices {$i,j,...,k,l$}. The sum is over all such permutations or Wick contractions." How exactly does one reach this result?


title: Particle mixing and indistinguishability

text: Neutral kaons have two flavor combinations: $\mathrm{d}\bar{\mathrm{s}}$ and $\mathrm{s}\bar{\mathrm{d}}$. They can also be weak eigenstates: $\mathrm{\frac{d\bar{s} \pm s\bar{d}}{\sqrt{2}}}$. But are neutral kaons of different compositions indistinguishable? If yes, how is it so, since they are composed of different fundamental particles? If not, what will happen when neutral kaon oscillates? They are suddenly indistinguishable when one oscillates into another? That indistinguishability can vary over time?


title: Einstein's Explanation for gravity vs. Newtonian

text: I was trying to understand the Einstein's explanation for gravity (gravitational force), and while I am able to understand why two moving masses will be attracted, due to the curving of the space, I am not quite able to understand what would make an apple fall, i.e., how will Einstein model explain the gravitational force between two stationary objects? (Please correct me if I am wrong anywhere. I am a computer scientist; hence physics is not my forte! :D)


title: Do magnets work in outer space?

text: Is there any media where magnet lose its property?