title: All mirrors always shrink to 50% scale?

text: I have this geometric optics exercise here, in which a man is looking at himself in a mirror.   1. Determine the minimum height at which the bottom of the mirror must be placed so the man can see his feet (his eyes are at 1.7m).   2. How does this height vary depending on the distance from the mirror to the man's eyes?   3. What is the minimum length of the mirror so that the man can see his entire body (assume he is 1.8m tall). From the principle that the reflected angle is equal to the incident angle, we show that the minimum height of the bottom of the mirror is directly between the feet and the eyes (1), and that this does not depend on the distance to the mirror (2). The top of the mirror must be directly between the man's crown and eyes at the lowest, in order to reflect the top of his head, **therefore the mirror must be half the height of the man, at minimum**. Thus, the mirror reflects the man's image at a scale of 50%. But this doesn't depend on the distance from the mirror, and from what I can tell, doesn't even depend on the size of the mirror. These results, to me, indicate that no matter what kind of mirror you use or where you stand, your image should appear half way up the mirror, and at 50% scale.


title: Shooting a bullet at a system of blocks

text: So, I made this question up myself.... and I'm curious about the answer. It requires only secondary-school-level knowledge of physics: You have a surface (ground) with a certain coefficient of friction. On this surface is a rectangular block of mass $M_{1}$, which has length $L$. On top of this block is another block (but square) of mass $M_{2}$ which has side- length of $L/4$ ... the very left edge of the square block is perfectly placed at $L/2$ of the rectangular block. In other words, if the rectangular block has length $L$, than the left edge of the square block (placed on top of the rectangular block) is placed at $L/2$ of the rectangular block. In addition, the top side of the rectangular block that is making contact with the square block has coefficient of friction = 0. You decide to shoot a bullet (horizontally) of mass $m << M_{1}$ and $ m << M_{2}$ which travels with velocity $v_{i}$ right at the center of the rectangular block on the yz-plane. So: if the blocks are sitting in 2D (xy- plane)... then the bullet is hitting the center of the rectangular block at its LEFT edge. Let's say that the bullet penetrates the rectangular block and "becomes part of it." Now, this is the question: Find the expression for the bullet velocity, $v_{i}$, that will JUST make the square block to fall off the rectangular block; let's call this the threshold velocity. Imagine that the ground goes off to infinity.


Why is it difficult to differentiate between interference and diffraction? Is it because we don't clearly understand how both of these phenomenon takes place? My thoughts: From an answer to one of my earlier question here I could understand that light waves behave differently when it strikes a slit which is of several times bigger than its wavelength and when it strikes a slit of size comparable to it its wavelength.In the former case the different points on the slit acts as wave-front sources producing circular wave fronts which interfere to form the diffraction pattern but in the later case plane wave-front striking a slit after passing through it forms circular wave-fronts and two such wave-fronts from two different slits interfere to form the interference pattern. But, I'm not satisfied by that because I don't get any intuition about points on the slit acting as wavefront sources. So, what is actually happening?


title: Replacing diode in circuit analysis?

text: While analyzing circuits containing diodes in reverse bias I replace it with a pure insulator for simplifying things. Similarly can I replace a forward biased diode with a conductor and a varying resistance in series?


title: Why can't you escape a black hole?

text: I understand that the event horizon of a black hole forms at the radius from the singularity where the escape velocity is c. But it's also true that you don't have to go escape velocity to escape an object if you can maintain some kind of thrust. You could escape the earth at 1 km/h if you could maintain the proper amount of thrust for enough time. So if you pass just beneath the event horizon, shouldn't you be able to thrust your way back out, despite the >c escape velocity? Or does this restriction have to do solely with relativistic effects (time stopping for outside observers at the event horizon)?


title: optics biconvex and plano convex

text: What is the resultant focal length If a biconvex lens is cut into half and then the resulting 2 plano-convex lens arranged in such a way that the plane surface of one faces the curved surface of the other plano-convex lens considering the focal length of the **plano-convex** lens to be f.


title: Differential geometry of Lie groups

text: In Weinberg's _Classical Solutions of Quantum Field Theory_ , he states whilst introducing homotopy that groups, such as $SU(2)$, may be endowed with the structure of a smooth manifold after which they may be interpreted as Lie groups. My questions are:   * If we formulate a quantum field theory on a manifold which is also a Lie group, does that quantum field theory inherit any special or useful properties?   * Does a choice of metric exist for any Lie group?   * Are there alternative interpretations of the significance of Killing vectors if they preserve a metric on a manifold which is also a Lie group?


title: AGN accretion disk vs. torus

text: The torus is the donut of dust encircling the Active Galactic nucleus. The accretion disk is inside the torus. Is there a boundary between the two? At what point does a torus become an accretion disk? What are their differences?


title: Hysteresis Curve and how to implement it using Preisach model or other models

text: As a homework I need to draw a hysteresis curve (preferably an interactive one) using Matlab or any other programming language. The problem is I have trouble finding a good algorithm to do so. I need something that can be implemented easily and in a systematic way. The Preisach model seems to be the best way, but I don't really get it. Can you suggest a good model/algorithm or explain Preisach model for me? The main problem that I have with Preisach model is that I don't understand how exactly $\mu$ and other magnetic properties affect $\alpha$ and $\beta$ in this model.


title: How do we explain the phase change between high and low star formation rates?

text: During a recent talk I was at, someone, who models galaxy characteristics from dust amounts and spectral energy distributions (SEDs), quoted a fairly prompt change from 'high' to 'low' stellar formation rates. How do we explain this prompt phase change? (If at all!)


title: How to apply faraday's law to lc circiut

text: I am having problem on using Faraday's law correctly. When I try to apply Faraday's law to the LC circuit: ![LC circuit](http://i.stack.imgur.com/JCdRC.png)   1. Assume the top plate of capacitor is positively charged and bottom is negative. So no current flows clockwise.   2. Start from bottom plate, follow the current, apply the Faraday's law (Vc = Voltage between capacitor's two plate):              -Vc (Since I go against electric field inside capacitor)     +0  (Since there is no E-field inside inductor)     =   -Ldi/dt      So I have: Vc - Ldi/dt = 0 However, this get me a complete different answer from the formula derived by energy conservation(which is Vc + Ldi/dt = 0) I try to apply Kirchhoff's Law in the meaning of "potential drop":                   +Vc (Since capacitor provides a potential increase)         +/-Ldi/dt      I am stuck here, because when the current increases, there is a potential drop because the inductor fights against the increased current, but when the current increases, there is a potential increase because the inductor wants to support the weakening current. I cannot decide the sign before term Ldi/dt. Can you point out what's wrong here? Thx~~


title: Is there a method to estimate the atomic yield of a supernova of a given size?

text: Given a supernova with stellar mass $M$, is there a theoretical method to estimating the isotope yield? If so, what processes are taken into account, and how accurate can the estimate be? Would it be more accurate if more properties were known about the star (e.g. metallicity)?


As stated in the heading: Is $\frac{\partial}{\partial \Phi(y)} \Phi (x) = \delta(x-y)$ correct? Here denotes $\Phi(x)$ denotes a scalar field. And if yes, why? Any reference where I can read about this would be great.


title: How can I determine if my telescope mirror is parabolic?

text: This week I purchased the “Skywatcher Explorer 130P Newtonian Reflector Telescope”, specifically as it has the parabolic mirror. I do not have the outer packaging for sundry reasons. Therefore, after setting up the telescope, how can I check to see whether my telescope is actually a version with the parabolic mirror or the standard mirror. The manuals that came with it do not tell you this, nor I am versed in spotting one from the other just by looking at it. Are there any tests I can do, or is there a ‘quick check’ to ascertain the parabolic mirror is on the telescope?


title: anti-matter activation energy

text: In chemical reactions, there are usually energy barriers that must be overcome before a reaction can take place. It seems that such a phenomena also exists at the atomic level as well. For instance, a certain amount of energy must be provided to two nuclei to bring them close enough together to fuse and release more energy than was input. In general, I read things like "energetically favorable" or "a lower energy state is available". However, in examining the behavior of anti-matter, it seems almost as if there is no such barrier between matter and anti-matter: they annihilate immediately on contact, converting most or all of their matter into energy. It seems like such a sharp divide between total annihilation with virtually no extra energy added to the equation, versus almost total stability for either matter or anti-matter by itself. Why such the big difference? Why does anti-matter seem to have no constraints to "decaying" into energy in the presence of matter (and vice versa) like other processes do? I know there are certain quantities like charge which must be conserved but it seems like a pretty abstract thing to say we don't all spontaneously decay into photons simply because charge must be conserved. Why isn't there some kind of energy barrier between matter and anti-matter which must be overcome before they can react? If anti-matter had negative mass, then it would have negative energy, in which the lack of such a barrier would make more sense. However, I don't know of any evidence that suggests anti-matter has negative mass. My naive guess would be that since a particle and anti-particle have opposite spin, they sort of "slam into" each other and cancel out, causing the particle to "break apart" into a free wave. However, since my understanding of spin is that it isn't really the same kind of spin we would observe at the macroscopic scale, i.e. actual physical rotation, this seems like an unreal explanation to me.


title: Can we detect whether food has previously been heated in a microwave oven?

text: An acquaintance told me that she refuses to eat microwaved food because she read that the microwave _changes the molecules_ of the food somehow. Her statement is vague and her sources are dubious but I am curious. I know that microwaves tend to heat food unevenly, causing changes at the macro scale to the structure of the food, but I am more interested in changes at the micro scale which might effect the food's nutritional properties. Are there any known, detectable changes that occur when food is heated in a microwave oven? Given a sample of cold food, is there any way to determine whether it has been microwaved at some point in the past?


title: Why does the cross derivative of the partition function disappear here?

text: They state that the chemical potential in a canonical ensemble is given by: $$\mu = -kT \frac{\partial{\ln Z(N,V,T)}}{\partial{N}} \tag{1}$$ But if I use the definition of chemical partial (which I assume to be general): $$\mu = -T\frac{\partial{S}}{\partial{N}}$$ And I substitute $S(N,V,T) = k\ln Z+kT\frac{\partial{\ln Z}}{\partial{T}} $ I get almost (1), but with an extra term. I have no clue why this term should be zero.


title: Undamped oscillations. Why is the solution a linear combination of $\sin()$ and $\cos()$?

text: ![enter image description here](http://i.stack.imgur.com/UVOM5.png) $ma = mg - cx$, where $x(0) = x_0 = 0$ is the position in which there is no tension in the rope. $dx/dt = v_0$ for $t = 0$; $v_0$ is a known constant. The discriminant of the characteristic equation is negative, we have complex roots, and the solution is a linear combination of a $\sin()$, $\cos()$ and a constant. But, intuitively, everyone knows that $m$ would "swing up and down", which, I think, is a sinusoidal movement (intuitively!). What is the physical significance of a second sinusoidal function in the solution? Motivation of the question: I am trying to calculate the maximum tension in the rope, through the maximum acceleration. However, I find it unusually difficult to compute $\mathrm{max}(c_1\sin(\omega t) + c_2\cos(\omega t))$.


title: Force, constant velocity, and 0 velocity

text: When a body has a net force of 0 on it, one cannot say that it is at rest or in motion at a constant velocity. If a body is accelerating because of some force and I want to stop it, I can apply the same force in the opposite direction. Unfortunately, that seems to mean that the body will start moving at the velocity it had when the opposite force was applied. Does this mean that you cannot ever stop a body (generally, but specifically when the body has a constant acceleration) by applying an equal and opposite force? How then do you stop such a body? You have to apply a force that keeps increasing in amount until the body comes to rest?


title: Viscous Burgers equation physical meaning

text: The viscous Burgers' equation: $$ q_{t}+q\:q_{x}~=~\nu\:q_{xx}, \mbox{ where } \:\:\nu >0, $$ combines the nonlinear propagation of $q(x,t)$ and the diffusion.   * What is this equation for? (in modelling context)   * What is so special about it? (other than if $\nu=0$ then the solution can form a shockwave)


title: Why is it faster (as in proportion to volume) to boil 4 cups of water than to boil 2 cups?

text: I did an experiment where I boiled two cups (500ml) of water in a kettle, and it took 1:30 minutes to reach around 98 C, average. However, when I boiled 4 cups of water, (1L) it only took me 2:30 minutes, when I expected it to be 3:00 minutes. Does this mean that the more water I boil, the faster it will reach 100 C (proportional to its volume, of course)? The kettle and the thermometer used were cooled down first before boiling another batch of 24.5 C water, and I did a few trials. Can you tell me the reason for this? And is there an equation I can use, to figure out, for example, how long it would take to boil 6 cups?


title: What is the 'super' parameter of superconductivity and what is the role of Cooper pair?

text: Only thing I know about superconductors is that here the electrical current face zero resistance. My first question is what is 'super' (physical or mathematical entity) about a superconductor. Or more precisely if I start from the Hamiltonian for a system, which parameter I should try find out to see whether the system is superconducting or not? I read about the mechanism involving the formation of Cooper pair. I understand how they are formed, but don't understand why the pair formation is required for superconductors. Or in other words, how does the pair formation lead to a dissipationless current?


title: What is the general statistical definition of temperature?

text: Temperature in an isolated system is defined as: $$\frac{1}{T} = -\frac{\partial{S(E,V,N)}}{\partial{E}} $$ But I wonder how one can generalize this to a random system. Or for instance to a point in a system. Because in these books about statistical physics they talk often about "temperature gradients in a system", but for these to exist, temperature has first to be defined in every point (although I can't find a general definition). I hope someone can help me out.


title: Would this be a metric?

text: would a matrix $M$ with diagonal entries not necessarily equal 1, i.e. diag $M = (a,1,1,1)$ be a metric if $a \neq 1$ or $\neq 0$? I.e. in this case would this be like some sort of more general euclidean metric or just not a metric at all?


title: Trying to understand EM wave and photon

text: When electrical fields and magnetic fields couple together, it forms electromagnetic waves. And we can "quantized" it and see each "package" of it as photon. So can electrical fields and magnetic fields be each "quantized" and visualized as a particle (half a photon?)


title: Contracting the Riemann tensor issues, p540 hobson

text: I am stuck trying to work through something on p540 in Hobson (General Relativity: An Introduction for Physicists), one is supposed to use the variation of the full Riemann tensor and then contract it to get $\delta R_{\mbox{ }\mu \nu }$: $$\delta R^{\sigma}_{\mbox{ }\mu \nu \rho} = \delta\partial_{\nu}\Gamma^{\sigma}_{\mbox{ }\mu\rho}-\delta\partial_{\rho}\Gamma^{\sigma}_{\mbox{ }\mu\nu} +\delta\Gamma^{\sigma}_{\mbox{ }\tau\nu}\Gamma^{\tau}_{\mbox{ }\rho\mu}-\delta\Gamma^{\sigma}_{\mbox{ }\tau\rho}\Gamma^{\tau}_{\mbox{ }\nu\mu}=\delta\partial_{\nu}\Gamma^{\sigma}_{\mbox{ }\mu\rho}-\delta\partial_{\rho}\Gamma^{\sigma}_{\mbox{ }\mu\nu} $$ and then contract it to show $$\delta R_{\mbox{ }\mu \nu }=[ \partial_{\nu}(\delta \Gamma^{\sigma}_{\mbox{ }\mu \sigma}) - \partial_{\sigma}(\delta \Gamma^{\sigma}_{\mbox{ }\mu \nu}) ] $$ MY ATTEMPT: I thought to contract it by multiplying by $g_{\sigma \alpha}g^{\alpha \rho}$ $$g_{\sigma \alpha}g^{\alpha \rho}\delta R^{\sigma}_{\mbox{ }\mu \nu \rho}=\delta R_{\mbox{ }\mu \nu } $$ but then I get $$g_{\sigma \alpha}g^{\alpha \rho}(\delta\partial_{\nu}\Gamma^{\sigma}_{\mbox{ }\mu\rho}-\delta\partial_{\rho}\Gamma^{\sigma}_{\mbox{ }\mu\nu})=(\delta \partial_{\nu}\Gamma_{\mu\rho}-\delta \partial_{\rho}\Gamma^{\rho}_{\mbox{ }\mu\nu}) $$ Can anyone help me get the correct answer (4th line), I may have made a mistake or could be completely wrong I am not sure.


title: Many photons, one quantum field?

text: If a photon can be described as an excitation in a quantum field, is this the same field for all photons, or does each photon exist in its own field?


title: CP-symmetry and Ward identities and finite temperature

text: I have a few questions about Ward-identities which I summarize here. For each I am very greateful for answers and references to literature. Wikipedia states about Ward-identities: > _The Ward-Takahashi identity is a quantum version of the classical Noether's > theorem, and any symmetries in a quantum field theory can lead to an > equation of motion for correlation functions._   1. Taken seriously this means that Ward-Takahashi identities also exist for discrete symmetries like CP (if present in the Lagrangian) in contrast to a conserved Noether current. If so, I wonder if CP relations between amplitudes, such as $|\mathcal{M}|_{S_i\rightarrow S_f} =|\mathcal{M}|_{\bar{S}_i\rightarrow \bar{S}_f}$ which I can get by computing the amplitudes in perturbation theory and comparing, can be considered Ward-identities. $\bar{S}_{i,f}$ are the CP-conjugated states of ${S}_{i,f}$ respectively.   2. Since Ward-identities are considered to hold for the quantized theory, I wonder if they are invariant under the renormalization group running of parameters. If so, how does RGE running take these restrictions into account.   3. Finally I wonder if there is an established finite-temperature generalization.


title: Lagrangians for non-local equations of motion

text: Say I have a multicomponent field $X_a(x,t)$ such that I know it Fourier modes satisfy the following equation of motion, $(\delta_{ab} \partial_t + \Omega_{ab}(t))X_b(k,t) = e^t \int \frac{d^3p d^3q}{(2\pi)^3} \gamma_{abc} (k,-p,-q) X_b (p,t) X_c(q,t)$ where $\gamma_{abc}$ is a function of the 3 momenta and $\Omega_{ab}$ is a time dependent matrix.   * Now is it legitimate to say that in real space the equation is, $(\delta_{ab} \partial_t + \Omega_{ab}(t))X_b(x,t) = e^t \int \frac{d^3k d^3p d^3q}{(2\pi)^{3+1.5}} e^{ik.x}\gamma_{abc} (k,-p,-q) X_b (p,t) X_c(q,t) $   * Now if one wants to write a Lagrangian such that this would be its equation of motion then one is faced with the "Dirac equation" kind of problem where by an equation of motion which is first order in space-time derivatives can't be gotten from Bosonic varaibles and hence one has to think of the field as Grassmannian. But if I want to keep my $X$ Bosonic then one possible way to do it is to introduce an auxiliary field say $A_a(x,t)$ and write the action as, $S = \int dt d^3k \\{A_a(k,t)[(\delta_{ab} \partial_t + \Omega_{ab}(t))X_b(x,t) - e^t \int \frac{d^3k d^3p d^3q}{(2\pi)^{3+1.5}} e^{ik.x}\gamma_{abc} (k,-p,-q) X_b (p,t) X_c(q,t)]\\}$ Then it seems that the equation of motion w.r.t variation of $A$ gives the equation of motion for $X$   * But does this make sense? (isn't the auxilliary field now becoming interacting with X?)    * Now can one just go ahead and do the usual field theory of calculating n-point functins etc by adding to this $S$ above a source field for both $X$ and $A$ as say $S'= \int dt d^3x [K_a (x,t) A_a (x,t) + J_a (x,t) X_a (x,t) ]$ Or is this all a bit too naive and something subtle is being missed?


title: Is there a way to see light frequencies invisible to the human eye without the use of electronic sensors?

text: I wonder if it's possible to be able to see light frequencies that cannot be normally seen by human eye, without using sensors designed specifically for detecting a certain spectrum range, but rather using some combination of lenses or something else that could transform the light without causing a noticeable latency.


title: NMR rotating frame

text: I'm reading about a linearly polarized field (in the context of NMR). The field is given by $$ {\bf H_{lin}}=2H_1({\bf i}cos(\omega_zt)).$$ This can be created by having a pulse field plus its mirror image; i.e. $${\bf H_1}=H_1({\bf i}cos(\omega_zt)+{\bf j}sin(\omega_zt)).$$ The mirror image field in the lab frame is obviously the same as ${\bf H_1}$ expect $\omega_z \rightarrow -\omega_z.$ Now, my question is what is the expression for the counter-rotating "mirror image" field in the ROTATING frame? I've having some trouble working through this...


title: Trapping Light in a shell

text: I was curious about this. Is it theoretically (mathematically) possible to create a shell such that when a photon 'hits' the shell, the spacetime at the shell surface is such that the photon travels perpetually along the surface. From the outside, it would seem like a black hole since no information could ever leave the 'black shell'. As more and more photons hit the surface, the 'radius' of the shell as seen from the outside would increase since the energy of the shell is continuously increasing. Hawking radiation shouldn't cause an evaporation since the radiation escaping outside the shell would be balanced by radiation inside the shell (my thinking here is that the shell remains infinitely thin and that the apparent increase in radius from outside the shell is not an increase in shell radius, but rather that everything outside the shell is 'pulled closer' to it). Would the shell necessarily collapse on itself at some point? If not, would this not lead to an overall decrease in the entropy of the Universe and Ultimately a kind of reversal of the expansion of the Universe? Thanks


title: Newton first law of motion, braking suddenly in car

text: While driving down the road, if you hit the brakes suddenly, your body moves forward. Explain the reason for this.


title: Modeling non-quantum objects (in finance, sociology etc) using fermionic fields?

text: Please provide (if any) applications of fermionic field theory in non-physics macro contexts (finance, sociology etc). I see only bosonic fields being used mostly. The only (minor) application of fermionic fields that I have come across is in section 7.9.1 in Baaquie's Quantum Finance book where nonlinear forward interest rates are modeled using fermionic fields. Is there an intuitive reason for such asymmetry between bosonic and fermionic field usage at such scales?


title: Virial Theorem and the Energy in a Gas

text: I clearly am interpreting the Virial Theorem incorrectly, but I don't know how. In dipole gases, the molecules can exhibit five kinetic modes, while they can only experience 2 potential modes. Doesn't the Virial Theorem say the two must have an equal distribution of energy?


title: formula for mean free path in two dimensions

text: I'm running some simulations of particle collisions in two dimensions with discretised time and space. In essence, particles only collide if they occupy the same location (cell) at the same time step. The particles are in a 2D box and collide with both each other and the walls, we also assume a closed system (so no gravity etc). I want to use the mean free path $\lambda$ (the average distance travelled by a particle between collisions) to determine the best values for number of particles $N$, rms velocity $V_{rms}$, and box length $L$; as such I want to know how to calculate mean free path in two dimensions. So far I have been able to find a formula on Wikipedia for three dimensions: $$ \lambda=\frac{k_BT}{\sqrt{2}\pi d^2p} $$ where d is the diameter of the particle and p is the pressure.   which I can easily turn into: $$ \lambda=\frac{mv_{rms}^2}{2\sqrt{2}\pi d^2p} $$ but again this is for three dimensions not the two dimensions that I need. So the question is, can this be converted to two dimensions? or does a two dimensional form already exist? Any help with this would be greatly appreciated.


title: Wald problem 4 of chapter 4

text: I'm trying to derive equation 4.4.51 in Wald's GR book (the second order correction in $\gamma$ term for the Ricci tensor): ![enter image description here](http://i.stack.imgur.com/C0oki.png) where $g=\eta+\gamma$. So $g^{\mu\nu}=\eta^{\mu\nu}-\gamma^{\mu\nu}+O(\gamma^2)$. I'm not sure how to proceed, but I got stuck for quite a while: Take $$\Gamma^{\mu}\,_{\alpha\beta}={}^{\left(1\right)}\Gamma^{\mu}\,_{\alpha\beta}+{}^{\left(2\right)}\Gamma^{\mu}\,_{\alpha\beta}+\mathcal{O}\left(\gamma^{3}\right)$$ where $${}^{\left(1\right)}\Gamma^{\mu}\,_{\alpha\beta}=+\eta^{\mu\nu}\partial_{(\alpha}\gamma_{\beta)\nu}-\frac{1}{2}\eta^{\mu\nu}\partial_{\nu}\gamma_{\alpha\beta}$$ and $${}^{\left(2\right)}\Gamma^{\mu}\,_{\alpha\beta}=-\gamma^{\mu\nu}\partial_{(\alpha}\gamma_{\beta)\nu}+\frac{1}{2}\gamma^{\mu\nu}\partial_{\nu}\gamma_{\alpha\beta}.$$ Writing \begin{align} R_{\mu\rho} = & \partial_{\nu}\Gamma^{\nu}\,_{\mu\rho}-\partial_{\mu}\Gamma^{\nu}\,_{\nu\rho}+2\Gamma^{\alpha}\,_{\rho[\mu}\Gamma^{\nu}\,_{\nu]\alpha}\\\ = & \underbrace{\partial_{\nu}\,^{\left(1\right)}\Gamma^{\nu}\,_{\mu\rho}-\partial_{\mu}\,^{\left(1\right)}\Gamma^{\nu}\,_{\nu\rho}}_{^{\left(1\right)}R_{\mu\rho}}\\\ & +\underbrace{\partial_{\nu}\,^{\left(2\right)}\Gamma^{\nu}\,_{\mu\rho}-\partial_{\mu}\,^{\left(2\right)}\Gamma^{\nu}\,_{\nu\rho}+2\,^{\left(1\right)}\Gamma^{\alpha}\,_{\rho[\mu}\,^{\left(1\right)}\Gamma^{\nu}\,_{\nu]\alpha}}_{^{\left(2\right)}R_{\mu\rho}}\\\ & +\mathcal{O}\left(\gamma^{3}\right) \end{align} This is what I have so far: my attempt \begin{align} {}^{\left(2\right)}R_{\mu\rho} = & \partial_{\nu}\,^{\left(2\right)}\Gamma^{\nu}\,_{\mu\rho}-\partial_{\mu}\,^{\left(2\right)}\Gamma^{\nu}\,_{\nu\rho}+\,^{\left(1\right)}\Gamma^{\alpha}\,_{\mu\rho}\,^{\left(1\right)}\Gamma^{\nu}\,_{\alpha\nu}-\,^{\left(1\right)}\Gamma^{\alpha}\,_{\nu\rho}\,^{\left(1\right)}\Gamma^{\nu}\,_{\alpha\mu}\\\ = & \partial_{\nu}\left(-\gamma^{\nu\lambda}\partial_{(\mu}\gamma_{\rho)\lambda}+\frac{1}{2}\gamma^{\nu\lambda}\partial_{\lambda}\gamma_{\mu\rho}\right)\\\ & -\partial_{\mu}\left(-\gamma^{\nu\lambda}\partial_{(\nu}\gamma_{\rho)\lambda}+\frac{1}{2}\gamma^{\nu\lambda}\partial_{\lambda}\gamma_{\nu\rho}\right)\\\ & +\,^{\left(1\right)}\Gamma^{\alpha}\,_{\mu\rho}\,^{\left(1\right)}\Gamma^{\nu}\,_{\alpha\nu}\\\ & -\,^{\left(1\right)}\Gamma^{\alpha}\,_{\nu\rho}\,^{\left(1\right)}\Gamma^{\nu}\,_{\alpha\mu}\\\ = & \gamma^{\nu\lambda}\left(-\partial_{\nu}\partial_{(\mu}\gamma_{\rho)\lambda}+\frac{1}{2}\partial_{\nu}\partial_{\lambda}\gamma_{\mu\rho}+\partial_{\mu}\partial_{(\nu}\gamma_{\rho)\lambda}-\frac{1}{2}\partial_{\mu}\partial_{\lambda}\gamma_{\nu\rho}\right)\\\ & +\eta^{\alpha\beta}\left(\partial_{(\mu}\gamma_{\rho)\beta}-\frac{1}{2}\partial_{\beta}\gamma_{\mu\rho}\right)\eta^{\nu\lambda}\left(\partial_{(\alpha}\gamma_{\nu)\lambda}-\frac{1}{2}\partial_{\lambda}\gamma_{\alpha\nu}\right)\\\ & -\eta^{\alpha\beta}\left(\partial_{(\nu}\gamma_{\rho)\beta}-\frac{1}{2}\partial_{\beta}\gamma_{\nu\rho}\right)\eta^{\nu\lambda}\left(\partial_{(\alpha}\gamma_{\mu)\lambda}-\frac{1}{2}\partial_{\lambda}\gamma_{\alpha\mu}\right)\\\ = & \frac{1}{2}\gamma^{\nu\lambda}\partial_{\mu}\partial_{\rho}\gamma_{\nu\lambda}-\gamma^{\nu\lambda}\partial_{\nu}\partial_{(\rho}\gamma_{\mu)\lambda}+\frac{1}{2}\gamma^{\nu\lambda}\partial_{\nu}\partial_{\lambda}\gamma_{\mu\rho}\\\ & +\frac{1}{4}\eta^{\alpha\beta}\left(\partial_{\mu}\gamma_{\rho\beta}+\partial_{\rho}\gamma_{\mu\beta}-\partial_{\beta}\gamma_{\mu\rho}\right)\partial_{\alpha}\gamma\\\ & -\frac{1}{4}\eta^{\alpha\beta}\eta^{\nu\lambda}\left(\partial_{\nu}\gamma_{\rho\beta}+\partial_{\rho}\gamma_{\nu\beta}-\partial_{\beta}\gamma_{\nu\rho}\right)\left(\partial_{\alpha}\gamma_{\mu\lambda}+\partial_{\mu}\gamma_{\alpha\lambda}-\partial_{\lambda}\gamma_{\alpha\mu}\right)\\\ = & \frac{1}{2}\gamma^{\nu\lambda}\partial_{\mu}\partial_{\rho}\gamma_{\nu\lambda}-\gamma^{\nu\lambda}\partial_{\nu}\partial_{(\rho}\gamma_{\mu)\lambda}+\frac{1}{2}\partial_{\nu}\left(\frac{1}{2}\gamma^{\nu\lambda}\partial_{\lambda}\gamma_{\mu\rho}\right)-\frac{1}{2}\partial_{\nu}\gamma^{\nu\lambda}\partial_{\lambda}\gamma_{\mu\rho}\\\ & +\frac{1}{4}\eta^{\alpha\beta}\eta^{\nu\lambda}(\\\ & \partial_{\mu}\gamma_{\rho\beta}\partial_{\alpha}\gamma+\partial_{\rho}\gamma_{\mu\beta}\partial_{\alpha}\gamma-\partial_{\beta}\gamma_{\mu\rho}\partial_{\alpha}\gamma\\\ & -\partial_{\nu}\gamma_{\rho\beta}\partial_{\alpha}\gamma_{\mu\lambda}-\partial_{\nu}\gamma_{\rho\beta}\partial_{\mu}\gamma_{\alpha\lambda}+\partial_{\nu}\gamma_{\rho\beta}\partial_{\lambda}\gamma_{\alpha\mu}-\partial_{\rho}\gamma_{\nu\beta}\partial_{\alpha}\gamma_{\mu\lambda}-\partial_{\rho}\gamma_{\nu\beta}\partial_{\mu}\gamma_{\alpha\lambda}\\\ & +\partial_{\rho}\gamma_{\nu\beta}\partial_{\lambda}\gamma_{\alpha\mu}+\partial_{\beta}\gamma_{\nu\rho}\partial_{\alpha}\gamma_{\mu\lambda}+\partial_{\beta}\gamma_{\nu\rho}\partial_{\mu}\gamma_{\alpha\lambda}-\partial_{\beta}\gamma_{\nu\rho}\partial_{\lambda}\gamma_{\alpha\mu})\\\ = & \frac{1}{2}\gamma^{\nu\lambda}\partial_{\mu}\partial_{\rho}\gamma_{\nu\lambda}-\gamma^{\nu\lambda}\partial_{\nu}\partial_{(\rho}\gamma_{\mu)\lambda}+\frac{1}{2}\partial_{\nu}\left(\frac{1}{2}\gamma^{\nu\lambda}\partial_{\lambda}\gamma_{\mu\rho}\right)-\frac{1}{2}\partial_{\nu}\gamma^{\nu\lambda}\partial_{\lambda}\gamma_{\mu\rho}\\\ & +\frac{1}{4}\eta^{\alpha\beta}\eta^{\nu\lambda}(2\partial_{(\mu}\gamma_{\rho)\beta}\partial_{\alpha}\gamma-\partial_{\beta}\gamma_{\mu\rho}\partial_{\alpha}\gamma\\\ & -\partial_{\nu}\gamma_{\rho\beta}\partial_{\alpha}\gamma_{\mu\lambda}-\partial_{\nu}\gamma_{\rho\beta}\partial_{\mu}\gamma_{\alpha\lambda}+\partial_{\nu}\gamma_{\rho\beta}\partial_{\lambda}\gamma_{\alpha\mu}-\partial_{\rho}\gamma_{\nu\beta}\partial_{\alpha}\gamma_{\mu\lambda}-\partial_{\rho}\gamma_{\nu\beta}\partial_{\mu}\gamma_{\alpha\lambda}\\\ & +\partial_{\rho}\gamma_{\nu\beta}\partial_{\lambda}\gamma_{\alpha\mu}+\partial_{\beta}\gamma_{\nu\rho}\partial_{\alpha}\gamma_{\mu\lambda}+\partial_{\beta}\gamma_{\nu\rho}\partial_{\mu}\gamma_{\alpha\lambda}-\partial_{\beta}\gamma_{\nu\rho}\partial_{\lambda}\gamma_{\alpha\mu}) \end{align} In my last line, somehow it seems like too many terms cancel out to get what the book has, in particular, there aren't enough $\gamma_{\mu\rho}$ terms..


title: Source material desired for behavior of derivatives of action

text: I'm basically looking for concise commentary, and especially source material/ short discussion pertaining to the following, which I will (emphasizing loosely) state as follows: Suppose a given action $S$ is extremized, i.e satisfies the pertinent Euler Lagrange equation. The topic I'm looking for is something quantifying the behavior of it's derivative(s).... which I know is vague, but phrased vaguely because I'm not trying to pinpoint the discussion too narrowly. For simple example by analogy: Let's entertain the case where $S$ quantifies the arclength in Euclidean space. It's extremized as a 'straight' line, i.e. the shortest distance between two points (in that setup). Now, consider it's (full) derivative with respect to distance along the arc. It would 'seem' that this thing is also extremized. The arclength is a minimum. BUT, clearly in this case, the amount that $S$ changes with respect to a change in the parameter the Lagrangian is integrated over, is also a minimum at each point along the distance. However I'm looking for more broad discussion, proof, disproof, of treatment of cases in general. If $S\rightarrow $ minimum/extremum is $d S\over d\lambda$ also implied to be min/ extremum? Discussion for the converse of that statement would be interesting as well.


title: Does negative energy density (i.e. weak energy condition violation) create closed timelike curves?

text: I remember reading something about Stephen Hawking denying the fact you can't make CTC's (Closed Timelike Curves) without weak energy condition violation. If this is true, where do the light cones point to in the $t$ direction? ![enter image description here](http://i.stack.imgur.com/i2a6d.png) On the end of the right and left, the cone points up (future) but near the cylinder it tilts. In a region with negative energy density, do the cones tilt all the way to the $-t$ direction?


title: Virtual displacement and generalized coordinates

text: I have a doubt regarding the expression of a virtual displacement using generalized coordinates. I will state the definitions I'm taking and the problem. The system is composed by $n$ points with positions $\mathbf r _i$ and subject to $3n-d$ constraints of the form: $$\phi _j (\mathbf r _1, \mathbf r _2,...,\mathbf r _n,t)=0\qquad (1\leq j \leq 3n-d), \qquad (1)$$ that, deriving with respect to time, gives: $$\sum _{i=1} \frac{\partial \phi _j}{\partial \mathbf r _i} \cdot \dot {\mathbf r}_i=-\frac{\partial \phi _j}{\partial t}.\qquad (2)$$ A set of possible velocities $(\mathbf v_1,\mathbf v_2,...,\mathbf v_n)$ is one that satisfies the above system of $j$ equations (with $v_i$ in the place of $\dot r _i$), while a set of virtual velocities is one that satisfies the homogeneous system $$\sum _{i=1} \frac{\partial \phi _j}{\partial \mathbf r _i} \cdot \dot {\mathbf r}_i=0.\qquad (*)$$ Finally, a virtual displacement is given by the product of a virtual velocity by a quantity $\delta t$, with $[\delta t]=[T]$. Now if the allowed positions are determined by $d$ coordinates $q_i$, that is if:$$\mathbf r _i =\mathbf x _i(q_1,q_2,...,q_d,t),$$ my text says that a virtual displacement can be expressed by:$$\delta \mathbf r _i=\sum _{h=1} ^{d} \frac{\partial \mathbf x _i}{\partial q _h}\delta q _h,$$ that I suppose means: $$\delta \mathbf r _i=\sum _{h=1} ^{d} \frac{\partial \mathbf x _i}{\partial q _h}\dot q _h \delta t.$$ But this means that the set of vectors $$\mathbf v _i = \sum _{h=1} ^{d} \frac{\partial \mathbf x _i}{\partial q _h}\dot q _h,\qquad 1\leq i \leq n,$$ satisfies $(*)$, or: $$ \sum _{i=1} ^n \frac{\partial \phi _j}{\partial \mathbf r _i}\cdot(\sum _{h=1} ^d \frac{\partial \mathbf x _i}{\partial q_h}\dot q _h)=0.$$ Now I see that since the $\mathbf r _i$'s satisfy $(1)$ and hence $(2)$, we have:$$- \frac{\partial \phi _j}{\partial t}=\sum _{i=1} ^n \frac{\partial \phi _j}{\partial \mathbf r _i}\cdot(\sum _{h=1} ^d \frac{\partial \mathbf x _i}{\partial q_h}\dot q _h+\frac{\partial \mathbf x _i}{\partial t}),$$ so that:$$ \sum _{i=1} ^n \frac{\partial \phi _j}{\partial \mathbf r _i}\cdot(\sum _{h=1} ^d \frac{\partial \mathbf x _i}{\partial q_h}\dot q _h)=-(\frac{\partial \phi _j}{\partial t}+\sum _{i=1} ^n \frac{\partial \phi _j}{\partial \mathbf r _i}\cdot \frac{\partial \mathbf x _i}{\partial t}),$$ but why in the world should last term be zero? Is the above derivation wrong, or am I wrong in my definitions and should check them back?


title: Question about Charles' law:

text: Charles's law says that the volume of a given mass of a gas is directly proportional to its absolute temperature. This means if we increase one, the other one is automatically increased. So the question that my mind is not accepting is: if we increase volume, than temperature will also increased, but in my observations, volume increased the temperature decreased? How is that possible?


title: How to calculate gravity path integrals about an AdS background?

text: Suppose I have some Lagrangian of some higher derivative gravity coupled to a may be matter fields. Now I want to fluctuate it to quadratic order about an AdS background and calculate the 1-loop partition function. Can someone point to a reference where such a thing might be done? This sounds like something standard which someone would have done... I can't see how one can even write down the fluctuated Lagrangian to quadratic order...And even if I wrote that down, how does one impose that the background is AdS? Then the issues of gauge fixing and ghosts are only added complication.. * * * I can imagine calculating the first order variation in the Christoffel symbols and Riemann and Ricci tensors but that goes only so far.


title: I'm having trouble wrapping my head around force, and thus acceleration, diminishing with a higher velocity when using a constant power

text: Here is an hypothetical situation I thought about that I don't understand. Let's say I have 2 identical rockets with a fixed power. These rockets start to race in the same direction using the full power of their engines. So, by the equation $P=Fv$ we get that with the increase of speed of the rockets, the force exerted by each of them, diminishes. And we also know that $F=ma$, so with increased velocity, the acceleration decreases. At any given time, the rockets are travelling at the same speed and at the same acceleration one from another, so it looks from their perspective that they are in the same position. But let's say rocket B breaks down. And now rocket A starts accelerating away from it. At that moment, the acceleration of rocket A is $a_0$ $$a_A=a_0$$ $$a_B=0$$ $$a_{AB}=a_A - a_B = a_0$$ But A is travelling at $v_0$, while it's travelling at $0$ relative to B. $$v_A=v_0$$ $$v_B=v_0$$ $$v_{AB}=v_A - v_B=0$$ But then, by this logic... If $P=Fv$, and $F=ma$, then $a=F/m=P/mv$ $$a_{AB} = P/mv_{AB}$$ $$a_A = P/mv_A$$ Which are clearly different. So, what am I doing wrong? Where is the flaw?


title: Difference between conventional spiral/helix CT and multi-slice spiral CT

text: I'm currently learning about CT scanners, and come across conventional spiral/helix CT, and then multi-slice spiral CT. What I can get out of it is, that the multi-slice scanner have a lot more detectors than the other, which, if I'm not mistaken make it faster to do scans. So basically, I'm not sure what the benefits are, if any, in conventional spiral CT ? I guess, that due to collimators, the X-rays in conventional spiral CTs are not as many for every turn, since the width of the detectors are not as thick as in multi-slice, meaning, that extra X-rays would just go through a patient, but not detected. But all in all you have to do more turns, if you want to measure the same volume, so the X-ray dose should be around the same, or...? So am I missing something, or is there no advantages at all for conventional spiral CTs compared to multi-slice spiral CTs, or than maybe the price ?


title: Learning about super-symmetric quantum mechanics and integrable systems

text: I'm an undergraduate interested in theory. I recently asked one of my professors, a physicist in the particle theory group at my school, if he'd be willing to take on an undergraduate for a senior project. He told me to check back in at the end of the quarter, and that he might have a project related to super-symmetric quantum mechanics and integrable systems that may be feasible for me to try out then. In the meantime, I want to familiarize myself as much as possible with the relevant math and physics so that I can impress him when I ask again. Do any of you have any books/online sources that you recommend I take a look at?


title: Relation between solutions to Yang-Baxter equations, integrability and exact solvability?

text: Wikipedia mentions that there is an implication: Yang-Baxter solutions yield integrable models, what 1D systems concerns. In arbitrary dimensions, what is the relation, if any, between solutions to Yang-Baxter equations, integrability and exact solvability? If somebody could provide a no-go theorem or cases, where integrability was already ruled out, it would be great.


title: How Does The Macroscopic Wavefunction Build Up?

text: How does the macroscopic wavefunction (the order parameter) builds up from zero value to the a finite value when liquid He undergoes a transition from normal to the superfluid state? How does it physically happen that the whole system ends up in a single quantum state? Does it have anything to do with the overlapping of de broglie waves of different particles because as the temperature is reduced further and further the de broglie wavelength increases? Does this wavefunction grow in magnitude further and further as we lower the temperature? If yes, can it be understood as more and more atoms are coming to the same quantum state?


title: Russell-Saunders Term

text: The Russell-Saunders term states that: $$^{2S+1}L_{J}$$ Now, if I'm not mistaken, $L$ is the total orbital angular momentum, $J = L + S$, but what is $S$ exactly? I know it's the total spin angular momentum, but how do I get it? Is it just, lets say I have a system with three electrons, either $S = 1/2$ or $S= 3/2$, since electrons have $\pm 1/2$ spin, or am I totally wrong here?


title: Kinetic energy question

text: Thought experiment: I am starting to walk on a coridor... From the coridor's perspective I have a certain increase of kinetic energy. From my perspective the whole universe is gaining kinetic energy because it has a speed relative to me. The kinetic energy increase from the coridor's point of view is definitely not equal to the kinetic energy increase of the universe from my perspective. What am I doing wrong?


title: Matrix representation of a triplet state

text: The $SU(2)$ triplet state is typically given in the fundamental representation as a column vector, e.g. \begin{equation} \vec{\Delta} = \left( \begin{array}{c} \delta^{++} \\\ \delta^+ \\\ \delta^0 \end{array} \right) \end{equation} where I use this notation to be consistent with the references below. However, I'm reading through papers (in the context of the type II see-saw mechanism, but I don't think it has much relevance) and most references are using what they call the matrix representation of the $SU(2)$ triplet. My instinct was to assume that they were projecting the fundamental representation onto the Pauli matrices: \begin{equation} \Delta \rightarrow \vec{\Delta } \cdot \vec{ \sigma } = \delta ^{++} \sigma _1 + \delta ^+ \sigma _2 + \delta ^0 \sigma _3 \end{equation} However, this doesn't quite give me the correct result, which is given by, \begin{equation} \Delta = \left( \begin{array}{cc} \delta ^+ / \sqrt{2} & \delta ^{ + + } \\\ \delta ^0 & \- \delta ^+ / \sqrt{2} \end{array} \right) \end{equation} Is there some other way to get this "matrix representation"? Two sample references where they say they are using the matrix representation can be found here (pg. 124, equation 68) and here (pg. 4 equation 1).


title: Uniform Circular Motion: v = v_0 + (a_T)*t as a vector VS as a scalar equation

text: First, I'll explain my confusion, which may be enough for someone to clarify my misunderstanding. Second, to give context, I will give the homework problem (which I've solved successfully) which caused my confusion. Note: I'll use boldface to denote vector quantities. Also, I am in my third semester of Calculus. A particle accelerates around a track. Velocity is given by the below equation. $$ v = v_0 + a_Tt $$ When I substitute in values for this equation to solve homework problems, I use scalar values—in other words, I use magnitudes. I find $|\mathbf{a_T}|$ to use in the above equation. However, if the equation is written as a sum of vectors: $$ \mathbf{v} = \mathbf{v_0} + \mathbf{a_T}t $$ Then I take the magnitude of both sides . . . ($\theta$ is the angle between the positive $x$-axis and $\mathbf{a_C}$.) $$ \begin{align} |\mathbf{v}| &= |\mathbf{v_0} + \mathbf{a_T}t|\\\ &\text{. . . algebra . . .}\\\ &= \sqrt{|\mathbf{v_0|}^2 + 2|\mathbf{v_0}||\mathbf{a_T}|t + |\mathbf{a_t}|t^2}\\\ &\ne |\mathbf{v_0}| + |\mathbf{a_T}|t\ \ \text{*} \end{align} $$ But the starred equation is what I am using for homework problems and I get the answers with no problems. * * * http://i.imgur.com/vUDZuKb.png To summarize my solution:   1. Use $\mathbf{a}$ to find $|\mathbf{a_C}|$ and $|\mathbf{a_T}|$.   2. Substitute $|\mathbf{a_C}|$ into $a_C = \frac{{v_0}^2}{r}$ and solve for $v_0$.   3. Substitute $v_0$ and $\mathbf{a_T}$ into $v_1 = v_0 + a_T t$ and solve for $t_1$ when $v_1 = 0$. I also run into sign issues because I'm treating the value I get for tangential acceleration as the magnitude of a vector. This means that the value for tangential acceleration is positive, but it needs to subtract from velocity. There is a similar issue with the centripetal acceleration. The only way to deal with it properly seems to be to treat vectors counter clock-wise around the circle or pointing away from the circle as positive, and other vectors as negative . . . which is akin to treating the circle as a straight line almost . . . It feels like a hell of a lot of hand-waving, though I am sure that's because I don't understand what is happening.


title: Laser cutting/drilling of copper

text: In industry, it's common to cut steel using 50-200 W $\mathrm{CO}_2$ laser. Meanwhile, for making holes in PCB (35 $\mu$m of Cu), DPSS 355 nm laser is usually used and secondary $\mathrm{CO}_2$ to drill through glass. Why $CO_2$ cuts steel easily, but cannot be used to make holes in Copper? **UPDATE:** Will it help to add thin layer of black paint + pre-heat to 200 °C to get through damn 0.035 mm of Cu?


title: Practical applications for a Bose-Einstein condensate

text: What are the main practical applications that a Bose-Einstein condensate can have?


title: Solution for the Finite 2D Potential Well - Rotational Symmetry

text: I was searching for the eigensolutions of the two-dimensional Schrödinger equation $$\mathrm{i}\hbar \partial_t \mid \psi \rangle = \frac{\mathbf{p}^2}{2m_e}\mid \psi \rangle + V \mid \psi \rangle$$ where the potential is given by $$V(\rho, \varphi)=\begin{cases} V_1 & \rho < R \\\ -V_2 & \rho \geq R \end{cases}$$ using a space representation and cylindrical coordinates, $V_i \geq 0$. I would be happy if someone could point me to a reference or even give the solution here. Thank you in advance   Sincerely Robert ### Request to close the question As I can see in the comments, questions of this kind seem to be **inappropriate**.   The eigensolutions are given by something like $$\psi_m(\mathbf{r},t)=e^{\mathrm{i}(m\varphi-\omega_m t)}\begin{cases} a_m J_m (k_{m,1} \rho) & \rho < R \\\ b_m K_m (k_{m,2} \rho) & \rho \geq R \end{cases}$$ where the $a_m$ and $b_m$ can be calculated from the steadiness of $\psi$ and its spatial derivative in $R$. Furthermore, $k_{m,1/2} = \frac1\hbar \sqrt{\pm\,2m_e(\hbar\omega_m - V_{1/2})}$. I am sorry for any inconvenience.


title: Why electromagnetic waves propagating along x transfers to electron momentum along z?

text: Why EM waves having only x momentum transfers to electron z momentum? Electron begins oscillating along z, so will not radiate EM waves along z direction, to compensate its z momentum. It seems that it is violated conservation of momentum?


title: Magnetron limits

text: What are the practical limits on generated wavelength in a Magnetron? We know that Magnetrons could be used efficiently for generating microwaves for water heating, or for radar applications, but what are the achivable wavelength limits?


title: "Magnetic mnemonics"

text: Over and over I'm getting into the same trouble, so I'd like to ask for some help.   1. I need to solve some basic electrodynamics problem, involving magnetic fields, moving charges or currents.   2. But I forgot all this rules about "where the magnetic field should go if the current flows up". I vaguely remember that it about hands or fingers or books or guns, and magnetic field should go somewhere along something, while current should flow along something else... But it doesn't help, because I don't remember the details.   3. I do find some very nice rule and use it. And I think "that one I will never forget".    4. ...time passes...   5. Go to step 1.  The problem is that you have to remember one choice from a dichotomy: "this way or that way". And all the mnemonics I know have the same problem -- I got to remember "left hand or right hand", "this finger or that finger", "inside the book or outside of the book". Maybe someone knows some mnemonics, that do not have such problem? ![By Bruno Touschek](http://i.stack.imgur.com/SxrgK.gif)


title: Why is it easier to **just** rub charge onto something?

text: We have all experienced the mild **zaps** caused by static build up of charges due to triboelectric effect > Electric field strenth of air = $ 3 × 10^6 Vm^{-1} $ Now, suppose we feel the zaps after getting $0.5 mm$ distance closed to some other grounded or otherwise object, this would imply > $ V = 3.6 × 10^6 × 0.5 × 10^{-3} V = 1.8 × 10^3 V$ > > assuming capacitance between let's say hand and grounded object to be of the > sorts of parallel plate capacitor; then let the area be $ 1 cm^2$ and the > distance as before $0.5 mm$, this gives a capacitance of $0.2 × 10^{-12} F$ > this would give a charge of magnitude $0.36 nC$ > > Now if I had to charge a small ball of comparable size lets say $5 mm$ > radius. The needed potential can be calculated as $ V = > \frac{1}{4\pi{\epsilon_0}} \frac{0.36 × 10^{-9}}{5 × 10^{-3}}$ This would > give $ V = 648 V$ This required potential is near about 3 times the supply voltage we get at home(India, 220 V) So as it seems that, if we want to charge the ball, it would be easier to just the rub it really hard till it gets desired charge rather than attaching it to a high tension battery ! Why is it so ?


title: Is there literature on a continuous mass spectrum for the Higgs field?

text: Various masses for the Higgs field are compatible with experiment, but is it possible that the Higgs field is not observable because it has a continuous mass spectrum? Work in the 60s and 70s on free fields that have a continuous mass spectrum, often called generalized free fields, established that there is no simple particle interpretation if the mass spectrum is not delta function valued, which I take it can be described as due to destructive interference between the different mass components. [I'm taking that largely from R. F. Streater, _Rep. Prog. Phys._ , **38** , 771-846 (1975).] Is the possibility that the Higgs has a continuous mass spectrum to be found in the literature, or is there some sufficiently obvious anomaly or other reason why it is not considered in any detail?


title: Mie scattering intensity and spherical particles on

text: I am running an experiment where I shine infrared light on (almost) spherical particles on the micron scale (PM2.5 - PM10). I then look at the (90 deg) scattering properties to try and size the particles. I am looking for the theory of Mie scattering and its intensity, specifically how the scattered intensity depends on the surface area of the particles. Looking for suggestions of good literature on this, or just simple equations.


title: Proton-proton collisions

text: I have a question about proton-proton collisions at the LHC. Firstly, the 4-momentum $p^\mu=(E/c,\vec{p})$ can be represented as $p^\mu =(m_T \cosh \Psi, p_T \cos \phi , p_T \sin \phi, m_T c \sinh \Psi)$ where $\Psi$ is the rapidity, $\phi$ is the azimuthal angle and $p_T=\sqrt{p_x^2+p_y^2}$ the magnitude of the momentum in the transverse plane. Then there is $m_T$ which if you use the conservation of 4-momentum $m_T^2=m_0^2+p_T^2/c^2$. The question is about a collision between two quarks inside the protons which create a new particle with mass $m$ and rapidity $\Psi$, known. The protons in the beam have 4-momentum $p^\mu_1=(E_{\text{beam}}/c,0,0,E_{\text{beam}}/c)$ and $p^\mu_2=(E_{\text{beam}}/c,0,0,-E_{\text{beam}}/c)$ (i.e. travelling in opposite directions) quarks inside the protons carry the fractions $\xi_1, \xi_2$ of their respective proton's momentum. Can I assume here that the form of the 4-momentum for a given quark is $p^\mu_{\text{q1}}=\xi_1 p_1^\mu$? It seems that there could be a way that the quark could have transverse momentum which cancels with the other quarks within the system giving a net transverse momentum of zero? If I proceed as above (no transverse momentum) the invariant mass squared of the the quark-quark system before the collision is $m^2=4E_{\text{beam}}\xi_1\xi_2/c^2$ if this gives rise to a new particle with mass and rapidity known, can I constrain either of the $\xi$s? If the quarks before the collision had $p_T=0$ is this necessarily true for the new particle? Would this mean that $4E_{\text{beam}}\xi_1\xi_2/c^2=m_T^2c^2-p_T^2=P^2$?


title: Why do cold metal plate make less noise?

text: I have a metal gate in my home so I have noticed that when it is a hot day and if I hit it it makes a lot of noise but when there is a rainy day and the weather and the gate re cold and if I hit the gate it makes very much less noise. I am assuming that when there is a cold day the metal is much dense so the vibration travel through it to the ground more easily as compared to that on a warm day, am I correct. ![enter image description here](http://i.stack.imgur.com/8jh1H.png)


title: Relativistic genarization of Quantum Harmonic Oscillator

text: I am trying to find out relativistic description of a quantum harmonic oscillator. For a classical relativistic oscillator mass is a function of co- ordinates(http://arxiv.org/abs/1209.2876). $$m(x)=m-\frac{m\omega^{2}x^{2}}{c^{2}}$$ I want to get a quantum mechanical model for this particular hamiltonian $$H=\sqrt{p^{2}c^{2}+m(x)^{2}c^{4}}$$ First,I thought of re-writing this hamiltonian with the help of $\alpha$ and $\beta$ matrices.I got $$H=\vec{\alpha}.\vec{p}+\beta m(x)$$ But by adopting this prescription implies wave function should be $4$-component spinors. How can I get $4$-component spinors which would reduce to $1$-component wave function in non-relativistic quantum mechanics? If someone can suggest some other methods to solve for the hamiltonian that would also help.


title: Eddy currents: Explanation of the force slowing down a moving metal plate

text: Suppose a solid metal plate moves trough a homogenous magnetic field $\vec{B}$ as shown in the following image. You will see that the plate will slow down (as long as not the entire plate is in the homogeneous field). For example in the upper part of the plate in the picture there are eddy- currents induced because the magnetic field is changing in that region (the part of the plate is going to enter the homogenous magnetic field). To explain that the plate is slowed down often Lenz law is cited which states in this case that the magnetic field produced by the eddy-currents is opposes the changing magnetic field. But how can one explain that this leads to a force which slows down the plate in detail? In particular the situation is not so easy such that one could speak of magnetic north and south poles... Is there a general mathematical expression for the force in this case? (It's clear that one cannot calculate it already because I didn't give any technical details, but I just want a general equation for this situation which illustrates the concepts in this example from a mathematical point of view) ![Eddy Currents](http://i.stack.imgur.com/jkVLL.png) Do you have any good references about this point?


title: Normalization of photon wavefunction and beam splitters

text: I am having trouble with the normalization of photon wave functions, when passed through a beam splitter. Let me define the single photon state as $$|1\rangle = \int \text{d}\omega \phi(\omega) a^\dagger(\omega)|0\rangle$$ and the two photon state as $$|2\rangle = \int \text{d}\omega_1\text{d}\omega_2\phi(\omega_1)\phi(\omega_2) a^\dagger(\omega_1)a^\dagger(\omega_2)|0\rangle$$ These states are normalized if $\int \text{d}\omega |\phi(\omega)|^2 = 1$. I have used the operator $a^\dagger$ to denote the creation operator in the first of my two modes. The operator for the second mode is $b^\dagger$. Let me insert the two photon state into the first port of a beam splitter which takes $a \rightarrow \frac{a-b}{\sqrt{2}}$. My output state is then $$|\psi\rangle = \frac{1}{2}\int \text{d}\omega_1\text{d}\omega_2 \phi(\omega_1)\phi(\omega_2) [a^\dagger(\omega_1)-b^\dagger(\omega_1)][a^\dagger(\omega_2)-b^\dagger(\omega_2)]|0,0\rangle$$ I can compute the amplitude of the various output events. The amplitude for both $|2,0\rangle$ and $|0,2\rangle$ is $\frac{1}{2}$. The amplitude for $|1,1\rangle$ comes out to be $-1$. The state does not seem to be normalized. Why not? ### Further details I compute amplitudes in the following way. $$\langle 2,0|\psi\rangle = \frac{1}{2}\int \text{d}\omega_1^\prime\text{d}\omega_2^\prime\phi^*(\omega_1^\prime)\phi^*(\omega_2^\prime) \int \text{d}\omega_1\text{d}\omega_2 \phi(\omega_1)\phi(\omega_2)\langle 0,0 | a(\omega_1^\prime)a(\omega_2^\prime) [a^\dagger(\omega_1)-b^\dagger(\omega_1)][a^\dagger(\omega_2)-b^\dagger(\omega_2)]|0,0\rangle$$ Only the term corresponding to two photons in the first mode survives. $$\langle 0,0 |a(\omega_1^\prime)a(\omega_2^\prime) a^\dagger(\omega_1)a^\dagger(\omega_2)|0,0\rangle = \delta(\omega_1^\prime-\omega_1)\delta(\omega_2^\prime-\omega_2)$$ Therefore the above integrals collapse to $$\frac{1}{2}\int \text{d}\omega_1\text{d}\omega_2 |\phi(\omega_1)|^2|\phi(\omega_2)|^2 = \frac{1}{2} $$ Similarly $$\langle 1,1|\psi\rangle = \frac{1}{2}\int \text{d}\omega_1^\prime\text{d}\omega_2^\prime\phi^*(\omega_1^\prime)\phi^*(\omega_2^\prime) \int \text{d}\omega_1\text{d}\omega_2 \phi(\omega_1)\phi(\omega_2)\langle 0,0 | a(\omega_1^\prime)b(\omega_2^\prime) [a^\dagger(\omega_1)-b^\dagger(\omega_1)][a^\dagger(\omega_2)-b^\dagger(\omega_2)]|0,0\rangle$$ This time two different terms survive, one corresponding to the $\omega_1$ photon jumping to the second channel, the other corresponding to the $\omega_2$ photon jumping to the second channel. $$\langle 0,0 |a(\omega_1^\prime)b(\omega_2^\prime) a^\dagger(\omega_1)b^\dagger(\omega_2)|0,0\rangle = \delta(\omega_1^\prime-\omega_1)\delta(\omega_2^\prime-\omega_2)$$ $$\langle 0,0 |a(\omega_1^\prime)b(\omega_2^\prime) b^\dagger(\omega_1)a^\dagger(\omega_2)|0,0\rangle = \delta(\omega_1^\prime-\omega_2)\delta(\omega_2^\prime-\omega_1)$$ Therefore, the integrals collapse and we get $$2\times \frac{-1}{2}\int \text{d}\omega_1\text{d}\omega_2 |\phi(\omega_1)|^2|\phi(\omega_2)|^2 = -1$$ I have added the amplitudes of these two terms rather than the probabilities because the two photons have an identical spectrum. Is this incorrect for some reason? ### Further edited in details I have used the following unitary for the transformation of the field operators. $$\frac{1}{\sqrt{2}}\begin{bmatrix} 1 & -1 \\\ 1 & \phantom{-}1 \end{bmatrix}$$ This transformation can be found in any quantum optics book such as Scully and Zubairy. The normalization of the $|2\rangle$ state follows $$\langle 2|2\rangle = \int \text{d}\omega_1^\prime\text{d}\omega_2^\prime\phi^*(\omega_1^\prime)\phi^*(\omega_2^\prime) \int \text{d}\omega_1\text{d}\omega_2\phi(\omega_1)\phi(\omega_2) \langle 0| a(\omega_1^\prime)a(\omega_2^\prime) a^\dagger(\omega_1)a^\dagger(\omega_2)|0\rangle$$ Now note that there are four operators, each corresponding to a different frequency. The crucial commutation relation between them is $[a(\omega_1),a^\dagger(\omega_2)] = \delta(\omega_1-\omega_2)$. The braket is non-zero if I equate $\omega_1^\prime$ and $\omega_1$ and similarly $\omega_2^\prime$ and $\omega_2$. The braket reduces to $$ \langle 0| a(\omega_1) a^\dagger(\omega_1) a(\omega_2) a^\dagger(\omega_2)|0\rangle \delta(\omega_1^\prime-\omega_1) \delta(\omega_2^\prime-\omega_2) = \delta(\omega_1^\prime-\omega_1) \delta(\omega_2^\prime-\omega_2)$$ Then the integral above collapses to $$\int \text{d}\omega_1\text{d}\omega_2|\phi(\omega_1)|^2|\phi(\omega_2)|^2 = 1$$


title: Why the position of apparent horizon depend on choice of Sigma t of foliation?

text: Whether a 2d surface is a trapped surface is independent of choice of Sigma t of foliation. Why its outer boundary (apparent horizon) depends on choice of Sigma t of foliation?


title: Relationship between Mars and Earth rotation

text: Is it by pure random chance that Mars and the Earth have nearly the same day duration (Mars day is barely 40 minutes longer, which is just 3% difference), or there is some causal relationship between the two?


title: Laminar flow in low-permeability material. Which is faster, gas or a liquid?

text: I consider a simple low-permeability material $(k \sim 10^{-18}\;\text{m}^2)$. Now I either apply a gas (e.g. nitrogen) pressure or liquid (water) pressure of - let's say $10\;\text{MPa}$ - on one side. I'm assuming that the gas viscosity is about two orders of magnitude smaller than that of water, but on the other hand the gas bulk modulus is not constant and smaller. Is there a straightforward way to approximately estimate, whether the gas or the liquid will permeate faster through the host matrix? As simple as the question sounds I could not find a satisfactory answer up to now. While some lab results I know of suggest the gas to be faster, a simple Darcy-flow code showed the water to be faster. Thank you very much guys!


title: I think there is something wrong with this problem. How do you know if the vertical acceleration is zero?

text: Suppose a block with mass 3kg is on a frictionless table and a force of 15N pulls it from an angle of 38 degrees above the horizontal. 1) What is the apparent weight of the block? 2) What is the minimum force for the block to take off? **Solution Given** $\sum F_y = n - mg - F\sin\theta = 0$ $\sum F_x = F\cos\theta = ma_x$ Solve for n, the apparent weight or the normal force. n = 20.16N The minimum force to take off when n = 0 $mg = F\sin\theta$ $F = \frac{mg}{\sin\theta}$ $F = 47.75N$ **Question** Why on earth is the force on the vertical direction 0? Shouldn't it be $\sum F_y = n - mg - F\sin\theta = ma_y$ There is a horizontal acceleration component, but no vertical acceleration.


title: Does compression of matter increase gravity?

text: Black holes are thought to originate when a star burns out and collapses to a smaller physical size. This compresses it's matter, but would that increase it's gravity? And would it be enough to capture light? If that star's light could have been observed prior to it's collapse, how could it's burning out and collapse enable it to become a black hole and capture light?


title: Where is the atmospheric pressure equal to zero?

text: Consider the following question. ![enter image description here](http://i.stack.imgur.com/vFwCV.png) My calculation is as follows. \begin{align*} P_C = P_A + \rho g h\\\ P_A = P_C - \rho g h \end{align*} Because $P_C=P_0=\rho g h$ then $P_A=0$. Is it correct that the zero atmospheric pressure occurs at A? It seems to me a bit illogical because according to other resources: > The atmospheric pressure decreased by 1 cmHg if we climb 100 m upward. Thus the zero pressure should be very far above the point $A$.


title: When driving uphill why can't I reach a velocity that I would have been able to maintain if I started with it?

text: Consider these two situations when driving on a long straight road uphill:   1. Starting at a high velocity $v_h$, which the car is able to maintain.   2. Starting at a lower velocity $v_l$, and then trying to reach $v_h$ while driving uphill. In my experience I've noticed that in case 2 it is very hard, and sometimes impossible, to reach the velocity $v_h$, even though if the car had started in that velocity it would have been able to maintain it. This observation was confirmed by another person I know. If we do a simple analysis of the problem assuming the engine outputs some fixed power $P$, it seems that there should be no problem reaching $v_h$. **Is there something in the inner workings of the car (like the transmission or fuel injection for example) that would make it harder than expected to accelerate uphill?** For simplicity let's assume the car is always in the same gear.


title: How is the formula derived for finding the field of view given drift time and declination of a star?

text: How is the formula for field of view derived given the drift time of a star and its declination? $FOV = (drift\ time \cdot \cos(declination)\cdot360 ) / 86164$


title: Perturbation of coupled spin

text: I am given a system with Hamiltonian (all 1/2 spins) $$H_0=\alpha(S_1\cdot S_2)$$ I broke it down and found that there were four eigenstates: $|1,[0,\pm1]\rangle$ and $|0,0\rangle$. Each has an associated eigenvalues. I am then told that a weak magnetic field is applied giving a perturbation $$H_V=-\mu B_0\cdot(\sigma_1+\sigma_2)$$ I am supposed to use stationary state perturbation theory to find $E_n^{(1)}$. So I know that the perturbed energies are given as $$E_n^{(1)}=\langle \psi_n^{(0)}|H_V|\psi_n^{(0)}\rangle$$ I am a bit confused on what to use for the $\psi_n$ and then the $\sigma$ matrices. I know that the eigenstates are those given above, but I do not see how I can form a matrix from those and then I am unsure of what sigma would be in order to calculate the matrix element


title: Do photons interact only with single particles or can a single photon be absorbed by 2 or more particles simultaneously?

text: Say I have 100 particles in a vacuum that are spread out such that their movement caused by a time variant electromagnetic field does not have an effect on each other. That is the particles can not exchange any energy among themselves. The varying electromagnetic field must be experienced in quanta (a photon particle). Does this mean that it will take at least 100 photons worth of light to excite all the particles?


title: Michaelis-Menten derivation for 2 enzyme substrates

text: We know that the Michaelis-Menten derivation for the following reaction: $E + S \rightleftharpoons ES \rightarrow E + P$ However, what if the reaction took place in a different scenario whereby: $E + S \rightleftharpoons ES_1 \rightarrow ES_2 \rightarrow E + P$ What would be the derived Michaelis-Menten equation now? [ES2] is assumed to be at steady state assumption. What would be the derived Michaelis Menten Equation?


title: Solids with very large specific heats

text: I've noticed that for many solids, specific heat is around 1 Kj/Kg*K is there a theoretical explanation for this? What common solids have the highest heat capacity per mass and per volume?


title: How to understand the emergent special relativity in the superfluid?

text: The superfluid vacuum theory was proposed to understand some features of the vacuum (aether) from the emergence point of view. Although made up of non- relativistic atoms, the low-energy excitations of superfluid, i.e. the phonons, are said to be "relativistic". I know that the superfluid phonons do have the linear dispersion $\omega = c k$, and the low-energy effective theory is Lorentz invariant. Also the superfluid is inviscid, so that the aether has no drag force. But I still doubt that the special relativity can really emerge in the superfluid. Because I can not show the following statement: **The velocity of sound in the superfluid is the same for all inertial observers, regardless of their relative motion to the superfluid.** I believe this should be a fundamental feature of relativity, but I do not see how to prove it. The velocity addition formula for sound is not the same as that for light. So I am very confused when people say that the phonons are relativistic. Any understanding of the emergent relativity in superfluid would be appreciated.


title: Dopant concentration and changes in band gap energy

text: Thanks to this lovely website, I was able to pop out reasonable values for my band gap energies from a translucent material. As expected, I found a decrease in band gap energy due to my treatments. Now, I also need some numbers put to dopant concentration. My dopants, in this case, are reduced forms of the semi-conductor material I am treating, but I think the concept can be generalized to all dopants. So my query is as follows: If I know how the band gap changes between a doped and undoped material, is there a formula that will give me the amount or density of the dopants in the material? For example, say my band gap before doping was 3.0eV and after doping it was 2.5eV, can I pull out how many dopant atoms were added to my material?


title: Horizontal $E$-field for a charged conducting disk

text: For part of a simulation I am writing, I need to know the electric field emitted from a charged conducting disk. If the disk was laid out in the $x$-$y$ plane, I am interested in the field in that same plane, not vertically. The method for getting it in the axis of the disk (the $z$-axis) is easy, but I can't figure out how to do this. Does anyone know the equation or how to calculate it?


title: Is there a way to decrease the rate of nuclear Beta decay?

text: In that question and its answers it was mentioned that you could trigger radioactive decay by bombarding atoms with gamma rays of the right energy level (there may be other solutions I do not know about, but of course if you bombard with neutrons you can trigger nuclear reactions) I am mainly interested by beta-decay. Is it possible to _decrease_ the probability of the beta decay of some radioactive material by a physical treatment? Is the rate fairly independant of temperature and external magnetic field for instance?


title: Very High Power Light Beams

text: Consider a collimated light beam in a vacuum (I am being unspecific about the frequency, anything from radio to gamma). If the beam power/cross section was increased _indefinitely_ would new phenomena occur at certain energy densities? I gather that pair-production couldn't occur due to momentum considerations but would there be some interaction with the quantum vacuum? At sufficiently high power is there a self-gravitation effect of the beam, a point touched on by a previous question? What if the beam was reflected back on itself by a mirror? Note: the motivation for this question relates to theoretical limits for beam weapons.


title: How is voltage generated due to difference in temperature in a thermocouple?

text: How is voltage generated due to difference in temperature between the two junctions of a thermocouple? What happens to the electrons at the junctions of the two materials and how do they behave when they are immersed in hot and cold medium?


title: Commutating Annihilators with a beamsplitter

text: I am reading Nielsen and Chuang on P. 291, for anyone interested in the origin of my question. Given an annihilator $a$ and its corresponding creator $a^\dagger$ such that $[a,a^\dagger] = 1$ and another annihilator $b$ with creator $b^\dagger$, an argument in a proof claims the following: Let $G = a^\dagger b - ab^\dagger$. Then, $[G,a] = -b$ and $[G,b] = a$. I don't see how these two relations hold. Can someone please point me in the right direction or prove them? Thank you SOCommunity!


title: Can one pump water without providing an external source of energy?

text: I have a farm situated right beside a canal through which we use to get water for our crops. I am not financially sound to buy an electric motor. The water is 15ft below the ground level. So can anybody suggest me a feasible way to get the water out of it by using air pressure or any simple instrument that helps me out? Here I attached an image for reference![enter image description here](http://i.stack.imgur.com/Y8LXW.jpg)


title: why ammonia is not a competitive energy carrier?

text: reading this article about Germany planning to store renewable energy as hydrogen gas, i'm left wondering why they deem so great this plan, compared with other alternatives like ammonia as a energy carrier for storage? Is there a comparison chart between the different processes to store electric energy in chemical format?


title: Are intergalactic voids inertial frames?

text: I know earth is a non-inertia frame, so is the solar system, the milky way, the local group and even our super cluster. But what about all the space, the void that is left outside of the superclusters. From wikipedia "Interspersed among superclusters are large voids of space in which few galaxies exist. Superclusters are frequently subdivided into groups of clusters called galaxy clouds." When we talk of the voids the only movement we can talk about is the space expansion, not even a real movement as I have learnt after several debates and discussions. So are these voids actually inertial frames? If they are not, why are they not?


title: I find a problem in the law of angular momentum conservation

text: Consider a system of spring mass as shown in the figure.entire loop is free but only one nail is there at the point A... Initially the mass is at rest and then released. Assume that the spring is initially stretched and is in equilibrium in point C. Now on the loop PQ (P=initial position of bead, Q=diametrically opposite) of the path the centre of the path is Point A. We are assuming that the loop is a massless, smooth rod. So the normal provided by the loop on the bead will pass through the axis. But the force of spring on the bead will be tangential to the path and the torque related to this force on the particle about the considered axis will be nonzero. Also no other forces are acting on the bead. So about the considered axis the angular momentum of the bead is changing. But if we analyse the entire loop+spring (massless)+bead system then there is no non-zero external torque acting on system. Since the only moving portion of the system is the bead angular momentum of the entire system is also changing. Note that the spring force is an internal force for the considered system. Where am I wrong ? ![Sketch of the problem](http://i.stack.imgur.com/WUIhb.jpg)


title: Probability density of electron orbital

text: Why the probability of density is higher in the area that is closer to the nucleus? I'm a high school student. I don't know much about wave functions.


title: Lattice pattern

text: I'm trying to find and outline a non-primitive conventional unit mesh, I'd also like to find any mirrors planes and rotional symmetry axes. ![enter image description here](http://i.stack.imgur.com/vNPjZ.png) I first thought it would be like this: , but I was told the answer was smaller in a different thread. ![enter image description here](http://i.stack.imgur.com/dge86.png) This was the solution left by LDC3 ![enter image description here](http://i.stack.imgur.com/NM7pS.png) Which one is correct? I don't know if his is correct because I thought the conventional unit mesh connected to points that looked the same as the original small pattern I picked(the small hexagon in the middle) Also tried the axis of symmetry, but am unsure if 8 lines from the topside or 6 for a hexagonal pattern should come out. https://imgur.com/Bkjwhih I think for hexagonal structures there should be more lines like this https://i.imgur.com/GopAzmM.png Any help is much appreciated.


title: mechanism of annihilation

text: Can the annihilation of matter and antimatter be explained by the electro-weak interaction? Can pair-production be explained in the same way?


title: More on the Feynman Path Integral Formula  in Brian Cox' Lecture and its Consequences

text: This is a continuation of this question about Brian Cox' lecture _Night with the Stars._ I know the main steps to get from $K(q",q',T)=\sum_{paths}Ae^{iS(q",q',T)/h}$ to $\Delta t > \dfrac{m(\Delta x)^2}{h}$ as stated below, but can you expand? (just read below) PART 1 The action function $S(q",q',T)$ is given by $ S = \displaystyle\int dt\left( \dfrac{1}{2} m v^2 -U\right)$. For a classical path that goes uniformly from one point to the other you have $v = \dfrac{\Delta x}{\Delta t}$ and so you get $S \propto m \left(\dfrac{\Delta x}{\Delta t}\right)^2\Delta t=m\dfrac{(\Delta x)^2}{\Delta t}$. What are the processes and steps taken to you get to $S \propto m \left(\dfrac{\Delta x}{\Delta t}\right)^2\Delta t=m\dfrac{(\Delta x)^2}{\Delta t}$? (explained clearly please). PART 2 $S/h$ appears as a complex phase term. To make it small we set $S/h < 1$, and we can then deduce that $\Delta t > \dfrac{m(\Delta x)^2}{h}$. What are the processes and steps taken to then get to $\Delta t > \dfrac{m(\Delta x)^2}{h}$?


title: Particles, waves and parallel wire filters. Transmission formula?

text: If I think of a photon as a particle, I think a parallel wire filter should transmit proportionally to the uncovered area. (and reflect proportionally to the covered area). Obviously polarization adds complexity by introducing a second factor to be considered. On the other hand, if I consider the photons as waves, I think the transmission should be less sensitive to the area covered. Both a fairly thick and a fairly thin filter should transmit most of the light in one polarization and reflect most of the light in the opposite polarization. Which is correct? To be more precise, consider a spectrum of filters with thick and thin wires. (on the same spacing). At one extreme end of the spectrum the thinest wires approach an empty space or vacuum, which transmits everything and reflects nothing. At the other end the thickest wires merge together and form a continuous surface, in effect a mirror which reflects everything and transmits nothing. My question is what happens in between these two extremes? Of course there are two answers, one for parallel polarization and one for perpendicular polarization. What are the two formulas for transmission as a function of wire thickness, or percent of space blocked? My guess is the answer for the transmitted polarization is close to a step function, only rounded off in the corners. Other possibilities is a linear function decreasing from one hundred percent to zero, or from fifty percent to zero. What are the two correct equations for the two complementary polarizations? Or more specifically, what is the correct formula for the transmitted polarization? For part credit, just give a reference or a convincing derivation showing whether this function is closer to a step function or closer to a linear approximation. For full credit, provide a reasonably accurate formula with source or derivation showing the full transition from zero to one hundred percent wire coverage. It should be accurate enough to get the round off in the corners right to a percent or two (at least 5%) or prove that there is no significant round off. A good answer will have either a derivation or a reference to a reputable source.


title: Configuration space of particles in the box

text: The notion of entropy says that we can count microstates that correspond to macrostate. But, I do not understand how this can be done. Does it imply that the state space is discrete (finite or countable)?


title: particle density operator in second quantization form

text: The particle-density operator is given by $n(\mathbf{x})=\sum_{\alpha}\delta^{(3)}(\mathbf{x}-\mathbf{x}_{\alpha})$, then how to derive its representation in terms of creation and annihilation operators $n(\mathbf{x})=\psi^{\dagger}(\mathbf{x})\psi(\mathbf{x})$?


title: A Kinematics Problem

text: > Problem: From the top of a building of height h = 100 m I throw a stone up > with velocity 10 m/s. What is the maximum height it reaches, and when does > this occur? First I solved for $h_{max}$, and found that its value is 105.1 m. Then, I solved for the time $t_{max}$ when the stone is at its maximum height in accordance with the initial conditions given. I used the following formula to find $t_{max}$ $h_{max}=-\frac{g}{2}t^2+v_0t+h_0$ where $v_0$ is the initial velocity, and $h_0$ is the initial height. I plugged in the values of all the variables and/or constants and ended up with a quadratic equation in $t$. Then, when I solved for $t$ using the quadratic formula, I got these two values $t_{max}=1s$ or $t_{max}=1.04s$. I know that it takes the stone 1s to reach $h_{max}$, but what about $t_{max}=1.04s$? Does this solution ($t_{max}=1.04s$) indicate something not apparent or should I ignore it? Note: I've taken the value of $g$ equal to $9.8 m/s^2$.


title: Mathematical definition of Bogomol'nyi–Prasad–Sommerfield (BPS) states

text: What is the mathematical definition of Bogomol'nyi–Prasad–Sommerfield (BPS) states, independent of any specific physical theory.


title: Can gravity be reversed?

text: I'm wondering if any set of hypothetical conditions could result in us gravitating in a different direction. Specifically a reversal of gravity on the surface of Earth (or at least part of it), so that for people there, the ground would be above their heads, and everything unattached to it would fall into the sky. At first I thought along the lines of a black hole passing by our solar system, but realized that could probably not suck things off the Earth's surface. Could anything other than a biblical rapture do this without breaking obvious physical laws? I know it's far fetched but thought I'd try anyway.


title: Fermion annihilation amplitude

text: Is there a physical reason why annihilation of 2 identical fermions with mass m to 2 scalars amplitude for $s=4m^2$ (fermions at rest) is zero? For example we can have 2 scalars annihilating in 2 scalars and the amplitude is non-zero if we have a quadrilinear term in the largangian $\phi_1^2 \phi_2^2$. Also a decay of a scalar to 2 fermions at $s=4m^2$ (m - fermion mass) is zero. I consider a Yukawa lagrangian $Y\bar{\psi}\psi\phi$, an annihilation in t and u channels at the tree level. I actually am interested in amplitude modulus squared and take an average over all spin states.


title: Is it better to build a space elevator from GEO down to the surface of the Earth?

text: Having just finished Arthur C. Clarke's "The Fountains of Paradise", Clarke seems to make a distinction between starting construction of a space elevator from geosynchronous orbit and working on our way down until we hit a point on Earth and constructing starting from Earth and going up. My question is, is there a difference between the two? Theoretically speaking, is starting from GEO and coming down different or even easier than starting from Earth and just building a really tall building and going up? It seems to me that starting from the surface and going up might be easier. We don't even have to restrict ourselves to the equator. Is there some fundamental difference or a deep physical reason why one works better than the other? Building a really really tall building/space elevator (36,000 km up) from the surface seems to work just as well because the top of the building/elevator will be "in sync" with its base on the surface, no? Effectively behaving like geosynchronous even away from the equator at higher latitudes? Its not like the top will be "moving" at different speeds than its base hence the elevator breaks. I thought only free-bodies such as satellites will have different speeds depending on their height in orbit because they are constantly "falling". If something is connected to the surface via a rigid solid body then it will move at the same rotational speed as its base point on the surface of the Earth. Thanks you.


title: potential energy of a dipole?

text: The very popular from of potential energy of the dipole is $-P.E$. But in the derivation of it, we have negelected the potential energy of the pair of charges constituting the dipole. will this not affect further treatment when we try to apply this to understand aligning of molecular dipoles in electric field? If yes, then how will the modified potential energy function look like?


title: Leakage rate Calculation

text: I have to calculate leak rate for helium/air from an enclosure which is made of fabric. This is similar to helium balloons. There are small holes in the fabric. How should i proceed ?


title: Proton Radius Puzzle, Is it possible that proton's radius differ depending on how you measure it? What does that even mean?

text: I just read this NewScientist article, and I was stunned by its results. So I found the original paper here on arxiv. In the introduction of the paper it is stated: > The recent determination of the proton radius using the measurement of the > Lamb shift in the muonic hydrogen atom [1, 2] startled the physics world. > The obtained value of 0.84087(39) fm differs by about 4% or 7 standard > deviations from the CODATA [3] value of 0.8775(51) fm. The latter is > composed from the electronic hydrogenate atom value of 0.8758(77) fm and > from a similar value with larger uncertainties determined by electron > scattering [4] What Randolf Pohl _et al_ basically did, is that they measured the radius of the proton using more accurate techniques. While they were expecting to just get a more accurate result(few more decimal places), their proton radius is clearly smaller than all other experiments before. How they did the experiment is described in the paper I pointed to. The conclusion of the paper is: > The proton radius puzzle is real. A physical parameter should not depend on > the method of extraction. Yet here we have two highly precise methods based > on electronic and muonic hydrogen atoms that disagree significantly. The > possible origins of the disagreement are the subject of much of this text. > To summarize: the possibilities are > >   * The electronic hydrogen experiments are almost, but not quite, as > accurate as stated (Sect. 2.2). >   * The QED calculations are almost, but not quite, as accurate as stated > (Sect. 4.2). >   * The two photon exchange term that depends on proton polarizability has > not been correctly evaluated (Sect. 4.3). >   * The electron and muon really do have different interactions with the > proton (Sect. 4.4), so that there is physics beyond the Standard Model. > > > None of these possibilities seem very likely, but all must be pursued. My question is, what are the possible implications of this new result? Does anyone know how significant are the potential flaws in the experiments mentioned? It is stated in the paper that one possible solution might be Beyond the Standard Model theories; assuming the experiment is correct, do we really need BSM theories to explain the new results? If so, which specific branches of BSM will benefit from these results?


title: Does the energy-time uncertainty principle require energy levels to have finite width?

text: The uncertainty principle also has the form: $\Delta$$E$$\Delta$$t>h/2\pi$ Now this should mean that the thickness of the lines we draw in the energy level diagrams to show energy change undergone by atoms and electrons cannot be a single straight line. Then that should mean that $E$ have a specific value so $\Delta$$E=0$ and so $\Delta$$t$ tends to infinity. Which is rather ridiculous cause that would mean that the particle(atom or electron) forever stays in some specified state and such transitions which are mentioned in the Bohr’s postulates, would be impossible. So such lines must be of certain thickness to illustrate the fact that $\Delta$$E$ should have a finite value, as in, E cannot be specified at a given time. Is my conclusion following the reasoning correct?


title: How would the solar system change if Earth was to just disappear?

text: Having just finished Arthur C. Clarke's "Childhood's End", my question is, what exactly will happen to the stability and configuration of our solar system if our planet Earth was to just disappear? I understand that numerically determining the stability and the configuration of our solar system in the long run (millions of years in the future) is a chaotic problem and we don't know what will happen in the long run. But in the short run, orbital dynamics are understood enough that we can keep satellites in space and predict the planets position and paths to a reasonable accuracy for the near future. So if Earth was to just _poof_ disappear all of a sudden, would the solar system destabilize where planets crash into the sun or escape the sun? Would the solar system stay mostly the same because the Earth is rather insignificant compared with the Sun and the giant planets? Or would the solar system stabilize into another different configuration? Or would it be a mixture where the inner-planets might change drastically/crash into the sun or each other but the outer giants will stay where they are? To be specific, I am talking about the Earth just disappearing (somehow)? I am not talking about the Earth exploding, shattering into a million asteroids/comets, or splitting in half with one half crashing into the sun and the other perhaps escaping the Sun, or colliding with another body, or creating another asteroid belt, nor it escaping perpendicular to the ecliptic plane nor escaping in any other way. Would a very tiny mass just disappearing have any effect on the solar system? Or are we really that insignificant in the grand scheme of things? In addition to answers, if any studies have been done, i.e. numerical or theoretical, I would love to see some references (published or otherwise). Thanks you.


title: Rest Mass, Where is it Stored?

text: Where is rest mass stored in a system? Reading Tipler, it says that a system of two 4 kg objects have a rest mass of 10kg, and atoms have rest masses smaller than their combined masses... Originally I thought the bonds in the nuclei would have given rise to rest mass, but even electrons which have no bonds have a rest mass. What _is_ rest mass, and where is it stored? Furthermore, the atomic bomb is cited at taking advantage of E=mc^2, but isn't it really just breaking the bonds in the nuclei? I feel like it's just taking advantage of potential energy from bonds, and while I know these bonds store mass, it seems very different from the quality of the rest mass of an electron.


title: Can weakness of gravity explore new dimensions

text: Since gravitational force is weakest force out of the four fundamental fources at the microscopic level. Is it possible that gravitational force is strong in a particular direction at a new microscopic dimension.


title: Cooling a cup of coffee with help of a spoon

text: During the breakfast with my colleagues, a question popped into my head: _**What is the fastest method to cool a cup of coffee, if your only available instrument is a spoon?_** A qualitative answer would be nice, but if we could find a mathematical model or even better make the experiment (we don't have the means here :-s) for this it would be great! :-D So far, the options that we have considered are (any other creative methods are also welcome): ## Stir the coffee with the spoon _**Pros:_**   * The whirlpool has greater surface than the flat coffee, so it is better for heat exchange with the air.   * Due to the difference of speed between the liquid and the surrounding air, the Bernoulli effect should lower the pressure and that would cool it too to keep the atmospheric pressure constant. _**Cons:_**   * Joule effect should heat the coffee. ## Leave the spoon inside the cup As the metal is a good heat conductor (and we are not talking about a wooden spoon!), and there is some part inside the liquid and other outside, it should help with the heat transfer, right? A side question about this is what is better, to put it like normal or reversed, with the handle inside the cup? (I think it is better reversed, as there is more surface in contact with the air, as in the CPU heat sinks). ## Insert and remove the spoon repeatedly The reasoning about this is that the spoon cools off faster when it's outside. (I personally think it doesn't pay off the difference between keeping it always inside, as as it gets cooler, the lesser the temperature gradient and the worse for the heat transfer).


title: Relationship between tones made by a piano

text: If a piano were to be tuned perfectly with the equal temperament system, what would be the relationship between standing waves from one note to the next? How would the frequency and wavelength of sound be affected?


title: Mars just collided with Earth! A question of eccentricity

text: One of the standard theories behind the formation of our Moon is the giant impact hypothesis, according to which Earth was struck by a Mars-sized body (about $10\%$ the mass of Earth) early in its history. The violent collision threw a large amount of material into orbit, and this coalesced to become the Moon. Now, we previously addressed the question of whether this would destabilize Earth's orbit around the Sun. The answer of course is that, no, orbits can't destabilize in Newtonian physics. But one might wonder how eccentric the Earth+Moon orbit around the Sun would be. After all, having a planet slam into you at several kilometers _per second_ seems like it may have consequences. For simplicity, let's assume the early Earth is in a perfectly circular orbit at $1\ \mathrm{AU}$. It is then hit by an object with the mass of Mars. **What would be the new eccentricity (and, why not, semi-major axis) after the collision?** Considerations:   * There are two important variables left unspecified - the impact velocity and the angle of impact. For the former, Wikipedia quotes $4\ \mathrm{km}/\mathrm{s}$, but an answer that shows how the result scales with this value would be best, since there is a great deal of uncertainty in it anyway. The same holds for the impact angle being $45^\circ$. That said, it seems reasonable to restrict ourselves to a single plane.   * Intuitively I feel a head-on collision will be a local maximum in post-collision eccentricity, since it will make the location of impact at $1\ \mathrm{AU}$ the aphelion. Similarly, a direct-from-behind collision will make that location the new perihelion and thus also will induce a greater eccentricity change than nearby angles. Is this intuition borne out by the math?   * Bonus points for justifying (or providing references that justify) the impact speed. One could ask how much energy is needed to separate all that mass, and how much of the pre-collision kinetic energy goes into this versus melting the surface and heating the mantle. This is mostly to put an upper bound on how much the impact could have perturbed Earth's orbit - it should be easy enough to argue that the collision wasn't any faster than, say, $30\ \mathrm{km}/\mathrm{s}$.   * When it comes to angular momentum (and energy), the situation is somewhat complicated by the fact that our orbiting object is not a classical point mass. The Earth can rotate, and the Earth+Moon system will clearly also have angular momentum. Can these extra degrees of freedom relieve the burden of the impact, leaving Earth in a still rather circular orbit?   * For the record, Earth's eccentricity around the Sun is only $0.0167$ today. Basically, the question is whether a Moon-forming impact could impart no more than about this much eccentricity. Of course, this value can change over time via long-range interactions with other planets - let's ignore that complication for now. References, graphs, equations, and good old-fashioned order-of-magnitude physics all welcome.


In the recent news, scientists at NASA have found “unprecedented” black hole cluster near Andromeda’s central bulge. I wonder why doesn't all these black holes merge and such each other in until just 1 remains? Is that even possible? Can one black hole suck in another black hole?


title: Modeling Water Surface Impact

text: I have a project that is to model the wave equation in a pond due to a pebble being thrown in. The project is quite simple as all we really have to model is the wave equation with an initial displacement. However, I was wanting to go a bit more in-depth and try to model it while knowing the initial conditions of the pebble. I have come up with a simple case of the water being in a perfectly cylindrical pond and the pebble being a perfectly spherical ball, but I have absolutely no idea of how to model the sphere's impact with the water. I know that the when the sphere initially makes contact with the water, there will be a brief moment of time that the surface tension acts as a normal force and this will cause some displacement of the water and what I assume will be a very fast starting wave speed. I'm stuck on this part of the problem, as the only way I can find to model the surface tension is the Young-Laplace equation, which doesn't even seem like it will work right here. I also know that after the sphere breaks the surface of the water, there is a period of transition where part of the sphere is in the water and then a point after which the entire sphere is in the water; I do not know how to model either of these. I know once the entire sphere is in the water, it can be modeled with fluid drag (or maybe Navier-Stokes), but I am not sure how that would translate into affecting the wave. Any ideas on this?


title: The work done by running in a rectangle

text: This may be quite off-topic but please help me. > _Is there anywork done when I run in a rectangle?_ I thought that the answer should be no. But my teacher says that we should calculate each side of the rectangle individually and then add it all together. Which one is correct and why?