title: Can the speed of light become complex inside a metamaterial?

text: The speed of light in a material is defined as $c = \frac{1}{\sqrt{\epsilon \mu}}$. There are metamaterials with negative permittivity $\epsilon < 0$ and permeability $\mu < 0$ at the same time. This leads to a negative refractive index of these materials. But do (meta-) materials exist with only negative $\epsilon < 0$ and positive $\mu > 0$ or vice versa? This would lead to a complex speed of light inside such materials. What would be the consequences of a complex speed of light? Could particles reach unlimited speed inside these materials? Would there still be Cherenkov radiation?


title: direction of Pockels Effect. Refractive index eigendirections

text: There is a linear electro-optic effect called Pockels Effect The brief is that refractive index changes due to electric field. If there is an anisotropy (like birefringence) and electric field is in perpendicular to optical axis then refractive index changes for different polarization of ordinary rays. It gets eigendirections 45 degrees to electric field. Why so? Why it gets min 45 degrees but not exactly towards direction of E?


title: Isometry group from information about the center of the group

text: I am reading this paper on Dyons and Duality in $\mathcal{N}=4$ super- symmetric gauge theory. The author finds the zero modes or a dirac equation obtained by considering first order perturbations to the Bogolomony equation for bps monopoles. He finds out that when the simple root used for embedding the SU(2) monopole is simple then the bosonic zero modes which I believe is the solution for the higgs field transform as $1 \oplus 0$, and when it is not a simple root it transforms as two doublets. Hope this is not required for my question. **Then he says that the zero modes are SENSITIVE to the center of SU(2), and hence the isometry group is SU(2)? What does sensitive to the center mean? Is center the subset which commutates with all elements of SU(2)? How can it tell me about the isometry group?**


title: How does this chair move?

text: I have a chair like this(shown in figure). I was kneeling on the part where we sit. I just gave a sudden push to the part where we rest the back, without moving my legs. I found that the chair is moving little in the direction I applied the force. I couldn't find any reason why this chair would move forward because I am inside this system and no external force is coming from outside. When I apply the force on it , equal amount of force I am applying there on the seat in the opposite direction, so there should not be any movement. I tried this experiment again right now. Now holding on the handles of the chair. Did the same thing. Again found the chair moving. Where I am wrong and how does the chair move? ![X](http://i.stack.imgur.com/NxRe7.jpg)


title: High power beam splitting

text: I'm trying to split a high power laser (1064nm, 20W, beam diameter @ 1/e^2 intensity: 3.2mm) into two beams and couple them after some additional optics into two SMF's (single mode fiber). So far I tried a PBS (polarizing beam splitter) cube and a NPBS (non polarizing beam splitter) cube. I can split the beam but the cubes seem to destroy the mode in the reflected port at high power. I conclude this because I couple approx. 20% less efficient in the SMF. Does anybody know how to split(50:50 would be sufficient although polarisation dependent would be preferred) a high power beam without aggravating the mode? Note1: Plate beam splitters are not preferable as they usually have the same dielectric beam splitter coating as cubes and the reflected port tends to interfere with reflections from the back side of the plate Note2: I'm aware of Glan-Laser Calcite, Glan-Thompson and Glan-Taylor polarizers but as they are, similar to PBS/NPBS cubes, two cemented prisms and I personally suspect this layer to be the reason. I would like to know if anybody has experience with them concerning high power.


title: How should I draw this free-body diagram?

text: > The horizontal force P acts on the rim of the homogeneous cylinder of radius > R and weight W. Determine the smallest coefficient of static friction that > enables the cylinder to start rolling up the 30◦ incline. > > ![image](http://i.stack.imgur.com/4JquL.jpg) Which FBD would be correct? should it be a vertical normal force with friction force going on left, or should it be the normal force facing perpendicularly the 30 degree include while the friction is along the 30 degree line also. which is the proper fbd?


title: Turn-on delay time for Laser diode

text: Do you know any simple explanation on the reason why the turn-on delay time on a laser diode is reducing while we increase the bias current? Turn on delay,is the time that the laser needs from the time that one applies the current until the time that the light goes out of the laser.This time is strongly depended to the input current density,the higher the bias current it is the less the turn on delay it is. That I don’t understand is the physics behind it,how that interaction occurs. Is it something obvious, because I am trying to find a simple explanation and I can not. Best Regards, George


title: Evolution principle of the physical laws

text: I wanted to know if there is a physical theory that considers that the laws of physics undergo an evolutionary process. That see the law of physics or the absence of them, as something dynamic, and that with time they slowly converge to something we know today. A kind of simulated annealing of the physical laws.


title: Can Helium Disappear from Earth?

text: Can Helium disappear? As we know Helium is lighter than air, so basically Helium fly off from Earth. Is it possible that in the future we will run out Helium?


title: Determinant of Dirac operator in flat space?

text: How would you evaluate \begin{equation}|iD\\!\\!\\!\\!/-m|\end{equation} Where $D_{\mu}=\partial_{\mu}-ieA_{\mu}$. I have an idea of how to do this without the gauge field, because it's essentially \begin{equation}|i\partial\\!\\!\\!/-m|=|\partial^{2}-m^{2}|^{1/2}\end{equation} and A. Zee's book covers this is some detail. From what I can tell, you use $\ln|A|=\mathrm{tr}\ln A$. The trace is in momentum space \begin{equation}\frac{1}{2}\int \frac{d^{4}k}{(2\pi)^{4}} \ln (k^{2}-m^{2}-i\epsilon)+C\end{equation} How do you do this in the presence of a gauge field? I've gotten this far \begin{equation}|iD\\!\\!\\!\\!/-m|=e^{\frac{1}{2}\mathrm{tr}\ln \left(D^{2}-(e/2)\sigma^{\mu\nu}F_{\mu\nu}+m^{2}\right)}\end{equation} With $\sigma^{\mu\nu}=(i/2)[\gamma^{\mu},\gamma^{\nu}]$. But I have no idea how to proceed.


title: Where do electrical charges go, on a nonconductor dielectric when we make it charged?

text: My question is about electrically nonconductor dielectrics. We know such materials don't possess free charges.They have atoms bound together and every atom has specific numbers of electrons turning around its nucleus. When we make a dielectric charged: where does this charge go? what keeps the charge fixed on the structure? Are they bound to specific atoms? If your answer is "yes",so What makes that atom specific to possess the charge? (same questions about positively charging.)


title: Applications to the Van der Pol equation?

text: What are some applications to the Van der Pol equation? Are there any physical examples?


title: Electric Field inside a hollow ball, excentred of a homogeneous charged ball

text: Q: We have a homogeneous charged ball with radius R which contains a ball-shaped hollow (with radius := r and distance from center of the bigger ball (M) to the center of the hollow (N) := b). What’s now the field force inside the hollow, depending on b? ———— I first thought about calculating E(Ball R, without a hollow) - E(of the hollow r, displaced ) to use the superpostion principle, but then i would have no idea how to calculate the latter of those… would be happy about any hints!


title: According to Hubble's Law, how can the expansion of the Universe be accelerating?

text: Scientists today think the expansion of the universe is accelerating. According to Hubble's law, objects further away are moving faster than objects closer to us. The further away an object is, the further back in time we are seeing, so in the past objects moved faster (is this sentence correct?). So because objects moved faster before than they do now, surely that's deceleration not acceleration? my textbook says "if Scientists today think the rate of expansion of the Universe were decreasing then distant objects should appear different to Hubble's Law predictions: universe is accelerating. More distant objects would seem to be receding faster (since expansion was faster in the past)"


title: polarization of the lower mode gaussian beam

text: In most introductory analysis of Gaussian beam optics, Helmoltz scalar optics is assumed. Hence polarisation is ignored. But I'm not clear what are the possible orientations for the lower transverse mode $T_{00}$ when full vectorial electric and magnetic fields are allowed. Assuming cylindrical coordinates, where the beam propagates in the axial coordinate z. The intensity of the fundamental mode is axially symmetric; but How does look the electric field orientation for a linearly polarized gaussian beam? is it always aligned toward a fixed direction? it has axial symmetry as well?


title: Gas giant evolution over lifetime

text: In various types of stars, one can learn how they evolve differently, depending on factors such as their size and chemistry. Some stars have a short lifetime and others much longer. But, what is known about the lifetime and evolution of gas giants. Do they just remain stable until some external source, such as the burn out of the sun, disrupts this?


title: Determinant for a coupled fluctuation Lagrangian

text: Lets consider a bosonic physical system in variables $t, x$ and $y(x)$ ($x$ dependent) with a classical Lagrangian $L$. To first order in fluctuations $x \to x+\xi_1$ and $y \to y+\xi_2$ the fluctuated action reads $$S_{fl}=\int dt dx \sqrt{g} ~{\vec \xi}^T~{\tilde O}~{\vec \xi}$$ Where the differential operator ${\tilde O}$ is given by $${\tilde O}=\begin{pmatrix} \nabla^2+M_1(x) & -\frac{1}{\sqrt{g}}\partial_x +M_{12} \\\ \frac{1}{\sqrt{g}}\partial_x +M_{12} & \nabla^2+M_2(x) \end{pmatrix}$$ Following the standard procedure, one has to compute the functional determinant of ${\tilde O}$ to obtain the effective action for the fluctuations. Now, since the system is coupled (off-diagonal elements are present), the computation of $\det[{\tilde O}]$ will be rather hard, if not impossible: $$\det[{\tilde O}]=\det\begin{pmatrix} \nabla^2+M_1(x) & -\frac{1}{\sqrt{g}}\partial_x +M_{12} \\\ \frac{1}{\sqrt{g}}\partial_x +M_{12} & \nabla^2+M_2(x) \end{pmatrix}\\\=\det\left(\left(\nabla^2+M_1(x)\right)\left(\nabla^2+M_2(x) \right)-\left(-\frac{1}{\sqrt{g}}\partial_x +M_{12}\right)\left(\frac{1}{\sqrt{g}}\partial_x +M_{12}\right)\right)$$ However, formally, before attempting to compute the determinant, one can multiply out all the terms in the fluctuation Lagrangian $L_{fl}={\vec \xi}^T~{\tilde O}~{\vec \xi}$, partially integrate once in the off-diagonal terms and rearrange the fields $\xi_1 \xi_2$ such that if one puts all the terms back into matrix notation, one finds: $$S_{fl}=\int dt dx \sqrt{g} ~{\vec \xi}^T~{\tilde O}~'~{\vec \xi}$$ where now $${\tilde O}~'=\begin{pmatrix} \nabla^2+M_1(x) & 0 \\\ \frac{2}{\sqrt{g}}\partial_x +2M_{12} & \nabla^2+M_2(x) \end{pmatrix}$$ If one proceeds to compute the determinant now: $$\det[{\tilde O}~']=\det\left(\left(\nabla^2+M_1(x)\right)\left(\nabla^2+M_2(x) \right)\right)\\\=\det\left(\nabla^2+M_1(x)\right)\det\left(\nabla^2+M_2(x) \right)$$ one finds a much simpler problem to solve. Now my question - is this "simplification" actually allowed, or did I miss some subtlety which forbids such a rearrangement of bosonic fields at the Lagrangian level?


title: Consequences of Quantum Mechanics

text: First of all, this question is going to seem a a bit of philosophy but know that vague and purposeless wandering is certainly not what i'm trying to propose here.   Also, the reason i didn't post in philosophy communities is that they certainly know a lot less ( if anything ) of quantum mechanics than most of you here. My question : I heard several times that the results of quantum mechanics (double-slit experiment for instance ) challenge our logic. One example of that is the famous physicist Lawrence Krauss. He keeps saying that after our discoveries of quantum mechanics, "logic" is becoming flawed.He wears a 2 + 2 = 5 T-Shirt to support his case. Does anyone know exactly are physics reffering to when they introduce the word "logic" to say it shouldnt be taked for granted ? Would "logic" be common- sense ? would "logic" be sensory experience ? if yes, then i'd highly agree. While we can't experience aurally frequencies out of 20-20khz range we are certain that they do exist. On the other hand, if "logic" reffers to our deductive and inductive logic then i dont understand.   How could we say that the result of lots of year of deductive and inductive logic ( from Thales to Newton to our modern Science ) points out that deductive and inductive logic is flawed.   Mathematics also rely heavily on deductive logic.Mathematics is also the language in which physics is formally expressed.   Denying or not taking for granted even our "deductive logic" would be not taking for granted Mathematics in which case would end Physics.


title: Understanding Quantum Physics

text: I have very little background in physics, and none in quantum physics, but I've been reading about how sub-atomic particles behave probabilistically, so I was wondering, is it possible (even though the probability would be unimaginably small) that all the particles which make up my body are located somewhere 100 light years away? Or have I misunderstood the concept?


title: Boundary conditions of relativistic wave solutions?

text: If you take Einstein's field equations, \begin{equation} R_{\mu\nu}-\tfrac{1}{2}g_{\mu\nu}R = -\kappa T_{\mu\nu}, \end{equation} and you insert the metric \begin{equation} g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}, \end{equation} then you get a field theory in flat spacetime for the propagation of the perturbation $h_{\mu\nu}$. This is 'linearised GR' and it's a first place to start for a description of gravitational waves. The equations result from the linearisation process are a set of wave equations of the form \begin{equation} \square^{2}\bar{h}_{\mu\nu} = - 2\kappa T_{\mu\nu}. \end{equation} Here, $\bar{h}_{\mu\nu} = h_{\mu\nu} - \tfrac{1}{2}\eta_{\mu\nu}h$ is called the 'trace-reversed' perturbation, and $h=\eta^{\mu\nu}h_{\mu\nu}$ is the trace. If you want to solve this animal, then you can follow the analogous problem in electromagnetism and use a retarded Green's function. You can guess the form of the Green's function and with a little refinement; this is how I have solved it. However, I recently read that one can solve this equation _without guesswork_. Apparently, you have to impose the boundary conditions, that \begin{equation} \displaystyle\lim_{t\to-\infty}\left[\frac{\partial}{\partial r} + \frac{\partial}{c\partial t}\right]\,r\bar{h}_{\mu\nu}=0 \end{equation} where the limit is taken along any surface $ct + r = constant$, together with the condition that $r\,\bar{h}_{\mu\nu}$ and $r \partial_{\rho}\bar{h}_{\mu\nu}$ be bounded by this limit. The book I saw this in said that the physical meaning of this is that there's no incoming radiation from past null infinity. What am I looking for?   * What exactly are these three conditions telling us,   * Why these conditions have the form that they do, and   * What is means that a surface has constant $ct + r$.


title: Domain wall and kink solutions from solitions equations

text: A general solition equation can be obtaion from scalar field theory $$\varphi(x) = v\tanh\Bigl(\tfrac{1}{2}m(x - x_0)\Bigr),\tag{92.6}$$ where $x_0$ is a constant of integration when we drived this solution from Lagrangian. **Comparing to the sine-Gordon theorywith this article , can we convert our equation to the sine-Gordon equation?** Because this equation represented for 2 dimensions (1 space dimension and 1 time dimension) then how will I get Kink solutions and domain wall equation from the above equation. Any elaboration would be helpful for me. This is a continuing post of Solving the soliton equation without energy. **ADDITION: If I want to visualize the solition equation (92.6) by graph then what would be the best process?**


title: Clarification what exactly this question wants regarding force

text: I would like to ask you if the question asked > _What is the minimum force parallel to the plane, which can be prevent the > block from slipping down the plane?_ I just did the calculation and and got the answer correctly but I didnt understand why i calculate like that to get the minimum force for that situation. And also, for what situation should I use static friction coefficient and kinetic friction coefficient? That's all thanks. I am studying for my final now so I need clarification not just by doing it right, but also knowing what I'm doing. I'm hoping for further explanation. :) So basically this is the plane and the block.(The drawing is quite epic as I'm using 'Paint'.) ![enter image description here](http://i.stack.imgur.com/SLyeJ.png)


title: Hilbert action's invariance under general coordinate changes

text: In an article, when considering invariance of the Hilbert action under a general coordinate change this formula appears for how the metric changes $$\delta{}g_{\mu\nu}=\partial_{\mu}\xi^{\rho}g_{\rho\nu}+\partial_{\nu}\xi^{\rho}g_{\rho\mu}+\xi^{\rho}\partial_{\rho}g_{\mu\nu}.$$ I have (maybe naively) tried using the tensor coordinate change formula considering $x^{\alpha}~\to~ x'^{\alpha}=x^{\alpha}+\xi^{\alpha}$ (formula I guessed, it isn't specified in the article I am reading) but I don't get there. So, how does this formula come up?


title: Is nuclear fusion via underwater vacuum possible?

text: An under water explosion creates a bubbly which quickly collapses on itself. The action takes less that 1/100 of a second for bubbles less than a foot across. This creates a point of extremely focused energy at the middle point where the bubble collapses. In theory, this point focuses enough energy to trigger nuclear fusion. Could the same effect be achieved by suspending a metal sphere under water which contains a near complete vacuum. If the sphere were opened uniformly along all of its surface area, the water rushing should create the same effect as the explosion, focusing all of the energy on a point at the center of the sphere. IF not, then what are the limiting factors (surface tension maybe?) In my mind, It seems the major limiting factor is the shrinking surface area of the bubble as it collapses. as the surface area shrinks, the water molecules along the surface of the bubble would resist the change of surface area (some would need move away from the surface, in the oppisit direction of the collapse), causing the collapse rate to slow.


title: Does Light Ever Cease Once Created

text: I was reading an article on Huffington Post (link to article) about the Multiverse theory. In the article it said this: > The big news last week came from the Background Imaging of Cosmic > Extragalactic Polarization 2 (BICEP2) experiment at the South Pole, which > saw imprints in the cosmic microwave background— **the oldest light in the > universe** , dating from shortly after the big bang... I am far from scientific, actually just a web dev, but I have an interest in learning and reading about science, so please explain your answer in a way I can understand it. My question is in regards to the bold type in the quote. Does light ever actually end, go away, or just cease to exist once it is created? Will this "oldest light in the universe" ever completely cease to exist, or does light stay around indefinitely.


title: What are the prerequisites to study topological quantum computation/topological phases of the matter?

text: I am an undergraduate student and I would like to approach the subject of topological order with focus on topological quantum computation, I know (very) little QFT and basic algebraic topology (if that matters...). To give a reference frame, I'd like to understand papers like http://arxiv.org/abs/quant-ph/9707021v1 or Preskill's great presentations: http://online.kitp.ucsb.edu/online/exotic_c04/preskill/, and later to take it as my field of research for a PhD. So my questions are: 1) What kind of studies are useful/needed for this field? I assume Quantum Computation/Information are, but what else? Quantum Field Theories, Topology, Quantum Optics...? 2) Which are univerities or research centers active in the field (preferably in Europe)? 3) Any reference suggested? (Better if beginner-focused)


title: Is the acceleration at the event horizon of a black hole independent of the magnitude of its radius?

text: Is the acceleration of a free-falling observer at the Schwarzschild radius of a black hole the same regardless of the numerical value of the radius (i.e. outside of a black hole, is the acceleration of the free-falling observer only a function of his distance from the Schwarzschild radius)?


title: Magnetic flux density a small distance off axis from a current loop

text: I'm currently studying physics at University and struggling with the problem mentioned, it's on a past paper I'm trying to do. I have calculated the B field on axis as (sorry don't know how to format this yet) $$B = \mu \frac{Ia^2 }{2(a^2 + z^2)^{3/2}}$$ In the $z$ direction. The question then asks: > At a small distance ($\rho\ll a$) off axis near the point $A=(0,0,z)$, the > diverging magnetic flux density has a component perpendicular to the axis. > Find this' ![Diangram of setup](http://i.stack.imgur.com/A9iqn.png) Should I use $\nabla\cdot\mathbf B=0$ then try and solve? Not really sure where to go from here.


title: Friction influencing the motion of a mass and non-inertial frame of reference

text: So... recently a Newtonian-mechanics related exercise has raised in me many questions basically ragarding which forces influence a given system and how, I know they might sound bit too "noobish" but I just had to get them out of my chest (yes it really gave me something to think about in the last day). perhaps you should take a look at the above mentioned exercise to better understand what I am trying to figure out ![image of the system](http://s21.postimg.org/ek3mmimw7/Nuova_immagine_bitmap.png) So, I'm trying to figure out how this system works... the exercise says (please forgive me for all the grammatical errors you will eventually find): > # Problem > > a cube with a mass $\ M=50Kg$ is standing on a plain surface and can move on > it without friction. On that cube is standing another cube with a mass $\ > m=10Kg$, at a given distance $\ d=0.5 m$ from the upper-left corner of the > cube. Initially, when everything is stationary, a force $\ F=100N$ is > applied to the bigger cube horizontally (I assume that the force $\ F$ is > constant); at the moment $\ t=2s$ the smaller cube falls. Calculate the > friction coefficient between the two cubes. # Variables used: > $\ μ$= friction coefficient > > $\ g$= gravitational acceleration on earth =$\ 9.81 m/s^2$ > > $\ a_M$= acceleration of the bigger cube ($\ M$) > > $\ a_m$= acceleration of the smaller cube ($\ m$) > > $\ F_f$= friction force So, I assume that the bigger cube carries the smaller one which moves with a constant, negative acceleration until it falls (please correct me if I'm wrong), with equation of motion $\ x(t)= x_0 + v_0t + 1/2 at^2$ so $\ 0.5m = - 1/2 at^2$ ------> $\ a= -0.25 m/s^2 $ now... $\ F- μ mg = Ma_M$ in other terms (and again please correct me if I'm wrong): $\ F$ is partially dissipated by the friction between the cubes, and the bigger cube moves under the influence of a force which equals to the applied force $\ F$ minus the friction force $\ F_f= μ mg$ now the universe collapses: the solution says: > $\ μmg = ma_m$ > > $\ a_r = a_M - a_m = [ μg (m + M) -F] / M$ which leads to $\ μ=0.15 $ But... what leads to those last two equations?


title: How can I derive this Hamiltonian?

text: I have a Lagrangian $L$, a momentum $p$ and a Hamiltonian $H$: $$L=\frac m 2(\dot z + A\omega\cos\omega t)^2 - \frac k 2 z^2$$ $$p=m\dot z + mA\omega\cos\omega t$$ $$H=p\dot z - L=\frac m 2 \dot z^2 - \frac m 2 (A\omega\cos\omega t)^2 - \frac k 2z^2$$ And I want to calculate $\frac {\partial H} {\partial z}$ and $\frac {\partial H} {\partial p}$. I understand from this question that I need to algebraically manipulate $H$ to express it in terms of $p$ and $z$. The answers there suggested trying to express $\dot z$ in terms of $z$ and $p$, and presumably I need to express $t$ in terms of $z$ and $p$ as well. But that seems like its going to lead into very nasty territory... first of all, I've got quadratics, so that's not an invertible function. Second of all, for $t$, if I have to use inverse trigonometric functions, then it'll only be valid over a particular range of the variable. Could I get some pointers on how to tackle this calculation? Is inverting the various relations really the way to go?


title: About the geodesics in general relativity

text: I'm learning general relativity from the book " Einstein's General Theory of Relativity - Øyvind Grøn and Sigbjorn Hervik". The field equations are derived by the Hilbert - Einstein action and are written in the form $$ R_{\mu\nu}-\frac{1}{2}R\,g_{\mu\nu}+\Lambda\,g_{\mu\nu}=kT_{\mu\nu}$$   where $k$ is acostant that should be determined. The authors find this $k=\frac{8\pi G}{c^4}$ by imposing the well known conditions of "Newtonian Limit". The first requirement is the following > The particles in free fall moving along geodesics induced by $g$. This is a postulate? By the Einstein equivalence principle we know that locally we can eliminate the effects of gravity and so living in a (flat) Minkowskian space-time but there is not mention of the above statement. I agree on the fact that once we have a curved space-time, in presence of gravitational fields, it is natural to require that the free falling particles move along the geodesics, but this can be proved or it is postulated?


title: The entropy of a static granular mixture

text: Regarding static granular mixtures (pile of sand - particles with different sizes), is there an expression that will give me the entropy of such a mixture? I am interested in finding the entropy of the system initially and once it has been screened ie once the larger particles get sorted out, giving two piles (I think that ths is a reversible process, correct me if I am wrong). Would this mixing entropy be consistent with a bulk entropy such as $\Delta S = m c_p ln(T)$ ? Could I use this bulk entropy and the mixing entropy together in some free energy expression? I have looked for some references but I get nothing that I can access. Thanks


title: Gauss's Law understanding

text: In the case of a point charge $q$ at the origin, the flux of $\vec{E}$ through a sphere of radius r is, \begin{equation} \oint \vec{E}\cdot d\vec{a} = \int \frac{1}{4 \pi \epsilon_0 }(\frac{q}{r^2}\hat{r})\cdot (r^2 \sin\theta d\theta d\phi \hat{r})=\frac{1}{\epsilon_0}q. \end{equation} I want to know that, how the above equation has the value of $\frac{1}{\epsilon_0}q$?


title: Synchronization of clocks and observations of time in special relativity

text: I have thought up a situation that I cannot understand with my understanding of special relativity. I don't know general relativity, but as the situation doesn't involve gravity or acceleration, I'm not sure if it is needed. * * * Imagine there are 2 digital count up timers, A and B, separated by 100 light days in the same frame of reference. There is a person at each timer. Both timers are turned on at the same time (this could be done via a signal at a point C which is the same distance away from both A and B). At this point, a person at A will see the timer at B being 100 days less than the timer at A (due to the time light takes to travel from A to B). Vice versa for a person at B. * * * Now imagine a ship is traveling at 0.99c through A to B. The ship also has a timer on it, which is initially off. ![timer setup](http://i.stack.imgur.com/D1chg.png) When it passes past A the pilot notes timer A is at 1000 days. The pilot then starts the ship's timer. Eventually, the ship will go past B. My questions are: 1) What time will people A and B see on the their timer and the ship's timer when the ship passes past B? 2) What time will the ship see on A and B when it passes past B? * * * Here are the issues I have with this: From B's perspective, the ship passes past A when B's timer is 1100 days. If you ignore relativity, you would expect a ship traveling at 0.99c to take `100 / 0.99 = 101.01` days to get to B. Wouldn't that mean from B's perspective, it only takes 1.01 days for the ship to get from A to B? I thought length contraction would explain that, but A and B aren't moving relative to each other. S would appear to be moving faster than the speed of light, which doesn't make sense. From A's perspective, S will experience time 7.088 slower than A. So when it passes B, the ships timer will only be `101.01 / 7.088 = 14.25` days. Taking 101.01 days to get to B, it will see B's timer being 1101.01 days. It will see this time when it's timer is 1201.01 days. That means it will take 201.01 days for the ship's timer to advance by 14.24 days, a time dilation of `201.01 / 14.24 = 14.12`, which is different to 7.088? I must be double counting somewhere, I don't know.


title: Emergence of $SU(2)\times SU(2)$ at the self-dual point in bosonic string theory

text: I want to understand the derivation of the equations 8.3.11 in Polchinski Vol 1. I can understand that at the self-dual point the Kaluza-Klein momentum index $n$, the winding number $w$, and the left and the right oscillator number $N$ and $\tilde{N}$ are related as, $$(n+w)^2 + 4N = (n-w)^2 + 4\tilde{N} = 4$$ Now one can see that this has two "new" sets of massless states, $$n = w = \pm1, N=0, \tilde{N}=1; n = -w = \pm 1, N=1, \tilde{N}=0$$ and $$n= \pm 2, w=N=\tilde{N} =0; w = \pm 2, n=N=\tilde{N}=0$$   * But after that I don't get the argument as to how the 4 states in the first of the above set is represented by vertex operators  $$:\bar{\partial}X^\mu e^{ik.X}\exp[\pm 2i\alpha '^{-1/2}X_L^{25}]:$$ and $$:\partial X^\mu e^{ik.X}\exp[\pm 2i\alpha '^{-1/2}X_R^{25}]:$$ What is the derivation of the above?   * Also now looking at these 4+4 states how does the argument for existence of a $SU(2)\times SU(2)$ symmetry follow? The argument in the paragraph below 8.3.11 is hardly clear to me.


title: Do black holes play a role in quantum decoherence?

text: Sorry for such a vague question but I could have sworn I read somewhere that Hawking proposed the reason we might see a classically appearing universe is due to the possible role of black holes in quantum decoherence. Is there such an idea by him or another physicist? If so, can anyone point me to some names where I can read up on more?


title: How does a "hammer thrower" that we see in the Olympics, impart so much momentum

text: How does a "hammer thrower" that we see in the Olympics, build so much momentum into the club? It's sort of like the golf swing, the more momentum, primarily in the club head, the further the ball will fly, not accounting for the "Magnus effect". So, how does a strong hammer thrower, create so much power?


title: Hubbard-Stratonovich transformation and mean-field approximation

text: For an interacting quantum system, Hubbard-Stratonovich transformation and mean-field field approximation are methods often used to decouple interaction terms in the Hamiltonian. In the first method, auxiliary fields are introduced via an integral identity, and then approximated by their saddle-point values. In the second method, operators are directly replaced by their mean values, e.g. $c_i^\dagger c_jc_k^\dagger c_l \rightarrow \langle c_i^\dagger c_j\rangle c_k^\dagger c_l + c_i^\dagger c_j \langle c_k^\dagger c_l\rangle$. In both methods, order parameters can then be solved self-consistently to yield the decoupled Hamiltonian. Are these two methods equivalent? If not, how are they related?


title: What are the length and time scales in turbulence?

text: I haven't been able to understand what are does someone mean by length and time scales, while talking about turbulence. Can someone explain it to me or give me a link where i can find a good explanation.


title: Relativistic momentum finding

text: Kinetic energy of an object $mc^2$. I want to evaluate the momentum. We know Kinetic energy, $$E_k = \frac{p^2}{2m}$$ $$mc^2 = \frac{p^2}{2m}$$ $$ p = \sqrt{2}mc$$ Note that the momentum is the relativistic mometum. Is there anything I did worng?


title: Pole and Barn Paradox

text: I'm having trouble reconciling two different versions of the Pole and Barn paradox. Version 1: Consider a pole 10 m long and a barn 5 m long with a front and rear door. A runner carrying the pole (frame S') moving with respect to the barn and a farmer (frame S) runs into the barn. The farmer will see that the pole fits into the barn if the runner is moving at a speed: gamma = Lp/L = 10/5 v = 0.866c But according to the runner, the barn is moving towards him. The barns length contracts to: L = Lp/gamma = 5/gamma = 2.5 m How is it possible for a 10 m pole to fit in a 2.5 length contracted barn? Basically by drawing a spacetime diagram it becomes apparent that the pole doesn't fit. From the farmers frame, the front of the 5 m length contracted pole gets to the rear door of the barn simultaneously as the end of the pole enters the front door. But in the runners frame, these events are not simultaneous and the pole does not fit into the barn (the front end of the pole leaves the back door before the end of the pole enters the front door). Version 2: Consider the same situation, but the rear door is replaced with an armor plate. From the farmers frame, when the pole fits into the barn, he shuts the front door (in the next instant, assuming the pole doesnt break, the pole must bend or break through the armor plate). From the runners frame, the front end of the pole hits the iron plate, with 7.5 m of the pole still outside of the barn. If information travels down the pole at speed of light c, it would take 10/c to reach the back of the pole. The barn on the other hand must reach the back of the pole in 7.5 / (0.866*c), which is less time than the time it takes for the info to travel down the pole. Thus, the runner is in agreement with the farmer, and the 10 m pole is contained within the 2.5 m barn. Question: How is it that the pole fits into the barn in version 2, but not in version 1? Question: My book says, as a part of version 2, "since this is relativity, the runner must come to the same conclusion in his rest frame [as the farmer] as the 2.5 m barn races towards him at v = 0.866c." But in version 1, the runner didnt come to the same conclusion?


title: Deriving the Lorentz force from velocity dependent potential

text: We can achieve a simplified version of the Lorentz force by $$F=q\bigg[-\nabla(\phi-\mathbf{A}\cdot\mathbf{v})-\frac{d\mathbf{A}}{dt}\bigg],$$ where $\mathbf{A}$ is the magnetic vector potential and the scalar $\phi$ the electrostatic potential. How is this derivable from a velocity-dependent potential $$U=q\phi-q\mathbf{A}\cdot\mathbf{v}?$$ I fail to see how the total derivative of $\mathbf{A}$ can be disposed of and the signs partially reversed. I'm obviously missing something.


title: What're the relations and differences between slave-fermion and slave-boson formalism?

text: As we know, in condensed matter theory, especially in dealing with strongly correlated systems, physicists have constructed various "peculiar" slave- fermion and slave-boson theories. For example, **For Heisenberg model,** we have Schwinger-fermion $\vec{S_i}=\frac{1}{2}f^\dagger_i\vec{\sigma }f_i $ and Shwinger-boson $\vec{S_i}=\frac{1}{2}b^\dagger_i\vec{\sigma }b_i $ approaches, with constraints $f^\dagger_{1i}f_{1i}+f^\dagger_{2i}f_{2i}=1$ and $b^\dagger_{1i}b_{1i}+b^\dagger_{2i}b_{2i}=2S$, repectively. **For $t-j$ model,** there are slave-fermion $C^\dagger_{i\sigma}=b^\dagger_{i\sigma}f_i$ and slave-boson $C^\dagger_{i\sigma}=f^\dagger_{i\sigma}b_i$ methods, with constraints $f^\dagger_{i}f_{i}+\sum b^\dagger_{i\sigma}b_{i\sigma}=1$ and $b^\dagger_{i}b_{i}+\sum f^\dagger_{i\sigma}f_{i\sigma}=1$, repectively. **For Hubbard model,** we have slave-fermion $C^\dagger_{i\sigma}=b^\dagger_{i\sigma}f_{1i}+\sigma f^\dagger_{2i}b_{i\sigma}$ and slave-boson $C^\dagger_{i\sigma}=f^\dagger_{i\sigma}b_{1i}+\sigma b^\dagger_{2i}f_{i\sigma}$ methods, with constraints $\sum f^\dagger_{\alpha i}f_{\alpha i}+\sum b^\dagger_{i\sigma}b_{i\sigma}=1$ and $\sum b^\dagger_{\alpha i}b_{\alpha i}+\sum f^\dagger_{i\sigma}f_{i\sigma}=1$, repectively. **And my questions are:** **(1)** Whatever it's spin or electron system, the slave-fermion and slave- boson constructions have very similar forms, simply interchanging bosonic and fermionic operators in one we would get the other one. So is there any deep connection between these two formalism? Does this similarity have something to do with **supersymmetry**? **(2)** From the mathematical point of view, both the slave-fermion and slave- boson constructions are correct. But physically, when should we use slave- fermion methods and when should we use slave-boson methods? What are the differences between these two approaches when we deal with a particular physical model?


title: Expectation Value of a Dynamical Variable

text: In quantum mechanics, we generally take about "expectation values of dynamical variables". However, by the postulates of quantum mechanics, every dynamical variable in quantum theory is represented by its corresponding operator. Is it therefore, incorrect to talk about "expectation value of an operator" (rather than "expectation value a dynamical variable")? Is is just semantics or is something more going on here? In other words is it incorrect to write: $$<\widehat{A} > =\int \psi \widehat{A}\psi^{*}\mathbf{ d^{3}r}$$ instead of $$<A > =\int \psi \widehat{A}\psi^{*}\mathbf{ d^{3}r}?$$


title: What keeps objects made of the same plastic apart?

text: I have two objects here which are made of the same plastic. When I hold them together, they remain separate objects: I can pull them apart with no resistance. How does each atom/molecule “know” which object it belongs to? Why do the objects not fuse?


title: Question about Bose-Hubbard Model

text: By using an optical lattice, how can one change the interaction term $U$? And how is the superfluid phase achieved in the hard-core boson regime? Why are these phases identified as superfluid or mott insulator?


title: Difference between Higgs contribution to electroweak boson mass, and fermion mass?

text: Thanks in advance to anyone who takes the time to answer this, and apologies in advance for what is probably (yet another) question here due to unfamiliarity with The Math. I had a chance recently to visit the CMS detector at CERN (sheer luck) and it's made me curious to better understand how the Higgs field interacts with the weak gauge particle field vs. the fermion field. I've read several descriptions that connect the former with electroweak symmetry breaking and the latter with "a different mechanism" which seems to be a Yukawa coupling (which I understand to be an interaction between a scalar field --the Higgs field --and the fermion field.) Damned if I really understand the difference though --do both involve virtual Higgs bosons?


title: Do processes $P\propto\frac{1}{V^2}$, $P\propto\frac{1}{V^3}$, $P\propto\frac{1}{V^4}$, etc., exist in the real world?

text: Is there any real process in which $PV^n=C$ where $P,V$ stands for pressure, volume respectively. $C$ is a constant and $n$ is a positive integer? I am familiar with Boyle's law that states that $P\propto\frac{1}{V}$ when the temperature is constant. But according to the first equation, since $n$ is any positive integer, there are systems where $P\propto\frac{1}{V^2}$ , $P\propto\frac{1}{V^3}$ etc. Do such systems in which pressure is inversely proportional to square or cube of the volume really exist? Can anyone explain with example? Does this have any application in Engineering? (I found this in an engineering textbook.)


title: Equations of motion from the Standard Model

text: For some time now I have been wondering if you could not derive any sort of equations of motion from the Standard Model: $$\mathscr{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+i\bar{\psi}D\psi+\bar{\psi}\phi\psi+h.c.+\vert D\phi\vert^2-V(\phi).$$ Since it is a Lagrangian shouldn't we be able to use the Euler-Lagrange equation to find some equations of motion? Since I don't understand the theory myself this might already have been done, or is being done by physicists. However that does not impact my curiosity.


title: Step by step algorithm to solve Einstein's equations

text: I cannot completely understand what is a _regular_ method to solve Einstein's equations in GR when there are no handy hints like spherical symmetry or time- independence. E.g. how can one derive Schwarzschild metric starting from arbitrary coordinates $x^0, x^1, x^2, x^3$? I don't even understand the stress-energy tensor form in such a case - obviously it sould be proportional to $\delta(x - x_0(s))$, where $x_0(s)$ is a parametrized particle's world-line, but if the metric is unknown _in advance_ how do I get $x_0(s)$ without any a priory assumptions?


title: Potential of a dipole with actual physical extension?

text: I think everybody here knows the equation that gives the potential of a point like dipole, but how does the field look like if you have e.g. a metal sphere with radius $R$ and a certain dipol moment, how does this potential look like?


title: Why aren't we surrounded by Black holes?

text: The Bekenstein bound is a limit to the amount of entropy a thermodynamical system can have. The bound is given by the following expression: \begin{equation} S \leq \frac{2 \pi k R E}{\hbar c} \end{equation} where $k$ is Boltzmann's constant, $R$ is the radius of a sphere that can enclose the given system, $E$ is the total mass–energy including any rest masses, $ħ$ is the reduced Planck constant and $c$ is the speed of light. The equality is reached for Black Holes. Now, a system is in thermodynamical equilibrium when the entropy of the system is in a maximum and the constrains of the system( like pressure, volume, etc.) are satisfied. In our daily live, when we consider thermodynamical systems the bound is never achieve; only thermodynamical systems at the scale of astronomical objects seem to satisfy it. Why is the equality only achieved at certain scales?


title: Possible abuse of notation in statistical mechanics

text: I know that it often occurs that we need to take a derivitive with respect to $\beta$ in statistical mechanics. However, I think it is poor style to use equations with both T and $\beta$ in them especially since in most of the theory we take $\beta = \frac{1}{T}$. I see this abuse of notation frequently in textbooks, how to correct this?


title: Transmutation with cosmic radiation possible?

text: Can cosmic radiation (alpha radiation) transmute the material of a space craft, particular carbon, titanium and aluminum? Where can i find transmutation tables or formulas to calculate the possibility of a transmutation and the outcome?


title: How many seconds is a temporal meter?

text: Is there a proof that time is a 4th dimension? If it is, then why not measure it in units of the previous three? Logical right? How many seconds is a temporal meter?


title: When to use ideal gas law in fluid mechanics?

text: The ideal gas law (aka the equation of state) is given by $$ p/\rho_N = k_BT, $$ where $\rho_N$ is number density. When am I allowed to use this to describe a fluid?


title: Riemann curvature tensor in first order perturbation theory as a Lie derivative of Riemann curvature tensor in zero order

text: I am having a difficulty solving my homework so I was hoping I could get some help, so here it is. It is about gravitational waves and first order gravitational perturbation theory, I have to prove that under the gauge transformation: $$h_{ab} \rightarrow h_{ab}+ \nabla{a} \xi_b + \nabla_{b} \xi_a$$ the curvature tensor: $${R^{i(1)}}_{klm}=\frac{1}{2} (\nabla_{l} \nabla_{m}{h^i}_k +\nabla_{l} \nabla_{k}{h^i}_m-\nabla_{l} \nabla^{i}h_{km}-\nabla_{m} \nabla_{l}{h^i}_k-\nabla_{m} \nabla_{k}{h^i}_l+\nabla_{m} \nabla^{i}h_{kl})$$ changes by: $$\delta {R^{(1)}}_{mnrs}=\xi^t \nabla_{t} {R^{(0)}}_{mnrs}+{R^{(0)}}_{tnrs} \nabla_{m} \xi_{t} - {R^{(0)}}_{tmrs} \nabla_{n} \xi_{t}+{R^{(0)}}_{mntr} \nabla_{s} \xi_{t}-{R^{(0)}}_{mnts} \nabla_{r} \xi_{t}$$ and hence is not gauge invariant. $h_{ab}$ is the metric perturbation of the first order which changes because of infinitesimal coordinate transformation, $x^{a} \rightarrow x^{a}+\xi^{a}$. This is given as exercise 9.6. in T.Padmanabhan, Gravitation - Foundations and Frontiers. I have tried all sorts of manipulations with covariant derivatives, but it all resulted in a bunch of asymetrical expressions with no connection with the solution which is the Lie derivative of Riemann curvature tensor in zero order.


title: A few questions about the Fermi Level/Energy

text: My first question is, how is the Fermi Energy for a material actually determined? I know this derivation, but it seems to say that the Fermi Energy is just based on the electron density (and maybe some effective mass) of the material. Is that really all that determines it? Secondly, I'm trying to figure out how the interfaces of various materials work in terms of their bands, but it's not clear to me exactly what _must_ be true in all cases (vs what is often but not necessarily true, or what is theoretically but rarely practically true). For example, Anderson's Rule starts by aligning the "vacuum levels" of the two materials, but then this article says that it's not a great idea to use the vacuum level, and the Anderson's Rule article says it's just not that accurate a rule, anyway. Similarly, it seems like the Schottky-Mott Rule isn't very successful either. Additionally, I've read somewhere that the Fermi Level (the electrochemical potential, the sum of the chemical potential and electric potential) has to be continuous everywhere in both of the materials, so that results in the _chemical potentials_ (i.e., the $T \neq 0$ Fermi Energies, which were normally different in the two materials) lining up, and that happens by having an _electric potential_ difference across them. But this picture from wikipedia then seems to suggest that either what I just said is wrong, or the label should really be "Fermi _energy_ " (or chemical potential) in their definitions. Which is it? So, what can I always depend on and know is true in these situations? Thank you!


title: Peierls Argument for Absence of Long Range Order

text: I'm really confused about the argument in Cardy's book for why there can't be long range order in 1D for discrete models. Let me just copy it out, and hopefully someone can explain it to me. He takes an Ising-like system as an example. We start with the ground state with all spins up, and we want to see if this state is stable against flipping the spins in some chain of length $l$. This chain has two domain walls at the endpoints, so we get an energy change of $4J$. Then the claim is that there is an entropy of $\log l$ associated with this chain, since "each wall may occupy $O(l)$ positions." If this were true, we would get a free energy change of $4J-\beta^{-1} \log l$, and this would imply that the ground state is unstable to flipping very long chains. The only part I'm not on board with is the claim about the entropy. I would say that if $L$ is the length of the system, then we have $L$ places to put the chain, so we get an entropy of $\log L$. Certainly as $L\to \infty$ this gives no long range order, as expected. So, is the entropy $\log l$ or $\log L$? (Incidentally, I'm perfectly happy with his argument in 2D...)


title: Time evolution of the worldlines of 2 particles

text: Suppose I have a lab frame that is freely falling in a gravitational field of the Earth -- assume non-homogeneity-- and a uniform constant electric field. There are 2 test particles in the frame -- both of mass $m$, but one is of charge $e$ and the other neutral. They are initially separated by by a vertical distance $h$. I would like to model how their distance evolves. Could anyone help me? * * * Things I've thought of (but may not be entirely right): I shall assume that the interaction between the particles is negligible. Then the geodesic equation for the neutral particle is $$u^a\nabla_a u^b=0$$ where $u^a$ is its 4-velocity. The worldline of a charged particle is $$u'^a\nabla_a u'^b=\frac{e}{m}F^b{}_au'^a$$ where $F^b{}_a$ is the electromagnetic tensor. And then...?


title: Walking problem physics

text: 1. How could you measure someone's walking speed at a constant rate?    2. What types of errors could you potentially come across?    3. And what are some different ways of modelling walking motion?


title: What is coherence in quantum mechanics?

text: What are coherence and quantum entanglement? Does it mean that two particles are the same? I read this in a book called _Physics of the Impossible_ by Michio Kaku. He says that two particles behave in the same way even if they are separated. He also says that this is helpful in teleportation. How can this be possible? Could somebody please explain?


In Fresnel Diffraction at a circular aperture the central image according to 'Optics' by brij lal and subramanyam will be bright if odd number of full half-period zones can be constructed.But(according to me) an odd number of half-period zones would correspond to 0 amplitude i.e. the central spot will be dark as each half-period zone differs from the next by a phase of pi. I'd like to know if I'm wrong and why!


title: Dispersion-less media

text: As far as I know, vacuum is the only dispersion free medium for electromagnetic waves. This makes me wonder if there are any other dispersion free media for these waves? (Experimentally established or theoreticaly predicted) If there are none for electromagnetic waves, are there any for other kinds of waves?


title: List of the basic quantum mechanical variables

text: Is there a list of basic quantum variables/attributes that all quantum particles have? Ex. An electron has charge, position, speed, momentum, etc. Is there a complete list of these variables? I would figure not all quantum particles share the same set of variables? A photon has position and an electron has position but a photon does not have a charge and an electron does, though I guess it is said a photon has neutral charge.


title: What is "field size" when referring to perspective?

text: I'm trying to get my head around perspective. If you look at the animation in Wikipedia  : ![enter image description here](http://i.stack.imgur.com/5jHlj.gif) You can see that the cube looks very distorted, then the edges go more parallel. From my understanding, the distorted version is when the camera is very close to the cube, and the less distorted version is when the camera is further away. But if the camera is further away, why does the cube not appear smaller? Is this something to do with keeping the "field size" constant? Is the field size a flat area? Or part of the area of a sphere? It looks as though you can see more stuff in the background and foreground when you're zoomed out compared to zoomed in, so how can you know that you're keeping the "field size" constant?


title: Electromagnetic radiation lenses

text: This will probably be a very basic question, but looking for a simple answers. What I know   * the _visible light_ is a form of electromagnetic radiation with a defined wavelength.   * the full spectrum contains radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays and gamma rays. For the visible light exists many types of lenses and/or mirrors - where the refraction changing the direction of waves - e.g. focusing lenses, made of transparent material like glass. Do such lenses exist for other types of electromagnetic radiation? E.g. Are there lenses for radio waves or for X-rays? What materials are they made of ? What about the mirrors (e.g. gamma ray mirror) ? If such mirrors/lenses do not exist, then why not? Can someone point me to some basic articles tolearn about this?


title: Right hand rule to measure direction of electric field

text: The diagram: a positive plate is on the left of the page and a negative plate is on the right. The velocity of the particle is going down the page. So, with a right hand rule, I would point my thumb in the direction of the velocity (down) and my palm in the direction of the force. But what's the direction of the force? I thought that for a positive charge it would be going from the positive to the negative charge (left to right on the page), but it appears by an answer key that it's actually going from the left to the right, even for a positive charge... This would mean the magnetic force is pointing out of the page. Help!


title: Delivered/Reflected Power by Drive on a Hamiltonian System

text: Imagine a SHO with a drive F(t). (or in general a Hamiltonian system) What is the power delivered to the system and can we talk about the power reflected? is i am imagining say a MW oscillator cavity fed by a transmission line. My 3 attempts at the question: 1) One definition of power I know is $P = \dot{U}$, where $U=\int F(x,t) \cdot dx(t)$ is the energy. so that the power delivered $\dot{U}=\frac{d}{dt} \int F(x,t)\cdot dx(t)$ which is a funcky chair rule derivative?? This is where I get stuck on #1 2) Another definition for power i know is $P = F(x,t)\cdot v(t)=F\cdot\frac{\partial \mathbb{H}}{\partial p}$   This may also explain how to do the "chain rule in #1": ie $\dot{U}=\frac{d}{dt} \int F(x,t)\cdot dx(t)=\frac{d}{dt} \int F(x,t)\cdot \frac{dx(t)}{dt}dt=\frac{d}{dt} \int F(x,t)\cdot v(t) dt=F(x,t)\cdot v(t)$ so maybe it is this simple? Where does this definition of Power come from / why would it be more fundamental? 3) Another approach I thought of is take the Hamiltonian term of the drive: $\mathbb{H} $ ~ $F(t)x $ and the power on this energy, should be: $P=\dot{U} = \frac{d}{dt} (F(t)\cdot x(t))=F\cdot v + \dot{F}x$ So here there are TWO terms! the power from part (2) $F\cdot v$ and a new term, which I cannot account for: $\dot{F}x$ What is this term and why doesn't this approach work? Is this the reflected power? I am sorry for this basic question, but I hope it will help clear up some fundamental things. :)


title: What is the sun's spectral series?

text: My physics book says that six colors can be distinctly seen in white light: red, orange, yellow, green, blue, and violet. Does solar light only use these six wavelengths and mix them additively, or does it use a range of colors from red to violet? And what effect does the scattering of blue light from the sky have on the solar spectrum?


title: De broglie equation

text: What is the de Broglie wavelength? Also, does the $\lambda$ sign in the de Broglie equation stand for the normal wavelength or the de Broglie wavelength? If $\lambda$ is the normal wavelength of a photon or particle, is $\lambda \propto \frac{1}{m}$ true? Can a wave to decrease its amplitude or energy with the increase of mass? (My chemistry teacher told me that all matter moves in the structure of a wave and because of de Broglie's equation matter with less mass shows high amplitude and wavelength while matter with great mass shows less amplitude and wavelength. So as a result of that we see that objects like rubber balls, which have a great mass relative to the electron, move in a straight line because of its mass. However, I haven't found any relationship between wavelength and amplitude.)


Why is a proton assumed to be always at the center while applying the Schrödinger equation? Isn't it a quantum particle?


title: The direction of the Lorentz Force

text: We have a square loop (I believe it's called) in a uniform magnetic field between the 2 poles of a permanent magnetic field (green is N, red is S). P is connected to the positive pole of a voltage source and Q is connected to the negative pole (so there is a current): ![enter image description here](http://i.stack.imgur.com/uUiEq.jpg)   * _Show the direction of the Lorentz force_ **My idea:** The magnetic field lines point towards S from N. There is no Lorentz force at b, because the angle between the current and the field lines is $180^o$. Using the left hand rule we find that at h the lorentz force goes 'into' the paper and at the right side of h the Lorentz force goes out of the paper. **Problem:** My book says that the magnetic field lines face from N to S, which causes opposite answers. Why is this? I'm thinking it has to do something with $P$ and $Q$, but I'm not quite sure..


title: Can $10^{23}$ stars be treated with methods of statistical mechanics?

text: Statistical mechanics is used to describe systems with large number of particles ~$10^{23}$. The observable universe contains between $10^{22}$ to $10^{24}$ stars. Can we treat those many stars as a statistical mechanical system (for which one can define an entropy, temperature..etc)?   .


title: Military-grade U-235 uses

text: In the news it is often mentioned that some countries are going to enrich uranium to "military grade" (i.e. 80%+) and that it is possible to use it for a nuclear bomb. 1) Is that correct that there are no nuclear warheads in service made of U-235, as plutonium ones are much smaller & much more efficient (i.e. burn most of it's fissile fuel unlike U-235 cannon-type ones)? 2) Is that correct, that the only current military uses for 80%+ U-235 are naval nuclear reactors and in rare occurrences - case and/or X-Ray "reflector" in Teller–Ulam configuration as it is slightly better neutron breeder compared to more commonly used U-238 and allows to slightly reduce mass of "high-tech" plutonium charge at the same yield?


title: How to calculate Temperature Humidity Wind Index?

text: I would like to know how to calculate Temperature Humidity Wind Index (THW Index)? I know how to calculate Heat Index and Wind Chill. I am asking this because my weather station Davis Vantage Pro2 calculates THW index but I could not find any information on NOAA. This is what it says under Help for my weather station: > The THW Index uses humidity, temperature and wind to calculate an apparent > temperature that incorporates the cooling effects of wind on our perception > of temperature.


title: Why are magnetic lines of force invisible?

text: We can very well feel the magnetic field around a magnet, but we can't see it. Why is that so? Also, can we cut a portion of the field and use it?


title: Kinetic energy of a photon and Schwarzschild radius

text: I have read here, that $\frac{1}{2}mv^2$ must not be applied on a photon ever. If i want to calculate escape velocity $v_e$ i need to use $\frac{1}{2}mv^2$ because we say that kinetic energy _(positive)_ must be same or larger than gravitational potential _(which is negative)_ in order for an object to escape. It is done like this: $$ \begin{split} W_k + W_p &= 0\\\ \frac{1}{2}mv_e^2 + \left(-\frac{GMm}{r}\right) &= 0\\\ v_e &= \sqrt{\frac{2GM}{r}} \end{split} $$ Than we say if there is a black hole inside certain radius we call Schwarzschield radius $r=R_{sch}$ not even light can escape because its escape velocity is smaller than needed to escape. At the border of the sphere with $R_{sch}$ light can barely escape, so it must hold that $c$ equals escape velocity $v_e$. So we write down the equation below and derive $R_{sch}$. $$ \begin{split} c &= \sqrt{\frac{2GM}{R_{sch}}}\\\ R_{sch} &= \frac{2GM}{c^2} \end{split} $$ This is a well known equation, but it is derived allso using $\frac{1}{2}mv^2$ for light (photons). This is in contradiction with 1st statement in this post. So is the last equation even valid???


title: block slides on smooth triangular wedge kept on smooth floor.Find velocity of wedge when block reaches bottom

text: Find the velocity of the triangular block when the small block reaches the bottom: ![enter image description here](http://i.stack.imgur.com/WXQHF.png) Here is what I did:   The final velocity(at the bottom)of the small block of mass m is $\sqrt{2gh}$ along the plane of the incline with respect to the triangle (due to uniform acceleration $g\sin a$ covering distance $\frac{h}{\sin a}$). Let the velocity of the triangular wedge be $V$. Since there is a net external force in the vertical direction, linear momentum is conserved only in the horizontal direction. Then,the velocity of the small block with respect to ground is $$ \Bigl(\sqrt{2gh} \cos a\ + V \Bigr) ,$$ we are not considering the direction of $V$, which intuitively should be leftward, but we take rightward. Afterward we should get a negative sign indicating the left direction. Applying conservation of linear momentum in the horizontal direction we get $$ MV + m \Bigl(\sqrt(2gh) \cos a\ + V \Bigr) = 0 .$$ Thus we find that $$ V = \frac{-m \Bigl(\sqrt(2gh) \cos a\ \Bigr)}{m+M}. $$ However, my book mentions that the answer is something different. I wouldn't like to mention it here because I do not want reverse-engineering from the answer. Please help and explain where I may be wrong.


title: Dichroism in uniaxial crystals

text: I need a same help with it. Some books where i can find a real math explanation of this effect will be good help!! simple exp of this effect will be good too)


title: What is the inertia caused by angular momentum when twisted on it's rotating axis?

text: I would like to provide a more thorough answer to this question here http://aviation.stackexchange.com/q/3709 but I realized I don't know enough about angular momentum. If an airplane wheel is rotating at 100 rpm, and the wheel weighs 10kg, with a diameter of 50cm and a uniform mass (approximations applicable to a standard small aircraft), what is the difference in force necessary to bring the plane to 20 degrees of bank as opposed to when the wheels are stopped? I know that this involves calculating angular momentum, which I have at 5kg*m/s per wheel, so 10kg*m/s total, I'm just not sure how I would quantify the affect of this angular momentum when trying to bank the aircraft 20 degrees over a course of 5 seconds (replicating first turn in the airport pattern). I bet the following terms are involved: $\sin(20), 5s, 10kgm/s.$ Not sure if it's relevant, but we can assume the aircraft wheels are suspended 1 meter below the aircraft.


title: Thermal AdS and the Hawking Page phase transition

text: I have some difficulty understanding the concept of pure thermal radiation, as described in Hawking and Page's paper on the Hawking-Page phase transition. The four-dimensional thermal AdS solution (with cosmological constant $\Lambda<0$) is given by $ds^2=f(r)d\tau^2+\frac{1}{f(r)}dr^2+r^2d\Omega^2$, with $f(r)=1+\frac{r^2}{L^2}$, $L^2\equiv {-3/\Lambda}$ and the imaginary time $\tau$ is periodic in the inverse temperature $\beta$. Apparently this describes thermal radiation. > How should I see this thermal radiation? Does it consist of a gas gravitons, > since the Einstein-Hilbert action has no other fields than the metric > tensor? Or does it consist of other particles and should I add other fields > to the action to describe these? Then in Hawking and Page's paper, it is stated that: " _The dominant contribution to the path integral is expected to come from metrics which are near classical solutions to the Einstein equations. Periodically identified anti-de Sitter space is one of these and we take it to be the zero of action and energy. The path integral over the matter fields and metric fluctuations on the anti-de Sitter background can be regarded as giving the contribution of thermal radiation in anti-de Sitter space to the partition function Z. For a conformally invariant field this will be: $\log Z= \frac{\pi^4}{90}g\frac{L^3}{\beta^3}$._ " Here $g$ is the effective number of spin states. > Since they state thermal AdS is taken as the zero of the action and energy, > does this mean that the $\frac{\pi^4}{90}g\frac{L^3}{\beta^3}$ comes from > loop corrections? And where does the expression come from? From the above expression the free energy $F_{AdS}$ of thermal AdS follows. Later on in the paper, the temperature $T_1$ at which the free energy $F_{BH}$ of the (stable) black hole solution becomes negative is determined. It would seem to me that the phase transition occurs when $F_{AdS}=F_{BH}$. But rather than calculating the temperature at which this occurs, it is stated that for $T\gtrsim T_1$ the black hole solution will have lower free energy, hence will be thermodynamically favorable. So it looks to me that $F_{AdS}$ is neglected. > Why can we neglect $F_{AdS}$? Is there a parameter in the expression for > $\log Z$ above which is very small? (Or is it just $\hbar$?) Any help is appreciated, I'd be happy with even one answer to one of the questions in the blockquotes.


title: I need to find a formula between the weight of an object and the volume of the buoy that can float it on water

text: So my problem is that i am working on an art instalation that needs to float. The idea is that i need to find a formula between the volume (V) of the buoy and the total weight of the whole instalation.


title: Should Brillouin zone be a continuous object rather than a discrete one in the thermodynamic limit?

text: For example, just consider a 1D atom chain with $N$ sites and lattice constant $a=2\pi$, under periodic boundary conditions, the crystal momentum reads as $k=\frac{n}{N}\frac{2\pi}{a}=\frac{n}{N}$, with $n=0,1,2,...,N-1$. In the thermodynamic limit $N\rightarrow \infty$, there would be infinitely dense $k$ points within the region $[0,1)$, and the Brillouin zone( **BZ** ) is defined as the _continuous_ real number interval **BZ** $=\left [0,1 \right )$. But $k=\frac{n}{N}$ is always a rational number, why not define **BZ** to be a _discrete_ set as **BZ** $=\mathbb{Q}\bigcap \left [0,1 \right )$ ? where $\mathbb{Q}$ represents the set of all rational numbers. Thank you very much.


title: Introductory examples of AdS/CFT duality

text: I would like to know, what are the simplest/starting/basic examples that are typically used to introduce students to how AdS/CFT really works? (not the MAGOO paper, as I am not sure it has concrete examples that a beginning student can work through) For example, are the PhD. papers of Maldacena used for this purpose? What about the papers on holographic entanglement entropy (like the ones by Faulkner and Hartman and Shinsei Ryyu and Takayanagi)? Or the papers on worldsheet derivation of the duality like http://arxiv.org/abs/hep-th/0703141 (and its predecessors) and http://arxiv.org/abs/hep-th/0205297 and the Gopakumar-Vafa papers of '98?


title: Quantum Rigid Rotor Perturbation

text: As the title says, I have a rigid rotor with a perturbation given below $$H=\frac{L^2}{2I}-\alpha B L_z.$$ So I know that the eigenvalues of $H$ will be $\ell(\ell+1)/2I -\alpha B m$ where $m$ is our projection component. I am then asked to find the probability that the system will be in the $\ell$th state if it started in the ground state of $H_0$. This is $$P=|\int_0^t \langle lm|V|00\rangle e^{i \omega_{fi}t'}dt'|^2$$ Which gives me 0 for the probability that it is in any excited state. Is this correct?


title: What exactly is an anti-neutrino?

text: According to the the definition of anti-particles, they are particles with same mass but opposite charge. Neutrinos by definition have no charge. So, how can it have an anti-particle?


title: Should you run when under rain?

text: When it's raining, would you get less wet if you run or more wet? I think you will get less wet, because rain is coming down in a constant volume and if you run you will have a shorter amount of time you will get less volume I'm not too sure though, some people say when you run you are running into a more volume of rain because rain is not all coming up to down, it will hit you more horizontally


title: The role of the affine connection the geodesic equation

text: I apologise in advance that my knowledge of differential geometry and GR is very limited. In general relativity the equation of motion for a particle moving only under the influence of gravity is given by the geodesic equation: $$ \ddot{x}^\lambda + \Gamma^\lambda_{\mu\nu}\dot{x}^\mu\dot{x}^\nu =0. $$ I am looking for a conceptual description of the role of the affine connection, $\Gamma^\lambda_{\mu\nu}$ in this equation. I understand that it is something to do with notion of a straight line in curved space. Comparing it to the equation for a free particle according to Newtonian gravity: $$ \ddot{x}_i = -\nabla\Phi, $$ Then it kind of looks like the affine connection is our equivalent of how to differentiate, except that our scalar field is now some kind of velocity?


title: Transforming an equation to the co-vector version

text: Ok, this question is more a result of my lack of knowledge of how to manipulate equations involving index notation rather than about physics... I have the geodesic equation with $U^\lambda\equiv\dot{x}^\lambda$:- $$ \dot{U^\lambda} + \Gamma^\lambda_{\mu\nu} U^\mu U^\nu $$ And I want to transform to the co-vector $U_\mu=g_{\mu\lambda}U^\lambda$. Can I simply multiply each vector by $g_{\mu\nu}$? Like so:- $$ g_{\mu\lambda}\dot{U^\lambda} + \Gamma^\lambda_{\mu\nu}g_{\mu\alpha}U^\alpha g_{\nu\beta}U^\beta $$ Or do I need to use $g^{\sigma\nu}g_{\nu\mu} = \delta^\sigma_\mu$ to rewrite $U_\mu=g_{\mu\lambda}U^\lambda$ and then sub it in? **Edit: Here's my attempt at the sub in method** So using $g^{\lambda\mu}g_{\mu\lambda} = \delta^\lambda_\lambda$ to rewrite $U_\mu=g_{\mu\lambda}U^\lambda$ as $U^\lambda=g^{\lambda\mu}U_\mu$. (Is this even correct?). Then differentiate:- $$ \dot{U}^\lambda=\dot{g}^{\lambda\mu}U_\mu + g^{\lambda\mu}\dot{U}_\mu $$ Can I assume that the differential of the metric wrt time is going to be zero? Obivously this is not going to be true in general since massive bodies move! But generally in simple problems would this be taken as true?


title: Chaos theory deterministic or non-deterministic?

text: ![enter image description here](http://i.stack.imgur.com/DOu2r.png) While i was studying about chaos theory, i stumbled upon this, When a ball confined in a square, and at the center is located a circle, is to bounce elastically, the path of the object deviates significantly. thereby causing chaos. I think this is equivalent to a sinai billiard. I couldn't understand the motion completely, but to start of with a simple case let two balls be located at the y-coordinate at the bottom middle,and the x-postion be with an uncertainty of $\pm \epsilon$ as shown in the above figure. The balls start moving with a velocity purely in the y-direction. What was claimed was that after successive bounces, THe ratio of the bounces could be determined.That is the ratio of the distance the two balls are from each other, for example After the $5th$ bounces from the square and $10th$ bounces from the square,their distance ratio could be determined. I really have no ideal how to begin with this problem. I know this is chaotic system and the initial errors in the position of the ball affects the later outcome, but i don't know how to proceed. Pleas help? And of course $\epsilon$ can be taken to be very small.


title: Bullet entering target

text: I've been given this question: > A bullet entering a target with an initial velocity of $ u $ loses $\frac > {u}{n}$ of its velocity after penetrating a distance $ a $ into the target; > how much further will it penetrate? My question is, doesn't some information about the nature of the bullet's acceleration need to given? How otherwise would I formulate the kinematic equations?


title: Why does local gauge invariance suggest renormalizability?

text: I'm reading Gauge Field Theories: An Introduction with Applications by Mike Guidry and this particular remark is not obvious to me: > _A tempting avenue is suggested by the QED paradigm, for if a local gauge > invariance could be imposed on the weak interaction phenomenology we might > expect the resulting theory to be renormalizable._ [Guidry, section §6.5, p. > 232] Is there an obvious argument for this _"local gauge invariance suggests renormalizability"_ remark? I should add that I still tend to get lost in the streets of renormalization when unsupervised, i.e. I'm not familiar enough with the entire concept to have any real intuition about it. (references on renormalizability that might help are of course also welcome)


title: Mathematical description of electron configuration

text: Does there exist a mathematical model for determining the electron configuration of an atom? I mean the theory which would generalize the notion behind the informal elements of the Aufbau principle.


title: Perturbation Theory Problem

text: I am self studying for upcoming exams and I am stuck on the end of a problem related to perturbation theory. Here is the problem ![enter image description here](http://i.stack.imgur.com/CrBoC.png) I can't manage to do the very last part -- showing the exact change in all of the energy levels. I have tried to factor the new Hamiltonian to get a change in the energy levels but it's not coming out. Has anyone got any ideas?


title: Do black holes accelerate in spin as they obtain more mass?

text: It is known that - When a star collapses during the formation of the black hole, the black hole obtains the spin of the star which it collapsed from... What I'd like to know is, If this spin _accelerates_ as a result of angular momentum (if any), What effects could this _rapid_ rotation have on the black hole, its gravity or anything else around it?


title: What happens to the $2\pi$ factor when calculating Raman-shifts in units of wavenumbers?

text: So from the classical theory, you find a formula for a dipole in a planar electromagnetic wave, where there will be two cosine terms with a frequency (actually angular velocity in $[rad/s]$, as the argument of a trigonometric function should be dimensionless and it is multiplied by $t[s]$) $\omega_i \pm \omega_n [rad/s]$ corresponding with the Stokes & anti-Stokes Raman shifts. ($i$ for impinging and $n$ for denoting the $n$th normal vibrational mode) The wikipedia page states that $\Delta \omega_{\pm} = \frac{1}{\lambda_i}\pm \frac{1}{\lambda_n}$ Now here is something that alarms me: this is _not_ the same $\omega$ (though it is proportional to it), it can't be. The units are all wrong. On top of that, even if you would just accept that assignment to $\Delta \omega$, then it's not a wavenumber, as the wavenumber $k$ is defined as $k=\frac{2\pi}{\lambda} [rad/m]$ I understand the cm$^{-1}$ part (magnitudes are then usually between 0-2000, which is perfect), but what happened to the $2\pi [rad]$ factor? Is the "Raman shift" actually in terms of "inverse wavelength" ? Do you have to multiply $\Delta \omega$ by $2\pi [rad]$ to get the $\text{actual}$ wavenumber? Extra remark: nowhere in the wikipedia-article is implied what $\omega$ should be (which is something _I_ (possibly wrongly) assume), though I find it a bit confusing to use the symbol commonly known as the angular velocity ($\propto$ frequence) for denoting a difference in wavenumbers (usually $k$)


title: Is the energy of a photon continuous/discrete?

text: I was struggling today with this question: does a free photon have a continuous energy spectra? Free means in no context of any energy system (eg. an atom, em field). Although I'm asking myself if the quantization of the electromagnetic field is omnipresent and will always make the energy discrete? Edit: This leads me also to the question: if we have 2 energy levels (like in hydrogen: ground state and first excited) the uncertainty principle tells us, that the energy isn't quite exact defined: $\Delta E\Delta t \ge \hbar$ . Therefore the final energy of the emitted photon won't have a discrete energy, since it would be sth. like $E_{photon} = E_{0} + \Delta E$ ?!


title: Magnetic field-pulsed microwave transmission line

text: Here's the reference: > The researchers showed that a magnetic field-pulsed microwave transmission > line containing an array of superconducting quantum interference devices, or > SQUIDs, not only reproduces physics analogous to that of a radiating black > hole, but does so in a system where the high energy and quantum mechanical > properties are well understood and can be directly controlled in the > laboratory. But my problem is - I'm unable to understand the above statement . Can anyone able to put the above statements in a simple manner


title: The issue on existence of inverse operations of $a$ and $a^{\dagger}$

text: I have asked a question at math.stackexchange  that have a physical meaning. > _My assumption: Suppose $a$ and $a^\dagger$ is Hermitian adjoint operators > and $[a,a^\dagger]=1$. I want to prove that there are no inverse operators > for $a$ and $a^\dagger$._ I thought that this assumption purely mathematical, but I have no answers there. Maybe I am missing something? I will clarify that $a$ and $a^\dagger$ are just Creation and annihilation operators for quantum harmonic oscillator.


title: What is paramagnetic current-current correlation?

text: I know what paramagnetism is. But first I want to know about the paramagnetic current and then the above-mentioned correlation? Actually, I am working on a paper on superconductivity where I have seen the term: Scalapino, White, and Zhang.


title: Where does a spinning figure skater's energy go when she slows down?

text: Today in physics class we were talking about angular momentum and rotational kinetic energy. My teacher used the classic example of a figure skater spinning on ice - when she pulls her arms in, her angular momentum is conserved and her angular velocity increases, meaning that her rotational kinetic energy also increases. Of course, this increase in energy must come from somewhere - in this case, it comes from the figure skater doing work on her arms and pulling them in toward her body. Then I started wondering - if the figure skater slows her rotation by extending her arms, she decreases her rotational KE. Where is her energy going? Or to put it another way, what force is doing work on the figure skater in order to decrease her energy?


title: How gravitation effect on tides

text: I know that tide is caused by the gravitational pull of moon but what I don't know is how it effects on water. I have actually these doubts.   1. Why does gravity of the moon creates tides only in water?   2. Are there other things (other than water) where tides are created in earth( I have heard that, in some moons of Jupiter tides(of ground) can be found on the surface due to Jupiter's gravity)?   3. If we take a bowl of some length; lets say 30 cm diameter and fill it with water and keep it in a full moon night. Whether it will create tide?   4. If moons gravitational pull can cause tides in seas, then why a sailor can't feel the gravitational pull of moon?


title: Will an emf be induced across the ends of a stationary metal rod placed in a time-varying magnetic field?

text: Consider a thin metal rod placed in a magnetic field whose direction is constant but whose magnitude is changing with time, with the length of the rod perpendicular to the direction of the magnetic field. The rod is stationary, so there is no motional emf. If the rod were part of a conducting loop, there would be an emf induced in the loop as the magnetic flux associated with the loop would change with time. But if I connected an ideal voltmeter (with infinite resistance) across the ends of the rod when it is _not part of a conducting loop_ , would the voltmeter show any deflection? If yes, what would be the magnitude of this emf?


title: Rydberg quasimolecules & stark states?

text: I found this image :![enter image description here](http://i.stack.imgur.com/2ua1x.jpg) on the internet and I traced it back to this article ,I wanted to use it as part of an architectural visualization for my project(architecture) but for this to happen I need to understand what is it about ,I understand what a "probability density" is ,and some "good enough" concepts of quantum mechanics "Without the hard math", but I couldn't find anything about stark states , all I could find on Wikipedia and the internet "Google" was stark effect but nothing that points out what this visualization "image" is about ?


title: Doesn't Conservation of angular momentum contradict the need for a centripetal force?

text: If an object is rotating in space, then there has to be a centripetal force acting upon it to constantly change its direction. I thought if an object begins to rotate in space, it would slow down since there is no centripetal force that keeps it changing direction, but that is contradictory to the conservation of angular momentum law.


title: What is Maupertuis' principle good for?

text: The strength of Hamilton's principle is obvious to me and I see the advantage. Now, for conservative systems we also have Maupertuis' principle that says: $$ \delta \int p dq =0$$ and I am not sure how to derive an equation of motion from this? Is this of any use in practical computations? So, can one apply this principle for example to the harmonic oscillator?- I have never seen anybody using it. Further, I read in Goldstein's classical Mechanics that the variation in Maupertuis' principle is not the one in Hamilton's principle, since we have constant Hamiltonian and changing time, whereas Hamilton's principle has constant time and varying Hamiltonian (in general). I am a little bit wondering about this, since you could easily get Maupertuis' principle from Hamilton's principle: $$ \delta \int L dt = \delta \int p \dot{q} - H dt = \delta \int p \dot{q} dt = \delta \int p dq =0,$$ if $H$ is constant. Can anybody here explain to me, why we have to use a different variation and how one can use this principle?


title: Simulating diffusion from bulk to individual particles

text: I have a 2 compartment simulation. The first compartment simulates reactions using ODEs. The second compartment uses Brownian motion. I want to be able to have molecules from the ODE compartment diffuse into the Brownian compartment. It would seem I need to answer this question: What physics/maths governs the rate of diffusion out of a volume? Can I simply use Fick's law?


title: Band-limited double-step Fresnel diffraction for images with radio waves?

text: Is it possible to use band-limited double-step Fresnel diffraction to assemble a holographic image with radio waves? If not is there a simular principle?


title: Question on expression in "J.S.Bell : On the Einstein Podolsky Rosen paradox"

text: I have a question on the article `J. S. Bell, On the Einstein Podolsky Rosen paradox, Physics 1, 195, 1964.` (link) My question concerns the expression (3) of the article, at page 196. I don't understand what is the reasoning that leads to this expression of the expectation value... I think I miss something but I don't know what. This is what I understood from now on : $\vec{\sigma_1}$ and $\vec{\sigma_2}$ are the spins of the two particles that move apart and must be exactly opposite according to quantum mechanics when measured in a direction of the component $\vec{a}$. First, did I understand well and do we really have $$A(\vec{a},\lambda) = \vec{\sigma_1}.\vec{a} = \pm 1 \\\ B(\vec{b},\lambda) = \vec{\sigma_2}.\vec{b} = \pm 1$$ then ? If not, what does $A(\vec{a},\lambda)$ and $B(\vec{b},\lambda)$ correspond to ? A sort of $sign$ function or something like in the next section? Secondly, why $$ <\vec{\sigma_1}.\vec{a}\; \vec{\sigma_2}.\vec{b}> = -\vec{a}.\vec{b}$$ Is it because $\vec{\sigma_1}$ and $\vec{\sigma_2}$ are opposite ? Thanks !


title: Conceptually, what is negative work?

text: I'm having some trouble understanding the concept of negative work. For example, my book says that if I lower a box to the ground, the box does positive work on my hands and my hands do negative work on the box. So, if work occurs when a force causes displacement, how does negative work happen? Are my hands displacing anything?


title: Physical interpretation of superstrings

text: The scalar fields $X^\mu$ in bosonic string theory have a clear physical interpretation - they describe the embedding of the string in spacetime. Adding fermionic fields on the worldsheet is a generalization for sure, gives fermions in the spectrum, has a smaller critical dimension and no tachyons, that's all good - but I don't see how they can have any physical interpretation as nice as the above for the scalars - isn't everything about how a string moves in spacetime described by the $X^\mu$ part?


title: Asymptotic Completeness, generalized free fields, and the relationship of thermodynamics with infinity

text: Asymptotic completeness is a strong constraint on quantum field theories that rules out generalized free fields, which otherwise satisfy the Wightman axioms. If we were to take a limit of a list of continuous mass distributions $\rho_n(k^2)$ that approaches a distribution in some topological sense, however, is there anywhere an analysis of how the behavior of the $\rho_n(k^2)$ would approach the behavior of the free field? The following statement seems too bald (from the review "Outline of axiomatic relativistic quantum field theory" by R F Streater, Rep. Prog. Phys. 38, 771-846 (1975)): "If the Källén-Lehmann weight function is continuous, there are no particles associated with the corresponding generalized free field; the interpretation in terms of unstable particles is not adequate". Surely as we take the support of a generalized Lorentz invariant free field to be arbitrarily small we could expect that the behavior, at least as characterized by the VEVs, which constitute complete knowledge of a Wightman field, would eventually be arbitrarily close to the behavior we would expect from a free field? Classical thermodynamics has a complicated relationship with infinity, in that the analytic behavior of phase transitions does not emerge unless we take an infinite number of particles, but the behavior of very large numbers of particles nonetheless can approximate thermodynamic behavior rather well. By this elementary analogy, it seems premature to rule out generalized free fields. It also seems telling, although weakly, that the Källén-Lehmann weight function of an interacting field is nontrivial in quasiparticle approaches. Being able to derive an S-matrix requires that a theory must be asymptotically complete, however real measurements are always at finite time-like separation from state preparations, with the interaction presumably not adiabatically switched off at the times of the preparation and measurement, so that something less than analytically perfect asymptotic completeness ought to be adequate. EDIT: To make this more concrete, the imaginary component of the mass $1$ propagator in real space at time-like separation is $I(t)=\frac{J_1(t)}{8\pi t}$. If we take a smooth unit weight mass distribution $$w_\beta(m)=\frac{\exp(-\frac{\beta}{m}-\beta m)}{2m^2K_1(2\beta)}\ \mathrm{for}\ m>0,\ \mathrm{zero\ for}\ m\le 0,$$ for large $\beta$ this weight function is concentrated near $m=1$, with maximum value $\sqrt{\frac{\beta}{\pi}}$. For this weight function, the imaginary component of the propagator in real space at time-like separation is (using Gradshteyn&Ryzhik 6.635.3) $$I_\beta(t)=\int\limits_0^\infty w_\beta(m)\frac{mJ_1(mt)}{8\pi t}\mathrm{d}m= \frac{J_1\left(\sqrt{2\beta(\sqrt{\beta^2+t^2}-\beta)}\right) K_1\left(\sqrt{2\beta(\sqrt{\beta^2+t^2}+\beta)}\right)}{8\pi tK_1(2\beta)}.$$ Asymptotically, this expression decreases faster than any polynomial for large $t$ (because the weight function is smooth), which is completely different from the asymptotic behavior of $I(t)$, $-\frac{\cos(t+\pi/4)}{4\sqrt{2\pi^3t^3}}$, however by choosing $\beta$ very large, we can ensure that $I_\beta(t)$ is close to $I(t)$ out to a large time- like separation that is approximately proportional to $\sqrt{\beta}$. Graphing $I(t)$ and $I_\beta(t)$ near $t=1000$, for example, and for $\beta=2^{20},2^{21},2^{22},2^{23},2^{24}$, we obtain ![Graph of $I\(t\)$ and $I_\\beta\(t\)$ near $t=1000$](http://i.stack.imgur.com/Pk3pG.png) $I(t)$ and $I_\beta(t)$ are very closely in phase, as seen here, until $t$ is of the order of $\beta^{2/3}$ in wavelength units. We can take $\beta$ to be such that this approximation is very close out to billions of years (for which, taking an inverse mass of $10^{-15}m$, $\sqrt{\beta}\approx \frac{10^{25}m}{10^{-15}m}=10^{40}$), or to whatever distance is necessary not to be in conflict with experiment (perhaps more _or_ less than $10^{40}$). This is of course quite finely tuned, however something on the order of the age of the universe would seem necessary for what is essentially a stability parameter, and the alternative is to take the remarkably idealized distribution-equivalent choice $\beta=\infty$ as usual. ~~[I would like to be able to give the real component of the propagator at time-like and space-like separations for this weight function, however Gradshteyn &Ryzhik does not offer the necessary integrals, and nor does my version of Maple.]~~ EDIT(2): Turns out that by transforming Gradshteyn&Ryzhik 6.653.2 we obtain $$R_\beta(r)\\!=\\!\int\limits_0^\infty\\!\\!w_\beta(m)\frac{mK_1(mr)}{4\pi^2 r}\mathrm{d}m= \frac{K_1\left(\\!\sqrt{2\beta(\beta-\sqrt{\beta^2-r^2})}\right) K_1\left(\\!\sqrt{2\beta(\beta+\sqrt{\beta^2-r^2})}\right)}{4\pi^2 rK_1(2\beta)},$$ which _is_ real valued for $r>\beta$. As for $I_\beta(t)$, the approximation to the mass $1$ propagator at space-like separation $r$, $R(r)=\frac{K_1(r)}{4\pi^2 r}$, is close for $r$ less than approximately $\sqrt{\beta}$. For the real component at time-like separation, ~~it is almost certain that~~ one ~~simply~~ replaces the Bessel function $J_1(...)$ by $Y_1(...)$.


title: Physical interpretation of different selfadjoint extensions

text: Given a symmetric (densely defined) operator in a Hilbert space, there might be quite a lot of selfadjoint extensions to it. This might be the case for a Schrödinger operator with a "bad" potential. There is a "smallest" one (Friedrichs) and a largest one (Krein), and all others are in some sense in between. Considering the corresponding Schrödinger equations, to each of these extensions there is a (completely different) unitary group solving it. My question is: what is the physical meaning of these extensions? How do you distinguish between the different unitary groups? Is there one which is physically "relevant"? Why is the Friedrichs extension chosen so often?


title: What goes wrong when one tries to quantize a scalar field with Fermi statistics?

text: At the end of section 9 on page 49 of Dirac's 1966 "Lectures on Quantum Field Theory" he says that if we quantize a real scalar field according to Fermi statistics [i.e., if we impose Canonical Anticommutation Relations (CAR)], the quantum Hamiltonian is no longer any good because it gives the wrong variation of the creation operator $\hat{\eta_{k}}$ with time. Unfortunately, I can't make anything go wrong, so would someone show my mistake, or explain what calculation I should do to understand Dirac's remark. Here's my calculation. The quantum Hamiltonian is, $$ \hat{H}=\int d^{3}k |k|\hat{\eta_{k}}\hat{\eta_{k}}^{\dagger} $$ and the Heisenberg equation of motion is, $$ \frac{d\eta_{k}}{dt}=-i[\eta_{k},H]_{-}=-i\int d^{3}k'|k'|(\eta_{k}\eta_{k'}\eta_{k'}^{\dagger}-\eta_{k'}\eta_{k'}^{\dagger}\eta_{k}) $$ where the hats to indicate operators have been left out and $[A,B]_{-}$ is a commutator. Now assume that the $\eta's$ obey Fermi statistics, $$ [\eta_{k}^{\dagger},\eta_{k'}]_{+}=\eta_{k}^{\dagger}\eta_{k'}+\eta_{k'}\eta_{k}^{\dagger}=\delta(k-k') $$ and use this in the last term of the Heisenberg equation, $$ \frac{d\eta_{k}}{dt}=-i\int d^{3}k'|k'|(\eta_{k}\eta_{k'}\eta_{k'}^{\dagger}+\eta_{k'}\eta_{k}\eta_{k'}^{\dagger}-\eta_{k'}\delta(k-k'))=i|k|\eta_{k} $$ In the above equation, the first two terms in the integral vanish because of the anticommutator $[\eta_{k},\eta_{k'}]_{+}=0$ and the result on the right is the same time variation of $\eta_{k}$ that one gets quantizing using Bose statistics: nothing seems to have gone wrong.


title: How to find acceleration given position and velocity?

text: Sorry for this very simple question but I am still very new to the laws of motion. I am dealing with 2-dimensional vectors in my programming environment and I'm following these slides to learn about simple integrators. Near the end of the slides for the 4th Order Runge Kutta Integrator he is calculating acceleration like this:               a2 = acceleration(p2, v2)      However, I'm not quite sure where that function is defined in the slides. I'm sure the answer is very simple but for all prior slides the acceleration was always constant.


title: How to apply a Hadamard gate?

text: How to apply a Hadamard gate to 3 qubits? by example how to apply $H$ to $(1/\sqrt{2})(\left|000\right> \+ \left|111\right>)$?


title: Expansion in solid spherical harmonics on the lattice

text: I'm interested in calculating scattering processes (e.g. Coulomb scattering of an electron beam by a single ion) in the context of lattice quantum field theory, and wonder if there is something like the expansion of a plane wave in spherical harmonics on the lattice? (I mean in discrete space modeled by a three dimensional, finite, cubic lattice.) So I am looking for an orthonormal basis for complex valued functions on a finite lattice, where the angular and radial variables would be (approximately?) separated, as this is the case for the solid harmonics. Thank you for your help!


title: Imaginary time and string theory

text: Is imaginary time an extra dimension? In other words, are time and imaginary time considered two separate dimensions? If so, does imaginary time appear (as a separate dimension) in string theory (thus contributing to the 11 dimensions in M-theory)?


title: Full wave rectification

text: In the construction of a full wave rectifier, why is there central tapping in the secondary winding of the power transformer? What's the pure reason behind it? If there was no central tapping, what would be the change?


title: How can I add an acceleration vector to a velocity with a different direction?

text: I am in my last year of high school and am struggling with some homework. I'm sorry if this question is incredibly stupid, but I simply can't find the answer in my notes. If I have an object with a velocity of 10ms-1 travelling at a bearing of 090 degrees (to the right) with no acceleration, and then all of a sudden it spontaneously gains an acceleration of 2ms-2 at a bearing of 180 degrees (downwards), how do I calculate the velocity (magnitude and direction) of the object for every second after the object gains this acceleration? Thank you very much for your help, I really want to be able to understand this!


title: How are symmetries defined mathematically?

text: I have started working on differential geometry very recently. I am little bit familiar with mathematical concepts such as manifolds, differential forms and associated concepts. As I was speeding through I caught interest in Lie groups and whose invaluable connection with the symmetries. I would be happy if anybody answers to these few questions.   1. I want to know what is a symmetry from the mathematical point of view?    2. How are $su2$, $su3$, and other symmetries first formulated?    3. What was the motivation?    4. How do we identify internal symmetries and corresponding conserved physical quantities?


title: localized field quanta?

text: In the quantization of the Klein-Gordon field (for example) we interpret the quanta of the fields with definite momentum and energy as particles but are they also localized in space? Shouldn't a physical particle in quantum mechanics be represented by a wave-packet in the peaked about some point is space? Is the scenario same in quantum field theory? If yes, how can we write such a state mathematically? In a nutshell, are the field quanta _point particle_?


title: Lorentz transformation for electric and magnetic fields

text: How do derive the following transformation rule (J.D. Jackson third Edition 11.10) for electric and magnetic field? $$\vec E' = \gamma \left( \vec E + \vec \beta \times \vec B\right) - \frac{\gamma^2}{\gamma +1} \vec \beta \left( \vec\beta \cdot \vec E \right ) \tag{1}$$ $$\vec B' = \gamma \left( \vec B - \vec \beta \times \vec E\right) - \frac{\gamma^2}{\gamma +1} \vec \beta \left( \vec\beta \cdot \vec B \right ) \tag{2}$$ I know if $\beta$ is along positive x axis, the transformation of fields are given by $$\begin{align*} E_1' &= E_1 \\\ E_2' &= \gamma (E_2 - \beta B_3)\\\ E_3' &= \gamma (E_3 + \beta B_2) \\\ \end{align*}\tag{3}$$ and for magnetic fields by $$\begin{align*} B_1' &= B_1 \\\ B_2' &= \gamma (B_2 + \beta E_3)\\\ B_3' &= \gamma (B_3 - \beta E_2) \\\ \end{align*} \tag{4}$$ here is how I think of it so far, taking $\beta = (\beta_1, 0, 0)$ and taking $x$ component of $(1)$ should give me first of equation $(3)$ but I get $$E_1' = E_1 \left( \gamma - \frac{\gamma^2}{1+\gamma} \beta_1^2\right) = \gamma E_1 $$


title: Fermi theory of beta decay

text: What are the facts that one should consider and incorporate in going from non- relativistic Fermi theory of beta decay to its relativistic generalization? In other words, how would the Lagrangian or Hamiltonian of the interaction be modified? Can someone suggest me a book or a link that discusses relativistic Fermi theory?