Straightforward questions about calculating SUSY F-terms

So in the Lagrangian for a SUSY theory we have the F-terms, which I have seen written (e.g., in Stephen Martin's SUSY primer) as $F^*_i F^i$ where $F^i = \frac{\partial W}{\partial \phi^i}$. I have a few questions about this. 1) What do we do in the case of a gauge multiplet? For instance, in the MSSM we have $Hu = (Hu^+,Hu^0)$. Let's look at the term $y_{jk} Q_j \bar{U_k} Hu$ in the superpotential, where $Q$ is the left-handed SU(2) squark doublet. Should we take $F^{Hu} = \frac{\partial W}{\partial Hu} = y_{jk} Q_j \bar{U_k}$? Or should it rather be two terms, like $F^{Hu^+} = \frac{\partial W}{\partial Hu^+}$ $F^{Hu^0} = \frac{\partial W}{\partial Hu^0}$? 2) If it is $F^{Hu} = \frac{\partial W}{\partial Hu}$, then where Martin has $F^*_i$, should it really be $F^\dagger_i$? Is that implied by the raised and lowered indices? 3) In either case, why is it not a problem that we will get terms like ($y_{33}Q_3\bar{U_3})^*y_{22}Q_2\bar{D_2}$? I'm worried about whether such terms will form gauge invariants and whether they imply vertices that should not be there. I know quarks can mix, but this looks like it describes something like a vertex where a 3rd-gen up-type quark-antiquark pair directly annihilate to a 2nd-gen down-type quark-antiquark pair. Am I misreading this? Thanks so much for any help! This seems like a straightforward thing to answer but I haven't seen it addressed in the primer, etc.


Symmetry Breaking and Vacuum Expectation Values

Consider a real scalar field $\phi$ in a theory with a Lagrangian $$ \mathcal{L}:=-\frac{1}{2}\partial _\mu \phi \partial ^\mu \phi -V(\phi ), $$ where $$ V(\phi ):= -\mu ^2\phi ^2+\frac{\lambda}{4!}\phi ^4, $$ where both $\mu$ and $\lambda$ are positive real numbers. We see that the potential has a couple of non-zero minima: $$ V'(\phi )=-2\mu ^2\phi +\frac{\lambda}{3!}\phi ^3=0\Rightarrow \phi =0,\pm 2\mu \sqrt{\frac{3}{\lambda}}=:\pm V_0 $$ (It turns out that $\phi =0$ is a local max, and $\phi =\pm V_0$ are local mins; check the second derivative.) As the usual story goes, we must define a new field $\psi :=\phi -V_0$ and re- write the theory in terms of this $\psi$ to get the appropriate Feynman rules of the quantum theory. If I did my algebra correctly (the details aren't exactly relevant here anyways), this substitution gives us $$ \mathcal{L}=-\frac{1}{2}\partial _\mu \psi \partial ^\mu \psi -2\mu ^2\psi ^2+\mu \sqrt{\frac{\lambda}{3}}\psi ^3+\frac{\lambda}{4!}\psi ^4-6\frac{\mu ^4}{\lambda}. $$ (Our Lagrangian no longer admits the symmetry $\psi \mapsto -\psi$, hence the term "symmetry breaking".) Th question arises: Why is this substitution special? This form of the Lagrangian has some nice properties (namely that the potential has a local min at $0$), but surely there are some other substitutions that could given us some other nice properties as well. What about those? My understanding of this was the following: The LSZ Reduction Formula, among other things, requires a priori that the fields one is working with _have vanishing vacuum expectation value_. Thus, when applying the LSZ formula, we must be working with $\psi$, not $\phi$, and so the appropriate Feynman rules can be read off only when the Lagrangian is written in terms of $\psi$. I have just recently discovered a problem with this explanation, however. Before, I was under the impression that $\langle 0_\pm |\phi |0_\pm \rangle =\pm V_0$ (this theory evidently has two physical vacuums, whatever that precisely means), so that the definition of $\psi$ forces $\psi$ to have vanishing expectation value, so that the LSZ formula can be applied. However, I recently learned that $\pm V_0$ is _only an approximation_ to $\langle 0_\pm |\phi |0_\pm \rangle$, which implies that $\psi$ _only approximately has vanishing vacuum expectation value_ , which means that LSZ doesn't technically apply. It seems that the proper substitution is in fact $\psi :=\phi -\langle 0_+|\phi|0_+\rangle$. There are several problems I see with this:   1. The Lagrangian re-written in terms of $\psi$ should have a small, but non-zero, _linear_ term in $\psi$.   2. The Feynman rules I've been using all along that arise from the substitution $\psi :=\phi -V_0$ are only approximation.   3. The coefficients that arise from the 'proper' substitution $\psi :=\phi -\langle 0_+|\phi |0_+\rangle$ are going to be written in terms or something that can (to the best of my knowledge) only be calculated perturbatively (namely $\langle 0_+|\phi |0_+\rangle$), but we need to know these coefficients to obtain the Feynman rules to begin with (resulting in a 'circularity' problem). How does one go about resolving all these issues? (Disclaimer: I asked a very similar question here not quite a year ago, but my understanding of the situation has improved since then, and as is usual, my improvement of understanding has only brought forth many more questions regarding this, so I felt it was appropriate to address the issue once again.)


Why can't I use Bell's Theorem for faster than light communication?

I read this description of Bell's theorem. I understand he's restating it slightly, so there may be incorrect assumptions there, or I may have some. I think Bell's theorem _should_ lead to FTL communication, and I'll try to lay out my assumptions as to how, and propose an experiment. So:   1. A and B are entangled photons.   2. With 100% probability, if I measure A at X°, and you measure B at X°, our measurements will be opposite (mine will go through iff yours is absorbed). 3. So this is the experiment:   1. Generate A and B, and send B lightyears away.   2. Whoever's manning the experiment at B does the following:  **1.** If they want to send binary 1, measure B at 20°. **2.** If they want to send binary 0, don't do anything.   3. Now whoever is at A measures A first at 0°, then (if it goes through), at 40°.   4. The results should be (it seems to me):     1. If **1** above, then B was absorbed with 50% probability and was transmitted with 50% probability. In either case, the probability of seeing A get transmitted at 0° and get absorbed at 40° is less than or equal to 11.6% (because the probability of case 1 or 2 in the article is 50% for the measurement at 20° times either the probability of A transmitted at 0°, or B transmitted 40° (which under assumption 2, is the same as A **not** transmitted 40°) so the probability of both must be less than or equal to 11.6%).     2. If **2** above, then the probability of seeing A get transmitted at 0° and get absorbed at 40° is 20.7%. So A can determine a binary message from B, violating the no-communication theorem. If done with enough photons, you can send much longer messages. I know it is far likelier that I've made a mistake here than that I've disproved the no-communication theorem, so what's wrong? If anyone doesn't understand the experiment, ask and I'll try to clarify.


Probability of having energy $E$ when $E$ is bounded

For a canonical ensemble the probability of a system to have energy $E$ is $P(E)=e^{-\beta E}$. For that we consider the that the system can have any energy between $0$ to $\infty$. What will be the probability if the system can have energy only within certain range say, $E_{min}\leq E \leq E_{max}$ ($E_{min},E_{max}>0$)? **Derivation for canonical ensemble** For a canonical ensemble I follow this derivation. Consider the total energy of system and reservoir ($E_1+E_2$) is constant and then optimize the total number of possible configuration of system and reservoir, i.e. $\frac{\partial}{\partial E_1} (\Omega_1 \Omega_1) =0$ (Statistical Mechanics, R K Pathria). This along with the relation $\frac{\partial E_2}{\partial E_1}=-1$ gives $\frac{\partial}{\partial E_1} \ln \Omega_1 = \frac{\partial}{\partial E_2} \ln \Omega_2 = \beta$. Assuming all the configurations are equally probable, the probability is given by $P(E_1)=\Omega_1^{-1} = e^{-\beta E_1}$. But this doesn't imply $P(E)=0$ for $E>E_{max}~or~E<E_{min}$. The reason is probably that the boundedness condition ($E_{min}\leq E \leq E_{max}$) is not used in the derivation. How to obtain the correct probability factor if the energy is bounded?


Difference between deltaV and specific impulse

When reading any literature regarding space propulsion, I keep getting about these terms deltaV and ISP or specific impulse. I know specific impulse is supposed to be the velocity the propellant mass leaves the exhaust, but I'm not sure what deltaV means. For instance, I've read that from Earth low-orbit to Mars orbit we require a deltaV of 6.1, what units are those? Also, I would expect ISP to be in units of $\frac{meter}{second}$, but most texts use units of second.


the action of $\bar \psi \psi$ on a one-particle state

Suppose, in the quantized theory of Dirac field, we start with some one- particle state $|\vec p\rangle\equiv a^\dagger_{\vec p}|0\rangle$ in the Fock space. What is the action of the operator $\bar\psi\psi$ on this state? What is the intermediate physics? First I want to understand how $\psi$ changes one-particle state $|\vec p\rangle\equiv a^\dagger_{\vec p}|0\rangle$ and then how does the term $\bar \psi$ changes the resulting state after that?


Inequality of Clausius

I'm having some trouble understanding the inequality of Clausius. My treatment of the inequality applies to an engine operating in a cycle between two temperatures (i.e. like a Carnot engine). The inequality states that for an irreversible process: $$ (1)\ \int _{closed\ path} \frac {dq_{irr}}{T}<0 $$ From what I've been reading, $ {dq_{irr}} $ is the heat added to the system (the engine) and $ T $ is the temperature of the reservoirs, $ T_{res} $. If this is the case, then $-{dq_{irr}}$ is the heat added to the reservoirs, $ dq_{irr,res} $, and $$ (2)\ \int _{closed\ path} \frac {-dq_{irr}}{T_{res}}=\int _{closed\ path} \frac {dq_{irr,res}}{T_{res}}>0 $$ I read somewhere that if the reservoir is infinite then any heat added is actually occurring infinitely slowly. Is this true? If this is the case, then $$ (3)\ dq_{irr,res} = dq_{rev, res} $$ $$ (4)\ \int _{closed\ path} \frac {dq_{rev, res}}{T_{res}}=∆S_{res}>0 $$ This seems reasonable. Since the engine itself is operating in a cycle, $ ∆S_{sys} = 0 $. This therefore satisfies the second law of thermodynamics: $$ (5)\ ∆S_{univ}=∆S_{sys}+∆S_{res}=0+∆S_{rev}>0 $$ Does this all look right? $ (5) $ seems to say that the entropy change for an irreversible process is 0 for the case of the system itself, but is > 0 for the case of the surroundings. Is this only the case for the situation I have described? I am also unsure about $ (3) $. If the heat is being added to the reservoirs infinitely slowly, then wouldn't the same be true of the system? This would make $ ∆S_{sys} = -∆S_{res} ≠ 0 $ and is obviously not correct.


Why maximum energy transfer at natural frequency even if max amplitude occurs below $f_0$

This is a paragraph from my book: "For a damped system, the resonant frequency at which the amplitude is a maximum is lower than the natural frequency.However, maximum transfer of energy, or energy resonance always occurs when applied frequency is equal to natural frequency" This doesn't make intuitive sense to me. I understand that if there is damping, maximum amplitude occurs below $f_0$... so, shouldn't maximum energy be transferred at this driving frequency as amplitude maximum instead of at $f_o$?


Can "vacuum be brought" from outer space?

Instead of creating vacuum in chambers on earth, why can't vacuum be brought from outer space in chambers? Outer space pressure ranges from $10^{-6}$torr to $10^{-17}$torr very very low. Is it possible at all?


Second Law of Thermodynamics, heat and quantum-level explanation

1. I heard somewhere that the second law of thermodynamics is stating that some heat (energy) becomes unusable for work (at the scale of the whole universe). Why is it like that? Can't heat be converted into some energy forms? I do understand that entropy is the measure of spontaneity, but I am having hard time understanding this.   2. How can we depict heat in relativistic (classical) level and quantum level? Is there heat energy that is not mediated by any form of particle?


Confused about the concept of time and time dilation

I am having a hard time understanding what is time. If scientists define time as a multiple of caesium frequency, then time itself is dependent on motion, so what if I have a number of particles that are at 0 Kelvin? Do these particle experience time by the fact it exist in the fabric of spacetime? And if so, is this the fundamental reason why time dilation exists? Because you cannot exceed the speed of light therefore your movement is restricted, thus the time dilates as your movement is restricted?


Why would classical correlation in Bell's experiment be a linear function of angle?

Sorry if it's a newbie question, but I have trouble understanding the following part in the Wikipedia's explanation for the Bell's theorem: > With the measurements oriented at intermediate angles between these basic > cases, the existence of local hidden variables could agree with a _linear > dependence_ of the correlation in the angle but, according to Bell's > inequality (see below), could not agree with the dependence predicted by > quantum mechanical theory, namely, that the correlation is the _negative > cosine_ of the angle. Experimental results match the curve predicted by > quantum mechanics. Accompanied with the following plot showing corellation as a function of angle: ![quantum mechanics predicts correlation to be a cosine function](http://i.stack.imgur.com/zCAMO.png) But, the thing that troubles me is that I can't find the proof to that reasoning. A projection of a line onto an axis is the cosine of its angle, right? To clarify, if I slightly modify the image from that article, let's say you measure particles A and B using parallel detectors: ![Parallel case](http://i.stack.imgur.com/O616g.png) The results are perfectly correlated (well, anti-corellated, but that's due to spins being different), as expected. Now, if you deflect the B measurement device slightly ($\pi/8$, since that's where the "large classical error" happens), we can represent it like this: ![pi/8 angle](http://i.stack.imgur.com/XeSgj.png) As a complete newb, my knee-jerk reaction would be that detector B would detect a projection of $cos(\pi/8)$ length, compared to the detector B, as shown in the detail: ![Detail \(projection at the detector B\)](http://i.stack.imgur.com/PY8nL.png) Why does the classical theory predict linear correlation, then?


Gauge Field Tensor from Wilson Loop

It is possible to introduce the gauge field in a QFT purely on geometric arguments. For simplicity, consider QED, only starting with fermions, and seeing how the gauge field naturally emerges. The observation is that the derivative of the Dirac field doesn't have a well-defined transformation, because: $$n^\mu \partial_\mu \,\psi = \lim\limits_{\epsilon\rightarrow 0}\big[\psi(x+\epsilon n)-\psi(x)\big],$$ i.e. the derivative combines two fields at different spacetime points (having different transformation rules). We need to introduce a parallel transporter $U(y,x)$ that transforms as $$U(y,x) \rightarrow e^{ig\alpha(y)}U(y,x) e^{-ig \alpha(x)},$$ such that we can adapt the defintion of the derivative into a covariant derivative, that transforms in a well-defined way: $$n^\mu D_\mu \,\psi = \lim\limits_{\epsilon\rightarrow 0}\big[\psi(x+\epsilon n)- U(x+\epsilon n,x)\psi(x)\big].$$ From geometric arguments, it is straightforward to show that the parallel transporter is a Wilson line: $$U(y,x) = \mathcal{P}\,e^{ig\int\limits_x^y dz^\mu\, A_\mu(z)},$$ which introduces a new field, namely the gauge field $A_\mu$. See e.g. Peskin & Schroeder Chapter 15 for more detail. However.. Where the interaction term $\bar{\psi} A_\mu \psi$ emerged in a natural way, I totally don't see how the kinetic terms emerge. The standard way to proceed, is to consider a Wilson loop (a Wilson line on a closed path), and use Stokes' theorem: $$\text{exp}\left\\{ig\oint_\mathcal{C}dx^\mu\, A_\mu \right\\} = \text{exp}\left\\{ig\int_\Sigma dx^\mu \wedge dx^\nu\,\left(\partial_\mu A_\nu - \partial_\nu A_\mu \right)\right\\},$$ where of course $\partial_\mu A_\nu - \partial_\nu A_\mu \equiv F_{\mu\nu}$. In Peskin & Schroeder, they then consider a small rectangular loop, and see that in the limit $\epsilon\rightarrow 0$, $F_{\mu\nu}$ is invariant. But what's the point? I mean, the transformation law for $A_\mu$ is easily calculated from the definition of the Wilson loop: $$A_\mu\rightarrow A_\mu+\partial_\mu \alpha,$$ making $F_{\mu\nu}$ invariant by definition: $$F_{\mu\nu}\rightarrow \partial_\mu A_\nu - \partial_\nu A_\mu +\square \alpha-\square \alpha. $$ I would have liked to see a calculation, starting from a particular loop parameterisation, that naturally leads to the correct kinetic terms in the Lagrangian, as was the case for the interaction term. In other words $$\text{exp}\left\\{ig\oint_\mathcal{C}dx^\mu\, A_\mu \right\\} \leadsto -\frac{1}{4}\left(F_{\mu\nu}\right)^2,$$ but I have no idea how to do it. Or is the idea simply 'look I have found some quadratic derivative terms that are invariant, now let me fiddle a bit and put its square in $\mathcal{L}$'? If yes, then why do Peskin & Schroeder bother calculating a loop parameterisation (p484), if using Stokes' theorem would have been enough to find $F_{\mu\nu}$ somewhere? **EDIT** Based on the comments, a few clarifications. This is not a question about why we need kinetic terms or how they should like. I perfectly know how to construct the SM Lagrangian the standard way. You have a vector field $A_\mu$, you need kinetic terms so you use some Klein-Gordon like structure, adapt it a bit (because of gauge behaviour of $A_\mu$) and so on. Standard QFT. But this is all _manual_ , almost a bit trial and error, putting in terms because you know they work the way they should. It is like making the Lagrangian as a puzzle: just put in those pieces that fit. No problem with that, it works and is a widely accepted way of doing physics, but from a theoretical point of view, it is not so elegant. But we can do it in a more elegant way. If we start only from a Dirac field and the Dirac equation, then purely by mathematical and geometric arguments, the gauge field appears, as discussed above. The question is if we can also make the kinetic terms for the gauge field appear purely based on mathematical and geometric arguments. If you believe the QFT book by Peskin & Schroeder, you can, starting from a Wilson loop (as discussed above see p484-494). But then you end up with a factor $\epsilon^4$, the loop area squared, in front of the field tensor. You could restrict to a class of loops with area $S=1$, but there are two problems with that:   * In the calculation, the exponent and integral where expanded, using the fact that $\epsilon <\\!\\!< 1$. This contradicts our restriction $S=1$.   * This restriction invalidates the elegance somewhat, as now the loop area has to be fine-tuned. I expected to find something like 'in the limit $\epsilon\rightarrow 0$, the field tensor is what remains'. So, is it not possible then to get the kinetic terms for the gauge field purely from geometric arguments? If the answer is no, it doesn't make sense to brag about the natural emergence of the interaction term. It is not elegant when the interaction term emerges naturally, but you have to hand-pick the kinetic terms. For one, how do you prove that the naturally emerging field (interaction terms) and the one you put in (kinetic terms) are the same field?


Hooke's Law: why is only one force considered?

I have a question about Hooke's Law springs. What I learned is that a Hooke's Law spring gives a force F pushing left and right and each force equals $k \cdot |l-l_o|$ Let's say there is a mass block resting on a spring on a table. The amount of force on the mass block by the spring is thought to be $F$. But I am thinking that that is only considering the force of the spring pushing against the block. Shouldn't we consider the $F$ the spring is pushing against the table as well?


Which of these two different forms of spin-orbit interaction is correct?

I am seeing the spin-orbit interaction in two different ways:   1. $\lambda [\mathbf{p} \times \nabla V]\cdot \sigma$   2. $\lambda [\nabla V \times \mathbf{p}]\cdot \sigma$ I don't see how these two expressions can be equivalent though; in (1) there will be derivative operators from momentum acting on the gradient whereas in (2) this will not be the case. Which one of these is correct?


Moving electron - finding the wavefunction

On our modern physics class my professor did a problem: > Write down a wavefunction of an electron which is moving from left to right > and has an energy $100\text{ eV}$. At first i said: "Oh i know this!" and solved the case like this. **My solution:** The energy $100eV$ must be the kinetic energy of the electron. So i said ok this kinetic energy is very small compared to the rest energy and i can say that $pc \ll E_0$ which means i have a classical limit where: \begin{align} E=\sqrt{{E_0}^2 + p^2c^2}\\\ E=\sqrt{{E_0}^2 + 0}\\\ \boxed{E=E_0} \end{align} So now i can write the general wavefunction for a free right -mooving particle like this: $$\psi=Ae^{iLx}\quad L=\sqrt{\tfrac{2mE}{\hbar^2}}$$ So if i want to get the speciffic solution i need to calculate the constant $L$ and then normalise the $\psi$. Because $E=E_0$ i calculated the constant $L$ like this: $$L=\sqrt{\frac{2mE_0}{\hbar^2}}$$ while my professor states that I should do it like this: $$L=\sqrt{\frac{2mE_k}{\hbar^2}},$$ where $E_k$ is the kinetic energy of the electron. Who is wrong? I mean whaaaaat? The constant $L$ is after all defined using the full energy and not kinetic energy...


Why are reflected objects blurry?

While this may seems like a strange question, I'll try to explain it the best I can. Like 75% of Americans, I wear glasses and contacts to correct my worsening vision. I'm not far sighted so objects in the distance appear blurry. I took my glasses off to look at my iPad. After doing so I noticed in the reflection of the glass, my lamp. The lamp is in the distance and looks blurry to me. But on my iPad's glass it was blurry unlike the rest of the objects on screen, including the glass which I could see clearly because I'm near-sighted. It is strange to me why the reflection was blurry even though it was close up. How can this be?


Is it possible to recover the old Bohr-Sommerfeld model from the QM description of the atom by turning off some parameters? Can we use Ehrenfest's theorem (or some other scheme) to reduce the QM model to the Bohr-Sommerfeld model? If not, why not? The issue seems to be significant because it might shed some deep conceptual issues.


Differential or integral form of the conservation equations?

Is there a 'rule' for when it is best to use either the differential or integral form of the continuity and momentum equations in calculations?


Why planes have propellers in front but watercraft have them behind?

Why do propeller airplanes mostly have their propellers in front (of the fuselage or wings) while ships and boats mostly have them at the back? I realize that there are aircraft with pusher configurations but they are relatively rare.


What is the difference between phase difference and path difference?

I have learnt that path difference is the difference between the distance travelled by two waves meeting at a point. If that is path difference,then how will one know what is phase difference and how can one calculate or determine it?


Biconvex vs planoconvex lens

As I understand, less spherical aberration is obtained when a collimated beam is focussed with a planoconvex lens as opposed to a biconvex lens. What would be a situation when a biconvex lens should be used? When should one use a biconvex lens vs a planoconvex lens?


Special Relativity Question: Doppler shift

Imagine an observer watching a moving rocket carrying on it both, a light source and a clock. If on the rocket the clock is synchronized with the frequency of the light being emitted then will not a stationary observer see not only the clock on the ship slow but then necessarily the wave length of the light increase. Again, imagine those on the rocket are measuring the distance of one wavelength comparing it with some high precision ruler, then will not the observer watch the ruler contract and then necessarily the wavelength of the light along with it, a contradiction to the original conclusion?


On the coordinate independence of general relativity

I've been having a bit of trouble with the idea of coordinate independence in general relativity. Let me start with a simple example that I think illustrates my question conceptually: Say you have two objects, A and B. A is at rest and B accelerates directly away from A. From a mathematical point of view, there is absolutely nothing which distinguishes the two objects - A is accelerating away from B as much as B is from A. To distinguish the two, we need something extra - something empirical which tells us that B feels a force acting on it and A does not. In this sense, there are infinitely many coordinate motions which might be the truly unaccelerated ones, and they can only be distinguished by the _physical_ fact that they feel no forces. So, at some point, doesn't any physical law, even if it is written in such a way that it is satisfied in all coordinate systems, have to make reference to truly unaccelerated frames, or something similar? In other words, in order to connect an abstract mathematical description of motion, which in some sense can be transformed in any way, to physical observations, doesn't one have to anchor the mathematical reference somehow to physically meaningful concepts like unaccelerated frames? Now, as I've been studying the Einstein field equations, I've wondered about the following: if you compare the equations for empty, flat space, in inertial coordinates versus some weird accelerating coordinates they'd be exactly the same - the space must be Ricci flat, or $R^{\alpha \beta} = 0$. Of course, there are many solutions of $R^{\alpha \beta} = 0$ (including non-flat ones, but ignore those for a second). Without some physical knowledge, wouldn't it be impossible to say which coordinates lead to $g_{\alpha \beta} = \eta_{\alpha \beta}$, and which coordinates lead to $g_{\alpha \beta}$ being some complicated (still Ricci flat) function of the coordinates? I imagine that this means that while the field equations must be satisfied, they alone are not enough to say what someone will observe - one must also know how his coordinates relate to locally inertial ones. So I now ask: by the "coordinate independence of General Relativity", does one really just mean that the expression for curvature makes no reference to a coordinate system - that the expression for curvature is coordinate independent, and so the law relating curvature and mass-energy, the Einstein field equation, is valid in all coordinate systems? And yet, even though the law must be satisfied, in order to know what one will observe, it's necessary to also know which coordinate systems are (locally) inertial? Finally, if you're feeling up to it, is this a real point I'm making in general? Don't the mathematical laws of physics all have to "give up" and some point and make reference to the solely empirically-defined concept of unaccelerated motion? **EDIT** : I don't think I made myself totally clear - let's just say this. If I asked you to solve Einstein's field equations in flat space in the coordinates $x^\mu$, wouldn't you, in general, have to say that there's not enough information? How could you possibly know whether $x^\mu$ were inertial coordinates and $g_{\mu \nu} = \eta_{\mu \nu}$, or if $x^\mu$ were some weird accelerated coordinates, and $g_{\mu \nu}$ were the same flat metric but written in these accelerated coordinates? Both are solutions of the Einstein field equations. In this way, aren't the field equations underdetermined? Don't they have to be supplemented with information about how the coordinates relate to locally inertial coordinates?


Is the Chern-Simons integral of gauge fields over black hole singularities zero?

Suppose we have an evaporating black hole and a nonabelian Yang-Mills theory with a $\theta$ topological term. This counts the total number of instantons minus antiinstantons. Consider the total number of instantons minus antiinstantons inside the black hole, i.e. between the event horizon and the singularity. This is equal to the difference between Chern-Simons integral between the singularity and event horizon. According to black hole complementarity, we replace the interior with a stretched horizon. Do we have to add an effective Chern-Simons term to the stretched horizon action? If so, is this equivalent to saying the CS integral over the singularity has to be zero? Otherwise, we get decoherence by CS singularity value, converting pure states to mixed states?


How is this a condition for a Majorana zero mode?

EDIT: Nevermind, I figured it out. When you apply the expression on a state, you can clearly see what it does: The new state created by applying the Majorana operators must have the same energy as the original one. So when applied on a zero-energy state, this creates a degeneracy and the so-called Majorana zero mode. * * * I am currently trying to grasp this lecture by Frank Wilczek about Majorana zero modes on an 1-dimensional wire. In equation (31), he states that a general condition for such a zero mode is $$ \left[ H, \sum_{j=1}^{2L} c_j \gamma_j \right] = 0, $$ where the $c_j$ are coefficients and the $\gamma_j$ are the Majorana operators for the sites of the chain. I'm not sure I understand why this is a general condition for a zero mode if the two operators commute. Can anyone shed some light on this? Thanks in advance!


Help in Basics for Capacitance

How does capacitor store a charge? I'm a little bit confused, as I know that a capacitor stores charge but how does it do so?


Positivity of Total Gravitational Energy in GR

I read the following statement in the introduction to an article: > _Over the last 30 years, one of the greatest achievements in classical > general relativity has certainly been the proof of the positivity of the > total gravitational energy, both at spatial and null infinity. It is > precisely its positivity that makes this notion not only important (because > of its theoretical significance), but also a useful tool in the everyday > practice of working relativists._ Since I haven't been involved in GR for about 30yrs, I've missed this. Could someone briefly explain how these results are stated and give some references if possible.


Fock picture of bosonification in condensates

I want to understand how bosonification in a condensate must be interpreted in the Fock states picture Say i have uncoupled fermions in a set of states $E_1$, $E_2$ ... over the vacuum $E_0$. They occupy all the levels up to the Fermi energy. Now, i introduce some coupling between them so, they become effectively boson pairs. Are the $E_i$ energy states out of the picture for these bosons? My random guess is that fermions would couple in states like $$ | E_1 \times E_2 \rangle - | E_2 \times E_1 \rangle $$ $$ | E_1 \times E_2 \rangle + | E_2 \times E_1 \rangle $$ $$ | E_3 \times E_4 \rangle - | E_4 \times E_3 \rangle $$ $$ | E_3 \times E_4 \rangle + | E_4 \times E_3 \rangle $$ And so on, but this is the aspect i'm not sure and i want clarification What are the states where the boson pairs are condensing into??


Could you make a bar of metal glow by forcing the charge density on the surface to oscillate?

Since light is just a changing electromagnetic wave through space, could you create light by changing a charge density at a certain frequency?


Why equivalence principle is principle and not law?

We can prove that the inertial mass and the gravitational mass should be the same (equivalence principle) from the $f=mg=ma$ then $g=a$, so we have equivalence _law_! But why we said equivalence _principle_?


What makes laminar cascade break?

Near my house there is a mall that have a **cascade, which has a pratically constant flow, and doesn't seem to have perturbations** (at least near the edge where water falls), between its two levels. What botters me is thar that laminar flow that seem unperturbed, break in the middle after falling for a height of approximately 1.1m (it falls another 0.5m, before being colected by a pool, and, I think, repumped up). I don't know exactly how to explain that behavior but I think it must have to do with waves propagating on that filament of water and that make it **break after falling a height $h$**. Part of that flow is directed forward, and part is directed backwards (see the front view of the sketch below). From this view I can conclude also that there is a perfect left-right simetry (in relation to the vertical red line). ### Update1: I incluided also what I expect from the velocity vectors due to viscosity (with purple arrows on the sketch), similar of what we can see on Flow velocities due to viscosity of the Laminar flow text. Unfortunately I don't have a real picture, or vídeo, of the local. * * * ## What is causing this effect? How can it be explained? * * * ![Cascade Views - Sketch](http://i.stack.imgur.com/2YgkR.png)


Calculation of pressure from flow rate of water

Anybody kindly help me to find how to calculate pressure in bar from flow rate. I have a pipe and from that I am transferring water at a constant flow rate of 5ml/min. At this flow rate, with a 0.5 cm diameter pipe, what will be the pressure generated by water?


Independancy of angular momentum with respect to origin for rotation about center of mass

Why is angular momentum for rotation about the center of mass independent of origin of the coordinate system?


Find the force used to press a book against a leaned desk so it doesn't slip

I know there are several threads concerning this issue, but none of it helped me, so I decided to open a new one. So, the task is to find a force $F_{\mathrm{push}}$ on a desk which is leaned for an angle of $\varphi=45^{\circ}$ I know that for a vertical system you calculate that $F_{\mathrm{push}}=F_{\mathrm{friction}}=k F_{\mathrm{\mathrm{normal}}}$ and that's how it is calculated, but for this particular case I do not know... I also know that $F_{\mathrm{static}} = mg\cos(\varphi)$ and $F_{\mathrm{dynamic}} = mg\sin(\varphi)$, so I guess that the $F_{\mathrm{friction}} = -kmg\cos(\varphi)$ Please help me, are any of these formulas correct?


I lost a factor of two in the electromagnetic field tensor

I apologize for this simple question, but I lost a factor of 2 and can't find it anymore, so now I'm looking on the internet, perhaps one of you has some information about its whereabouts. :-) Consider the electromagnetic field tensor $F_{\mu\nu}$ which corresponds to a differential form $F =\frac12 F_{\mu\nu} dx^\mu \wedge dx^\nu$. The wedge product of $F$ with itself can be expressed as $$ F \wedge F = \frac14 F_{\mu\nu}\mathcal F^{\mu\nu} $$ where $\mathcal F^{\mu\nu} = \frac12 \epsilon^{\mu\nu\sigma\tau}F_{\sigma\tau}$ is the dual field tensor. Writing $A = A_\mu dx^\mu$ for the vector potential, we have $$ \int_V F \wedge F = \int_V (d A) \wedge F = \int_V d(A \wedge F) = \int_{\partial V} A \wedge F$$ But it appears to me that the latter integral is equal to $$ \dots = \int_{\partial V} dx^\mu\ A_\nu \mathcal F^{\mu\nu} .$$ Yet, at the same time, we can write $$ \int_V dx^\alpha \frac12 F_{\mu\nu}\mathcal F^{\mu\nu} = \int_V dx^\alpha (\partial_\mu A_\nu)\mathcal F^{\mu\nu} = \int_V dx^\alpha \partial_\mu (A_\nu\mathcal F^{\mu\nu}) = \int_{\partial V} dx^\mu\ A_\nu \mathcal F^{\mu\nu} .$$ Clearly, that cannot be. > I have lost a factor of two. Where is it?


Are these two quantum systems distinguishable?

Suppose Stanford Research Systems starts selling a two-level atom factory. Your grad student pushes a button, and bang, he gets a two level atom. Half the time the atom is produced in the ground state, and half the time the atom is produced in the excited state, but other than that you get the exact same atom every time. National Instruments sells a cheap knockoff two-level atom factory that looks the same, but doesn't have the same output. In the NI machine, if your grad student pushes a button, he gets the same two-level atom the SRS machine makes, but the atom is always in a 50/50 superposition of ground and excited states with a random relative phase between the two states. The "random relative phase between the two states" of the NI knockoff varies from atom to atom, and is unknown to the device's user. Are these two machines distinguishable? What experiment would you do to distinguish their outputs?


Doubt regarding inclined plane in feynamn lectures

I recently read something in the feynman lectures which I find difficult to understand. Imagine an inclined plane on the top of which is a smooth pulley. There is a weight hanging from the pulley and another weight at the base of the inclined plane on the other side. Feynman says since the weights are just balanced, the system is reversible and the weights can move up and down? What does feynamn mean here and why is the word reversible significant here? Also if the weights are balanced how can they move up and down?


I am a beginner to physics and would need an explanation on a statement in a book "Karttunen, _Fundamental astronomy_ ". In a section named _"Rayleigh diffraction by circular aperture"_ , author states: > The sum of the amplitudes of the waves from a small surface element is > **proportional** to the area of the element $\operatorname{d}x$ > $\operatorname{d}y$ And then author writes down the equation for amplitude vector $\boldsymbol A$ $$ \boldsymbol A = \operatorname{d}x \operatorname{d}y (\cos{\delta} \boldsymbol{\hat{i}} + \sin{\delta} \boldsymbol{\hat{j}}), $$ where $\cos{\delta} \boldsymbol{\hat{i}} + \sin{\delta} \boldsymbol{\hat{j}}$ is a direction of an amplitude vector $\boldsymbol A$ and vectors $\boldsymbol{\hat{i}}$ and $\boldsymbol{\hat{j}}$ are unit vectors. **And my question is, how does the author know that amplitude is proportional to element $\operatorname{d}x$ $\operatorname{d}y$ ???**


Supersymmetry in Quantum Mechanics

I was reading _Supersymmetry in Quantum Mechanics_ and got stuck in the various mathematical terminology like "Graded-Lie Algebra", "Super Algebra". Is there any good lecture notes concerning these topics, explaining the basic mathematics needed to understand supersymmetry. I know basic analysis, topology, and first course in group theory, but never encountered ring theory, modules etc.


Definition of Kinetic energy

In class we had that $ T= \frac{1}{2}T_{ij}v_iv_j$ where we used the Einstein summation convention. Hitherto we only discussed examples where the kinetic energy was dependent of the square of one coordinate and our teacher assured us that this is the case in most examples. Now I was wondering whether anybody if you knows an example where the kinetic energy depends in ONE term on different $ v_i , v_j$?


Conjugate Variables and Fourier Transforms in Classical Physics

Let q be a generalized coordinate with a conjugate momentum p and a potential resulting in a periodic motion of q. What is the meaning of the Fourier transform of q(t) over its period? Can this be interpreted as the distribution of momenta? I know this is the case in QM but what happens in the Classical case? For instance for a particle in a box (classical) with an initial velocity of p/m, the motion is a triangle wave. The Fourier transform is a squared sinc function. As the triangle wave gets smaller, the F transform spreads out and vice versa. Very nice but the only two momenta for the particle are p and -p as the particle either moves up with constant speed or moved down at constant speed.


Quick question on sketching wavefunction in well

![enter image description here](http://i.stack.imgur.com/kjjrv.png) Usually for an infinite well, the sketch for n=3 level is this: ![enter image description here](http://i.stack.imgur.com/I6Nqf.png) Now I think if one side of the potential barrier is higher, the particle will be more likely to spend time on the left side than the right side, so the wavefunction should have higher amplitudes on the left (skewed to the left): ![enter image description here](http://i.stack.imgur.com/iepnV.png)


Why are all basic physical measurements ratios?

Why is it that "all basic physical measurements [...] are ratios"? This question comes from the book _Mathematical Physics_ by Donald H. Menzel. Here is the explanation given: > **1\. The significance of an observation.** Any direct observation of a > physical nature ordinarily results in a number expressing the magnitude of > the measured quantity. The simpler measures are those of lengths, of masses, > or of times. More complicated ones may be of velocities, of energies, or of > angular momenta. The number, by itself, does not indicate what is being > observed; its magnitude depends on the upon the type of measuring scale > employed. We may express lengths in centimeters, miles, or light years; we > may define masses in grams, tons, or in units of solar mass. _All basic > measurements are, therefore, ratios._


Higgs VEV in terms of measurements on an ensemble?

Let $A$ be a Hermitian operator corresponding to some observable. If we prepare $N$ identical systems in the state $\psi$ and measure this observable in each system, the average of the measurements (for large $N$) will be $\langle \psi | A | \psi \rangle$. In the standard model, $\langle 0 | \phi^0(x) | 0 \rangle = v/ \sqrt{2} = 174\ GeV\ \ \forall x$. Questions:   1. What observable, if any, corresponds to $\phi^0(x)\ $?   2. Is $v/ \sqrt{2}$ really the average of a large number of measurements of identical systems in the vacuum state? (For precision, you could modify question 1 to "what observable corresponds to $\int d^4 x\ f(x) \phi^0(x)$," where $f:\mathbb{R}^4 \to \mathbb{R}$ is a smooth function localized about some $x_0$.)


Christoffel symbol for Schwarzschild metric

I know that the christoffel (second kind) can be defined like this: $$\Gamma^m_{ij} = \frac{1}{2} g^{mk}(\frac{\partial g_{ki}}{\partial U^j}+\frac{\partial g_{jk}}{\partial U^i}-\frac{\partial g_{ij}}{\partial U^k})$$ but I don't know how $U^i$ is defined (specifically for the Schwarzschild metric.


Does the existence of dualities imply a more fundamental structure?

I was wondering if the existence of some kind of duality in physics always implies the existence of some underlying more fundamental structure/concept? Let me give a few example from history:   1. Wave-particle duality$~\Rightarrow~$ Existence of quantum particle.    2. Heisenberg's Matrix Mechanics $\&$ Schrodinger's wave formulation of QM $~\Rightarrow~$ Existence of Dirac formulation of QM.   3. Magnetic field $\&$ electric field $~\Rightarrow~$ Existence of an electromagnetic theory. Similarly, can one conclude that   1. for example, from AdS/CFT correspondence,   2. or more generally, because there is an holographic equivalence between quantum gravity in $D+1$ dimensions, and QFT in $D$ dimensions,  then there must be a more fundamental underlying structure that incorporate both sides of the correspondence?


Blandford-Znajek process: Why/how does the current flow along the magnetic field lines

Related: How would a black hole power plant work? _I have put a bit of commentary enumerating my confusions in parentheses_ I read in _Black Holes and Time Warps_ (Kip Thorne), that quasars can generate their jets from four different processes. These all involved the accretion disk, but there was one which doesn't make quite as much sense. It was called the Blandford-Znajek process, and it involved magnetic field lines carrying current. The process was visualized in two ways. A black hole, with magnetic field lines, is spinning. In the first visualisation (viewpoint actually), the magnetic field lines 'spin' along with the black hole, and nearby plasma is anchored onto the field lines by electrical forces (where did the electrical fields come from?). The plasma can slide along the field lines but not across them (why?). Since the field lines are spinning, centrifugal forces will fling them up and down the field lines, forming jets. The other viewpoint is this, and it makes even less sense (to me that is, I haven't had a formal education in GR): The magnetic fields and the swirl of space generate a voltage difference across the field lines (Why? How?). The voltage carries current across the magnetic field lines (why are the _field lines_ behaving like wires?). This current travels across plasma, which accelerates it, creating the jets. Now the main thing that doesn't make sense, is that magnetic field lines are behaving like wires. Why would they? I suspect the answer lies hidden somewhere in the equivalence of EM waves in different frames, but I can't think up any convincing argument from that side. If the answer involves GR equations, you don't need to solve it here (wouldn't make sense to me), but if you have to, just refer to the equation and what you did to it, along with the final result. Thanks!


Classical electrodynamics formulated in terms of forces

The Newton's law of universal gravitation is described in terms of a force, which is produced by an action at a distance. It also can be described using the concept of a field, and that would be an equivalent formulation. Now, classical electrodynamics is formulated in terms of fields in a more complicated way that Newton's gravitation. My question is: Is there an equivalent formulation of classical electrodynamics in terms of action at a distance that is completely equivalent to the formulation in terms of fields (Maxwell's equations)? I would tempted to think that it is not possible, since in Maxwell's equations you may have electromagnetic waves without the presence of charges, and I don't know how to do it only in terms of forces. However, I'm not entirely sure, that's why I'm asking.


Point group symmetries and unit cell

I was wondering if the unit cell (of a given lattice) had to have every point group symmetries of the lattice it defines ? I guess there is no unique way to define a unit cell and that it may not have all point group symmetries. However, is it possible to define a unit cell that has all point group symmetries ?


Does the Chandrasekhar Limit scale for a Black Hole?

No physicist/astrophysicist I; All I know about the Chandrasekhar limit is that it apparently limits the mass a star may survive, beyond which it degenerates to a neutron star, or a black-hole. Does something similar apply to a black-hole? Is there an upper limit to black-hole mass? Can it acquire infinite mass? Is there a limit beyond which it may no longer acquire mass? Does a black-hole die? Is there a corresponding time-frame applicable?


What is the meaning of the word "particle" in particle physics?

I want to use Matt Strassler's definition of the word "particle" as a specific example: Matt Strassler writes: > (1) "...all the elementary “particles” (i.e. quanta) of nature are quanta of > waves in fields..." My first question is, in this sentence, is the word "quanta" short for "quanta of waves in fields?" Sentence (1) simply defines the word "particle" to be another name for "quanta". I looked up quanta and quanta means > a wave that for a given frequency, have minimum energy and amplitude. So, a quantum is a wave. Is this correct? From this analysis, I conclude that, in particle physics, a particle is a wave, because a particle is defined as a quantum. In mathematical terms: $quantum \equiv wave$ or $particle \equiv wave \equiv quantum$ So, in particle physics, the word particle, has the properties of a wave, and it has no other properties. A particle, in particle physics, for instance, is not a spherical object with finite radius. This is confirmed by Matt Strassler, who writes (2) "...the word “particle” in particle physics has the same meaning as the word “quantum”..." Sentence (2) is incomplete, because the word "particle" has several other meanings besides being another name for quanta. This is very confusing to non- physicists because the word "particle" has been defined many times and physicists pick and choose a meaning that saves the context. My questions are: 1) The word "quantum" is a perfectly good word meaning a wave with minimum amplitude; why are physicists insisting on calling a quantum a particle? 2) Is there a unique definition of the word particle in physics not dependent on context (not casuistic) and that all physicists can agree upon? If you cannot answer any of these, can you clarify if "particle" is another name for "quantum".


Does zero change in magnetic flux always imply zero emf induced?

If you have a uniform B field, with a finite piece of wire inside it. Assuming the B field spans all space and the wire cannot leave the field. Are you able to create an emf by moving the wire ? I say there is only one possible way, and that is if you rotate it so from its starting perpendicular to the field position it does parallel to the field. That would create a change in flux. But are you able to create emf if the wire stays perpendicular to the wire?


Correlators at large N and large N factorization

I am having this very basic problem. In e.g Maldacena's AdS/CFT review (0309246) (page 5), he has defined operators as $\mathcal{O}=N\,{\rm tr}[f(M)]$ for some matrices $M$ and got the connected correlators as $\langle\mathcal{O}\mathcal{O}\rangle_c\sim N^2$ and $\langle\mathcal{O}\mathcal{O}\mathcal{O}\rangle_c\sim N^2$. Whereas in the MAGOO review (9905111) (page 14-15 e.g) we see some operators defined as $G$ which is added to the action as $S\rightarrow S+N\sum g_jG_j$ which gives the correlators as $\langle\prod_j G_j\rangle\sim N^{2-n}$. So that three points and higher vanish at large N. I thought this one is generic. If so I wanted to understand how is the first conventions/case are different and their difference (has it something to do with connected/disconnected diagram? etc.). Related to that I wanted to know about large N factorization. For a particular model if I calculate disconnected diagrams, I see that at large N I indeed get this factorization. I learned that its like WKB in $\hbar\sim 1/N$. But I wanted to feel it more generally in possibly a mathematical+insightful way if possible. Thanks for the help. Edit: If it has a long answer please at least let me know which way to think or any references where I can get some ideas. Thanks again.


Does Mansuripur's Paradox involve fictitious charges?

Mansuripur's Paradox involves a magnet moving at relativistic speeds in an external electric field. _Additional:_ thanks to Retarded Potential, who found the original paper. If I understand correctly, the fictitious charges on either side of the magnet should cause a torque, which of course does not actually exist. Apparently this is conventionally resolved by adding an (equally fictitious) internal angular momentum. Mansuripur appears to be arguing that a better resolution would be to add a (fictitious) additional term to the Lorentz force law. Am I right in thinking fictitious charges are involved? If so, why are they introduced, i.e., what is the advantage in doing so? If not, where does the torque really come from?


What's the difference between these two formulas and how are these called?

I just want to know the differences between these two formulas: $h = h_0 + v_0 t ± \frac{1}{2} g t^2$ and $y = y_0 + v_{0y} t + \frac{1}{2} g t^2$ Also, how are these called in English?


Radioactive Decay

> **Problem** :Nuclei of a radioactive element $\Bbb X$ having decay constant > $\lambda$ , **(** decays into another stable nuclei $\Bbb Y$ **)** is being > produced by some external process at a constant rate $\Lambda$.Calculate the > number of nuclei of $\Bbb X$ and $\Bbb Y$ at $t_{1/2}$ I tried to create an equation for rate of change of the number of nuclei a: $$\dfrac{dN_{X}}{dt}=\Lambda-N_X\lambda $$ I did that because in simple decay $\dfrac{dN}{dt}=-\lambda N$ holds and here it's also being produced by rate. But after integration should we write $$ln\Bigg(\dfrac{\lambda N_X-\Lambda}{\lambda N_0-\Lambda}\Bigg)=-\lambda t$$ or $$ln\Bigg(\dfrac{\lambda N_X-\Lambda}{N_0}\Bigg)=-\lambda t$$ First one because limit was on $N: (N_0\to N)$ And next what to substitute for $t$ (ie. what is $t_{1/2}$? $ln2/\lambda$ or something else?) Also how to do it for $\Bbb Y$? Just write $$\dfrac{dN_Y}{dt}=\lambda N_x $$?


How is North Korea planning to test a nuclear device underground?

I am reading the news about North Korea planning to test a nuclear device **underground**, and while fascinating, I have no idea what this phrase actually means. Could somebody explain, in absolute layman's terms, how an underground nuclear test actually works, what the dangers are, and how North Korea will use the results?


Resistance from voltage and charge

I'm sure this is pretty basic, but I'm not very good at electricity. > At both ends of a conductor we apply 12 volts. For one minute an electric > charge of 72 C passes through the conductor. What is the resistance of the > conductor?


Why is number of collisions equal to number of scattered particles times number of scatterers?

In Landau, Lifshitz "The Classical Theory of Fields" $\S12$ "Invariant cross- section" the following is said1: > ... according to the usual definition of the cross-section $\sigma$, the > number of collisions occuring in volume $dV$ in time $dt$ is $$d\nu=\sigma > v_{\text{rel}}n_1n_2dVdt,$$ where $v_{\text{rel}}$ is the velocity of > particle 1 in the rest system of particle 2 ... I'm trying to make sense of this formula, and here's what I got so far: $\sigma v_{\text{rel}}n_1dt$ is total number of scattered particles in time $dt$, and $n_2dV$ is total number of scattering centers. Why isn't total number of scattered particles equal to the number of collisions? Is it considered that there's more than one collision per scattered particle? If yes, then why should it appear that each scattered particle collides with _each_ of scattering centers (only in this case would we just multiply by number of scattering centers)? 1\. In English version which I cite the formula lacks a $dV$ factor. I put it here according to original Russian book.


Do algorithms have an intrinsic time direction?

This article says > There is no intrinsic time direction in Newton's mechanics nor in the > differential equations of the new physics. My question is, do other types of mathematics, say a cellular automata as an example, have the intrinsic time direction that the equations lack? With a CA, all steps have to be completed sequentially. **UPDATE** : Here's what I mean. On this page, it describes projectile motion. It uses one approach with equations, and another approach with an algorithm. Very clearly both approaches yield the same answers. The biggest difference between them is how time works. Question: Is having "an intrinsic time direction" the same thing as "time having no choice but to march forward in the algorithm"?


Slowing of photons in a EB Condensate

Help me understand this. I have heard tell that they are now able to slow the forward progress of a photon traveling through an EBC to somewhere near 35mph. Is this actually the physical slowing down of the photon, or is it the slowing down of the absorption and emission process? If the photon is actually slowed down, does it gain mass? Or am I miss interpreting this all together. Thanks


Heat reflection on distant planets

I was watching a documentary last night on the first planet discovered outside our solar system. The first one apparently is a giant gas planet that orbits very close to its sun over a very fast period (~1 week per orbit), and it's so hot all metal elements vaporise and form parts of it's atmosphere (so the current theory states anyway). Out of curiosity, when they estimate the surface heat of the planet (they give big numbers), did they take into account reflection of heat from it's metallic atmosphere, and internal reflection from heat bouncing off the surface and it's atmosphere (like global warming) when estimating surface temperature of the planet?


Homework Question Transformation Energy

A $1400kg$ car is approaching the hill shown in the figure at $14.0m/s$ when it suddenly runs out of gas. ![enter image description here](http://i.stack.imgur.com/RDU61.png) What is the car's speed after coasting down the other side? I think I have to use this equation $mg(\Delta h)=\frac 12 m(\Delta v)^2 $ $(1400)(9.81)(\Delta h)= \frac12 (\Delta v)^2$ I am not sure if this is correct. If it is, what do I plug in for $\Delta h$?


Can a wave propagate in an elastic fluid in the absence of volume forces?

A motion (wave) $\mathbf{x}: \mathcal{B}_0 \times [t_0,t_1] \to \mathcal{E}:$ such that   $q-o = \mathbf{x}(p,t)=(p-o)+\mathbf{a}_0 cos(\mathbf{k}_0\cdot(p-o) - \omega_0 t)$ can propagate in an elastic fluid with constitutive equation:   $\mathbf{T}=-\pi(\rho)\mathbf{I}$ not subject to any volume force? What restrictions on polarization vector $\mathbf{a}_0$ and the vector $\mathbf{k}_0$ are needed? $\mathbf{a}_0$ and $\mathbf{k}_0$ should be parallel? What restrictions on $\pi(\rho)$ are needed? I can write the balance equation of linear momentum, in an eulerian description I suppose: $\rho \mathbf{a} = \mathrm{div}_q \mathbf{T} = -\nabla_q \pi = - \pi' \nabla_q \rho$ But it is easier to compute the acceleration in a lagrangian description:   $\ddot{\mathbf{x}}(p,t) = -\omega_0^2 \mathbf{a}_0cos(\mathbf{k}_0\cdot(p-o) - \omega_0 t)$ At this point computing $\nabla_q\rho$ is not easy I think. There is some mistake here? There is a better and simpler way to prove the results? _Thanks and Please excuse my bad english_


Reflectance and phase shift vs incident angle and wavelength

Plot the refectance and phase shift on reflection $\varphi$ for p and s polarized light versus incident angle $\theta$ for various wavelength $\lambda_1=.., \lambda_2=..,..$ in the simple case of reflection from a bare substrate given the complex refractive index $N=..$ of the substrate. Since the reflectance is $$R=(\frac{1-\eta_{s,p}}{1+\eta_{s,p}})(\frac{1-\eta_{s,p}}{1+\eta_{s,p}})^* $$ where $\eta_{s,p}$ is $Ncos\theta$ and$\frac{N}{cos\theta}$ for s and p polarized light. Also, the phase shift on reflection is:$$\varphi=\arg(\frac{1-\eta_{s,p}}{1+\eta_{s,p}})$$ I can see the dependence of $R$ and $\varphi$ on the incident angle, but how can i plot it at different wavelength ? i don't see the dependence of $R$ and $\varphi$ on wavelength at all.


Böotes Void and Dark Galxies

Wouldn't it be possible that Böotes Void, a space we generally consider to be "empty" be filled with a number of undetectable dark galaxies?


What is (or where can I discover) the Burke Potential?

I have very much enjoyed William L. Burke's _Applied Differential Geometry_. Reading around on the web it seems that he discovered something which is called the (retarded) Burke Potential, but I have not been able to find any references to it. Can anyone point me at a reference or explain succinctly what it is?


Is there a way to define a function of a meniscus curvate?

If you cut a thin slit in thin opaque material and then put it into water and pull it out, the meniscus will be formed in the slit. For my research I need to know if it is analytically possible to define the curvature of meniscus( is it hyperbola, parabola, circle or something else). I need the function of curvature so I will be able to make calculations of how the light passes through the meniscus. I know there are numerical methods to make an approximate function, but I want to know if there is any other way.


Time taken for collision

We have three particles at the vertices of equilateral triangle of side $d$. At $t=0$ they start moving in such away that at all instant of time each of them has a speed $v$ towards adjacent one. We have to find the time after which they will collide. I know that you can solve this by calculating from reference frame of any one particle, but I want a way to solve this by calculating the total distance traveled first in ground frame and then dividing by speed.


tritium beta decay - probability of being in 1s state

Hydrogen-like wavefunctions have the form: $$R_{10} = \left(\frac{Z}{a_0}\right)^{\frac{3}{2}} 2\space e^{-\frac{Zr}{a_0}}$$ $$Y_{00} = \frac{1}{\sqrt {4\pi}} $$ where $a_0 = \frac{4\pi \epsilon_0 \hbar^2}{\mu e^2}$ Suppose Tritium $^3$H that's in its ground state beta decays to $^3$He suddenly. I want to find the probability the helium atom is in the 1s state. Firstly, is the reduced mass $\mu$ the same for both helium and $^3$He? I assumed it is $m_e$. Wavefunction of helium is $R_{10}' = \left(\frac{2}{a_0}\right)^{\frac{3}{2}}2\space e^{-\frac{2r}{a_0}} $. Wavefunction of tritium is $R_{10} = \left(\frac{1}{a_0}\right)^{\frac{3}{2}}2\space e^{-\frac{r}{a_0}} $. To find the probability, do I simply overlap the two and square them? $$\langle (1,0,0)_H| (1,0,0)_T \rangle = \left(\frac{1}{a_0}\right)^3 (2)^{\frac{3}{2}} (2)^2\left(\frac{1}{4\pi}\right) \int e^{-\frac{3r}{a_0}} r^2 sin\theta \space d\theta d\phi $$ $$ = \frac{2^{\frac{9}{2}}}{27}$$ Thus probabiliy is $\frac{2^9}{27^2} = \frac{512}{729} $?


Deriving the Electromagnetic Tensor

The electromagnetic tensor is given as: ![enter image description here](http://i.stack.imgur.com/b7xa5.png) How do you derive this? And how come there is a partial derivative in front of $A_\mu$? Do you multiply the derivatives or what?


How would a physicist move?

Let's assume that Johnny Physicist has decided to move from his poor dingy second story apartment, into his much deserved home. Without making modifications to the existing structures? What would be the most work efficient way of moving all of his possessions? [Lets assume that the apartment is outside facing, and has stairs leading up to it] [Also lets assume that his possessions are a mix of carryable boxes, and large items]


What is the leading order Feynman diagram for nucleon-anti-nucleon annihilation into two mesons ($\psi^{\dagger} \psi \to \phi\phi$)?

I am working with a standard basic scalar Yukawa theory. I.e. the only interaction term is $-g\psi^\dagger\psi\phi$, where the $\phi$ field quanta are the mesons, the $\psi$ field quanta are the nucleons and the $\psi^\dagger$ field quanta are the anti-nucleons. I have tried fiddling around with various combinations and have looked at some relevant papers, but I have not found a solution which does not invoke other interaction terms. Is this process possible?


Two-loop regularization

Working out some quantum field theory computations, I have to find out the value of the two-loop Feynman integral $$ I(p)=\int\frac{d^4p_1}{(2\pi)^4}\frac{d^4p_2}{(2\pi)^4}\frac{1}{(p_1^2+m_1^2)(p_2+m_2^2)[(p-p_1-p_2)^2+m_3^2]}. $$ This integral is rather common and so, its value at small $p$ should be already well-knwon. But I was not able to find out it in literature. I would also appreaciate to see all the procedure to get the right value with whatever regularization procedure one likes. Thanks beforehand.


Particle Physics : Conservation laws

As a whole, is conservation of a Lepton number valid, or is it basically the generalization of conservation of electron and muon numbers?


How is the direction of the instantaneous acceleration determined?

I know from the text book that the direction of velocity at any point on the 2D path of an object is tangential to the path at that point and is in the direction of motion. But how would one determine the direction of instantaneous acceleration? In case, the only force influencing is gravity, direction of instantaneous acceleration is downward towards the earth's center. but what in the other cases?


Parity of spin states

Since orbital angular momentum commutes with the parity operator and since both are hermitian it is possible to build a common basis. These are the spherical harmonics, whose parity is known. Now, we also know that since the spin operator commutes with the parity operator we can build common eigenkets, but I have never been told which is the parity of a spin state. Moreover, when calculating the parity of, for example, a particle I always multiply the contribution of the orbital angular momentum by the intrinsic parity of the particle, and the spin part is never mentioned. I guess that the spin contribution is in that intrinsic parity. So i would like someone to tell me about the parity of the spin states and how they are related to the intrinsic parity of particles(if in fact my guess is right)


What is anti-screening?

In his book "The Lightness of Being" Frank Wilczek (page 48) writes about screening and anti-screening. In screening, a bare charge attracts virtual particles of opposite charge which lessen the effect of the bare charge. He writes that anti-screening is the "opposite of screening" with no further physical explanation. This is an attempt to explain why two quarks attract weakly at close range and more strongly with distance. Is there any physical explanation of this anti-screening?


How can heat turn into light

I am confused about how hot surfaces can radiate light to their surroundings. When I shine a light on a surface the light turns to heat spontaneously, and when I leave that hot surface it radiates light spontaneously. To me this suggests that the process isn't driven by entropy increasing because it is two directional. Could anyone explain how this process complies with the second law of thermodynamics?


If I were standing on the surface of the sun, how far could I see?

Assuming it would be possible to handle the intense heat... Is the surface even? If not, would average surface elevation be an appropriate definition of "surface"?Does the sun have an atmosphere that would degrade visibility? How far apart are small solar flares that would block the horizon? What affect would the heat have on the light rays? Assume I am 2 meters tall.


Will we see interference in a double slit experiment in these three cases?

First case:   1. A detector registeres particles going through the slits, but doesn’t reveal the information to conscious observes   2. A detector irreversibly erases the information about particles before they hit the film   3. The particles hit the film   4. A person observes the film. Second Case:   1. A detector registeres particles going through the slits, but doesn’t reveal the information to conscious observes   2. The particles hit the film   3. A detector irreversibly erases the information about particles before they hit the film   4. A person observes the film. Third case:   1. A detector registeres particles going through the slits, but doesn’t reveal the information to conscious observes   2. The particles hit the film   3. A person observes the film   4. A detector irreversibly erases the information about particles without anyone ever knowing what it was


What is sound and how is it produced?

I've been using the term "sound" all my life, but I really have no clue as to what sound exactly is or how it is created. What is sound? How is it produced? Can it be measured?


What Hermitian operators can be observables?

We can construct a Hermitian operator $O$ in the following general way:   1. find a complete set of projectors $P_\lambda$ which commute,   2. assign to each projector a unique real number $\lambda\in\mathbb R$. By this, each projector defines an eigenspace of the operator $O$, and the corresponding eigenvalues are the real numbers $\lambda$. In the particular case in which the eigenvalues are non-degenerate, the operator $O$ has the form $$O=\sum_\lambda\lambda|\lambda\rangle\langle\lambda|$$ **Question:** what restrictions which prevent $O$ from being an observable are known? For example, we can't admit as observables the Hermitian operators having as eigenstates superpositions forbidden by the superselection rules. **a)** Where can I find an exhaustive list of the superselection rules? **b)** Are there other rules? **Update:** **c)** Is the particular case when the Hilbert space is the tensor product of two Hilbert spaces (representing two quantum systems), special from this viewpoint?


Faraday's law in a ring

What role does the induced emf in Faraday's law play in generating current in a ring in which the magnetic flux is changing ?


Torque of a winch with pulleys

I'm working on a project that involves using a remote-controlled electric winch to raise and lower my bicycle using a pulley system. The problem is that I don't know how much torque will be required of the electric motor in this system. A somewhat rudimentary diagram is as follows: ![enter image description here](http://i.stack.imgur.com/dR3xC.png) I haven't bought the winch yet, but it'll probably be more like 2-3 cm radius rather than 5 cm as labelled, so I'll assume 3 cm. My calculation is as follows: The pulley system has a 4:1 mechanical advantage, so the weight on the rope is ~50 N, and then $$T = rmg = (3\,\mathrm{cm})(50\,\mathrm{N}) = 1.5\,\mathrm{N\,m}$$ Which means that this motor should do the job. Is my reasoning correct?


Any projects to build a telescope in space?

Like in the title. With E-ELT we are pushing the limit of what we can build on the ground. With JWST we are pushing the limit of what can we launch into space. So instead, why don't we launch tools and materials to build a telescope and polish its mirror in orbit? This has several benefits. The environment is vacuum and microgravity, so very large components can be moved with small and light tools. The structure can be very light and fragile, bacause it doesn't have to survive the launch. It can be huge, because it doesn't have to fit into a rocket. Errors can corrected before it is commisioned. I am sure, I'm not the first one to have this idea, but I failed to find any existing projects (upcoming or future) on the web. Do you know any such projects being considered?


Superconductor: What form of paramters (like London penetration depth) to use?

I am dealing with thin film and LN2 temperature, and am trying to calculate values such as the Pearl Length, but am trying to find clarification as to which value of $\lambda_L$ to use in this case. e.g. $\lambda_L(t,T)$ with T = 0 or T = 77K? I have other formula, such as the kinetic inductance for a CPW, but the articles I am pulling them from are somewhat unclear which forms of the parameters they are using. Any help is appreciated.


How to get Intensity over a range of wavelengths from a Total Intensity

This is a theory vs experiment problem for me I think in that - as usual - I think I know the theory, but when it comes time to run an experiment, I can't actually extract squat. So, here is my light source: I am using the Cold White LED (MCWHL2). If you scroll down a bit, you'll see the LED normalized intensity vs. wavelength profiles for all the LEDs. The black line on this graph is my LED. As you can see, it meanders instead of being a simple Guassian-like thing. I measure the intensity of this light by using this power meter: Scientech Astral AD30 meter - which displays power and energy - with an AC2501 detector. Specs are here . So I'm calculating total intensity by dividing by the area of the beam. Now, here's my question: say I get a reading of $ 10\rm\frac{mW}{cm^2} $. If I want to know what is the intensity of the output only over the range 400nm to 455 nm, based on the Intensity vs Wavelength graphs and the characteristics of the meter and detector, what do I do? Now, Planck's equation for a black body was where I first looked. My procedure was to find the total intensity for the light temperature (6500 K) and integrate from 0 to infinity. I then numerically integrated from 440 nm to 455 nm and divided the two. Multiplying this by the output on the detector gave a ridiculous solution. Given the intensity profile - not really black bodyish - the ridiculous results make sense. It was at this point that I started searching and found nothing but Planck stuff. Thus, I humbly come before all of you. Please help! There must be some sort of standard procedure for those without specialized light analyzing equipment! Thanks! Sam


Cauchy Problem for Boltzmann Equations

One of the first profound analysis about the solutions of the Boltzmann Equation was given by DiPerna and Lions in the late 1980s. You can find one of their main papers here: http://www.jstor.org/discover/10.2307/1971423?uid=3737528&uid=2&uid=4&sid=21102076873573 If you do not have access to the paper then use this link: http://www.4shared.com/office/eeIqgvxB/MathematicsOn_the_Cauchy_Probl.html I was reading through this and similar papers and one of the things I do not understand is the _interpretation_ of the solution. The paper delivers more or less a proof for the existence of a global solution of the Boltzmann Equation, but what can we do with it? If you have an idea how to interpret it I would be very thankful. P.S: I put a question about the proof of one of the Lemmas on http://mathoverflow.net/questions/132871/cauchy-problem-for-boltzmann- equations


What is the deepest cause of the such high specific heat capacity of water?

Yes, I know about the hydrogen bridges. But I think, it isn't the deepest cause. Anyway, they are only second-order bindings, although quite strong. I think, somehow should have the water a significant number of "hidden degrees of freedom", or some like internal structure with a lot of degrees of freedom. But it is only my naive intuition. So, what is the deepest cause?


How exactly do I calculate this correlation function?

I found a research paper (from 1977) that has a particular equation I need to reproduce. The paper essential calculates dynamic light scattering correlation functions. The full equations I need to work out is this: $$ f(q,\tau) = \frac{\int_{-1}^{+1}(1-2w^2\nu^2) \exp(-w^2\nu^2)\ d\nu \times \int_{0}^{2\pi} A[q,\mu(t)]A^*[q,\mu(0)]\ d\psi}{\int_{0}^{2\pi} |A[q,\mu(0)]|^2\ d\psi} $$ Essentially I need to calculate this for values of the delay time $t$. The bit I am not sure how to do is the $\int_{0}^{2\pi} A[q,\mu(t)]A^*[q,\mu(0)]\ d\psi$. The actual functional form of A isn't that important, but it consists of spherical Bessel functions and the term $\psi$ which is to be numerically integrated over. Now, where I am stuck is what do I actually do to calculate that function for the delay times. I have tried a lot of things but I just cant get it looking like they have in the paper. Here is what I have tried:   * Calculate $\int_{0}^{2\pi} A[q,\mu(t)]$ for all delay times, and then use a built in autocorrelation function to obtain the results.   * Take my first $t$ value (say 0.01 sec), and my second $t$ value (say 0.02 sec) and then calculate $\int_{0}^{2\pi} A[q,\mu(0.02)]A^*[q,\mu(0.01)]\ d\psi$... etc for all t values (so $t0$ lags 1 behind t.   * A few other things. As you can see I am a bit confused, if anyone has any insight as to how I would calculate this by hand that would be extremely helpful. * * * ### Edit: The full functional form of $A(q,\mu(t))$ is $$ A(q,\mu(t))\propto \frac{3j_1(u)}{u} + F^3 \frac{m_2-m_1}{m_1-1}\frac{3j_1(f u)}{fu} %Old version: \frac{F^3 (m_2-m_1) (3 j_1(F \mu(t)))}{(m_1-1) (F \mu(t))}+\frac{3 j_1(\mu(t))}{\mu(t)} $$ where $j_1$ is a spherical Bessel function of the first kind, $u=q\sqrt{a^2\mu^2(t)+b^2(1-\mu^2(t))}$, $F=1-\frac ta$, and $\mu(t)=\cos(\psi)\sin(\theta)\sin(\alpha(t))+\cos(\theta)\cos(\alpha(t))$, for $\nu=\cos(\theta)$ and $\alpha(t)=\omega t+\phi$. Yes, I am after $f$ vs $t$ for a $q$ of my choice. I am just not sure I am computing the integral properly for the values of $t$. For example, lets say my $t$ of interest was $0.1, 0.2,0.4, 0.8, 1.6$ (just an example). For the very first $t$, would I simply do, $$ \int_{0}^{2\pi}A[q,\mu(0.2)]\times A[q,\mu(0.1)] $$ Then for the next $t$ $$ \int_{0}^{2\pi}A[q,\mu(0.4)]\times A[q,\mu(0.1)] $$ or $$ \int_{0}^{2\pi}A[q,\mu(0.4)]\times A[q,\mu(0.2)] $$ And dont I need to average over the same delay time $\tau$? So for $\tau 1$ its $$ \int_{0}^{2\pi}A[q,\mu(0.2)]\times A[q,\mu(0.1)]\\\ \int_{0}^{2\pi}A[q,\mu(0.4)]\times A[q,\mu(0.2)]\\\ \int_{0}^{2\pi}A[q,\mu(0.8)]\times A[q,\mu(0.4)]\\\ \int_{0}^{2\pi}A[q,\mu(1.6)]\times A[q,\mu(0.8)] $$. I am using Mathematica to evaluate these functions. As you can see I am rather confused. Thanks so much for your help. The paper I am referencing is: > Rotational—translational models for interpretation of quasi-elastic light > scattering spectra of motile bacteria. Michael Holz and Sow-Hsin Chen. > _Appl. Optics_ **17** no. 20, pp. 3197-3204 (1978).


Does General Relativity require that Spacetime must be a orientable?

> **Possible Duplicate:**   >  Can spacetime be non-orientable? Apart from the constraints put on the topology of spacetime by QFT (Parity For example), if the global topology of a universe is that of a Mobius band and at one point there is a gravitational field pointing towards a star and a passenger travels along the band and returns to the same point. Then he will find the gravitational field pointing away from the star causing an anti- gravity effect. Does this mean that space-time must be orientable?


Is thermal/heat energy the same thing as internal energy?

I learned that internal energy of an object is the sum of the kinetic energy and potential energy of the molecules of the object. Is this energy the same thing as the heat energy the object contains?


One-loop $\phi^4$ theory in $d = 3$

I'm trying to calculate the 1 loop correction to the propagator in massless $\phi^4$ theory, in $d = 3$, just for fun. The diagram just looks like a straight line with a circle touching tangently to it. I'm not sure whether to take $ m = 0$ straightaway or to compute it for $m > 0$ before taking $m \to 0$, because the two methods give different answers. The diagram is $\sim \int d^3k \frac{1}{k^2 - m^2}$. If I set $m = 0$ straightaway, then I get $\sim \int k^2 dk \frac{1}{k^2} \sim \Lambda$, where $\Lambda$ is the cutoff. On the other hand, if I do the integral, using the results in the appendix from Peskin and Schroeder \begin{align} \int d^dk \frac{1}{k^2 - m^2} \sim \frac{\Gamma(1-d/2)}{(m^2)^{1-d/2}} =\Gamma(-1/2)(m^2)^{1/2}. \end{align} $\Gamma(-1/2)$ is finite, so taking $m \to 0$ this diagram is $0$. So one one hand it's UV divergent, on the other hand it's 0. Which method should I be using? Are the theories where $m=0$ and $m \to 0$ not the same? I also seem to have missed out the $i \varepsilon$ prescription in the propagator in the first case - if I include that in and evaluate the diagram before sending $\varepsilon \to 0$, then the diagram is $0$ too. Or perhaps it does not matter too much because the mass renormalization term can take care of either divergence and such discrepancies will not show up in calculations of any observables anyway? Thanks.


What's an "Action" and what does the Lagrangian equation mean exactly?

How and why would a particle take the shortest path? $L=KE-PE$? What's the $KE-PE$ mean in English? I understand the 'mechanics' but not the idea itself. Please explain simply, I do know Calculus but the simpler the better. Please don't say because that's how nature is, because I get that a lot.


Ways of verifying the origin of a meteorite?

Assume you have a small sliver of a lunar or martian meteorite (or an object asserted to be so). Without using any special scientific equipment, is it possible to verify (or give a high probability) that it is indeed of lunar or martian origin? Are there any visual or other tests one can do to confirm the source? _(If this question is off-topic or not quite right, feel free to close.)_


Alternative symmetries for the Maxwell Lagrangian?

I'm wondering about how to show that $A_a\rightarrow A_a+\alpha\partial_0A_a$, with $\alpha$ infinitesimal, is an infinitesimal symmetry of $\mathcal L=-\frac14F_{ab}F^{ab}$. \begin{equation} F_{ab}\rightarrow F_{ab}+\partial_a(\alpha\partial_0A_b)-\partial_b(\alpha\partial_0A_a)=F_{ab}+\partial_0(\alpha F_{ab}). \end{equation} \begin{equation} \Rightarrow \mathcal L\rightarrow\mathcal L-\frac12F_{ab}\partial_0(\alpha F^{ab})=\mathcal L-\frac14\partial_0(\alpha F_{ab}F^{ab}). \end{equation} We require $\delta\mathcal L=0$ for it to be called "a symmetry of $\mathcal L$", right? Or do we only require $\delta\mathcal L=$ a total derivative? Either way, I don't seem to get what we want.


Phase factor for nearest neighbor hopping in the Haldane Model

In Haldane's model, he imagines a staggered magnetic field in graphene where the net flux through a unit cell is zero. To model this, he has a phase factor in next nearest neighbor hopping on the honeycomb lattice. Why does he not consider a phase factor for hopping between nearest neighbors as well?


What happens to electrons when they are detected?

I know photons are absorbed and annihilated when they are detected but a) when electrons are detected, are they absorbed and annihilated? and b) if electrons are ever absorbed, what happens when they are? Are photons omitted?


Condition for circular orbit

I am a little confused about the condition for circular orbit. Goldstein's Classical Mechanics has the condition for circular orbit as $$f'=0\tag1$$ where $f'$ is the effective force. I understand that the reason for this requirement is that the corresponding $V'$is at an extremum. However, setting $f'=0$ yields $$f(r)=-\frac{l^2}{mr^3}\tag2$$ and the one of the equations of motion becomes $$m\ddot r=0\tag3$$ This can't be right as it is saying that the net force on the object is zero, and yet the object is supposedly in a circular orbit. Am I interpreting equation (3) wrong?


Frequency modulation using visible light

Currently I am doing research into visible light communication from an Embedded Systems background (MSc), but I am struggling to relate the concepts of modulation of radio to visible light. I already apply modulation to a working VLC system, but the underlying system still puzzles me. My goal of this post is to create an understanding of the underlying fundamental concepts, so if there are flaws in my assumptions below please point me to it. **Radio** In RF communication, two modulation techniques are used: amplitude modulation and frequency modulation. All other modulation techniques are variations on these types. These modulation techniques change the electromagnetic field in terms of coherent photons with a specific frequency and amplitude (source). **Visible light communication** In VLC, amplitude modulation is the most common technique (changing the intensity), but frequency modulation is also used. As frequency relates to color, some research uses RGB LEDs to do this in combination with color filtered photodiodes. **(1)** However, and this is where the problem is, some research applies frequency and phase modulation using a single white LED. They use a photodiode as a receiver which is sensitive to intensity but not to color variations, so how is frequency modulation then possible? **(2)** It seems that VLC has a third kind of main modulation technique in comparison with RF, is this correct? **(3)** If so, what makes light so different from RF then, as it also is electromagnetic radiation? And can this third type be applied to RF?


Why not drop $\hbar\omega/2$ from the quantum harmonic oscillator energy?

Since energy can always be shifted by a constant value without changing anything, why do books on quantum mechanics bother carrying the term $\hbar\omega/2$ around? To be precise, why do we write $H = \hbar\omega(n + \frac{1}{2})$ instead of simply $H = \hbar\omega n$. Is there any motivation for not immediately dropping the term?


Fermi statistics and Berry phase

When the positions of two fermions are exchanged adiabatically in three- dimensional space, we know that the wave function gains a factor of $-1$. Is this related to Berry's phase?


Producing photons with same frequency, different amplitude wave

I don't understand how two photons of the same _frequency_ can have different _amplitudes_ , neither how to produce them. I know that classically the square of the amplitude is proportional to the energy, but photons aren't classical particles. My understanding is that a photon's energy is $h\nu$ - what does the square of its amplitude represent, then? Are there bounds to the amplitude of an EM wave? Take two waves of amplitudes $A_1$ and $A_2$ and frequency $f_0$. If $A_2 = 2A_1$, can the wave with amplitude $A_2$ be said to carry/be two $A_1$ photons of frequency $f_0$?


Explanation of homogeneity of space and time by giving examples?

while reading landau lifshitz i came across these three terms:-   1. homogeneity of space.   2. homogeneity of time.   3. isotropy of time. it will be a great help for me if someone can explain it to me by giving examples .


Energy of the electron-muon reaction

Lets see the reaction: $e^- \mu^- \to e^- \pi^- \nu_\mu \;\;\;\;\;\;\;\;\;\;\; {(1)}$ I suppose, that this reaction occurs as follows $e^- \mu^- \to e^- \mu^- \pi^+ \pi^- \to e^- \pi^- \nu_\mu$ Is it possible at energy less than 2*140 MeV? The same is for the analogous proton-muon reaction $p^+ \mu^+ \to p^+ \pi^+ \bar{\nu}_\mu \;\;\;\;\;\;\;\;\;\;\;{(2)}$ Once more reaction: $p^+ p^+ \to p^+ p^+ \pi^- \pi^+ \to p^+ n \pi^+ \;\;\;\;\;\;\;\;\;\;\;{(3)}$ What are the experimental data? * * * **P.S.** This question is important enough. The main solar reaction is $p^+ p^+ \to d^+ e^+ \nu_e $ If this reaction occurs as follows $p^+ p^+ \to p^+ p^+ e^- e^+ \to p^+ n \nu_e e^+ \to d^+ e^+ \nu_e$ then it would require the energy of more than 2*0.511 MeV to take place. Cross section of this reaction will be much less, so the main solar reaction should be $p^+ p^+ e^- \to d^+ \nu_e $ * * * * * * **Update 20.02.11** Why I think so? I suppose that new particles are created in pairs particle- antiparticle. So the reaction $e^- e^- \to e^- e^- \pi^- \pi^+ $ requires the energy of more than 2*140 MeV to take place As well as the reaction $e^- e^- \to e^- e^- \pi^- e^+ \nu_e $ from symmetry considerations, since there is a decay $\pi^+ \to e^+ \nu_e$ And the result is $e^- e^- \to e^- e^- \pi^- e^+ \nu_e \to e^- \pi^- \nu_e $ contrary to $e^- e^- \to e^- \nu_e W^- \to e^- \pi^- \nu_e$ with the minimum reaction energy of 140 MeV The same is valid for the reactions (1) (2) (3) So what is the **experimental data** on minimum energy of these reactions?


How to derive Fermi-Dirac and Bose-Einstein distribution using canonical ensemble?

My textbook says that microcanonical ensemble, canonical ensemble and grand canonical ensemble are essentially equivalent under thermodynamic limit. It also derives Fermi-Dirac and Bose-Einstein distribution from grand canonical ensemble. My question is then: How to derive Fermi-Dirac and Bose-Einstein distribution using canonical ensemble. The expression of canonical ensemble $$\rho\propto e^{-\beta E}$$ seems to imply Boltzmann distribution only.


Something I don't understand in Quantum Mechanics

I've just started on QM and I'm puzzled with a lot of new ideas in it. 1.On a recent lecture I've attended, there is an equation says: $\langle q'|\sum q|q\rangle \langle q|q' \rangle =\sum q \delta(q,q')$ I don't understand why $\langle q'|q\rangle \langle q|q' \rangle =\delta (q,q')$ Can you explain this equation for me? 2.Actually, I'm still not clear about the bra-ket notation. I've learnt the bra and the ket could be considered as vectors. Then what are the elements of the vectors? Thank you very much!


Question on inflation as a phase transition

I have just finished watching the following video http://www.youtube.com/watch?v=beQ9fZ0jVdE where Laughlin, Gross and some students discuss e.g. about inflation. The following question is risen: Is inflation a phase transition (e.g. of the geometry of space time)?


Does 'maximum principal stress fracture criteria' only apply in tensile loading?

I'm working on 3d simulation of solid fracture, primarily for graphics applications. I have one question about the Rankine fracture criteria, which is widely used in the computer graphics community. The maximum principal stress fracture criteria, aka the Rankine criteria, states that the fracture is initiated when the greatest tensile principal stress in the solid reaches a critical magnitude: $$\sigma{}_{1 max} = \sigma{}_{TS}$$ where the subscript TS stands for tensile strength. My question is: Does this criteria only apply in tensile loading? Is it that this criteria can only model the 'Mode I' fracture? Note: the 'Mode I' fracture means the 'Open' fracture, as can be seen here: Fracture wikipedia.


Can superdeterminism resolve contextuality, entanglement and Shor's algorithm in quantum mechanics?

Superdeterminism is the idea that the apparent freedom for the choice of experimental apparatuses and their settings are nothing but an illusion. Contextuality is the dependence of the properties of a system upon the choice of measuring device. However, what if there is no "free" choice of measurement settings, i.e. the measurement settings aren't a freely variable independent degree of freedom? Bell inequalities require counterfactual polarization settings, but what if the polarization setting could only have been one way and not any other? As for the ability of Shor's algorithm to factorize large integers and compute discrete algorithms, maybe the answers gotten at the end retrocausally determine the numbers fed into the quantum computer in the first place so that it couldn't have been any other number? Are there any absolutely fatal objections?


Fluid Mechanics Piston Problem

![enter image description here](http://i.stack.imgur.com/OEbe7.png) **The actual problem:** A hydraulic car lift has a reservoir of fluid connected to two cylindrical fluid filled pipes. The pipe directly below the car has a diameter of 1.8 m. the pipe on which the plunger acts has a diameter of 0.045m. the plunger is depressed a distance of 1.5m. How much does the car rise? **My attempt:** Since the piston movess downaward thorugh a displacement Δx1 equals the volume of liquid pushed up on the right as the right piston moves upward through a displacement $\ π (\frac{0.0225m^2}{4}) (1.5m) = π (\frac{0.9m^2}{4})(x)$ x=0.00094m **My thoughts:** My attempt is wrong but I am confused as to why. Since $(A1)(Δx1)=(A2)(Δx2)$ , I thought that if I used the depressed distance of 1.5m for $Δx1$, then I would be able to find the hight of the car. I understand the diameter of the pipe on the left is very small and unrealistic if you ask me (typical text book problem). **My question** Any hints to how I can look at this problem differently? Maybe I am looking at the problem wrong. Any hints would be appreciated!


Visualizing Tension in FBD -- Rope With Mass

Asked to find the tension of a point that is halfway up a rope (where the rope has mass), given the system: ![enter image description here](http://i.stack.imgur.com/OQWpl.png) The solution of the problem (FBD included) is as follows. In this diagram (Tm = tension at mid-point, and Tt = tension at top). Question(s): (1) Why is Tt incorporated into the FBD? (2) Why does Tm point downward? ![enter image description here](http://i.stack.imgur.com/4OG2i.png) Also if it helps the FBD's for the individual components are given here: ![enter image description here](http://i.stack.imgur.com/ZvJeq.png) Thanks!


Gravity and electromagnetism

If light bends due to the curved spacetime,would not the act of bending light warp space? How does one describe curved light?


Radiation heat transfer

I'm looking at the radiosity equations for heat transfer http://en.wikipedia.org/wiki/Radiosity_(heat_transfer)#Radiosity_method Specifically, I'm hesitant to accept the equation: $$ \dot Q_i = \frac{ A_i \epsilon_i }{ 1 - \epsilon_i } ( \sigma T_i^4 - J_i )$$ My analysis is as follows: power goes out via Stefan-Boltzmann term, which is given by a term like $$ - A_i \epsilon_i \sigma T_i^4 $$ and radiation goes in via absorption of all emissions from nearby surfaces, that is: $$ A_i \epsilon_i H_i = A_i \epsilon_i \sum_j F_{ij} J_j $$ where the radiosity $J_j$ expression is correct on the article, and we are assuming Kirchoff's law of thermal radiation to hold. But the above implies that the actual expression for the net heat transfer is: $$ \dot Q_i = - A_i \epsilon_i ( \sigma T_i^4 - H_i ) = - A_i \epsilon_i ( \sigma T_i^4 - \sum_j F_{ij} J_j ) $$ Thoughts? Is my derivation correct, or is the one on the article?


A Sphere of Black Holes

Imagine a sphere of black holes surrounding a piece of space. Will this piece be separated from the rest of normal spacetime (at least for some time, till these black holes finally attracted themselves). So, seen from the outside, we have a black hole, but with a non-singular interior. Is this possible?


Why is classical physics not valid for a harmonic oscillator in its lowest energy state?

I am reading Born's interpretation of wave function in quantum physics by Eisberg & Resnick and I am not able to understand this description about comparison between the classical and quantum validity for an oscillator at lowest energy.![enter image description here](http://i.stack.imgur.com/aDFYf.png) ![enter image description here](http://i.stack.imgur.com/5Hjwt.jpg) ![enter image description here](http://i.stack.imgur.com/z64Pc.jpg)


Fractionalization and the structure of spin rotation group?

As we know, the phenomena of fractionalizations in condensed matter physics is fantastic, like fractional spin, fractional charge , fractional statistics, .... And one key point is that the quasiparticals must be created or annihilated by _pair_. On the other hand, consider the groups $SU(2)$ and $SO(3)$, they are the rotation groups for _half-integer_ and _integer_ spins, respectively. And we know that $SU(2)/\mathbb{Z}_2=SO(3)$, which means that each element in $SO(3)$ can be viewed as **one pair** $(U,-U)$, where $U\in SU(2)$ (otherwise put: the coset $\left\\{U, -U\right\\} \subset SU(2)$ in the quotient group $SU(2)/\mathbb{Z}_2$ is our element in $SO(3)$). So I wonder that whether is there any underlying connection between the _pair_ nature of quasiparticals in topological phase in physics side and the _pair_ structure relating $SU(2)$ and $SO(3)$ in mathematics side? Thank you very much.


Applicability of the concept of voltage in electrodynamic circuits

In electrostatics, we have $$\nabla \times \vec{E} = 0$$. Hence, we can define a scalar potential $V$, where $$\vec{E} = -\nabla V$$. We know from Faraday's law that $$\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$$. In situations involving constant currents(D.C.) the rate of change of the magnetic field is zero, so we can continue to use the concept of scalar potential, also known as voltage in circuit theory, for analysis of DC circuits. However,in practice, in the literature, we find the concept of voltage being used to study alternating current circuits as well. Further, it is also used in the study of high frequency alternating current circuits used for generation of electromagnetic waves. In such situations $$ \oint \vec{E}.\vec{dl} \neq 0$$. So strictly speaking, we cannot use the concept of voltage to study alternating current circuits, especially in the high frequency applications. Yet, in practice we find that it continues to be used in the literature. So is it really justified to use this concept outside of its realm of validity, and why or in what way?


Is continuous evolution from one eigenstate of operator $O$ to another $O$-eigenstate possible?

Eigenvectors associated with distinct values of an observable are orthogonal, according to quantum mechanics. Does this entail that a quantum system cannot continuously evolve from one eigenstate into another, for ANY observable? At first, that seems strange: it seems like a particle should be able to "travel" in the sense of continuously moving from one eigenstate of position to the next. Another example: can't a particle "speed up" (i.e. go continuously go from one velocity eigenstate to the next)?


What is the world's biggest Schrodinger cat?

How big is it by a truly quantum measurement? I am thinking of comparing Science magazines "Breakthrough of the Year" (BYOT) with the Zeilinger buckyball. The BYOT is a piezoelectric mechanical oscillator (PO) the size of the cross section of a human hair. It is placed in a superposition of its ground state and its first excited state. The well known buckyball experiment is a two slit experiment using buckyballs. (A third candidate might be a macroscopic Josephson junction oscillator conducting both ways at once.) I have made some basic calculations. For instance, the BYOT contains about 10^14 atoms compared to 60 or 72 atoms in the buckyball. By this measure the BYOT is bigger by a factor of about 10^12. On the other hand, the two slits are separated by 50 to 100 nanometers, or 10^-7 meters. In its first excited state, the top of the BYOT PO moves about 10^-15 meters per cycle, according to my calculations. By this measure the buckyball wins by a factor of about 10^8. Calculating energy of the moving parts, I find a much closer horserace, but the buckyball is about 100 times bigger. However, none of these calculations is at all quantum mechanical (QM). ArXiv lists at least five papers proposing truly quantum mechanical measures of the size of a macroscopic Schrodinger cat. The most recent is Lee and Jeong http://arxiv.org/abs/1101.1209 which references the other four. Can someone competent (or expert) in QM apply one or more of these quantum measures to the BYOT and the buckyball and tell me which is larger? TIA. Jim Graber


Vertical circular motion followed by projectile motion

If you have a stone tied to a string, and it is given a certain velocity at the lower most point such that it completes a quarter circle but then the string becomes slack, and it undergoes projectile motion. The problem is, is it necessary that this parabola will be symmetric about the verticle? If so, is it impossible that the stone passes through the center of this circle during downward motion? Please give a detailed solution Thanks in advance


Spring Damper System : Recoil Reduction

I am designing a man-"portable" 30mm sniper rifle. Although it may be classified as a anti-materiel cannon. I have the mass of the round in total, and the mass of the projectile leaving the barrel, as well as the pressure inside the barrel. I have the weapon's weight as well. Here is my source : http://www.pmp.co.za/index.php?page=mediumcalibre6 Now, my problem.   1. How do I calculate the force that this bullet imparts upon the rifle?   2. If I use a spring-damper system to reduce the force on the shooter, how do I actually determine the force on the shooter? If you do not find the need to supply me with a direct answer, which is understandable, could you please redirect me to a website/book that might help me?