title: why is total electron energy of an electron in metal negative?

text: In my textbook, it says that any electron bound in metals, modelled as some potential well $U$, has negative total electron energy, as shown below in the figure. Why is the total electron energy negative? And how can this be possible? Secondly, the (b) part (it is below the (a) part) of the figure is the graph of the potential energy seen by electrons. I am curious why the part $x<0$ has $-U$ as its potential energy seen by electrons. The textbook stated that the metal works(or is modelled) as the potential well depth $U$. So why is it suddenly $-U$? ![enter image description here](http://i.stack.imgur.com/4PiK2.png)


title: Linear acceleration vs angular acceleration equation

text: I'm learning about angular velocity, momentum, etc. and how all the equations are parallel to linear equations such as velocity or momentum. However, I'm having trouble **comparing angular acceleration to linear acceleration**. Looking at each equation, they are not as similar as some of the other equations are:   * Anglular acceleration = velocity squared / radius   * Linear acceleration = force/ mass I would think angular acceleration would take torque into consideration. How is Vsquared similar in relation to force, and how is radius's relation to Vsquared match the relationship between mass and force? ![angular velocity](http://i.stack.imgur.com/degWv.png) ![Angluar accel](http://i.stack.imgur.com/I9AED.png) I suppose the root of this misunderstanding is how I'm thinking of angular acceleration, which is only an vector representing an axis's direction, and having a magnitude equal to the number of radians rotated per second. I also am confused on what exactly 'V' (tangential velocity) represents and how it's used. Is it a vector whose magnitude is equal to the number of radians any point on a polygon should rotate? What is the explanation?


title: How does AC current flow in an open circuit?

text: In common house hold wiring we have the hot lead, neutral and ground. If the hot lead in electrical wiring contacts earth ground (perhaps though a short circuit in the chassis of a device) then the current shorted to earth. This however doesn't appear to me to be a complete closed circuit. My intuition tells me current should flow, but my academic knowledge says that it shouldn't because we have an open circuit condition. So how does the current therefor flow? I've heard the argument that because the neutral wire is earth grounded at the service panel (aka breaker box), that it is actually a complete circuit. So in my example, current travels though earth and back up into neutral at the service panel. However IF at the service panel, the neutral was not connected to earth ground (so the neutral left floating), then we definitely don't have a closed circuit, yet my intuition tells me somehow current would still flow into earth in a short circuit condition. I've looked up a similar question like this here on stackexchange. In that question, current flow is examined between earth and mars in an incomplete circuit. In that scenario the answer looked at the planets somewhat like two capacitor plates. That would not apply in my question however


title: Divergent sum in lightcone quantization of bosonic string theory

text: I had the following question regarding lightcone quantization of bosonic strings - The normal ordering requirement of quantization gives us this infinite sum $\sum_{n=1}^\infty n$. This is regularized in several ways, for example by writing $$ \sum_{n=1}^\infty e^{- n \epsilon } n = \frac{1}{\epsilon^2} - \frac{1}{12} + {\cal O}(\epsilon^2) $$ Most texts now simply state that the divergent part can be removed by counterterms. David Tongs notes (chap. 2 page 29) specifically state that this divergence is removed by the counterterm that restore Weyl invariance in the quantized theory (in dimensional regularization). I would like to see this explicitly. Is there any note regarding this? Or if you have any other idea how one would systematically remove the divergence above, it would be great!


I would be most thankful if you could help me clarify the setting of primary constraints for constrained Hamiltonian systems. I am reading "Classical and quantum dynamics of constrained Hamiltonian systems" by H. Rothe and K. Rothe, and "Quantization of gauge system" by Henneaux and Teitelboim. Consider a system with Lagrangian $L(q,\dot{q})$ and define the momenta $p_j = \partial L/\partial{\dot{q}_j}$, with $j$ from 1 to $n$ (for $n$ degrees of freedom), and the Hessian $W_{ij}(q,\dot{q}) = \partial^2 L/\partial{\dot{q}_i}\partial{\dot{q}_j}$. Let $R_W$ be the rank of the Hessian $W_{ij}$. Assume that $L$ is singular and $R_W < n$. In order to ask the question, I'm now quoting how the Rothes describe the setting of primary constraints on pages 26 and 27 of their book. > Let $W_{ab}$, ($a,b = 1,...,R_W$) be the largest invertible submatrix of > $W_{ij}$, where a suitable rearrangement of the components has been carried > out. We can then solve the eqs. $p_j = \partial > L(q,\dot{q})/\partial{\dot{q}_j}$ (1) for $R_W$ velocities $\dot{q}_a$ in > terms of the coordinates $q_i$, the momenta $\\{p_a\\}$ and the remaining > velocities $\\{\dot{q}_\alpha\\}$: $\dot{q}_a = > f_a(q,\\{p_b\\},\\{\dot{q}_\beta\\})$, with $a,b = 1,...,R_W$ and $\beta = > R_W + 1,...,n$. > > Inserting this expression into the definition of $p_j$, one arrives at a > relation of the form $p_j = h_j(q,\\{p_a\\},\\{\dot{q}_\alpha\\})$. For $j = > a$ ($a = 1,...,R_W$) this relation must reduce to an identity. The remaining > equations read $p_\alpha = h_\alpha (q,\\{p_a\\},\\{\dot{q}_\beta\\})$. But > the rhs cannot depend on the velocities $\dot{q}_\beta$, since otherwise we > could express more velocities from the set $\\{\dot{q}_\alpha\\}$ in terms > of the coordinates, the momenta, and the remaining velocities. This is where the presentation from Rothe stops, and my concern is that equations of the form $p_\alpha = h_\alpha (q, \\{p_a\\},\\{\dot{q}_\beta\\})$ (2) with all $\\{\dot{q}_\beta\\}$ present can still be possible, and yet one cannot solve for more velocities from the set $\\{\dot{q}_\alpha\\}$ in terms of the coordinates, the momenta and the remaining velocities if the conditions stipulated in the Implicit Function Theorem are not met, for not all equations of the type (2) can be solved implicitly for $\\{\dot{q}_\alpha\\}$. Therefore it is not proven that there are $(n - R_W)$ primary constraints of the form $\phi_\alpha (q,p) = 0$. Henneaux and Teteilboim even state that these $(n - R_W)$ constraints of the form $\phi_\alpha (q,p) = 0$ are functionally independent, but give no justification to this statement. I would be most thankful if you could help clarify my above concern and also if you could clarify the statement by Henneaux and Teitelboin as to the fact that the constraints are functionally independent. Thank you!


title: Is General Relativity applicable for all coordinate systems?

text: My understanding was that relativistic physics can be expressed in any _inertial_ coordinate system, but not arbitrary systems. That is, no experiment can determine if we are "still" or "moving" at a constant velocity; but we can determine if we are accelerating, or moving in a circle (which by definition involves constant acceleration perpendicular to the current velocity). Thus, we can clearly state that the Earth is _orbiting_ , and can't view it as relativisticly stationary. But, to my shock, I recently came across this text http://books.google.com/books?id=lWEmNBaHCJMC&pg=PA211&dq=einstein+infeld+physics+ptolemy+copernicus&hl=en&ei=dWZ_TubbKqn20gH8hNjSDw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCwQ6AEwAA#v=onepage&q&f=false (which has Einstein as a coauthor) which, on page 212, seems to say that although special relativity requires an inertial coordinate system, general relativity does not! And that therefore we can state that the Earth is stationary and the Sun orbits it! I would reject this as pseudoscientific bunk, if not for the authors of the book.


title: Can a planet have a larger diameter than Jupiter?

text: I have heard it said that if mass were added to Jupiter, then only its mass and density would increase but the diameter would stay about the same. Is this the case for Jupiter or some property of Jupiter specifically, or is 150,000 km pretty much the largest diameter that a non-fusing object can achieve? I am specifically interested in the case of a physically larger object (planet) orbiting a smaller but more massive object (star).


title: Why can't this speed be measured?

text: > Superman and Supergirl were playing catch. When Superman is moving with a > speed of 0.800c relative to Supergirl, he threw a ball to Supergirl with a > speed of 0.600c relative to him. > > a. What was the speed of the ball as measured by Supergirl? I tried to solve number 1. heres my approach. let s be the supergirl frame of reference while s' be the frame of reference by superman, heres my diagram               (<--)superman     (--->)ball     supergirl(stay)      so s' is going to the left, so it has negative velocity (-0.8c). vx' is going to the right (+0.6c) . now (vx) velocity of ball with respect to supergirl is (-.13c) using lorentz transformation, vx is negative, so it means its going to the left and it will never reach supergirl. so it appears that answer for letter a is, its impossible? thanks


title: What is the "shape" of atomic/subatomic particles?

text: I apologize in advance for my ignorance if this is a question with an obvious answer... I am not experienced in this field. But are such particles in the universe points with a charge, or are they very small spheres with a charge? Or does it not even matter in the end? This isn't homework, it's just curiosity.


title: Why does the amplitude of a ripple tells us that it is a particle?

text: The quote below is from Matt Strassler's blog: > a particle is a ripple with many crests and troughs; its amplitude, relative > to its overall length, is what tells you that it is a single particle. If I understand correctly, what he calls "ripple" is "probability wave". Why is it that the amplitude of a probability wave is the sign of "a single particle"?


title: How do I use the electro-optic tensor?

text: I would like to calculate the performance of an optical phase modulator, in which a varying electric field across a crystal modulates the effective refractive index of light passing through the crystal. The effect depends on the crystal orientation, polarization (i.e. direction of the E-field of the optical beam), and the direction of the applied electric field. The dependence on these vectors is encoded into the electro-optic coefficients, which form the electro-optic tensor. My question: How do I apply these coefficients to give the magnitude of the electro-optical effect (in radians of optical phase per applied volt of potential difference) when the light is polarized in direction $i$, propagating in direction $j$, with an applied electric field in direction $k$? In short, how do I _use_ the electro-optical coefficients? For concreteness, I am interested in lithium niobate. A datasheet gives the coefficients as: ![enter image description here](http://i.stack.imgur.com/2Tynp.png)


title: A particle of charge $-e$ orbits a particle of charge $Ze$, what is its orbital frequency?

text: A point particle $P$ of charge $Ze$ is fixed at the origin in 3-dimensions, while a point particle $E$ of mass $m$ and charge $-e$ moves in the electric field of $P$. I have the Newtonian equation of motion as $$- \frac{Ze^2}{4 \pi \epsilon_{0}} \frac{\vec{r}}{r^3} = m \ddot{\vec{r}}$$ I derived this from Newton's law and Coulomb's law. I then went on to show that the particle moves in a plane by showing it has a constant normal vector. Can anyone help me find this if the orbit is circular about $P$ what it's orbital frequency is -in terms of the constants we have? I would be extremely grateful. I have attempted this by parameterizing the orbit, differentiating with respect to time and substituting into the equation of motion, however all I get are two unsolvable differentials equations.


title: Mechanics question (a block on top of a block)

text: This question is very similar to this one here. > A block of mass $m_1$ is placed on another block of mass $m_2$ lying on a > smooth horizontal surface.   >  The coefficient of friction (static and kinetic) between $m_1$ and $m_2$ is > $\mu$. > > Find the acceleration of the blocks if the force applied to $m_1$ is $5N$, > given that $m_1 = 2kg$ and $m_2 = 4kg$ and $\mu=0.2$. I can prove the result in the link, and obtain the critical force as $6N$, so for $5N$, they will have same acceleration, which will be equal to $\frac{5}{6}$. But when I take the the general case, draw free body diagrams, I get these equation - $$ m_1 a_1 = F - \mu m_1 g \\\ m_2 a_2 = \mu m_1 g $$ This gives the answer as $a_1 = \frac{1}{2}$ and $a_2 = 1$ The only place where I think I could have made a mistake would be in case of force of friction. I am taking the maximum value of friction ($4N$), but since the force applied is greater, I presume this can happen. Can anyone tell me where I go wrong?


title: An optical system with image from a convex lens forming behind a concave lens

text: I am having a problem drawing optical ray lines for optical systems. Namely, one which have a formation of image of the first lens beyond the second lens. Take the picture below as an example: ![enter image description here](http://i.stack.imgur.com/sSrZi.jpg) In this picture, the image from the first (convex) lens focuses at 80 cm, which is beyond the position of the second (concave) lens. I don't think I have any problem calculating the image distance, etc., but I don't know how to draw the ray lines, similar to the picture below: ![enter image description here](http://i.stack.imgur.com/i2rjd.jpg) Thank you in advance!


title: Can someone explain to me inertia?

text: I'm a little new to physics and I'm not quite getting the concept of inertia. Can someone help me?


title: Diffeomorphisms, Isometries And General Relativity

text: Apologies if this question is too naive, but it strikes at the heart of something that's been bothering me for a while. Under a diffeomorphism $\phi$ we can push forward an arbitrary tensor field $F$ to $\phi_{*}F$. Is the following statement correct? If $p$ is a point of the manifold then $F$ at $p$ is equal to $\phi_* F$ at $\phi(p)$, since they are related by the tensor transformation law, and tensors are independent of coordinate choice. () I have a feeling that I'm missing something crucial here, because this would seem to suggest that diffeomorphisms were isometries in general (which I know is false). (*) However if the statement isn't true then it menas that physical observables like the electromagnetic tensor $F^{\mu \nu}$ wouldn't be invariant under diffeomorphisms (which they must be because diffeomorphisms are a gauge symmetry of our theory). In fact the proper time $\tau$ won't even be invariant unless we have an isometry! What am I missing here? Surely it's isometries and not diffeomorphisms that are the gauge symmetries?! Many thanks in advance.


title: Normal modes of oscillation: how to find them

text: Are normal modes the eigenvectors of the matrix $(\omega ^2 T- V)$ where $T$ is the matrix of kinetic energy and $V$ is the matrix of potential energy? Is it the only way to express them? How can I express them using the coordinates that I have choosen at the beginning of the exercise?


title: Does clipping to $m$ guarantee a maximum peak-to-peak amplitude $m$?

text: There is a technique called clipping in sound synthesis. It is explained on Wikipedia . I make music and like this technique: You make an extreme "fat" noise, maybe with a lot of resonance, but then it sounds somehow too loud. Then you can use clipping to make the sound less loud. Now, I ask myself whether clipping to a value $m$ really guarantees that there is no "particle" oscilating with more peak-to-peak amplitude than $m$. Let's say the speaker has reached the part where clipping shall start. Then it suddenly stands still. However, due to inertia, won't the particles in air keep the movement, and thus reach a higher amplitude? Practical application of the question is whether clipping kind of protects your ears. For me, clipped sounds often sound louder than non-clipped. E.g. a square wave (which is close to a clipped very loud sine wave) sounds louder than a sine wave, both with the same amplitude. Why?


title: Computing microstate probabilities based on Boltzmann distribution for chemical systems - Is it rigorous?

text: One approach to predicting the folded structure of a polymer (DNA, RNA, protein) is to compute the probability that any particular part of the polymer $x_i$ is "paired" with another part of the polymer $x_j$, per molecule of polymer in a solution at equilibrium. This pairing probability is often computed by enumerating all conformations where $x_i$ is paired with $x_j$, summing over the free energy of all such conformations, and then evaluating a Boltzmann distribution. More precisely, to compute the probability that a polymer adopts a particular conformation $s$, per molecule of polymer in solution at equilibrium, one computes: $$P(s) = \exp(-G_s/RT)/Z$$ where $G_s$ is the free energy of conformation $s$ and $Z$ is the partition function a.k.a normalization factor needed so that $P$ is a probability distribution. The question: Is this a rigorous way to compute $P(s)$? The reason I doubt the rigorousness of the above computation is that I can get widely different results for $P(s)$ by making very slightly different assumptions about the polymer in question. * * * Why I believe $P(s)$ is ill-defined: First, let's take a very simple polymer with just two components and two states: UNPAIRED and PAIRED. Say that the chemical potential of UNPAIRED is $-1$ kcal mol$^{-1}$ and the chemical potential of PAIRED is $-1$ kcal mol$^{-1}$. Then $P($UNPAIRED$)$ is $1/2$, which seems very reasonable. But, let's say that we add a little bit more complexity to our model for the polymer and split PAIRED into 5 states, which are each very slightly different versions of PAIRED (but they are, in fact, different). These new states, PAIRED$_{i}$ for $i = 1, 2, ... 5$ still have chemical potential $-1$ kcal mol$^{-1}$, and UNPAIRED still has chemical potential $-1$. Now, $P($UNPAIRED$)$ is $1/6$. Both models accurately describe the polymer, however the latter is slightly more precise. However, it seems that $P(s)$ is sensitive to such changes in the model. To what extent, then, is $P(s)$ rigorous?


title: Photons and Absorption

text: Hello and thanks for reading my question: Imagine we send one photon at an atom, and it happens to be the right frequency such that it gets absorbed fully by an electron in this atom. Obviously that photon is gone. However, in light that a photon is a perturbation of an E&M field, this implies that the E&M wave which represented the photon is gone. My question is how does the E&M wave dissapear? I have reasoned that it cannot dissappear instantaneously, as that would break SR. Moreover, the wave existed throughout the path between the electron which emitted it and the electron that is absorbed by. This leads to my confusion: if it does not dissappear instantaneously, then the photon still exists and is not absorbed. tl;dr: E&M Waves(Photons) are not localized, so how can they be absorbed.


title: From the top of a 400 m high rock by the sea, a ball is shot out in the horizontal direction. The ball speed is 100 m / s

text: From the top of a $400 m$ high rock by the sea, a ball is shot out in the horizontal direction. The ball speed is $100 m / s.$ a) After how much time the ball hits the water? b) At what rate does the ball hits the water?


title: What is the meaning of the negative vacuum expectation value of the Higgs field? Do we see it in nature?

text: In studying about the Higgs field and related, I find little mention of the equilibrium point at -V. I would like help conceptualizing what a negative vacuum expectation value is, ideally with respect to the Higgs field. Do you have any hint on how to observe or create it?


title: Text interpretation in Griffith's intro to QM

text: It says in Griffith's chapter 2.1, that: > $$\tag{2.14} \Psi(x,t)~=~\sum_{n=1}^{\infty}c_n\,\psi_n(x) e^{(-iE_n > t/\hbar)}$$ It so happens that _every_ solution to the (time-dependent) > Schrodinger equation can be written in this form [...]. By "solution" does he mean solutions on separable form > $\tag{2.1} \Psi(x,t)~=~\psi(x)f(t),$ which he stated to begin with? `


title: Experimental evidence for color-neutral gluons $( (r\bar{r}−b\bar{b})$ and $(r\bar{r}+b\bar{b}−2g\bar{g}) )$

text: Is there any experiment/measurement that would have a different outcome if one of the following scenarios is applied:   1. The two color-neutral** gluons would not exist   2. Those gluons would have very large masses ** we are talking about the 2 gluons not exchanging color, conventionally written as $(r\bar{r}−b\bar{b})/\sqrt2$ $(r\bar{r}+b\bar{b}−2g\bar{g})/\sqrt6$ EDIT: Since I am now aware that 'color-neutral' does not exist, the question only makes sense if stated as 'What would be the experimental evidence if color-symmetry was broken in some way? (such that 2 of the gluons were singled out)'


title: Fictitious Forces $\overset{?}{⇔}$ Constraint Forces (re: D'Alembert's Principle)

text: Are fictitious forces and constraint forces the same thing?


title: What's wrong with this solution to freshman textbook example for Ohm's law and Kirchhoff's current law?

text: I'm a freshman taking an introductory course on circuit theory and I've run into a bit of trouble working through a textbook example practice problem. Unfortunately, it only lists the answer and not the steps involved. ![textbook example problem](http://i.stack.imgur.com/EJAD9.png) In the above problem, it is given that $i_x=3\,A$ and the leftmost current is $8\,A$. From what I understand, the three parallel branches (all but that with the $v_x$ source) should have identical voltage drops; this suggests $(6\,\Omega)i_x=18\,V$ which is consistent with the given $i_x=3\,A$ -- so far, so good. I run into trouble when trying to apply Kirchhoff's current law to the top node connecting these three branches, however. Since the voltage drop across each branch is $18\,V$ it follows the current through the $R_a$ branch ought to be $(18\,V)/R_a$. As the other two branches are specified to be $8\,A$ and $3\,A$ it follows we have:$$13\, A=3\, A+\frac{18\,V}{R_a}+8\,A$$... from which I conclude $R_a=9\,\Omega$, rather than $R_a=1\,\Omega$. Try as I might I've been unable to figure out exactly where I've gone wrong. Which steps were incorrect? Any and all help is greatly appreciated. Thank you!


title: Do I have the meaning of the property temperature correct?

text: OKay my book just starts out talking about the vague definition we have for temperature and we ended up with the Zeroth law of Thermodynamics which states: > Two systems are in thermal equilibrium if and only if they have the same > temperature. So does that mean, the (brief) meaning of the property _temperature_ is **a measure of whether one system is in thermal equilibrium with another**?


title: Photonics: Slab As a Lens

text: The question can be found here: http://gyazo.com/fc4d26cd35e6ce368ad2a8ed504f1dcc The refractive index it references can be found here: http://gyazo.com/94fd2f3b5ea7da9226c3acd56b0024c1 I'm not really sure where to get started on the "showing" part of the question. I anticipate that Snell's Law will be used somehow. I'm not looking for a full answer, just a push in the right direction.


title: Can various wavelengths participate in C/D Interference?

text: My question is can multiple wavelengths or at least two different wavelengths interfere with one another? I know that they usually have to be the 'same' wavelength, but you'd think they can vary a bit from one another, so what is the relationship of wavelength/frequency to interference? This question popped up into my ol' mind grapes when reading this thread: Producing photons with same frequency, different amplitude wave Basically I originally thought that the energy of an EM wave was simply put(I took general physics mind you but I don't mind you throwing on some math if that's what it takes) by the relationship to Planck's constant and the frequency, but when googling what amplitude specifically meant (I assumed it was the intensity of the wave, w/e that is) I learned that it is the sq.root of E. Therefore it seemed logical that Planck's constant, the frequency and the amplitude where all related, but that still eludes me as well. While David Z's answer from the above link clarified this situation for me, it left me wondering: he says that the energy described by the amplitude, compared to the energy described by Planck's constant and the frequency, can be related to each other through the quanta(i.e. 'photon' in simpler cases) in monochromatic light. This got me thinking that if the location of a photon(or its crests/peaks) were important(as are its freq/wavelength), then can two waves of slightly different wavelengths influence each other? Thanks for the help!


title: One question about Weinberg's derivation of arbitrary spin fields expressions

text: In his book "QFT" (vol. 1) Weinberg writes the expression for an arbitrary spin massive field: $$ \hat {\Psi}_{a}(x) = \sum_{\sigma = -s}^{s} \int \frac{d^{3}\mathbf p}{\sqrt{(2 \pi)^{3}2 \epsilon_{\mathbf p}}}\left( k_{1}F_{a}(\mathbf p) \hat {a}^{\sigma}(\mathbf p )e^{-ipx} + k_{2}G_{a}(\mathbf p) {\hat {b}^{\sigma}}^{\dagger}(\mathbf p )e^{ipx} \right). $$ By assuming very long way of derivations (which consists of lorentz transformation law for fields) before that he concludes that $$ F_{A}^{\sigma}(\mathbf p ) = (-1)^{s + \sigma}G_{A}^{-\sigma}(\mathbf p). $$ How to prove this faster than Weinberg do it? Let's assume that equations of motion and degrees of freedom decreasing relations for corresponding field are known.


title: Meaning of "Simple" in Simple Pendulum and Simple Harmonic Motion?

text: I have gone through the Phys.SE question Why is simple harmonic motion called so?. From the 1st answer of this Question it seems to me that another type of " **Harmonic motion** " is " **Damped Harmonic Motion** ". But can you please explain what the other type of Pendulum is? Is this Compound pendulum or Damped Pendulum or Something else?


title: Thought experiment regarding an object approaching a mirror

text: Here's a thought experiment I came up with in class today when my mind drifted (I however highly doubt I'm the first to think about this since it is pretty rudimentary) : **Let's say superman approaches a (flat) mirror at 200,000 km/s. At what speed does superman's image approach him?** This has given me a big headache. I have pondered a lot of solutions, but none of them really convinced me. What is the correct solution to this thought experiment?


title: Difference between angular frequency and angular velocity?

text: What is the difference between angular frequency and angular velocity? I think one is used for SHM and the other for circular motion? Also can both be used for centreptal accelartion? I think angular freq=2πf and angular velocity=dΘ/dt? Please confirm or expalin. Are they the same thing for circular motion??


title: Gillespie's stochastic framework valid for particles in aqueous solution?

text: Gillespie proposed a stochastic framework for simulating chemical reactions which is predicated on non-reactive elastic collisions serving to 'uniformize' particle position so that the assumption of well-mixedness is always satisfied (see page 409 in the linked version). This is formulated from kinetic theory. A corollary to this is that a non-reactive collision between two molecules that are _able_ to react _does not induce local correlation_ , i.e., _two particles able to react with each other that just collided, but didn't react are no more likely to react with each other in the next dt than any other particle pair in the volume_. Gillespie's algorithm is commonly used in biology where biochemical species are modeled in the _aqueous_ environment of cells. Is this valid, and if so why? It seems the validity may depend on an assumption of Boltzmann- distributed velocities which may or may not be valid in aqueous phase. I recently asked about this (question here), however, the question was deemed a duplicate even though there was some disagreement among the answers.


title: How does a plasma globe work?

text: I am trying to simulate a plasma globe using computer. But I do not know how to model the motion of its filaments. My first guess is to construct a potential field. And the filaments are the paths minimize the works in this potential field. By altering this potential field, the filaments get animated. I am wondering if this model is correct. And what is this potential field going to be? Is it an electric field?


title: Is the current situation possible for recreaction in a lab?

text: I want to convert cellulose to another form (crystalline => amorphous). Cellulose requires a temperature of 320 °C and pressure of 25 MPa to become amorphous in water.(Is the pressure achievable in the lab?) Is this doable in the lab and how should I begin on reproducing that situation (which tools would I need) ? I'm guessing that a bunsen burner would produce high enough temperatures (but I'm not quite sure about the pressure) EDIT (info) : I know that this is a chemical process, but aren't you physicists better in figuring out the 'realisticativity' of pressures?


title: Are there any modern textbooks on statistical mechanics which don't ignore Gibbs' analysis of the microcanonical ensemble?

text: I have lately been reading Gibbs' book _Elementary Principles in Statistical Mechanics_ , and I'm surprised how much in that book seems to have been ignored by later textbook writers. In particular, Gibbs looks carefully into the question of whether in the microcanonical ensemble we can sensibly define entropy and temperature, and his answer, to paraphrase, is "meh, kind of". Gibbs pays close attention to exactly deriving properties of systems with any number of degrees of freedom (not just macroscopic systems), and shows that the canonical and grand canonical ensembles give perfect explanations for thermodynamic equations, _even extending them to small systems_. This is not so for the microcanonical ensemble:   * There are two valid definitions of microcanonical entropy (phase space "surface" entropy (Shannon/Boltzmann entropy) and phase space "volume" entropy, to use the modern terminology). These are both valid since they become equivalent in the thermodynamic limit.   * Neither definition, when differentiated against energy, leads to a satisfactory value of temperature, because two microcanonical systems, when they are combined and allowed to exchange energy, do not maintain the same temperature that they had before. (Except in the thermodynamic limit.)   * For a system with 1 or 2 quadratic degrees of freedom (e.g. particle in a 1D or 2D box), these temperatures behave super strangely. As in, being always negative or always infinite, and the system becoming colder with increasing energy. Again, the thermodynamic limit escapes these problems. It seems like the only value of the microcanonical ensemble is in the thermodynamic limit where these two expressions of entropy and temperature become equivalent, and start to behave properly. On the other hand, in a lot of textbooks the microcanonical ensemble is taken to be somehow the "fundamental" ensemble, even though the ensembles become equivalent in the thermodynamic limit anyway! So, putting the question in a slightly different way, are there any modern textbooks on statistical mechanics which are as careful as Gibbs, or are learners simply expected to read Gibbs for the "real thing"? I would guess that proper treatment of small systems is a big deal for, e.g. nanotechnology.


title: Particle sliding on a semisphere - solving the differential equations analytically

text: I have the following problem: > A particle of mass $m$ is at rest at the top of a semispherical surface of > radius $R$. After an impulse, it starts to slide along a meridian with speed > $v_0$. _The surface has dynamic friction coefficient $\mu$._ At what angle > of colatitude does the particle lose contact with the surface? I know how to solve the problem in the case $\mu=0$, but in this case things seem to be more complicated. These is my attempt at a solution. Let $\theta$ be the colatitude, $N$ the surface´s reaction. By $\vec F=m\vec a$ we get:$$N=m(g \cos \theta-R\dot \theta ^2),$$ and $$R\ddot\theta=g(\sin \theta-\mu \cos \theta)+\mu R \dot \theta ^2.$$ In the case $\mu = 0$ I would multiply last equation by $\dot \theta$ to obtain the angular velocity in function of the angle (which is equivalent to use the kinetic energy theorem) and then set $N=0$ in the first equation. It´s easily seen that this strategy doesn´t work for the general case. Is it possible to solve these differential equations analytically?


title: Calculate shadow path&size + object size based on distance

text: What are the variables I need to take in account if I want to calculate shadow path and size?   I will try to simplify my questions. I have a $2D$ circle (radius: $28$) which means the size of the circle (in box) is $56\times 56$ at specific point (lets say $(300,300)$). I tell the $2D$ circle to go in specific path, lets say from ($300,300$) to ($300,200$). I want to tell its shadow what path to do based on the $2D$ circle path.   ofcourse the size of the ball need to be calculated aswell..   I guess I need some extra variables because I am talking about $3D$ space in $2D$.. (lets say the "sun" is always above the ball so you don't need to calculate its degrees from the "sun", only its distance) I am pretty sure it's possible but I have no idea how to calculate and what extra variables I need... Basically, something like that: http://jonobr1.github.io/Physics/examples/cloth.html Hopefully you understood what I want, if u didnt just tell me which part and I will try to re-explain or something.   Thanks


title: Where can I find how to do data analysis for a magnetic hysteresis loop?

text: I am given a data set that has a hysteresis loop but there's a diamagnetic signal that I understand needs to be subtracted, but I don't know the how this is done. I remember finding the slope of the loop at the flat ends and using this in some way but that's how far my memory serves me.


title: Is speed squared constant?

text: q. > A heavy particle is projected at speed $U$ at an angle $\alpha$ to the > horizontal. The particle is subject to air resistance which is > experimentally found to vary proportionally to the square of the speed. Show > that > > $$\vec{\dot{v}} = -\frac{g}{V^2}\lvert\vec{v}\rvert\vec{v} - g\vec{j},$$ > > where $V$ is the terminal velocity of the particle. If $\alpha = > \frac{\pi}{2}$ (so that the particle is projected directly upwards), find > the maxium height reached and the time taken to reach it. What is the speed > of the particle when it returns to the horizontal? I'm on the part where "If $\alpha = \frac{\pi}{2}$ ..." seeing as $\alpha = \frac{\pi}{2}$ it would be acceptable to think of motion in one dimension, so from the "show that" equation I get $$\ddot r = \dot v = \dfrac{-g}{V^2}v^2 - g $$ Now I'm curious here, can I just integrate w.r.t to t? Usually in vector form v is a function of t, and I'm asuming it is here aswell, but is this allowed: $$\ddot r = \dfrac{-g}{V^2}v^2 - g $$ $$\dot r = \left ( \dfrac{-g}{V^2}v^2 - g \right )t + C$$ If so, why? If not, why not? Furthermore, I get a complex solution to the terminal velocity, what does this mean - As the terminal velocity can't be 0, what can we conclude from that? edit: solving $\dot v = -kv^2 - g$ where $k = \dfrac{g}{V^2}$ I get $\dfrac{dv}{kv^2+g} = -dt$ and solving this I get $v = \dfrac{\sqrt{g}}{\sqrt{k}} arctan(C\sqrt{k}\sqrt{g} - \sqrt{k}\sqrt{g}t)$ where C is a constant, integrating this again I get $r = \dfrac{1}{k}\log (sec(c\sqrt{k}\sqrt{g} - \sqrt{k}\sqrt{g}t)) + C_1$ then for $v = 0$ I get $ t = C$ so $r = C_1$, fro r = 0 I get $t = \dfrac{C\sqrt{k}\sqrt{g} - arcsec(e^{-Ck})}{\sqrt{k}\sqrt{g}}$ so $v = \dfrac{\sqrt{g}}{\sqrt{k}}tan(arcsec(e^{-Ck})) = \dfrac{\sqrt{g}}{\sqrt{k}}\sqrt{1 - \frac{1}{e^{-2Ck}}}e^{-Ck}$


title: Does space expand locally without restriction in freefall to central region of black hole?

text: It is often stated that a distant observer will observe a freefalling object as experiencing an infinite journey to the event horizon, but that a traveler in the local frame will experience a finite-time journey to a singularity. Is this however true? During the inward journey to a black hole (non-rotating for simplicity) the traveler will accelerate towards the speed of light, while travelling through an increasing gravitational field. At some point the velocity based local space expansion (special relativity) will be overtaken by gravitational space expansion (general relativity), after which point space will continue expanding as observed locally. This point is when the local space expansion (correlating to the externally observed length contraction) from an external frame point of view has exceeded the speed of light. The traveler will then be experiencing standstill in a singularity-free space undergoing an accelerating expansion. After some (local) time, this space will have many characteristics that are similar to the expanding universe in which it is embedded.


title: Motion in a Paul trap: $2n$th harmonic with larger amplitude than $n$th harmonic

text: Using a Paul trap, we captured the motion of a light charged particle (based on a rotating potential applied by AC current). Our rotational frequency was 50 Hz, and so when used FFT on the data, we were expecting this frequency to have largest amplitude. Yet , the double frequency, 100 Hz, had as much as twice the amplitude that the 50 Hz had. how would one explained such an observation?


title: S-Matrix in $\mathcal{N}=4$ Super-Yang Mills

text: This is a general question, but what is meant when people refer to the S-Matrix of $\mathcal{N}=4$ Super Yang Mills? The way I understood it is the S-Matrix is only well defined for theories with a mass gap so we can consider the asymptotic states to be non interacting and then apply the LSZ formalism. The idea breaks down for general CFTs and the observables should just be the correlation functions. Based on what I've seen in the literature, this does not seem to be the case and people talk about the S-Matrix for $\mathcal{N}=4$ Super Yang Mills, a superconformal field theory. Is it that we consider a deformed CFT so there exists a gap in the spectrum and take the limit as the deformation goes to zero? Or is there a way to define an S-matrix in an exactly conformal theory? Edit: For anyone who finds this question the following reference (the introduction at least) is of use in showing how the normal logic breaks down: http://arxiv.org/pdf/hep-th/0610251.pdf


title: Commonalities among string theories?

text: So string theory is a developing theory with presumably different variants. What is the commonality among all these variants? What makes the next modification of a current string theory another string theory, as opposed to a completely new theory? More specifically, is there a general mathematical structure that is common to all the theories that are considered to be string theories?


title: Decomposition of representations of the Virasoro algebra under $sl(2)$

text: The Virasoro algebra has a finite $sl(2)$ sub-algebra generated by $L_{-1}$, $L_0$ and $L_{+1}$. Let's consider a unitary highest weight representation of the Virasoro algebra with conformal weight $h>0$ and central charge $c>1$. How does this representation decompose under the $sl(2)$ sub-algebra? It is clear that there is an invariant $sl(2)$ sub-module of weight $h$ consisting of the highest weight state $|h\rangle$ and the descendants $L_{-1}|h\rangle$, $L_{-1}L_{-1}|h\rangle$, $\dots$ However, there are many other descendants, such as $L_{-2}|h\rangle$, $L_{-3}|h\rangle$, etc. What $sl(2)$ modules will these states sit in? In particular, is the full $sl(2)$ representation completely reducible, so that it can be written as a sum of irreducible $sl(2)$ representations? Can anything general be said about the multiplicity of the $sl(2)$ representations appearing in the decomposition?


title: How would we see a near-lightspeed object emitting light?

text: Consider an object travelling near the speed of light relative to us (let it be a spaceship or a star), which is emitting light (consider it monochromatic resulting from a two level electronic transition). Regarding the different relative time in the object itself (since it's travelling at high speeds) and the time of an observer (say in Earth), what would happen to the number of photons that we would see emitted by that object, i.e. **photons/second** , as it increases speed ? I believe that we would see that the object would be emitting less and less photons (per unit time) as its speed increases because the **time for radiative recombination** , i.e. the time **required** to produce a photon, would pass slower. Is this true? _**If yes:_** It is possible that we may fail to detect matter when we look to the outer space and think that nothing is there? Relating the fact that no radiation detectable from that point and that the relativistic mass is especially high, could it, or why cannot, this be related to dark matter? **Note:** I am NOT referring to change of energy caused by red-shift, I am referring to change of the number of photons per unit time!


title: How thick does steel have to be to be able to withstand 300 bar (sphere)

text: How thick does the material in a sphere have to be to withstand the (inner)pressure of 300 bar if the material is steel? (With an inner-radius of 2cm) Atmosphere pressure = same as 0 meter above sea level Is there some "golden rule" like sqrt(bar)cm thickness times stress-constant?


title: Water, can it break through temperature?

text: If water is heaten up to ridicilously high temperatures, is it possible for the atoms in the molecules to lose their bonds? And if it is possible, isn't this some kind of chain-reaction?   Like you heat up water --> steam, keep heating it, then wamm you get 1 oxygen and 2 hydrogen atoms, and at that temperature they most likely will self combust creating more heat to make more steam turn into oxygen & hydrogen etc. Extra question: If you got 1 mole oxygen and 2 mole hydrogen in a container, will it turn into 1 mole water? Or does the pressure matter? Or does the temperature matter?


title: Writing Dirac Mass Term For Massive Neutrinos

text: How does one write down the following Dirac mass term for a collection of "massive" neutrinos? \begin{equation} -[\overline{(\psi_R)}M_D\psi_L+\overline{(\psi_L})M^\dagger_D\psi_R] \end{equation} I can write down the mass term as $-\bar\psi M \psi$, where M is a diagonal matrix with mass eigenvalues, $\psi=(\psi_1,...,\psi_n)^T$ is a vector in flavor space and defining adjoint as $\psi=(\bar \psi_1,...,\bar\psi_n)$ and any $\psi_i=\begin{pmatrix} \psi^i_L\\\ \psi^i_R \end{pmatrix}$. My confusion is, with these I can show the above expression except I get only $M_D$ and not $M_D^\dagger$, but in the literature of neutrino mass people write it this way.


title: Does any material glow, under appropriate conditions?

text: Given a high enough temperature, or a high enough electrical field, can we make every material emit visible light?


title: QED Vertex Factor/Rule

text: On page 303 in Peskin&Schroeder they give the vertex factor as $$V = -ie\gamma^\mu \int d^4x$$ ![enter image description here](http://i.stack.imgur.com/sdI3O.jpg) while on page 304 they write $$V_\times = -ie\gamma^\mu\int d^4x A_\mu(x).$$ ![enter image description here](http://i.stack.imgur.com/U3cWY.jpg) Why are the diagrams pictorially different (one of the have the funny cross at the end of the photon line)? Why do the gauge field appear in $V_\times$ and not in $V$? Are $V$ and $V_\times$ identical?


title: Rotating reference frames

text: I'm trying to understand the equations that govern velocity in a rotating reference frame... \begin{equation} v_i = (\frac{dr}{dt})_r + \Omega \times r . \end{equation} I'd like to build a simple simulation of a rocket taking off from earth with some constant inertial velocity, say: $v_i=[1,0,0]^T$. I assume some $\Omega$ value to represent the rotation of the earth about z, say $\Omega=[0,0,1]$. And then solve for the percieved velocity in the rotating frame: \begin{equation} (\frac{dr}{dt})_r = v_i - \Omega \times r . \end{equation} What I expect, after integrating velocity into position, would be an outwardly rotating spiral showing the relative position of the "rocket" to an observer in the rotating earth frame. What I see, from a simple simulink sim, is quite different. My sim: ![enter image description here](http://i.stack.imgur.com/G9NuJ.png) The output: ![enter image description here](http://i.stack.imgur.com/WA74E.png) Thoughts?


title: Can an extremely small force lift a mountain in a hydraulic system? If not, then why not?

text: ![enter image description here](http://i.stack.imgur.com/xTugm.jpg) As you see in the picture, a car can be lifted with a small force F1. Here it is shown that 'F1 . A1 = F2 . A2'. What will happen if A1 is of the size of a pin's cross-sectional area. Can a child lift a truck in such a way, using a safety pin? If not, and I feel not, then what is it that will prohibit this from happening? Will the metal structure break? Assume that the child does not hurts itself with the pin as it pushes it...


title: Minimal vs. Non-minimal coupling

text: What is the difference between Minimal vs. Non-minimal coupling in General Relativity? A brief introduction to Minimal Coupling in General Relativity could be useful too.


title: Contradictory boundary conditions in electrostatics problem?

text: Consider the following problem: A conducting cube of side $a$ is grounded. Inside there's a horizontal (i.e., perependicular to the $z$ axis) sheet with uniform surface charge density $\sigma$. The objective is to find the electrostatic potential $\phi$ inside of the cube, using Laplace's equation $\nabla^2 \phi = 0$. I'm not going to ask about the math since that's not the problem. My issue is that I have a conceptual problem with the boundary conditions. First, the potential should be zero at the faces of the cube (where $x$, $y$, $z$ are each $0$ or $a$). Second, there should be a jump discontinuity in the $z$ component of the electric field $E_z = - \partial_z \phi$ proportional to $\sigma$. Now suppose we stand on the charged sheet, very close to the edge of the cube. Since the sides are equipotential surfaces, the electric field will be perpendicular to them. This means that $E_z$ will be zero on both sides of the sheet, which contradicts the fact that there should be a jump of $\sigma$. Does my reasoning make sense? I can back it up with math if needed; the general idea is that we can write $\phi = \sum_{n,m} \sin(\frac{n\pi}{a}x) \sin(\frac{m\pi}{a}y) f(z)$, but there's no way to write $\sigma$ as a sum of sines, since any such sum will necessarily be zero at $x,y = 0, a$. _Edit:_ The previous paragraph is wrong (nonuniform convergence of series!), but the physical side of question still stands.


title: Variational derivatives of strongly connected diagrams functional in gauge theory

text: # Background In Jorge C. Romao's "Advanced Quantum Field Theory", at the end of page 218, Eq (6.266) reads: $$\tag{1} \left.\frac{\delta^{2}}{\delta \omega^{b}(y)\delta A_{\mu}^{c}(z)}\left[ \frac{\delta \Gamma}{\delta K_{\mu}^{a}(x)}\frac{\delta \Gamma}{\delta A^{\mu , a}(x)} \right]\right|_{\text{sources} = 0} = \left.\frac{\delta^{2} \Gamma}{\delta \omega^{b}(y)\delta K_{\mu}^{a}(x)}\frac{\delta^{2} \Gamma}{\delta A^{c}_{\nu}(z)\delta A^{\mu,a}(x)}\right|_{\text{sources} = 0}. $$ Here $$\tag{2} S = \int \mathrm{d}^{4}x\left( -\frac{1}{4}F_{\mu \nu}^{a}F^{\mu \nu}_{a} - B_{a}f^{a} + \frac{1}{2 \varepsilon}B_{a}B^{a} - \bar{\omega}^{a}M_{ab}\omega^{b} + K_{\mu}^{a}\delta A_{a}^{\mu} + L_{a}\delta \omega^{a} \right), $$ where:   * $B_{a}$ is support scalar field which is needed for BRST transformations   * $f_{a}$ is the gauge condition for $A_{\mu}^{a}$ fields   * The last two terms in (2) are BRST-invariant terms $(K_{\mu}^{a}\delta A_{a}^{\mu} + L_{a}\delta \omega^{a})$ added to the "usual" action for the convenience of work with generating functional $Z[J]$ (see transition from (6.238) to (6.244) through (6.239) and (6.243)); $K_{\mu}^{a}, K_{i}$ are sources for BRST-transformed fields $\delta A_{\mu}^{a}$ and $\delta \omega_{a}$   * $\delta \varphi$ means BRST-transformation of $\varphi$ We have the partition functional $$\tag{3} Z[J] = \int D(A, \omega, \dots)e^{i(S + \int \mathrm{d}^{4}x (J_{\mu}^{a}A^{\mu}_{a} + \bar{\eta}^{a}\omega_{a} + \bar{\omega}^{a}\eta_{a}))}; $$ and generating functional for strongly connected diagrams $$\tag{4} \Gamma = W - \int \mathrm{d}^{4}x(J_{\mu}^{a}A^{\mu}_{a} + \bar{\eta}^{a}\omega_{a} + \bar{\omega}^{a}\eta_{a}). $$ The author writes this equation in a chapter about derivation of transverseness of full gauge boson propagator by using Slavnov-Taylor identity. # Question I don't understand why in $(1)$ cubic functional derivative terms like $[\delta^{3}\Gamma/\delta(\cdots)]$ are zero, and why the terms $$\tag{5}\frac{\delta^{2}\Gamma}{\delta A_{\nu}^{c}\delta K_{\mu}^{a}}\frac{\delta^{2}\Gamma}{\delta \omega^{b}\delta A^{\mu , a}}=0.$$ **Why do they vanish?**


title: If you dig a deep tunnel, will the rock sublimate?

text: If a tunnel is dug deep inside the crust (but before reaching places where the rock is liquid), how will the enormous downwards pressure manifest itself? Will the difference in pressure $(\rho_{rock}-\rho_{air})gh$ cause sublimation of the rock, in order to reach thermal equilibrium?


title: Place each foot on a scale: can you add the two to find your weight?

text: I frequent a blog from a British psychologist, and every Friday he likes to pose an interesting puzzle or riddle. The Monday after that he posts the answer. They're good fun, and IANAP but this week's answer made my it-might- not-be-quite-as-simple-as-that detector go off. My question boils down to this: let's say I have two identical scales, and I stand on the scales with one foot on each scale. The scales read W1 and W2. Does my weight equal W1 + W2?


title: Anomaly cancellation and fermion number violation

text: In the standard model, an axial $SU(3)$ currents has anomaly which after quantization leads to the fermion number violation. However, taking all the fermions into account we note that the anomalies cancel. Does the cancellation imply that the net fermion number violation is zero?


title: In an avalanche breakdown, where are the electrons that break free from?

text: In an avalanche breakdown, are the electrons that break free under the influence of the applied electric field from the depletion region or outside it? Also, under reverse bias, how exactly is the internal potential difference widened? (causal mechanism, microscopically) I've always seen it stated but never explained.


title: Higgs Boson revealed video

text: I'm newbie here, no background in physics, just trying to understand a bit about the Higgs Boson. I guess the key moment of this video (https://www.youtube.com/watch?v=RC- apKhWNDE) is when Fabiola Gianotti, Ph.D. shows the plots about the combined results (1:43-1:54). where does the graphics came from? what is the thick and thin line? which units are used for the axes? Thanks a lot.


title: QCD Parton Branching - Is the angle always negligible?

text: The GLAP equation describes the evolution of parton distribution under successive branchings. When branching takes place at a finite angle, the outgoing parton carries some finite transverse momentum. However, this is ignored in deep inelastic scattering calculations, where the parton momentum is always assumed to be longitudinal. I have trouble understanding why the angle should be ignored. The GLAP equation effectively comes from resummation of collinear branching to all orders in perturbation theory. So you can imagine 100 branchings, each at angle of $\pi/200$, resulting in an outgoing parton at an angle of $\pi/2$, i.e. perpendicular, to the velocity of the incoming nucleon. Why is this not a problem?


title: Use kirchhoff's second law to deduce the resistance R of the resistor

text: ![](http://i.stack.imgur.com/zqxcb.jpg) Using what I learned so far, i constructed an equation like this: $$30V=(I_{1}\times 20\Omega)+10V+(0.2A\times 10\Omega)+(I _{3}\times 10\Omega)+(0.5A\times R)$$ Which is clearly not solvable on its own. What am I missing from the picture?


title: Relationship between temperature and energy

text: What is the definition of temperature in relation to energy? I'm mostly interested in general dimensional terms. Is temperature the kinetic energy per mass? Or kinetic energy per volume?


title: Mass-Energy relation

text: Einstein mass- energy relation states $E=mc^2$. It means if energy of a paricle increases then mass also increases or vice-versa. My question is that what is the actual meaning of the statement "mass increases"? Is really the mass of the particle increasing or what?


title: Lorentz model and energy exchange

text: The Lorentz model, describing the electron of the atom as an harmonic oscillator forced by an oscillating electric field $\vec{E}$, shows that the dipole moment $\vec{D}$ obeys the following equation in the stationary regime: $$\vec{D}=\frac{q^2}{2m\omega_0}\frac{\vec{E}}{\omega_0-\omega-i\gamma_d}$$ where $\omega_0$ and $m$ are the natural frequency and mass of the electron, $\gamma_d$ is the dissipation introduced by hand to take account of the energy emitted by the electron when accelerated (and also its collisions). My trouble is that one finds that the energy transmitted by the electron to the field by unit of time, given by: $$P=-\vec{E}.\frac{d\vec{D}}{dt}$$ is always negative, implying that the electron cannot "feed" the field, however how much it is radiating. Can it be attributed to the fact that the electron energy is in first place coming from the field, and thus, our electron cannot give more than it receives? In what cases does the electron sources the background field?


title: Finding the acceleration of Block attached using tricky string setup

text: Below shown is a setup, and block B starts from rest and moves towards right with a constant acceleration. Does the acceleration differ for the blocks ? I am a bit confused because of the tricky string setup. If after time t, the velocity of A with respect to B becomes v, then what would be the acceleration of A ? My understanding is : After time t, Block B will be brought to rest instantaneously resulting in non-zero relative velocity which otherwise will continue to be 0. ![Block String Setup](http://i.stack.imgur.com/ErLPK.png)


title: How wide does a wall of ice need to be to stay in place?

text: Let us say that we have unlimited manpower to construct a huge wall of water ice e.g. 200 m tall (700 feet). -and that the wall is placed in a climate, where the temperature never (for your purpose) goes above freezing point. The wall is to have fairly steep sides, so let us assume a perfect rectangular cross section. A very narrow wall would obviously shatter at the bottom. A very wide wall would approach a natural ice cliff side and thereby be possible.   * How wide does the wall need to be?   * What properties of ice is relevant to calculate that?


title: Has BCS Cooper pair condensate been observed in experiment?

text: Feshbach resonance in s-wave scattering states a BCS Cooper pair condensation at B-field just above the resonance where the scattering length a <0\. Just wondering if the condensation has been observed in an experiment in any group? I was looking at a notes written in 2005 which claimed that this observation was the holy grail, and I seem not to be able to find a paper claiming that particular observation in experiment.. Many thanks :) Simon


title: Fresco in his "Future By Design" introduces drawing of underwater vessel and its front system of generating air bubbles. Is idea energy efficient?

text: I recently saw Fresco's Future By Design and noticed something miniature to investigate. My notice regards about one of his illustration he describes in the documentary. Here is the link on youtube, timecode added where it starts with stuff that cought my eye. http://www.youtube.com/watch?feature=player_detailpage&v=5zn8MRKOskw#t=2812s He introduces front system of generating air bubbles in one of his underwater vessel. It's usage would be for reducing resistance while the vessel is moving thru water. I recon he invested some time to devise this illustration and this system of air bubbles fits his high tech vision perfectly and my intention with this question is to kinda have his work evaluated xD Now my question: can this system be constructed to be energy efficient? i'am no engineer but i have an opinion that in reality, its invested/returned energy ratio has to be high and therefore ridiculous to be used.


title: An electron has no known internal structure, does that imply it has an unknown one?

text: I'm currently reading Alonso and Finn's Electromagnetism book. It explains that the spin contributes to the magnetic moment and is somewhat comparable to a rotation of the particle around its own axis. It says that the spin of a particle is caused by a certain internal structure, which makes sense in the aforementioned analogy. Right underneath the paragraph with the explanation of spin, it says "The electron has no known internal structure", but since it does have a spin, does that mean that we know the electron has an internal structure but we just don't know what it is?


title: Calculate TEC efficiency

text: I am currently doing a small experiment using a Peltier to cool 250ml of water. My aim is to achieve the cooling of 250ml from 23degC to 8degC in under 20minuets which I have done by strapping the Peltier to the underside of an aluminium cup and a large air cooled heat sink and fan to the hot side of the Peltier. The Peltier was consuming 4A at 6V The heatsink reached 31degC (average from top to bottom) I am now looking at the theory behind the efficiency of it. I have read that (currently) peltiers can only achieve a maximum efficiency of 15% to me that means for every watt of power in you get 0.15watts cooling power, the rest is wasted in heat and work. When I researched how they got to that figure it came up with co-efficiency of performance (COP). I took the figures from my experiment and applied the COP formula and got somewhere around 0.47COP as COP is a ratio that means I should be getting somewhere close to 50% efficiency. Can someone explain this to me please?? I would also like to rerun the experiment after tweaking some values to find its optimum performance, but I need to address the efficiency thing first.


title: Conformal maps and flow around corners

text: The complex potential $\Psi = \psi + i \phi$ ($\psi$ is the velocity potential, $\phi$ the stream function) is an analytical function for an irrotational inviscid flow, since its constituents in this case satisfy Laplace's equation. However, say I look at flow around a corner with some angle $\alpha$. How does the conformal map enter into all of this? Why is it useful?


title: Divergent path integral

text: What does it mean to have a divergent path integral in a QFT? More specifically, if $$\int e^{i S[\phi]/\hbar} D\phi (t)=\infty $$ What does this mean for the QFT of the field $\phi $? The field $\phi$ has action $$S[\phi]=\int\left(\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-V(\phi)\right)\mbox{d}vol$$ where we use Minkowski signature $(+,-,-,-)$.


title: Equal hourglass levels blindly

text: Is it possible to equal the sand level in both bulbs of an hourglass without looking into it? For example, if one bulb contains more sand than the other one, and more sand in would mean it flows faster, if the bulb is at top, we may just turn the hourglass over and over quite fast, until the both bulbs are balanced. But I guess that would not work, as some people mentioned the sand would run at quite equal speed independent of the amount of sand above it. Another idea would be to knot a string to the hourglass center to check the balance and turn it by hand to adjust the sand levels. However, as the sand does not rest in defined position (as water would do), it seems impossible to check for equal sand amount by checking the horuglass balance. Any further ideas?


title: Angular momentum eigenstates

text: My textbook says that if $L^2$ is the square of the angular momentum and if it's eigenstate is $|\alpha,\beta>$ then its eigenvalue is $\hbar^2\alpha$ i.e. $$L^2|\alpha,\beta>=\hbar^2\alpha|\alpha,\beta>$$ It then goes on to talk about the $z$ component of the angular momentum ($L_z$) saying that $$L_z|\alpha,\beta>=\hbar\beta|\alpha,\beta>$$ i.e. that $\hbar\beta$ is the eigenvalue of $L_z$. Here is where I need help later on it says that $\beta_{max}=l$ i.e. the greatest value $\beta$ can take is $l$ and that $\beta=m$ hence: $$L^2|l,m>=\hbar^2l(l-1)|l,m>$$ and that $$L_z|l,m> =\hbar m|l,m>$$ Firstly why is the $\alpha$ in the ket not the same as that outside but the $\beta$ is? and secondly why if $\alpha$ is bigger then or equal to $\beta^2$ (as my book says) does $\alpha := \sqrt{\beta_{max}}=\sqrt{l}$. Edit: Please look at my comment in neuneck's question


title: Photon emission and absorption by atomic electrons

text: Assume a photon is produced by an atomic electron making a transition down from a certain energy level to another. Can that photon only be absorbed by another atomic electron making _exactly_ the opposite transition? Is there any chance that the photon could be absorbed by an atomic electron undergoing a transition with a slightly different energy difference?


title: Energy difference between symmetric and antisymmetric wavefunctions

text: Is there any energy difference between a particle in a symmetric wavefunction and an identical particle in an identical potential but in a state with an anti-symmetric wavefunction? Or is it case-dependent? The full problem statement (from this assignment) is ![problem](http://i.stack.imgur.com/QAzs5.png)


title: Do atomic nuclei have fixed orientation and position with respect to the electron cloud?

text: Does the nucleus of an atom of any element (besides protium) maintain a particular orientation in relation to its electron cloud? Does it bounce around within its cloud? Can it be ejected from its cloud outright by the impact of a high-energy particle or sudden entry into a strong negative electric field?


title: Why do people say "Bosons are either gapped or condensated, except physical principle protected cases (Goldstone boson, photon)."?

text: Bosons are either gapped or condensated, except physical principle protected cases (Goldstone boson, photon, etc.). I read this in a paper (version1 of http://arxiv.org/abs/1404.3728v1, 1st page 1st column) and in other lecture slides (page 6 of http://discor2013.files.wordpress.com/2013/12/lee.pdf). This slide appears to talk about some mean field result of spin liquid state. This statement is not really pertinent to the mainstream of that paper. But it sounds rather intriguing and necessary to be further clarified. So could anyone make it clearer? Thanks!


title: Is Weyl invariance absolutely necessary for string worldsheets?

text: The Polyakov action for a string worldsheet has Weyl invariance. In the conformal gauge augmented with Weyl gauge-fixing, we can always impose a flat worldsheet metric in Minkowski coordinates. The residual gauge symmetries take on the form of conformal Virasoro and anti-Virasoro algebras. This is equivalent to a two dimensional conformal field theory. Later, we impose the Virasoro constraints. However, can't we come up with more general worldsheet actions which are invariant under diffeomorphisms but not Weyl transformations? The conformal gauge is still possible, but the worldsheet volume factor is dynamical. Residual diffeomorphisms taking on the same form as conformal transformations, minus the compensating Weyl transformation, still exist. They still take on the form of Virasoro algebras and Virasoro constraints. This is no longer a conformal field theory because we have a characteristic length scale on the worldsheet. Some of these theories are also modular invariant. Do they describe valid string theories?


title: $G$-parity in an electromagnetic decay

text: I am looking at the decay $\eta\rightarrow\pi^+\pi^-\gamma$ and I would assume that the decay itself (ignoring the $\pi\pi$ final state interaction that is obviously strong) is electromagnetic since there is a photon involved. In a paper by Stollenwerk et al. on this decay they write that the partial wave of the pion pair is dominated by $1^{--}$. I assume this means that only odd partial waves contribute and that the F wave is already suppressed enough to be neglected. I can only explain this via $G$-parity conservation, which would yield a photon isospin of $1$ and consequently $I_{\pi\pi}=1$, but for an apparently electromagnetic process I cannot see why that should apply. So my actual questions: a) Is there any way a process involving a photon can be strong? b) Are there types of electromagnetic processes for which $G$-parity is conserved?


title: Is it necessary for EM fields to be dependent & co-exist in static conditions?

text: I was having a discussion today with one my colleagues in the lab about the independence and co-existence of EM fields.$$$$ **My argument:**   1. In time-varying fields: EM fields are necessary dependent, and they have to co-exist. They get coupled through the time derivatives that appears in Maxwell's equations.   2. In static fields: EM fields are independent and each can exist separately or together.  $$$$ **His argument:**   1. In time-varying fields: (SIMILAR TO MINE)   2. In static fields: EM fields must co-exist. (He is mentioning the spin of the electron about its axis of rotation, and the rotation of the electron around the nucleus and how that induces a magnetic field.)$$$$ **Examples to clarify my question:**   * A point charge at rest (for simplicity), have a static E-field around it pointing in the radial direction. (is there a magnetic field?)   * A DC current flowing in a conductor produces an M-field around the conductor. (is there an electric field?)


title: What are the modern marvels in Physics that seem crazy but true?

text: What are the modern marvels in Physics which are crazy but true? Ideas which are ridiculed and dismissed in the beginning but passed the test of time?


title: Why do 2.4GHz frequencies offer greater range than 5GHz routers?

text: I would've thought that as 5GHz is a higher frequency, and it carries more energy, it would be able to pass through walls much more easily compared to a 2.4GHz frequency- similar to how short frequencies cannot pass through the atmosphere (and be received by satellites) but high frequencies are able to "penetrate" through due to their higher energy.


title: Emmy Noether's theorem in simpler terms

text: I'd like to understand Noether's theorem and its contents as to what it implies in a bit simpler terms. I am familiar with mathematics unto Calculus 1,2,3 and some linear algebra and group theory. I am familiar with that for each symmetry there is a conservation law.   1. Now is there a one-one correspondence between these two always?   2. What about laws like conservation of charge?    3. And conservation of energy?    4. And conservation of mass?    5. Considering our view of nature was changed as Special Relativity was introduced and thus, what earlier the theorem would have asserted as mass and energy to be originating from different symmetries then how did we reconcile these two as just one symmetry?


title: How to transfer mechanical power from the inside of a vacuum chamber to the outside while maintaining a seal?

text: In a vacuum chamber how would one transfer mechanical power (either rotation or linear) from inside to the external environment? I'm working on an idea for a new/different type of motor that would require an evacuated internal atmosphere and am wondering how to transfer the generated motion outside the case. I have thought about using rotating magnetic coupling on either side of a thinned wall section but don't think it would scale well for larger versions of the motor (perhaps powering a bike or car). I expect with a high enough budget (which I don't have) that an extremely high-tolerance mechanical seal could be machined but am wondering if there are any solutions that could sidestep the high tolerances by thinking laterally.


title: Momentum in Free Scalar Field

text: I have a few issues with making the transition between these: $\phi(x)=\int{\frac{d^3p}{2\pi^3}\frac{1}{\sqrt{2\omega_{\vec{p}}}}(a_{\vec{p}}e^{i \vec{p} \vec{x}}+ a^{\dagger}_{\vec{p}}e^{-i \vec{p} \vec{x}}})$ $\pi(x)=\int{\frac{d^3p}{2\pi^3}(-i)\sqrt{\frac{\omega_{\vec{p}}}{2}}(a_{\vec{p}}e^{i \vec{p} \vec{x}}- a^{\dagger}_{\vec{p}}e^{-i \vec{p} \vec{x}}})$. So from the Lagrangian of the real scalar field, we get $\pi(x)=\partial_0 \phi(x)$ which just means differentiating our $\phi(x)$ w.r.t time. But where does the $-i$ come from (as in the minus sign)? Also aren't the exponentials independent of time as $\vec{p} \vec{x}$ isn't a four vector? So now I'm wondering where the $\omega_{\vec{p}}$ has come from as well!


title: Ignore terms in the equation of motion

text: While working on a problem relative to particles motion, I ended up with the following second order nonlinear differential equation: $$\tag{1} \ddot{y}(t)+\frac{y(t)}{b^2}\left(a-\frac{\dot{y}(t)^2}{2}\right)=0,$$ where $[a]=m^2~s^{-2}$, $[b]=m$, $y$ is the particle's position, such that, $[y]=m$ and, consequently, $[\dot{y}]=m~s^{-1}$. In general I can't solve this equation so I thought that I could solve for a simplified case: when the quantity $a$ is much bigger than the particle's velocity, throughout the particle's motion. Such that Eq.$(1)$ is given by $$\ddot{y}(t)+\frac{a~y(t)}{b^2}=0,$$ the harmonic oscillator equation. My question is: can I do this? If not, do I need to scale Eq.$(1)$ first? I ask the last question because of what is done to get the Reynolds number for the Navier-Stokes equations...


title: How to calculate the partial entropy in a fully connected ising system

text: I'm trying to reproduce a calculation that should lead to the partial entropy in a fully connected ising model for the high-temperature range ($\beta < 2$) in the thermodynamic Limit ($N \rightarrow \infty$). Although this is a very tempting goal -- the partial entropy would lead to an analytic expression for the mutual information -- I'm still failing and after a very frustrating week my last hope is to find help this way. The model is described by the hamiltonian $H=-\frac{S^2}{N}$. Here $S:=\sum_{i=1}^N s_i$ is the sum of all spins and $N$ is the number of spins in the model. The partial entropy then is $H_A = - \sum _{p_A=0}^{N_A} \binom{N_A}{p_A} R(p_A) \log_2 R(p_A)$ with $R(p_A):= \sum_{p_B=0}^{N_B} \binom{N_B}{p_B} \frac{\exp \left[ N \beta \left( \frac{p_A}{N} + \frac{p_B}{N} - \frac{1}{2} \right)^2 \right]}{Z}$ where $N_A$ and $N_B$ are the number of spins the subsystems $A$ and $B$ respectively and $p_i$ are the number of spins of the subsytsem $i$ pointing in the $-x$ direction. What i'm failing now is to reproduce that this leads to $H_A = \tau N - \frac{1}{2 \log 2} \left\\{ \frac{\beta \tau}{2 - \beta} + \log \left[ \frac{2-\beta}{2-\beta(1-\tau)} \right] \right\\} + O(1/N), \quad N \rightarrow \infty,\ \beta < 2, \tau := \frac{N_A}{N}$. My last attempt was to apply the replica trick to get rid of the $\log_2$ in $H_A$ and then use some kind of saddle-point method for the resulting $n+1$-fold Integral to solve it in the thermodynamic limit. This looked promising on the first sight but i can't find a closed analytic expression for the determinant of the Hesse-Matrix. The analytic expression is formula #26 in "J. Wilms, J. Vidal, F. Verstraete, and S. Dusuel. Finite-temperature mutual information in a simple phase transition. _Journal of Statistical Mechanics: Theory and Experiment_ , 2012:P01023, 2012."


title: Pauli's Exclusion Principle

text: Can someone tell me how Pauli's Exclusion Principle gives stability to matter? I know two electrons cannot occupy the same energy state so that is why we cannot squeeze bulk matter after a limit and this principle is responsible for keeping us from collapsing into the ground while standing.But my question is if that is so we should not be able to dive into water but we can. Why? And one more cant electrostatic repulsion be responsible for the same? Owing to the fact that we are made made up of electrons and so is everything surrounding us so that could be one major possibility no? Is Pauli's Exclusion Principle necessary for explaining that?


title: Cheating to communicate beyond the horizons in expanding universe?

text: Let's say the Hubble constant is constant in the time interval we consider. So subjective "horizon" for any object has the same radius R. Let's say Bob and Charlie are just within the horizon of Alice, but on opposite sides: vectors AB and AC are exactly opposite. And then Bob accelerates sufficiently close to the speed of light, so as to keep the distance to Alice constant with time. Charlie does the same. Now Bob and Charlie stay within Alice's horizon for as long as they want. Now, Bob can send radio messages to Alice (He's withing her horizon, right?). But then Alice can receive from Bob and forward them to Charlie, effectively acting as a relay. So, this way Bob can communicate with Charlie, and this channel is a two-way. So, this way Bob beyond Charlie's horizon, and Charlie is beyond Bob's horizon. But there is a two-way communication possible between them. Does this actually work? And, if it does, how does "horizon radius" depend on the relative velocities of objects?


title: Rolling resistance

text: Today, my physics teacher asked the following question (and gave it as homework): > How can a moving body deaccelerate both in means of center of mass and > rotationally? I quickly answered that this happens due to rolling resistance and I gave as an example a car climbing a sand hill. Was I right? I would like to know! My (draft) proof: In the intersection points of the 2 surfaces the 2 surfaces don't have the same gradients ($\nabla S_p(x_i,y_i,z_i) \not= \nabla S_b(x_i,y_i,z_i)$) so there is a force applied to the solid by the plane wich can be analysed in two constituent forces then the one constituent force (Fy) parallel to the weight of the solid has the same magnitude with it so the total torque is: $$\sum \tau = \|\vec w \times \vec \ell_1 \| + \|\vec F_y \times \vec \ell_2\| = w \cdot \mathrm d(\vec w, \vec F_y) < 0$$ and the center of mass deaccelerates as well: $$\vec a = \frac{\sum \vec F}{m} = m^{-1} (\vec w_x + \vec F_x) \Rightarrow a < 0 $$ (as algebraic value) PS Fx is the other constituent force


title: Application of diffraction problem!

text: Here is a problem that I am working on, which is the applying the concepts of diffraction to the setting of the sun: > Air has a small, usually negligible index of refraction. It is 1.0002926. > This causes the Sun to actually be below the horizon when it appears to be > just on the verge of sinking below it. > > Suppose you are on the sea-shore watching the Sun apparently sinking into > the ocean. When only its upper tip is still visible, by what fraction of the > diameter of the Sun is that tip actually already below the surface? > > As an approximation, take the earth's atmosphere as being of uniform density > out to a thickness of 8.600 km, beyond which there is no atmosphere. This > means that, with the Earth's radius being 6400. km, your line of sight due > West along the ocean surface to the horizon will intersect this "upper > surface" of the atmosphere at about 331.9 km from your eye. (The diameter of > the sun subtends 0.5000 degrees at your eye). This is how I've attempted to model the problem, but I'm not sure if it's correct... ![problem](http://i.stack.imgur.com/mOsYQ.png) I'm not really sure how to approach this problem.... Can anyone please help me?


title: Velocity of real gas molecules?

text: It is known that the velocity of ideal gas molecules can be computed using Maxwell-Boltzmann law of distribution of molecular velocities, with average velocity given as: $\overline{v}=\sqrt{\frac{8RT}{\pi M}}$ What would be the corresponding average velocity for a real gas (which are not at STP)?


title: About the microscopic form of magnetocrystalline anisotropy

text: Currently people write magnetocrystalline anisotropy as $H_{an}=-K s_x^2$ from its classical counterpart: $H_{an}=-K ( \sin \theta)^2$ where $K$ is the anisotropy constant, but for spin 1/2, $s_x^2$ is just the identity matrix, which shows no anisotropy, so how to write the correct form of microscopic Hamiltonian? I also heard that magnetocrystalline anisotropy comes from spin-orbit interaction, is there some paper depict there relations? like deriving the Heisenberg Hamiltonian $S_i \cdot {S_j}$ from Coulomb interaction by Dirac?


title: What's the difference between two Hydrogen atoms?

text: If we are given two Hydrogen atoms, would the only difference between them would be their quantum state (Energy level or eigen value, and the corresponding Orbital or eigen state) and their location (say you are the origin and each of the Hydrogen atoms are located an arm's distance apart) This leads to my next question which would be what's the difference between two Hydrogen atoms separated arm's distance and a Hydrogen molecule? Wouldn't the only difference be that they are located very close to each other?


title: Sufficient conditions for the energy to be not conserved?

text: I'm almost embarrased to ask this question because I thought I was by now very confident with classical mechanics. Someone has stated that given a mechanical system with a Lagrangian $L$ s.t. $\frac{\partial L}{\partial t}=0$ where the kinetic energy $T$ is NOT homogeneous quadratic in the generalised velocities, one cannot infer that the total energy $E=T+V$ is NOT conserved. However, I think this is already enough to show that $\dot{E} \neq 0$. Let's assume that $L$ looks as follows: $L = \frac{1}{2}(\dot{q}^2 + 2 \dot{q}f(q))-V(q)$. Then, after plugging in the equation of motion $\ddot{q} = -V^{\prime}$, I obtain \begin{equation} \dot{E} = \dot{q} [\dot{q}f^{\prime}(q) - V^{\prime}(q)f(q)]. \end{equation} I can't see how one can make this vanish. To me it is clear that scleronomic constrains imply that $T$ is homogeneous quadratic in $\dot{q}$. Then, one has of course energy conservation. But, does $T$ being not homogeneous quadratic in $\dot{q}$ also imply that there is a rheonomic constraint? (Because then it is also physically clear, why $E$ is not conserved.) I'd be grateful for answers!


I have a quadratic relation inside of a Dirac delta function with the following relation \begin{align} \delta((x-x_1)(x-x_2)) = \dfrac{ \delta(x-x_1) + \delta(x-x_2) }{|x_1-x_2|}. \end{align} How do you handle a functional input in a Dirac delta function and prove these types of relations?


title: Why hasn't warm dark matter replaced cold dark matter as the standard model of cosmology?

text: The $\Lambda\rm CDM$ (cold dark matter with cosmological constant) is the current standard model of cosmology because the model comes with a long list of phenomena successfully explained by it. However, there are a remaining handful of problems which are not yet resolved in the context of $\Lambda\rm CDM$, for instance (with links to my picks for reasonably accessible and/or up to date technical papers):   * The too big to fail problem   * The cusp-core problem   * The missing satellites problem   * The disk of satellites problem$^{1}$ These (perceived? it is as yet unclear whether these issues can be resolved within the $\Lambda\rm CDM$ framework) problems have prompted many research groups to look at alternate theories. Some examine modifications or alternatives to general relativity - since much of the evidence for the existence of dark matter assumes GR, an alternate theory of gravity might make DM obsolete. For the purposes of this question, I want to retain the assumption that GR is correct. The other approach is to question the cold part of CDM. There is evidence ruling out hot (i.e. relativistic) dark matter, but so called "warm" dark matter (WDM) is an area of active research. There's also been some buzz about self-interacting dark matter (SIDM, i.e. interaction couplings within the dark sector). There are a number of papers claiming solutions to the $\Lambda\rm CDM$ problems in the framework of WDM or SIDM, or rather more conservatively the magnitude of the problems can be at least alleviated with alternate dark matter models. However, I assume that the scientific community hasn't fully embraced WDM because it has trouble in other areas where CDM is just fine. **What is/are the observations that WDM/SIDM/other-alternate-DM have trouble explaining that prevent them from replacing CDM as the standard model of cosmology?** Or is one of these alternate models now competitive with CDM and we just need compelling evidence that WDM solves the remaining problems in CDM before re- writing the textbooks, so to speak? $^{1}$Sorry this last reference is paywalled - a version should show up on arXiv soon, but it's very recent and from Nature, so for now you need a subscription...


title: Which cosmological dark matter theory is most compatible with current particle physics?

text: This occurred to me while I was writing Why hasn't warm dark matter replaced cold dark matter as the standard model of cosmology? but I think this deserves a question of its own. The background info for this question is covered in that link, so I'll just cut straight to the point. Is one of the current cosmological dark matter models of interest (cold dark matter, warm dark matter, self-interacting cold/warm dark matter) easier to reconcile with our current understanding of particle physics and the current measured limits on possible undiscovered particles?


title: Poincaré group on quantum Klein-Gordon field (C*-algebraic scenario)

text: on the same topic as this question, I have been trying to fool around with the free real K-G field in flat spacetime on the C*-algebraic scenario (Haag- Kastler axioms, Weyl quantization, etc). Since I'm talking about the free (linear) Klein-Gordon field, the C*-algebra is taken as the CCR algebra generated by the unitary Weyl operators $W(f)$ (with $f$ a test function) which can be looked as the exponential of the field operators, $\exp\big( i\Phi(f)\big)$. The action of the Poincare group usually is given for the field operators (Borchers algebra), with something as $\alpha_{(\Lambda,a)}\Phi(x) = \Phi(\Lambda x + a)$, as operator-valued distributions. Now I imagine that you can transport that to Weyl operators, something as $\alpha_{(\Lambda,a)} W(f) = \exp\big( i [\alpha_{(\Lambda,a)}\Phi](f)\big)$. My questions are   1. Is the expression for $W(f)$ correct?    2. Does the action on the Weyl unitaries extend to a nice action on the CCR-algebra? By *-automorphisms? Is it inner or outer or what?   3. Where can I read about it? I could use a "for dummies" reference... [EDIT: fixed the notation, as sugestion of user1504]


title: Does the momentum operator commute with the Pauli matrix?

text: I tried to calculate the effect of spin orbit coupling $H_s=\alpha(\sigma_xp_y-\sigma_yp_x)$ as in the Rashba effect. But I just found out that it is not hermitian. Some paper propose some way by adding the hermitian conjugate to ensure the hermicity. However, when I tried to add a h.c. term, the Hamiltonian turned out to be 0. $H_s=\alpha(\sigma_xp_y-\sigma_yp_x)=\alpha\left( \begin{array}{cccc} 0&\frac{\hbar}{i}\frac{\partial}{\partial y}+\hbar\frac{\partial}{\partial x}\\\\\frac{\hbar}{i}\frac{\partial}{\partial y}-\hbar\frac{\partial}{\partial x}&0 \end{array}\right)$ This matrix is just anti-hermitian and will end up to be 0 after adding up its hermit conjugate. Am I doing this correctly?


title: Can I take heat from the air and convert it to electricity?

text: Its a summer day and the air in my house has been heated up. I could switch on my air conditioning, but then I'd be using energy from the grid in order to reduce the amount of energy in my house. What I'd much rather do is capture the heat energy from the air, so cooling it down, and turn that heat energy into electricity. (So I could store it in a battery or sell it to the grid or use some other means of disposing of it.) Is such an apperatus possible within our present understand of nature? Why or why not?   (I presume not, as people would be selling kits.) Just to pre-empt a possible answer: I've seen similar conversations in the past and "inefficient" has been given as the answer as to why not. I don't understand this as an objection, because the heat energy is effectively free. The sun rudely went and gave me an excessive amount of heat energy without being asked to. Is converting heat to electricity really so ineffiecient that I'd be better off running air conditioning?


title: Koopmann von Neumann (KvN) Theory

text: I was just wondering does anyone have any informative sources apart from the obvious wikipedia articles regarding Koopmann von Neumann (KvN) theory? Or if its possible could someone explain the basic premise? I just cannot find anything on the web at all!


title: Understanding Calculus Notation in Physics

text: I have just started a first-year calculus-based physics course about electromagnetism and waves. I am having trouble understanding what calculus notation means in the context of physics. Here is a paraphrased example from Fundamentals of Physics 9th ed. by Halliday et al. that considers finding the electric field at point P (refer to the figure below): Let $ds$ be the arc length of any "differential" element of the ring. $dq=\lambda~ds$ is the total charge of the ring, where $\lambda$ is linear charge density. The magnitude of the field is given by $$dE=\frac{1}{4\pi\epsilon_0}\frac{dq}{r^2}=\frac{1}{4\pi\epsilon_0}\frac{\lambda~ds}{r^2}$$ Since the x-components of the field are balanced at point P, after the application of geometry, the equation becomes $$dE\cos{\theta}=\frac{z\lambda}{4\pi\epsilon_0 (z^2+R^2)^{3/2}}ds$$ The author then puts an integral sign in front of both sides and integrates around the circumference of the ring: $$E=\int dE~\cos{\theta} = \frac{z\lambda}{4\pi\epsilon_0 (z^2+R^2)^{3/2}}\int_0^{2\pi R}ds =\frac{z\lambda(2\pi R)}{4\pi\epsilon_0 (z^2+R^2)^{3/2}}$$ In math, I was taught that Leibniz's notation ($dy/dx$) is not a fraction. After a little bit of research, I found that it sometimes _acts_ as a fraction, even though it is _not_ a fraction when defined with limits. Nevertheless, what should I think of when I see $dq$, $ds$, and $dE$? What does "differential" mean as used in the text above? So far it has made sense for me to think of $dq$ as an "infinitesimal" charge (for example), even though I was taught calculus using limits (I know very little about infinitesimals). Secondly, the way the integral is taken seems informal and confuses me. I understand that it represents the sum of the electric field caused by all the charge around the ring, but how can you just put an integral sign in front of both sides and take the integral with respect to whatever $dE$ or $ds$ that happens to be there? Also, the integral on the left side is indefinite whilst the one on the right is definite. How should I view this kind of material? ![figure](http://i.stack.imgur.com/SesBB.png) I am unsure if this question belongs on the math stack exchange or this one.


title: Venturi in nozzle of water jet

text: I am trying to understand a waterjet system for maneuvering boats from a physics perspective. (See Marine Holland Parts if you are interested.) The system uses venturis in the nozzles. My question is WHY? ![Venturi in nottle](http://i.stack.imgur.com/mRMce.jpg) I imagine it has something to do with increasing the speed of the flow ejected from the nozzle to increase thrust. If so, why use a venturi which narrows and then opens up again rather than just a narrowing towards the exit? The speed of flow at the exit would be greater.


title: Mathematical interpretation of Poisson Brackets

text: Lets say we are working in a classical scalar field theory and we have two functional $ F[\phi, \pi](x)$ and $G[\phi, \pi](x)$. In most of the references, starting with two functional the Poisson bracket is defined as $$\\{F(x),G(y)\\} = \int d^3z \left( \frac{\delta F(x)}{\delta \phi(z)}\frac{\delta G(y)}{\delta \pi(z)} - \frac{\delta F(x)}{\delta \pi(z)}\frac{\delta G(y)}{\delta \phi(z)}\right) . $$ But as explained here the functional derivative $\frac{\delta F}{\delta \phi} $ is a distribution rather than a function, so the previous definition does not make much sense. I was wondering then, if the Poisson bracket can be interpreted as the convolution calculated in $(x-y)$ (in the sense of distributions) between the functional derivatives. This works in case of interest such as $\\{\phi(x), \pi(y) \\}$ but I'm not sure it can be applied for two generic functional (the dependence $(x-y)$ is not explicit). Is there a proof that the Poisson bracket is a convolution? More in general, can field theories be formulated in a formal way in the sense of distributions?


title: Thermal superconductivity

text: I have heard several times of the concept of "thermal superconductivity" (As opposed to "electrical superconductivity"), but I am unclear on exactly what that could mean. It turns out to be _really_ hard to google, since everything comes up with thermal effects on electrical superconductivity (critical temperature and all that). So, given that electrical superconductivity is when electrical current can flow without energy loss, what exactly is the corresponding concept for thermal superconductivity? Given that flows of heat are, well, already composed entirely of _heat_ , what energy-loss mechanism is there that could be minimized? Or am I just completely missing the point?


title: Interchaning a position between two reference frames?

text: > $\vec{r}_a$ is a positional vector from reference frame $a$. What is the > position of same point from reference frame $b$ ? If required, assume > position of origin of frame $a$ is $\vec{m}$ and unit point (i.e. $\langle > 1,1,1\rangle_a$ ) is $\vec{n}$ from reference frame $b$. I am studying Kleppner and Kolenkow and this is the first thing I asked myself. Unexpectedly its taking a while to figure it out. So help needed.


title: why is a resistor frequency independent

text: I had a doubt that why is a resistor, frequency independent? Since, as frequency increases the movement of electrons increases so heat increases which causes change in resistance. So my question is why a resistor offers same resistance to every frequency. Any Clarification on this is appreciated. Thanks.


title: What is the reason (quantum mechanical) behind Laminar Flow and the ability to return to near-original state in Laminar fluids?

text: Laminar flow here refers to the smooth disruption of fluid in layers, in a medium. A peculiar trait in a few examples of laminar flow is the ability to return to near-original state, as demonstrated at http://www.youtube.com/watch?v=p08_KlTKP50 . Why is this not considered as a violation for the laws of thermodynamics, for it appears to be a reversal of entropy. What is the (deepest possible) reason behind such flow and restructuring, and how is it different from turbulent flows?


title: Problem understanding free fall with water resistance

text: Please take a look at the picture below before read my question: ![enter image description here](http://i.stack.imgur.com/mRK1b.jpg) As you see in the picture the object A is floating on the water because the $F_{drag}\gt F_{grav}$ obverse the object B is sinking in the water because the $F_{drag}\lt F_{grav}$. But the problem is: How can the drag force $F_{drag}$ (Water resistance force) for the object A be greater than gravity force $F_{grav}$ while the speed of it equals to $v=0\,m/s$ (rest object) and thus the drag force $F_{drag}$ (Water resistance force) should be less than gravity force $F_{grav}$ and the object should sink ? Example: If the density of the fluid (water) $\rho=998\,kg/m^3$, speed of the object A $v=0\,m/s$ (rest object), the cross-sectional area $\mathrm A=3\,m^2$, the drag coefficient $C_d=0.47$ and the mass of the object A $m=0.005\,kg$. $F_{drag}=\frac{998\times0^2\times0.47\times3}{2}=0\,\mathrm N$ $F_{grav}=m\,a=0.005\times 9.8=0.049\,\mathrm N$ $F_{drag}\lt F_{grav}$ (object A should sink)


title: Are Field Lines an accurate depiction of reality?

text: Field lines are used for explaining a wide variety of phenomenon. But is it really an accurate depiction of reality? Is it more accurate to imagine a field in a different manner. For instance, using grey-scale colour to imagine intensity. For instance, for a positive point charge, instead of imagining infinite lines emanating from the point, we image concentric shells around the charge, of infinitesimally thin thickness and all of a different grey-scale colour. So if we imagine black to be the maximum strength(intensity) and white to be 0 intensity, you would effectively imagine an infinitely large sphere whose colour change from white at infinity to black at the centre. Rough approximation: ![enter image description here](http://i.stack.imgur.com/sv7Zs.png) The reason why I thought this might be more accurate was because we can do away with the whole problem (not sure if it is actually a problem!) of having gaps between lines that are filled with infinite other lines. It also seemed more natural to imagine this for inverse square law obeying phenomenon in 3d. But I lost my confidence when I saw iron filings on a sheet above a bar magnet actually taking up shapes of lines. I think that this may be what inspired Faraday and other at the time. But I think that that could actually be because of some attraction that a magnetised filings have on each other. ie, if you were to move an entire field line of filings to a 'line' between itself and an adjacent filing, it wouldn't move back, would it? So I'd like to know if this kind of thinking is a more accurate representation of reality? EDIT: As Daniel Knapp points out: in case of a uniform field, one cannot determine the direction using this technique. It has to be explicitly mentioned. However, I think that for more complicated fields, would this be more accurate? EDIT2: I think that using a similar diagram with 6 colours with suitable alpha values for up, down, 4 sides would be better. There will be atmost 3 colours blending together for any 2d slice, so it would represent it quite well, imho. I welcome comments regarding that, but the question has been answered and sadly, in the views of a few, this may not be the ideal place to have such discussion about improved field diagrams.


title: Special Relativity Problem

text: I am having trouble with the following problem: > Fry travels in a rocket ship towards Leela, at constant relative speed $v$: > Fry is delivering a pizza, which in its rest frame stays hot for exactly > another 2 minutes. If Leela measures that Fry is 27 million kilometers away, > then calculate the minimal value of $v$ for which the pizza is hot when > delivered. My approach was to use speed=distance/time and account for time dilation but I cannot figure out how to eliminate the Lorentz Factor. The answer is meant to be $3c/5$. Any pointers would be appreciated! Thanks.


title: Obtaining isotope stability

text: For a given isotope, one can obtain the binding energy using the semi- empirical mass formula. For example, ![From Codecogs](http://i.stack.imgur.com/U4is7.png) has a binding energy of 1782.8 MeV. From this information, how can the likelihood of the isotopes stability be found?


title: Will A Vertical Projectile Travel the Same Distance as A Horizontal Projectile

text: If a projectile is fired vertically from an initial height, $h$, above ground level, and an identical projectile is fired with equal initial velocity from the same initial height in the horizontal direction, will these projectiles travel the same distance? The distance of the vertically shot projectile is to be measured as the change in height from it's initial height to the apex of its trajectory. The distance of the horizontally shot projectile is to be measured as the horizontal distance traveled before the projectile touches the ground. Also, after the initial velocities are imparted to both projectiles, no other external force acts on either projectile, with the exception of gravity, for the duration of their flights.


title: Is it possible that all neutron stars are actually pulsars?

text: I'm assuming that what I've been told is true: We can only detect pulsars if their beams of electromagnetic radiation is directed towards Earth.   That pulsars are the same as neutron stars, only that they emit beams of EM radiation out of their magnetic poles. So, isn't it possible that neutron stars emit EM radiation in the same fashion as pulsars, just not in the right direction for us to detect it?


title: What current alternatives are there to the standard concordance model of cosmology?

text: The current "standard model" or concordance model of cosmology is Lambda CDM which includes late time acceleration due to a cosmological constant, cold dark matter as the missing matter component and an inflationary period at very early times. What alternative theories are also consistent with current observational data?


title: What ways are there to measure the local polarization of a laser beam?

text: Measuring the polarization of a laser beam is a simple enough task if the polarization is the same everywhere. You can even buy commercial polarimeters. How do you go about it if the light beam has different polarizations in different parts of the transverse plane? One example is a radially polarized beam. More generally, is there a good technique for sampling the local polarization (which might be linear, elliptical, or circular, anywhere on the Poincaré sphere) in one transverse plane?