title: Measurement of the amplitude

text: I want to ask how do we actually measure the probability amplitude that appears in Schrödinger equation. From what I read in quantum mechanics textbooks, it appears that after the measurement, the system "collapses" to its eigen-state. And we know this because the probability amplitude of events changed. For a typical event involving two states, previously we only know its amplitude is $|A_{1}+A_{2}|^{2}$, but now it is $|A_{1}|^{2}+|A_{2}|^{2}$, etc. My question is, how do we know that our measurement of the probability amplitude is accurate? How do we know that there is no uncertainity principle that makes $$ \Delta A_{1}*\Delta A_{2}\ge k $$ for example? Since the probability amplitude is one of the intrinsic quantum properties of the system, it seems to me any measurement should disturb it to some extent. For example, if the above hypothetical "uncertainty relation" holds, then we cannot say in principle what $|A_1+A_2|^{2}$ is (since it must be disturbed by our measurement), but only what $E|A_1+A_2|^{2}$ is based on experiment. But if $A_1,A_{2}$ is in certain range, we may not be able to distinguish $|A_1+A_2|$ and $|A_1|^{2}+|A_2|^{2}$ anymore unless we take a huge number of experiments. To elaborate it, my rough conception of the way people measure it is this: We make the experiment in identical situation $N$ times, and we assume via strong law of large numbers that the average frequency must approach the mean value. However, this assumption does not exclude the possibility that every time we measure the probability of event $A_1$, the accuracy of measuring event $A_2$ may be somehow influenced. Therefore in actual measurement, the probability we get is close to $E|A_1+A_2|^{2}$, but not really the real value if by measurement we caused a huge variance in the data. Let us for simplicity consider an even simpler case with only one event $A$. $A$'s probability amplitude is given by the complex number $a$. Suppose in actual measurement, we found if we take $N$ measurements at the $i$th time, then the sample average is about $a+(-1)^{i}b$. Then we can propose either $A$'s probability amplitude changes with time (like in a two state system), or our measurement somehow influenced $a$'s sample value. Suppose we are in the second case, how do we know what is the true probability amplitude $a$? If we only do $N$ experiments, we would only get a biased value. And if we do more, the chance of the bias accumulating is small but still not negligible if $b$ is really large.


title: Has any possible non-extensive entropy been classified?

text: How many non-extensive entropies do exist? Tsallis, Havrda-Chavrat, Renyi, Kaniadakis, Sharma-Mittal,... To be more precise, I am wondering if some classification like those of "finite" or Lie groups do exist for uniparametric, biparametric or multiparametric (generally non-extensive) entropy functions (of course, another issue would be their physical applications...)


title: When the Cosmic Microwave Background radiation cools, where does the energy go?

text: I understand how photons can change wavelength via gravitational redshifting, but that doesn't seem to be what's going on with the CMB radiation. I've heard it explained as happening because of the expanding universe, but I'm thinking that would have to imply that as the universe expands, lower wavelengths have higher energy, so to conserve energy, the CMB would have to redshift. Is that the case, or is something else going on?


title: Why use a hadron collider to search for Higgs Boson

text: If, according to relativity matter is energy condensed, wouldn't breaking open the endless Matryoshka dolls of matter to find the most fundamental particle which gives mass to matter be fallacious? Fallacious in the sense that searching for a particle composed of energy, energy being something immaterial forming the composition of matter, is searching for something that in itself has no physical existence therefore breaking down particles into sub- particulate matter would go on ad-infinitum because there would be no such "particle" to be found.


title: Why are Cooper pairs formed by electrons of opposite momentum and spin?

text: I understand that Cooper pair in low-temperature superconductivity are formed by electron-phonon interaction. Normally one then assumes that electrons of opposite momentum and spin are paired. This is what i dont understand. In my understanding two electrons with opposite momentum should seperate and thus not feel an attractive force due to electron-phonon interaction.


title: Why is Spin Less Classical than Position?

text: It is often repeated that "the spin observable is purely quantum and has no classical counterpart". What is actually meant by that? I see no principle difference between the set of spin observables and the set of position- momentum observables in that respect (of being purely quantum): both sets have non-commuting observables. The only interesting difference I see is that spin, having a discrete spectrum, can nonetheless transform continously in time between two orthogonally spin states thanks to the principle of superposition, whereas a classical coordinate (of phase space) must have a continuum of possible values for the value to evolve continously in time (So in this respect spin cannot be classical but position-momentum could ; but I don't think this is the answer since also energy is discrete for bound systems)


title: How do subgrid-scale parametrisations in climate models work?

text: Global Circulation Models typically have grids of 100-300km on a side. There are obviously lots of atmospheric processes that happen at smaller scales than this. Convection, cloud formation, the effect of mountains... How are these processes built in to the model?


title: Why is photon annihilation associated with the POSITIVE frequency component of the electric field?

text: I'm reading Glauber's paper "The quantum theory of optical coherence". In his work he does not introduce the annihilation and creation operators, but he refers instead to the positive and negative frequency components of the electric field (which probably are proportional to the former, I guess). He says that "...the positive frequency part $E^{(+)}(\mathbb{r}t)$, may be shown to be a photon annihilation operator..." and then cites Dirac book "The principles of quantum mechanics", 3rd ed. pp 239-242, but even from the reference I couldn't understand why the _positive_ frequency components is associated with the photon annihilation. PS with the capitalization I don't mean to yell but to stress on the term "positive".


title: Why does water make a sound when it is disturbed?

text: When I disturb a body of water, what causes the familiar "water moving" sound?


title: What determines the pitch of an explosion?

text: It's bonfire weekend here in the UK, and everybody is intent on shooting fireworks. I was wondering, what determines the pitch of the sound produced by an explosion? The sound has clearly a dominant mid-range pitch, around a few hundred hertz, and bigger fireworks also have a very low, but large component at around 20-50 hertz. What determines this pitch? The amount of explosive energy? The shape of the vessel? Something else?


title: Why did people expect the number of electrons emitted because of the photoelectric effect to rise with the radiation's intensity?

text: The number of electrons emitted because of the photoelectric effect   1. rises with the frequency of the radiation   2. ~~is not influenced by the intensity of the radiation~~ (As Chad points out, this is wrong, the electron's _energy_ isn't influenced by the radiation's intensity.) Why was this not expected when it was discovered?


title: Electric current in quantum mechanics?

text: Quick question: I want to find an expression for the (electric) current density of an electron, in quantum mechanics. Either a single electron or a general charge distribution $\rho$. Classically **j** =$\rho $ **v**. What should I use here? Maybe the electric charge multiplied by the probability current? Thanks.


title: Realistic vs Idealistic capacitance

text: I am doing an investigation into the differences of calculating capacitance using the well know formula for an idealistic parallel plate capacitor, based on the assumption of a uniformly distributed electric field:$C=\epsilon_0 \frac{A}{d}$ vs using numerical methods for calculating the capacitance of a realistic model i.e. with fringe fields (see here for a more detailed explanation of my methods). I am investigating how the capacitance changes as i move the plates further away from each other, and what i have found is that, based on the results I'm getting, as the plates are moved further away from each other the ratio of realistic/idealistic increases, implying that the realistic electric field can store more energy. My question is, is this correct and if not why? Thanks [Edit] I originally thought that this may be because the realistic formula is only valid for small d, however the ratio seems to increase linearly, rather than converging to a value and then dropping off leading me to think that this is not the problem? [Info for comments] $$C=\frac{\epsilon_0 L}{V} \sum_{bound} |\phi_{outer}-\phi_{plate}|$$ $$C_{\infty}=\frac{\epsilon_0 A}{d}$$ Dividing them, where $A=lL$ i.e the area of the plate, gives: $$\frac{C}{C_{\infty}}=\frac{d}{V l} \sum_{bound} |\phi_{outer}-\phi_{plate}|$$


title: integration for upper ocean mixed layer equation

text: I am trying to work through the following paper specifically trying to get from equation (1) to equation (6). Equation 1 states that $$ Q = \beta S e^{-\beta z} + 2B \delta(z) $$ where $\delta z$ is essentially equal to $\frac{1}{2}$. Thus $$ Q = \beta S e^{-\beta z} + 2B \frac{1}{2} $$ $$ Q = \beta S e^{-\beta z} + B $$ Equation 3 of the paper than specifies that $$ \frac{dT}{dt} + \frac{\partial}{\partial z} \left(\overline{W'T'} \right) = Q $$ Thus equation 3 can be written as: $$ \frac{dT}{dt} + \frac{\partial}{\partial z} \left(\overline{W'T'} \right) = \beta S e^{-\beta z} + B $$ The paper then describes that equation 4 is the integration of 3, which should produce: $$ \frac{dT}{dt} + \left(\overline{W'T'} \right) = S + B - Se^{-\beta z}$$ How do they get this outcome? Furthermore, how do we get to equation 6 in the paper: $$ h\frac{dT_{s}}{dt} + \Lambda \left( T_{s} - T_{h} \right) \frac{dh}{dt} = S + B-Se^{-\beta h} \approx S+B $$


title: How do electrons actually move in a circuit?

text: Last year, we were taught about electricity, about how electrons in a closed circuit. But as our teacher had not taught us about electric fields yet, she gave us a simplified model of motion of electrons in a circuit. (given below) ![enter image description here](http://i.stack.imgur.com/uTECB.png) She told us that, due to chemical reactions taking place inside the dry cell, accumulation of electrons occurs at the -ve terminal and cations at the +ve terminal. When a conductor is connected with the battery, due to accumulation of e- at the -ve terminal, there is mutual pushing between electrons, and this causes them to move away from the -ve terminal and towards +ve terminal. Also, cations attract electrons towards the +ve terminal.   **Q 1. Is this true that electrons are both attracted and repelled (by cations and other electrons respectively) in an electric circuit?      Q 2. Is this analogy correct? What exactly is happening Q 3. Is this (below) a better analogy of electron motion (field lines)? Can electric fields be altered by conductors such as a metal wire? Will most of the field lines be inside of the conductor, Therefore providing a better path for electron flux/flow? ** ![enter image description here](http://i.stack.imgur.com/dO7zn.png) This is not a Duplicate so please don't mark it as one. My real question is, are both analogies correct? Thank you! **Similar** : How does electricity flow in conductor when potential difference is applied? **Very similar** : Why does electrical current start to flow?


title: Feynman propagators for scalar fields

text: If there are few massless scalar field, are the propagators of those different massless scalar fields indistinguishable?


title: Position vs Time Function for a Falling Object Without Assumption of Constant Acceleration

text: If you have an asteroid for example and a planet, isolated from all other gravitational influence and initially at rest with respect to each other, how would you find a position vs time function for the asteroid and the planet falling toward each other?


title: Lagrangian formulation of the problem: small oscillations around an equilibrium

text: I'm having trouble understanding how some conclusions are made in my book. I'm studying from a coursebook based on Goldstein's "Classical Mechanics", here's what's written in my book, with my problems under each section: > Assume a system with conservative forces and holonomic, time independent > constraints, described by $n$ generalized coordinates $q_k$.   >  The Lagrangian is $L=T-V$, with the kinetic energy: > > $T=\sum_{kl}\dot q_k\dot q_l (\sum_i \frac{1}{2}m_i\frac{\partial\vec > r_i}{\partial q_k}.\frac{\partial\vec r_i}{\partial q_l}) = > \sum_{k,l}\frac{1}{2}m_{kl}(q1,...,q_n)\dot q_k \dot q_l$   >  ...   * I understand how to get the first part, but I don't understand the second equality. What is meant by "$m_{kl}$"? > ...   >  a polynomial of the second degree in the generalized velocities. The > potential energy $V(q_1, ..., q_n)$ only depends on the generalized > coordinates. > > The system has a point of equilibrium ($q_k^0=q_1^0, ... q_n^0$) when all > generalized forces are $0$ in this point: > > $Q_k=-(\frac{\partial V}{\partial q_k})_0=0$ > > If the system is in a point of equilibrium at a certain starting time, with > the starting velocities $0$, then the system will remain in that point of > equilibrium. In other words: $q_k(t)=q_k^0$ is the solution to the Lagrange > equations > > $\sum_l m_{kl}\ddot q_l+\sum_l\sum_m\frac{\partial > m_{kl}}{\partial{q_m}}\dot q_l\dot q_m=\sum_{lm}\frac{1}{2}\frac{\partial > m_{lm}}{\partial q_k}\dot q_l \dot q_m -\frac{\partial V}{\partial q_k}$ > > with these initial conditions.   * This last equation I don't understand at all, I know what the Lagrange equation is but when filling it in I don't see how you would get this result.


title: Controlled-measurement of a quantum register

text: Given a state vector $\left[\alpha,\beta,\gamma,\delta\right]$ which is not known _a priori_ , does there exist an operation, which I will call "controlled-measurement", which results in the ensemble $\left[\alpha,\sqrt{|\beta|^2+|\delta|^2},\gamma,0\right]$ with probability ${|\beta|^2}\over{|\beta|^2+|\delta|^2}$ and $\left[\alpha,0,\gamma,\sqrt{|\beta|^2+|\delta|^2}\right]$ with probability ${|\delta|^2}\over{|\beta|^2+|\delta|^2}$, and informs me in which of these two states is the system now? (Assume that the phase of the roots is inconsequential.) Failing this, does there exist an operation, which I will call "controlled- initialization", which operates on the same unknown state vector but deterministically results in the first case of the above ensemble? If not, what law of quantum mechanics is violated by this operation? If either operation is impossible, what law of quantum mechanics precludes it?


title: Crystal momentum and the vector potential

text: I noticed that the Aharonov–Bohm effect describes a phase factor given by $e^{\frac{i}{\hbar}\int_{\partial\gamma}q A_\mu dx^\mu}$. I also recognize that electrons in a periodic potential gain a phase factor given by $e^{\frac{i}{\hbar}k_ix^i}=e^{\frac{i}{\hbar}\int k_idx^i}$. I recall that $k_i$ plays a role analogous to momentum in solid state physics. I also recall that the canonical momentum operator is $P_\mu=-i\hbar\partial_\mu-qA_\mu$. Notice that when you operate with the momentum operator on a Bloch electron, $\psi(x)=u(x)e^{\frac{i}{\hbar}k_ix^i}$, you get $e^{\frac{i}{\hbar}k_ix^i}(-i\hbar\partial_i+k_i)u(x)$. My question is whether a parallel can be drawn between the crystal momentum, $k$, and the vector potential $A$. It seems they play a similar role quantum mechanically, but I have never seen Bloch's theorem described in terms of vector potentials. I suppose one does not even need a nontrivial vector potential for Bloch's theorem to hold. Still, crystal momentum and the vector potential play very similar roles in phase factors and I wonder whether there is any deeper meaning to that.


title: How does one explain this pattern generated by earthquake waving driving a pendulum?

text: How to explain this pattern generated by earthquake wave driving a pendulum? Specially, there are three groups of curves that look categorically different: 1) The group of curves outside the olive-shaped envelope   2) The group of intertwined curves at the center   3) The group of curves inside the envelope but outside (2). ![Earthquake Rose](http://i.stack.imgur.com/teASN.jpg)


title: Vector and Spinor Representation in Ramond-Neveu-Schwarz Superstring Theory

text: I am learning Ramnond-Neveu-Schwarz Superstring theory (RNS theory). I often find the following notation, especially in the closed string spectrum etc.: $$\mathbf{8}_s,\mathbf{8}_v $$ And it is noted that these are vector and spinor representations of something. I have two questions regarding these.   1. What are these representations of? Are they representations of $SO(8)$?   2. What do they actually mean? How do you represent something in vector/spinor notation.


title: Usage of the terms Raman, Stokes and anti-Stokes scattering

text: I am confused over the correct usage of the terminology for "Raman scattering", "Stokes scattering", "anti-Stokes scattering", or even "Stokes- Raman scattering" and "anti-Stokes-Raman scattering". Is Raman scattering the superior term for Stokes and anti-Stokes scattering? Or do they essentially mean the same, and people are just too lazy to always write "anti-Stokes-Raman scattering"? Could you post an explanation and maybe examples for the correct usage?


title: quantization of angular momentum

text: What is the most direct way of observation of quantization of angular momentum?


title: How would you describe a black hole to a complete layman?

text: Non-astrophysicist visitors to Black Hole, will most likely barely make it past the introduction before getting lost in the technical details. Moreover, the Wikipedia article does not teach laypeople how to think of a black hole, or give them a better intuitive sense than pop culture references to black holes, some of which are really awful!* So, please explain the concept of a black hole in a way that won't glaze over the eyes of your typical English or Business major. *For examples, and the pop-culture smearing of the concept of black hole together with other gravitational anomalies, see:   * Black Holes in Fiction.   * Portable Hole.   * Wormholes in Fiction.   * White Holes in Fiction.


title: Reverse Sun position algorithm?

text: To find the sun's position (elevation, azimuth) we can use many algorithms including the PSA algorithm. These algorithm share the fact that the input is the local time, longitude and latitude and the output is the elevation and azimuth. is there any algorithms that we can use to reverse this, the input would be the sun position and the output would be the local time ? thanks


title: Number of bosonic states in a $n$ particle system

text: Lets assume we have a system of three particles in which three energy-levels labeled by $$E_1, E_2, E_3$$ are occupied. We want to write down all states that are symmetric (bosonic). One symmetric state is easily found by using a permanent, but how do I know that it is the only one? How many bosonic states are there if there are $n$ particles involved?


title: Space Interferometry Mission (SIM) cancellation

text: NASA's SIM mission was recently cancelled despite \$600 million being spent already over the last ten years. Has this \$600m been completely wasted or has any new technology been developed that could be useful in the future?


title: Jiles-Atherton model of hysteresis

text: In my quest for finding a good algorithm to draw hysteresis curve,I have found Jiles-Atherton model which seems to be a rather new model. But I have trouble understanding some of the variables/constants that are used in the formulation.There seems to be a rather similar question here. What I have trouble understanding is that what are Mirr and Man? And how is it possible that M is defined using Mirr and Man, and yet in the definition of Man we use M ?


title: Probability current vs. direction of wave function

text: I did an exercise for my Quantum-Mechanics Lecture: Let $\hbar$=2m=1. A particle in 1 dimension has $j(x)=2\ Im(\overline{\psi} (x) \ \psi'(x))$ and it's to show that there are superpositions $\psi (x) = a_1 e^{i k_1 x} + a_2 e^{i k_2 x}$, where $k_1, k_2 > 0$, of waves which propagate to the right at x=0 but j(0)<0\. You can show that by calculating j(0) which leads to a non positive semidefinite quadratic form in $a_1,\ a_2$. (Remark: This superposition can not be normalized, but the exercise states that there are analogue waves which can.) I have troubles understanding that. How can the wave (and therefore the probability of the particle to be at position x) propagate to the right when the current is negative? Maybe someone could explain me how to think about this? Edit: The official solution of the exercise: "With $\psi'=i(k_1 a_1 e^{i k_1 x} + k_2 a_2 e^{i k_2 x})$ is: $\overline \psi(0) \psi'(0)=\sum_{i,j=1}^{2}i\ \overline{a}_i k_ja_j$ and $j(0)=\sum_{i,j=1}^{2}(k_i + k_j) \overline{a}_i a_j$ This quadratic form in $a_1, a_2$ is not positive semi definite because the determinant is given by $-(k_1 - k_2)^2 < 0$"


title: Deriving an equation involving Killing vectors

text: I'm currently studying Carroll's GR book _Spacetime & Geometry_, and ran into some trouble understanding the text. When discussing Killing vectors, Carroll mentions that one can derive $$K^{\lambda}\nabla_{\lambda}R=0$$ That is, the directional derivative of the Ricci scalar along a Killing vector field vanishes (here, $K^{\lambda}$ is a Killing vector). He remarks that the only necessary ingredients to derive this equation are: $$\cdot\;\;\text{The Killing equation:}\ \nabla_{(\mu}K_{\nu )}=0$$ $$\cdot\;\;\text{The Bianchi identity:}\ \nabla_{[\mu}R_{\nu\rho ]\sigma\lambda}=0$$ $$\cdot\;\;\ \nabla_{\mu}\nabla_{\nu}K^{\rho}=R^{\rho}_{\lambda\mu\nu}K^{\lambda}\longrightarrow \nabla_{\mu}\nabla_{\nu}K^{\mu}=R^{\mu}_{\lambda\mu\nu}K^{\lambda}=R_{\lambda\nu}K^{\lambda} $$ With $R_{\mu\nu\rho\sigma}$ the Riemann curvature tensor and $R_{\mu\nu}$ the Ricci tensor. Although I was able to derive all of the equations involved, I don't see how to put them together to get the sought-after result. The only way I saw to get started is the following: $$K^{\lambda}\nabla_{\lambda}R=\nabla_{\lambda}(K^{\lambda}R)-R\nabla_{\lambda}K^{\lambda}=g^{\mu\sigma}(\nabla_{\lambda}(R_{\sigma\mu}K^{\lambda})-R_{\sigma\mu}\nabla_{\lambda}K^{\lambda}) $$ which immediately runs me into trouble because I don't see how to simplify/manipulate either of the terms to even make use of any of the three 'ingredients'


title: A Calorimetry Problem

text: I have a question in calorimetry from an old competitive exam. The question is: > The temperature of $100$ grams of water is to be raised from $24 ^\circ$C to > $90 ^\circ$C by adding steam to it. Find the mass of steam required. I tried attempting the question by assuming that the added steam would convert back to water and thus lose an amount of heat calculated by the latent heat of vaporization. Additionally, to cool from 100 degrees to 90 degrees, the 'watered steam' will contribute some additional heat. I equated this to the heat gained by the 100 grams of water to reach 90 degrees from 24 degrees. But I am not getting the right answer! I would appreciate any help on this matter. Thank you :)


title: How does tension apply torque on a pulley?

text: How is tension in a string able to apply torque on a pulley? How does string itself able to apply a force on pulley? What is happening inside the pulley? The pulley has a mass $m$ and is a disc.


title: Photons angular momentum / spin

text: I have a textbook that says that photons have a spin of absolute value $\hbar$ and at some other point, they say that it has angular momentum of absolute value $\hbar$. Now, since they are different things, I was asking myself: Is it maybe right to say, that it is a spin that is transformed into angular momentum when the photon is absorbed by matter? I would be glad if anybody here could write down a few clarifying sentences about this topic.


Why can apparent horizon be computed based on its local geometry? In the paper titled _Black Holes, Geometric Flows, and the Penrose Inequality in General Relativity_ by Hubert L. Bray, has been written: > _Another very interesting and natural phenomenon in general relativity is > the existence of black holes. Instead of thinking of black holes as > singularities in a spacetime, we will think of black holes in terms of their > horizons. For example, suppose we are exploring the universe in a spacecraft > capable of traveling any speed less than the speed of light. If we are > investigating a black hole, we want to make sure that we do not get too > close and get trapped by the “gravitational forces” of the black hole. We > can imagine a “sphere of no return” beyond which escape from the black hole > is impossible. It is called the event horizon of a black hole. However, one > limitation of the notion of an event horizon is the difficulty of > determining its location. One way is to let daredevil spacecraft see how > close they can get to the black hole and still escape from it eventually. > The problem with this approach (besides the cost in spacecraft) is that it > is hard to know when to stop waiting for a daredevil spacecraft to return. > Even if it has been fifty years, it could be that this particular daredevil > was not trapped by the black hole but got so close that it will take one > thousand or more years to return. Thus, to define the location of an event > horizon even mathematically, we need to know the entire evolution of the > spacetime. Hence, event horizons cannot be computed based only on the local > geometry of the spacetime. This problem is solved (at least for the > mathematician) by the notion of apparent horizons of black holes. Given a > surface in a spacetime, suppose that it emits an outward shell of light. If > the surface area of this shell of light is decreasing everywhere on the > surface, then this is called a trapped surface. The outermost boundary of > these trapped surfaces is called the apparent horizon of the black hole. > Apparent horizons can be computed based on their local geometry, and an > apparent horizon always implies the existence of an event horizon outside of > it._ _Question_ : Bray has argued that event horizons cannot be computed based only on the local geometry of the spacetime because we cannot define the location of an event horizon. On the other hand we know that apparent horizon is always inside (or coincide) the event horizon. How can we define the location of apparent horizon? When spacecraft cannot define the location of event horizon, how can it define that of apparent horizon? Unlike the event horizon, why can apparent horizon be computed based on its local geometry?


title: Electromagnetic duality

text: A key aspect of modern physics is the mapping of theories or different descriptions of a theory into a one-to-one correspondence. As I am trying to further understand the electromagnetic field tensor, the Lorentz group and field theories, I have come across the recurring theme of duality. In terms of the free Maxwell theory could somebody explain/provide text or papers on the self-duality. Namely, discuss the mutual exchange of the electric and magnetic field, which leaves the Maxwell equations invariant. An example applying the Lorentz transformation to the Electric Field would be helpful.


title: Higgs boson mass and electroweak energy scale

text: Is it a coincidence that the mass of the Higgs boson is exactly half the electroweak energy scale?


title: Does shadow fading change if obstacles are fixed?

text: Given a RF sender and a receiver, suppose the obstacles between them are fixed, both their material and dielectric properties, does the path loss caused by their shadowing vary? If yes, what causes the variation? If not, why is it generally modeled as a log-normal random variable? We can assume there is only one path between the sender and receiver if this simplifies the question.


title: Partition Function as characteristic function of energy?

text: I'm going through a book on statistical mechanics and there it says that the partition function $$Z = \sum_{\mu_S} e^{-\beta H(\mu_S)}$$ where $\mu_S$ denotes a microstate of the system and $H(\mu_S)$ the Hamiltonian, is _proportional_ to the characteristic function $\hat p(\beta)$ of the energy probability distribution function. This allows us to make then the next step and conclude that $\ln Z$ is the _cumulant generating function_ with the nice result that $$\langle H \rangle = -\frac{\partial \ln Z}{\partial \beta}$$ and $$\langle H - \langle H \rangle \rangle^2 = \frac{\partial^2 \ln Z}{\partial \beta^2}$$ but I fail to see why $Z$ is proportional to characteristic function. Also, if I imagine that $Z$ is the characteristic function to the energy, then wouldn't I have to evaluate the derivative at $\beta = 0$? I know that the two formulas above can also be obtained by explicitly doing the calculation using the definition of $Z$ in the first line, but I'd like to generalize this result to the momenta and cumulants of all orders.


title: Is there a contradiction of the theory of relativity here? -- Length contraction and EMR amplitude

text: Suppose there is a laser beam powerful enough to burn through iron aimed at a piece of iron. You observe this event while you are in the same frame as the piece of iron and the laser-beam generator. In this frame, there is a certain part of space that you know that the light is traveling through. Now say that you get in a rocket that travels a few meters away from the laser beam in a direction perpendicular to eventually at a constant speed arbitrarily close to the speed of light. As you do so, the area that you knew the light was traveling through contracts in the direction that you are traveling in. Say that the amplitude that you think that the light has decreases to the point where the light would no longer be carrying enough energy to burn through the piece of iron. If you see the iron stop being burned by the laser, our universe is seriously weird. I don't think this would happen. Would you see the laser continue to burn through the iron even though it does not seem to you to have the energy necessary to do so? Would this mean that, instead of having the length contract, that there is such a thing as absolute distance? (If we suppose that the amplitude you perceive the laser to have remains constant regardless of what inertial frame you are in, then the laser would appear to have a constant amplitude even as objects around it continued to contract in the direction you are traveling through. This would mean that the laser would have to seem to affect more and more of space as you traveled faster and faster, so it seems it would have to seem to burn through more and more objects as you went faster, which doesn't seem right to me.) Does this contradict the theory of relativity? Is there an error in here somewhere? Thanks.


title: Can the sun cast a shadow?

text: The sun's light can cast the shadow of another object, but does it ever cast its own shadow?


title: What phrases describe collisions with coefficients of restitution less than zero or greater than one?

text: The coefficient of restitution describes the elasticity of a collision:   * 1 = perfectly elastic, kinetic energy is conserved   * 0 = perfectly inelastic, the objects move at the same speed post impact However, COR values > 1 and < 0 are also physically meaningful:   * COR > 1, a collision where the impact adds energy (e.g. an explosion)    * COR < 0, a partial collision where the objects partially pass through each other. Say, like this: ![apple and bullet](http://i.stack.imgur.com/glFbp.jpg) Are there colloquial terms that characterize these types of collisions? Perhaps explosion does work well enough for the first case, but I can't think of anything that adequately describes the apple case. (If inelastic is defined as COR != 1 it's probably broad enough but likely not very illuminating in practice.)


title: Optical element is heated by laser: is it possible to get oscillating heat distribution?

text: Imagine that we have thin optical element, which is irradiated by laser. Laser heats element, so there is some heat distribution in element. There is a heat sink through upper and lower element faces proportional to temperature. There is also a heat conduction in element. The most important characteristic of the model is that heat consumption depends not only on power of laser, but on heat of the element too and relation between current heat and heat change may be very complex. With given parameters after some time heat distribution in element usually sets to some stable state (with equal heat income and outcome in all element points). Stable heat distribution can vary depending on initial heat distribution. So, the question is: if laser radiation power distribution depends on element coordinates and heat consumption is related to temperature in current part of element, but all the parameters do not change in time, is it possible that heat distribution will not come to some stable state but instead will oscillate, or behave in more complex way?


title: Calculating Inertia Tensor with Parallel Axis Theorem

text: Say you have a solid you are approximating as n point masses at different points in a 3D space. Each point mass has a mass of 1. The origin is not the center of mass. All the points have location x,y,z >= 0. I have running totals Sx=sum(x), Sy=sum(y), Sz=sum(z), Sxx=sum(x*x), Syy=sum(y*y), Szz=sum(z*z), Sxy=sum(x*y), Sxz=sum (x*z), Syz=sum (y*z). My idea is to calculate the center of mass (Cx,Cy,Cz) = (Sx/n,Sy/n,Sz/n), calculate the inertia tensor around the origin (based on the other sums) and subtract something to get the inertia tensor around the center of mass, using the parallel axis theorem. The way I tried ended up with the matrix: (Syy+Szz-n*(Cy*Cy+Cz*Cz)),--------(n*Cx*Cy- Sxy),----------------------(n*Cx*Cz-Sxz), (n*Cx*Cy- Sxy),-------------------------(Sxx+Szz-n*(Cx*Cx+Cz*Cz)),------(n*Cy*Cz-Syz), (n*Cx*Cz-Sxz),-------------------------(n*Cy*Cz- Syz)------------------------(Sxx+Syy-n*(Cx*Cx+Cy*Cy)) The idea being, once I have an inertia tensor, its rather straightforward to go on to calculate angular velocity from an applied force. Is this result reasonable? I'm a bit out of my element. I don't fully understand how the parallel axis theorem relates to inertia tensors.


title: What is a "reference current"

text: Excuse me, I m not a physicist I am currently reading a paper about optimization on minimizing electrical energy in electrical vehicles. The paper talks about a "reference current" in a current regulator of the motor. I do not understand what this means, google sends me to pages that have nothing to do with the term. So I am not sure if this is actually a proper term in physics. The link to the paper is http://www.optimization-online.org/DB_FILE/2011/04/2997.pdf The term in question is in page 5 of the pdf, section 3, in the last sentence of the 2nd paragraph. Thank you


title: How fast does current flow? And the magnetic field?

text: A conductor carries 10 Amps, and generates a magnetic field $B$.   1. How fast does it take for that current to flow from the wire throughout the entire circuit?    2. And how fast does it take for the magnetic field to be created?


title: Materials about S-matrix and S-matrix theory

text: What is the best book or paper to learn about analytical structures of S-matrix and S-matrix theory? I already know books as _The Analytic S-matrix_ by RJ Eden, PV Landshoff, DI Olive, JC P and _Quantum Field Theory_ by S. Weinberg.


title: Would a $50\: \mathrm{Hz}$ spinning magnetic plate in front of a human generate harmful current?

text: We know that spinning magnetic plate will generate current inside any wire. so if we stand before 50Hz spinning magnetic plate, then, does that 50Hz spinning magnetic plate in front of us (human body ) generate current inside human body which will cause harm to body? By considering the fact, human body is not bad conductor of electricity.


title: Can the electroweak/strong forces, and/or quantum mechanics be thought of as geometric?

text: Can the electroweak and strong forces be written as geometric theories? - Why and why not? Can quantum mechanics in general? For example, the Kaluza-Klein theory explains the electromagnetic field as "twists" that include an extra dimension of space. (As written by Lubos Motl in a previous question).


title: How to get an imaginary self energy?

text: The Lehman representation of the frequency-dependent single particle Green's function is $$G(k,\omega) = \sum_n \frac{|c_k|^2}{\omega - E_n + i\eta}$$ where $n$ enumerates all the eigenstates of the system, $c_k$ is the overlap between $|\Psi_n\rangle$, the eigenstate to eigenenergy $E_n$, and the state $|k\rangle = c_k^\dagger | 0 \rangle$. And $\eta$ is a very small convergence factor, sometimes also written as $0^+$. The corresponding spectral function is $$A(k,\omega) = \sum_n |c_l|^2 \frac{\eta}{\pi} \cdot \frac{1}{(\omega - E_k)^2 + \eta^2)}$$ This means we have Lorenzian with width $\eta$ at the eigenergies of the system, i.e., for small $\eta$ we see that the spectral function has sharp peaks for all values of $\omega$ that correspond to the system's eigenergies. (Note that for $\eta \rightarrow 0$, the Lorenzian becomes a $\delta$-function). So far, so good. Now we can also write the Green's function using the **self energy**. If $\varepsilon(k)$ is the dispersion of the non-interacting particle, then the self energy is defined via $$G(k,\omega) = \frac{1}{\omega - \varepsilon(k) - \Sigma(k,\omega) + i\eta}$$ It is mentioned on many occasions that a self-energy with a non-zero imaginary part corresponds to a quasi-particle with a finite lifetime. In the spectral function, an imaginary self energy leads to a _broad_ peak, and the width of that peak is related to the imaginary part of $\Sigma(k,\omega)$. **Question:** At first, I have shown that the spectral function has infinitely sharp peaks at precisely the system's eigenergies. But later, I have shown that an imaginary self energy leads to _broad_ peaks. How can I reconcile these two facts? **Related Question:** If I find a "continuum" of states in my spectral function, how can I tell if these form a continuum of "true" eigenstates of the system or a bunch of decaying quasiparticles? The motivation: In the Holstein model, we have a single electron on a lattice in the tight-binding approximation interacting with Einstein phonons of frequency $\Omega$. In the non-interacting limit, we have the non-interacting electron band $\varepsilon(k) = -2t \cos(ka)$ and then, beginning at an energy $\Omega$ above $-2t$, a continuum of a states corresponding to an electron of momentum $k - q$ and a phonon of momentum $q$. These are "true" eigenstates of the (non-interacting) system. But if I then turn on the interaction of the type $$ \frac{g}{\sqrt{N}} \sum_q c_{k-q}^\dagger c_k (b_q^\dagger + b_{-q})$$ I still have a continuum an energy $\Omega$ above the ground state, but now these are not true eigenstates any more, since the additional phonon can decay via the interaction term. But when they are not true eigenstates any more, then they shouldn't show up in the Lehman representation and hence shouldn't contribute to the spectral function?


title: Transverse current equivalence in Coulomb gauge

text: I have a question concerning transverse (solenoidal) current in the Coulomb gauge. This current is the one that enables the radiation, since we have a wave equation for the vector potential: $\nabla^2 \vec A - \mu_0\epsilon_0\frac{\partial^2\vec A}{\partial t^2} = -\mu_0\vec{J}_{\perp}$ Where the transverse current can be written as $J_\perp = \nabla \times \nabla \times\int\frac{\vec J(\vec x')\textrm{d}^3x'}{4\pi|\vec x - \vec x'|}$ This transverse current extends to all space. We can solve the wave equation using Green's function. $\vec A(\vec x, t) = \frac{\mu_0}{4\pi}\int\frac{\vec J_\perp(\vec x', t')}{|\vec x - \vec x'|} \delta(|\vec x- \vec x'|/c-(t-t'))\textrm{d}^3x'\textrm{d}t'$ The point is that, when we want to keep only terms that will result in fields falling like $1/r$, neglecting $1/r^2$ and so on, it seems that we don't have to consider the complicated expression for the transverse current. It seems to be enough in some cases to keep the projection of the current perpendicular to the observation vector $\vec x$ and derivate the vector potential with respect to $t$, though this is not obvious, as $\vec J_\perp$ covers all the space. I checked two different cases -an instantaneous dipole at the origin and an instantaneously accelerating point charge, and the field that goes as $1/r$ is created, as expected, by currents transverse to the line of sight. However, I do not have a rigorous explanation to ignore the non-perpendicular currents in the integral for $\vec A$. I've asked my boss, checked some books and searched in the Internet, but I couldn't find any hint or proof. Do you know of any place? Thank you very much.


title: Shankar's Active/Passive Change of Basis

text: I'm working my way through Shankar's Quantum Mechanics (7th printing, and I'm doing it alone, so I apologize if I have core concepts completely wrong). He has a section on Active and Passive Transformations (which seems to be slightly different from what I know as Active & Passive Transformations, but that would be a different question :) ), and it reads like this: > Suppose we subject all the vectors $|V\rangle$ in a space to a unitary > transformation $$ |V\rangle \rightarrow U|V\rangle $$ Under this > transformation, the matrix elements of any operator $\Omega$ are modified as > follows: $$ \langle V'|\Omega|V\rangle \rightarrow \langle > UV'|\Omega|UV\rangle = \langle V'|U^\dagger\Omega U|V\rangle $$ It is clear > that the same change would be effected if we left the vectors alone and > subjected all operators to the change $$ \Omega\rightarrow U^\dagger\Omega > U$$ I was trying to wrap my head around this, so I was working through some examples I made up and realized that the formula $\Omega\rightarrow U^\dagger\Omega U$ only works when $U$ is unitary. In my own examples I worked it out for an arbitrary (viz. non-unitary) change of basis T to be $$\Omega\rightarrow T^{-1}\Omega T$$ Like I said, I worked this out myself, so it may be wrong, but it checks out for the unitary case because when $U$ is unitary, $U^{-1} = U^\dagger$. My question is: _what part of Shankar's logic above makes use of the fact that U is unitary?_ It seems like his logic holds fine even without that constraint, but then it gives us a formula that isn't true for non-unitary U.


title: Thermodynamics, bullet fired into ice

text: I'm beginning to study basic thermodynamics and I want to solve this exercise: It fires a lead bullet from $3g$ to $30^oC$ with a speed of $240\frac{m}{s}$ to a block of ice at $0^0C$. How much ice melts? (the bullet remains embedded in the ice). I think that the energy becomes kinetic friction: $$Q=K=\frac{1}{2}mv^2=86,4J$$ Then, I can find the mass of ice from the ecuation of latent heat: $$Q=mL_{f}$$ $$\implies m= 0,26 g$$ Is this correct? Do not influence the temperature of the bullet? Thanks for your help.


title: $\nabla({\bf u}^2)=2({\bf u}\cdot \nabla){\bf u} - 2(\nabla \times {\bf u}) \times {\bf u}$

text: Please see the next link: http://www3.kis.uni- freiburg.de/~peter/teach/hydro/hydro02.pdf In (2.13), he used: $$\nabla({\bf u}^2)=2({\bf u}\cdot \nabla){\bf u} - 2(\nabla \times {\bf u}) \times {\bf u}.$$ Is this is a formula? can someone let me some details/link about this formula? Any help appreciated!


title: A question from Ticcati's red QFT textbook.

text: From Ticcati's textbook, he asks to show that from the axioms of position operator we get that: $$ \text{e}^{-ia\cdot P} |x\rangle = |x+a\rangle $$ where the axioms are: $$ X=X^{\dagger} $$ If $\Delta_a$ is a space translation, then $U(\Delta_a)^{\dagger} X U(\Delta_a) = X + a$, where $U$ is the representation of a unitary operator, we know that $\text{e}^{ia\cdot P} X \text{e}^{-ia\cdot P} = X + a$. If $R$ is a space rotation then $U(R)^{\dagger} X U(R) = RX$. Here's what I tried so far to do with this: $$ \text{e}^{ia\cdot P} X \text{e}^{-ia\cdot P}|x+a\rangle =(X+a) |x+a\rangle = (x+a) |x+a\rangle $$ $$ X|y\rangle:=X \text{e}^{-ia\cdot P}|x+a\rangle = (x+a)\text{e}^{-ia\cdot P}|x+a\rangle=(x+a) |y\rangle $$ Now I want to show somehow that $|y\rangle=|x+2a\rangle$, but I don't see how, any hints? Thanks in advance.


title: What is the "inner" force behind attracion/repulsion?

text: First of all, I'm sorry for any grammatical big error - English is not my native language. I have a question that maybe does not have an answer beyond those which we already have, but maybe there is someone who can help me understand deeper than high-school level: what is exactly an electrical charge? I already know something is charged (negative or positive) when it attracts or repulses another object or particle. But, stoping at that point, the electrical force coming from a charged body seems like magic. I know it's analog to gravity: given at least 2 bodies with really big masses, they will one attract each other. So to friction/rubbing (similar forces to opposite sides)... But, why do all these happen? Sintetizing: what exactly produces electrical attraction or repulsion? How can a particle move itself in one or another direction depending its "charge"? Sorry for confusion... Thanks


title: List of freely available physics books

text: I'm trying to amass a list of physics books with open-source licenses, like Creative Commons, GPL, etc. The books can be about a particular field in physics or about physics in general. What are some freely available **great** physics books on the Internet? edit: I'm aware that there are tons of freely available lecture notes online. Still, it'd be nice to be able to know the best available free resources around. As a starter: http://www.phys.uu.nl/~thooft/theorist.html * * * jump to list sorted by medium / type * * * Table of contents sorted by field (in alphabetical order):   * Chaos Theory   * Earth System Physics   * Mathematical Physics   * Quantum Field Theory


title: Electric field at a point being an $n^{th}$ derivative of electric (or magnetic) field at some other point

text: This is a theoretical question for which i would like to know an answer with an example. I'd like to know if its possible to create a setup where the electric field at a point $P$ is $n^{th}$ derivative (for some number $n$) of the electric (or magnetic) field at some other point $Q$. Assumption : Continuous electric charge distributions exist, which means there is no need to base everything on an electron. PS: derivative is with respect to space or time. **EDIT : I am adding this in view of comments given below.** Assume that the derivative is with respect to time. My question is some what similar to this situation here, assuming an inductor connected to an ideal current source (AC), then the potential drop across the inductor is proportional to the derivative of the current through it.The only difference here is i need it in terms of fields (assuming any type of charge distributions (except distributions(math))) rather than in a circuit with lumped elements.


title: Does the Standard Model plasma develop a spontaneous magnetisation at finite temperature?

text: Reference: arXiv:1204.3604v1 [hep-ph] Long-range magnetic fields in the ground state of the Standard Model plasma. Alexey Boyarsky, Oleg Ruchayskiy, Mikhail Shaposhnikov. The authors of this paper claim that, at least in a certain regime, the standard model at finite temperature and density is unstable to the development of a long range magnetic field. They also claim that this field can be cosmologically significant but hasn't been accounted for in prior work. The argument runs as follows:   1. Finite temperature and density breaks Lorentz invariance to 3D rotational invariance.   2. This means that the photon vacuum polarization can develop a term $i \epsilon_{ijk} p^k \Pi_2(p^2)$ involving the 3-momentum $p^k$.   3. This term can be expressed as an effective Chern-Simons term in the Lagrangian, which they connect to the axial anomaly (I'm fuzzy on how this works).   4. Computing $\Pi_2(0)$ in an effective theory they show that for some range of fermion densities and temperatures the vacuum polarization develops a negative eigenvalue. The corresponding mode is unstable and condenses into a long range magnetic field. There are a number of technical points in the derivation that confuse me (mainly the chemical potential stuff) since I'm only just learning thermal field theory. But I have a few questions of a more general nature:   1. What is the status of this result? Does it hold up? Is there a precedent for this in other models? I've only found 2 citing articles on InspireHEP, though it is a fairly recent article so maybe this is no big issue?   2. How sensitive are the standard cosmological observables to a long range magnetic field in the early universe? Are there limits on the field present during, e.g. BBN?   3. Is this effect genuinely new? How could it be missed for so long?


title: Can someone please derive $T=2\pi\sqrt{l/g}$ or prove it without using calculus?

text: I don't know much calculus, but I want to know that how one derives the formula to find time period $T$ of a simple pendulum.


title: Electric charge and the distance

text: The strength of an electric field is: $E = 200\ \mathrm{N/C}$ The potential (of the test charge) is: $V = 600\ \mathrm{V}$ $\epsilon_r=1$ I need to calculate the distance between this point and the charge, the size of the charge. I think that the distance between this point and the charge is $E/V\implies E=V/m$, so it's $3$. To calculate the size of the charge, I used: $$ E=\frac{kQ}{r^2},$$ So, $$Q=\frac{E r^2}{k},$$ but I'm not sure what $k$ and $\epsilon_r$ are.


title: Would a submarine float in the atmosphere of Jupiter?

text: Would a submarine float in the atmosphere of Jupiter, or would it get crushed?


title: Gravitational wave energy

text: Electromagnetic energy can be related to it's frequency via $E=h\nu$. Is there a comparable relationship between gravitational wave energy and frequency?


title: effect of internal forces to a glider's descent

text: This is a continuation of a previous question, with more thinking applied, thanks to a helpful comment. Imagine a glider is airborne in a breezeless sky. The glider is 3 KM high, is traveling at 100 KM / hr, and is descending at a constant rate of 10 m per min. The glider weighs 200 kg. The sole occupant of the glider is a man weighing 100 kg. Assuming that nothing external to the glider would affect the rate of descent of the glider for a short time (such as a minute), what happens to the glider's altitude or rate of descent if the man jumps in the glider? He jumps straight up (perpendicular to the floor of the glider) to a height of 50 cm and drops back down to the floor. He jumps at the center of mass of the glider and lands in the same spot. Is there any effect to altitude or rate of descent by his muscles pushing off against the floor, and later landing on the floor? Is there any notable difference from the above action if the man jumps just as before, but as he reaches his maximum height of 50 cm, he grabs on a rope hanging from the ceiling that is attached at the center of the plane? He grabs on tight, instantly transferring his full weight to the rope. I'm mainly interested in a qualitative answer, but if you wish to produce a formulaic result, of course feel free. A comment was made suggesting that I consider a system with a trampoline resting on a scale with a man jumping on it. (Thanks to Mike Dunlavey.) Based on your analogy, here are my current thoughts about it. The weight on the scale with the man at rest is trampoline + man; when he pushes off, the weight would briefly increase by the amount of the force of his push-off, then would briefly be just the weight of the trampoline, and then would be roughly the weight of the trampoline + Man + downward force again. Applying this to the glider, I believe the effect on the glider would be a small drop in altitude due to the force of the jump. Then while the man is in the air, the now lighter glider would have a slight upward tilt due to it's continued speed, angle of attack of the wing, but temporarily without the man's weight. Then he lands in the glider, and the glider loses some more altitude and its angle tips a bit down too. I believe the net qualitative result, if the glider were seen from the side and its movement was accurately recorded, would be an interruption in a straight downward line. The first change would be a jog downward, then a small decrease in the slope of the line, then another jog downward and an increase in the slope of the line (making it a bit steeper of a descent). If the man grabbed the rope after jumping, then his weight would again be acting on the glider, causing a small increase in the downward slope of the plane, but this time only the man's weight and not the additional force of the man dropping by 50 cm would be applied to the glider. So, somewhat less of a downward jog in altitude than in the previous scenario. Does this reasoning seem sound, and the results approximately correct? Thanks for your help!


title: What is the status of Witten's and Vafa's argument that the QCD vacuum energy is a minimum for zero $\theta$ angle?

text: The argument, which I reproduce here from Ramond's `Journies BSM', is originally by Witten and Vafa in ($\it{Phys}$. $\it{Rev}$. $\it{Lett}$. 53, 535(1984)). The argument is that for $\theta = 0 $ (mod $2\pi$) the QCD vacuum energy $E(\theta)$ is minimized. Starting from the Euclidean path integral for QCD ( with just massive fermions charged under QCD, nothing else) in a volume V $ e^{- V E (\theta)} = \int \mathcal{D}A \mathcal{D} q \mathcal{D} \bar{q} exp \left( - \int d^4 x \mathcal{L}\right) $ where $\mathcal{L} = - \frac{1}{4 g^2} Tr (G_{\mu \nu}G_{\mu \nu}) + \bar{q}_i (\gamma^\mu D_\mu+ m_i) q_i + \frac{i \theta}{32 \pi^2} Tr(G_{\mu \nu} \tilde{G}_{\mu \nu} ). $ Integrating out the quarks we obtain: $ e^{- V E (\theta)} = \int \mathcal{D}A \det{(\gamma^\mu D_\mu+ m_i)} \mathcal{D} q \mathcal{D} \bar{q} exp \int d^4 x\left( \frac{1}{4 g^2} Tr (G_{\mu \nu}G_{\mu \nu}) - \frac{i \theta}{32 \pi^2} Tr(G_{\mu \nu} \tilde{G}_{\mu \nu} )\right). $ In pure QCD the quarks have vector-like couplings and so $\det{ (\gamma^\mu \mathcal{D}_\mu+ m_i) }$ is positive and real. For each eigenvalue $\lambda$ of $\gamma^\mu \mathcal{D}_\mu$ there is another of opposite sign. Thus $\det{ (\gamma^\mu D_\mu+ m_i) } = \prod_\lambda (i \lambda +M) = \prod_{\lambda>0} (i \lambda +M)(-i \lambda +M) = \prod_{\lambda>0} ( \lambda^2 +M^2 )^2 >0 $ Thus if $\theta$ were zero, the integrand would be made up of purely real and positive quantities. Now, the inclusion of the $\theta$ term with its i can only reduce the value of the path integral, which is the same as increasing the value of $E(\theta)$. It follows that $E(\theta)$ is minimized at $\theta = 0$. This is the motivation for axions where $\theta$ is promoted to a dynamical field which then relaxes to a vev of 0. Ramond adds there is `the slight caveat' that with Yukawa couplings, the fermion determinant may not longer be positive nor real. Witten and Vafa don't seem to make any similar caveat. Ramond's note about yukawas seems to invalidate the whole argument. Where does this leave axions as a candidate to solve the Strong CP problem?


title: Voltage and Current in transformers

text: In transformers, the ratio of the voltages equals the ratio of the turns - so double the output coil's turns and the output voltage doubles. Then, in order to conserve energy, current halves. This makes perfect sense in terms of P=VI, but what happened to V=IR? Doubling voltage and halving the current seems to completely contradict this basic law. That is, of course, unless the resistance in the output circuit changes, with R proportional to V^2 - but I don't see how this is possible. So how can a transformer obey both laws? Can resistance change or am I missing something else?


title: Avogadro's number

text: Could I get an explanation of Avogadro's number and how it relates to determining the mass of a substance? My chemistry textbook only serves to confuse me and the Wikipedia article is aimed towards a more wordy audience. I understand that Avogadro's number is $6.022 \times 10^{23}$, but I'm not sure how to relate that to determining the amount of atoms in a particular substance. Is it a direct proportion between the weight of C-12 and the weight of whatever substance you're attempting to determine the amount of atoms in?


title: 1D Wave mechanics (string)

text: Suppose I have a wave traveling to the right described by $e^{iw(t-{x\over c})}$. (It obeys the 1D wave equation). AND at $x=0$, there is a mass $M$ fixed to the string such that we have $M{d^2y\over dt^2}|_{x=0}= T{\partial y\over \partial x}|_{x=0_+}-T{\partial y\over \partial x}|_{x=0_-}$. How on earth do I find the reflected and transmitted waves?? Thanks.


title: What is the difference between material fatigue test with 1 sec cycle and 1 hour cycle?

text: Assume I have two pieces of metal. I want to test the material fatigue (e.g. how many cycles can the material stand before it will break). Onto one of them I apply a tension force every second. Onto the other one I apply a tension force every hour. Should the results have any difference?


title: Doubt on the analysis of $RC$ circuit

text: I'm studying $RC$ circuits and I'm in doubt on how to deal with one kind of situation. Well, first when we have simply a circuit with just one resistor and one capacitor in series, or a circuit that is equivalent to one of those simplest ones I know how to deal with it: we just use Kircchoff's loop rule and then it leads to a first order linear differential equation which can be easily solved: from here I obtain many results, like the final charge when charging is $Q\varepsilon$ and so on. However, there are some cases in which it's not possible to reduce to this simple case, and so I'm unsure on how to proceed. The example I've trying to tackle for instance is the same I've asked here: ![enter image description here](http://i.stack.imgur.com/cwNW7.jpg) Now, as I've been answered on that question I made it's not possible to reduce what's on the right to an equivalent resistor. I cannot work with impedance, just resistance and capacitance really. I've tried using Kircchoff's loop rule again, however the currents are different and I don't know how to tackle this. Can someone give a little help on how do we tackle this kind of problem? Thanks very much in advance.


title: Does the electron have spin in its own reference frame?

text: In our atomic physics class, we saw that the spin-orbit coupling term arises from the scalar product of the magnetic moment of the electron (proportional to its spin), and the magnetic field created by the nucleus rotating around the electron. But this implies that in the electron's reference frame, it still sees it's own spin? How is this possible?


title: How far is the range of an ultrasonic sensor as far as it can transmit?

text: How far can ultrasonic waves propagate in the air? (if the energy is enough), how far can ultrasonic waves propagate in the ground (concrete floor)? Are there any articles or experiments about attenuation of ultrasonic waves in different mediums?


title: What is the difference between Radiation and Electromagnetic Radiation

text: Are the two equivalent or is Electromagnetic Radiation a subset of Radiation. I am further confused by the fact that electromagnetic radiation includes both ionizing and non ionizing types of radiation. The closest answer I can think of is that Electromagnetic radiation is the set of all forms of radiation except alhpa and beta particles, because they are **not** massless photons. The only site which I found which supports this is this one. Could someone please give a definitive distinction of the two. I would also appreciate an exhaustive list of everything that is, and is not Electromagnetic Radiation.


title: Why don't alpha particles have magnetic moments?

text: As I understand, particles such as the neutron, whilst having no external charge still possess a magnetic moment due to the underlying charges of its components. By that logic why does the alpha particle have a magnetic moment of zero?


title: Confusing mirror problem

text: A piece of thin spherical shell that has a radius of curvature of 106 cm is silvered on both sides. The concave side of the piece forms a real image 79.5 cm from the mirror. The piece is then turned around so that its convex side faces the object. The piece is moved so that the image is now 35 cm from the piece on the concave side. How far was the mirror moved? Was it moved toward or away from the object? I'm not sure if the shell is the mirror or not, but this is what I have done: I took the radius of curvature divided by 2 to find the focal length. $f=53$. Then using $1/f=1/s+1/s'$: $1/53-1/79.5=1/s$, $s=159$. Now that I have $s$, I started working on flipping it to the convex side, which makes $f=-53$. I am not sure how to continue to find $s$ from here. In the end I believe taking the first object distance, 159, minus the second object distance should give me the distance the mirror has been shifted.


Why does the area of the plates affect the capacitance? Lets say I have a parallel plate capacitor with a charge of 10C and a potential difference of 5V. By the definition $C=Q/V$, the capacitance is 2 farads. Now if I increase the area of the plates, the charge definitely doesn't change. By the definition $V=kq/r$, the voltage doesn't change either. So why does the capacitance increase if we increase the surface area?


title: Why doesn't this metric cover all of de Sitter space?

text: This represents a confused attempt to work through a problem in Carroll's _Spacetime and Geometry_. Supposedly I should be able to use the geodesic equation, $$\frac{d^2x^\mu}{d\lambda^2}+\Gamma^\mu_{\rho\sigma}\frac{dx^\sigma}{d\lambda}\frac{dx^\rho}{d\lambda}=0$$ to show why this metric $$ ds^2=-dt^2+e^{2Ht}[dx^2 + dy^2 + dz^2] $$ fails to cover the entire manifold of de Sitter space. I am urged to combine these two equations to solve for the affine parameter $\lambda$ as a function of $t$, and then show that the geodesics of the space reach $t=-\infty$ in a finite value of the affine parameter, demonstrating what I sought to prove. I begin by parametrizing the metric in terms of the proper time, $\tau$ $(\lambda = \tau)$, to yield: $$-1 =-\frac{dt^2}{d\lambda^2}+e^{2Ht}[\frac{dx^2}{d\lambda^2} + \frac{dy^2}{d\lambda^2} + \frac{dz^2}{d\lambda^2}] $$ And from the geodesic equation, I derive: $$\frac{d^2 x^i}{d\lambda^2} = -2H\left(\frac{dt}{d\lambda}\right)\left(\frac{dx^i}{d\lambda}\right) $$ To solve any one of these three equations, say, for example, the first one, I make the subsitution: $$\nu = \frac{dx}{d\lambda}, t'=\frac{dt}{d\lambda} $$ then $$\nu'=-2H\nu t'\\\\\nu'/t'=\frac{d\nu}{d\lambda}\frac{d\lambda}{dt}=-2H\nu=\frac{d\nu}{dt}\\\\\frac{d\nu}{dt}=-2H\nu\Rightarrow\nu = C_1e^{-2Ht}$$ Substituting back into the metric, I'm left with: $$1 = \left(\frac{dt}{d\lambda}\right)^2 - e^{2Ht}\left(\sum_i C_i^2 e^{-4Ht}\right)$$ Define $\sum_i C_i^2 = \alpha$, yielding: $$ \frac{dt}{d\lambda} = \sqrt{1 + \alpha e^{-2Ht}} $$ I can't seem to find a nice analytical solution for $\lambda(t)$ that demonstrates that $t \rightarrow -\infty$ in a finite value of the affine parameter without relying on a computer, and this doesn't leave me with any nice physical (or mathematical) intuition for what's going on in the problem. Have I made an error somewhere, or is this, as Walter Cronkite (and my GR professor) would say, "the way it is"? Can anyone point me in the right direction, or show me how to proceed? Thanks in advance.


title: Speakers and Static from Hands

text: Why is it that when you touch the a bare male end of a speaker feed that the speaker makes hissing noises? Is it just (eddy?) currents running through you?


title: Find current in a nontrivial circuit

text: Let us assume that we have this circuit: ![Non trivial circuit](http://i.imgur.com/aMvsnUe.png) We know that: $$I_1 = I_2 + I_3$$ $$I_1R_1 + I_3R_3 = V$$ $$I_1R_1 + I_2R_2 + \dfrac{q_c}{C_1} = V$$ Therefore we can write: $$I_1 = k_1 - k_2*q_c$$ with: $$k_1 = \dfrac{V(1/R_2+1/R_3)}{1+R_1/R_2+R_1/R_3}$$ $$k_2 = (R_2*C_1(1+R_1/R_2+R_1/R_3))^(-1)$$ The question is, Can we write the equation: $$\dfrac{dq(t)}{dt}={k_1-k_2q(t)}$$ I doubt if $q_c=q(t)$ because $I_3$ does not flow through the branch which contains the capacitor, therefore: $$I_1 = I_2 + I_3 => q_1 = q_2 + q_3$$ I think $q(t)=q_1$ But if it is so $q_2 = q_c \neq q(t)$. How can I write down the differential equation in order to find $q(t)$?


title: How can I validate a numerical model using an analytical solution of the same scenario?

text: In the course of my research project I have been tasked with validating an acoustic model, however I am having trouble conceptualising what I am supposed to do. I have found an analytical solution for a given scenario:               A planar acoustic wave is incident on an infinitely thin sensor at some arbitrary angle.      **Analytical solution** I am able to determine the average pressure across the sensor for the given angle (this is time independent). **Numerical model** In my numerical model I have constructed the numerical analogue of the above situation, however my incident acoustic field is time varying. **What I would like to do** I would like to compare the average pressure across the sensor in the analytical and numerical solutions. I'm just not sure how I can compare a time varying spatially averaged pressure (numerical model) to the analytical solution. I'm pretty certain I'm missing something trivial here - but I cannot for the life of me seem to wrap my head around this. [edit] **Specific information about the scenario**   1. The amplitude of the plane, harmonic wave is unity and it is incident in the xz plane   2. The sensor is of finite diameter   3. The domain in the numerical solution is bounded by perfectly matched layers, however I have made the grid large enough (for now) that the acoustic wave does not interact with these yet. All boundaries perpendicular to the plane of the wave are rigid so as to (temporarily) avoid diffraction. To be more clear, as I understand it the analytical solution is computing the acoustic pressure across the sensor independently of time i.e. $$ P(x,y,z) = \int_{-\infty}^\infty ... $$ Whereas the numerical approximation computes it over some finite time i.e. $$ P(x,y,z,t) = \int_0^t ... $$ Given this difference, how can I compare one with the other?


Canadian astronaut Chris Hadfield's song Space Oddity is making news around the world today. It makes me wonder: **What are the limitations of performing music in space?** Clearly, there is no point to play music when there is no air. To make the question more meaningful, let's assume that we are inside a space craft where there is air. In such case: **Are there any musical instruments that would be affected by the lack of gravity (and in what ways)?** For example, Chris can be seen playing guitar and I assume it works just fine up there. The only example of a musical instrument that I could come up with which would not work, is a glass harp. Any other ideas?


title: Difference between ways of transmitting power

text: There are two ways to transmit the same amount of power, 1 amp at 1 million volts or 1 million amps at 1 volt. Conceptually what is the difference? How can I think about it conceptually? I would prefer it if an analogy were made.


title: Tension in a closed loop around an object, carrying stuff

text: Sorry when this sounds like a first year physics homework question, but it's actually not. It arose during a discussion regarding safe practices for rock climbing anchors... So here we go: Imagine you have a closed loop (a sling) around an object such as a tree. Let's for the moment assume the tree to be frictionless and a cylinder of radius $R$. You attach a load (via a carabiner, but it doesn't matter here) to that sling. It's pulling away from your tree with force $\vec{F}_l$. Now my question would be: What's the tension in the rope given the angle in the rope at the attachment point. Well... I know the carabiner in equilibrium isn't moving, so the force $\vec{F}_l$ must be balanced by forces coming from the sling. I can draw the parallelogram of forces. Let's assume I already know the angle that the sling forms at that point. My confusion stems from this: Because I have a closed loop, should there be some sort of "back action" from the tension in one leg of the loop on that on the other leg? The situation is clear when instead of a closed loop I have piece of rope attached to two points of attachment: Then I can work out the tension in each leg and know that this tension will be transferred to the points of attachment... Also, it should be obvious that the sling exerts force on the tree, but if I only look at tension, which runs in the direction of the rope, I can't really tell where that force is coming from. I must confess, back in my first year in physics I'm not even sure we properly did discuss more advanced situations regarding rope tension. It was always the good old frictionless pulleys introducing $90$ or $180$ degree turns... EDIT: Here's an attached image. The left one shows the setup I'm interested in, the right one shows a situation that I'd know how to solve... ![forces in slings](http://i.stack.imgur.com/9Ggky.png)


title: Floating a ping pong ball in the air just using a pen!

text: **Caution: Apparently this problem is harder than it seems!** * * * There is a well known phenomena, which I first learnt about when I was a 10 years old kid. You can levitate a ping pong(or whatever not-so-heavy) ball in the air by blowing throw an empty pen body or using a hair dryer. An interesting demonstration can be seen here. ![enter image description here](http://i.stack.imgur.com/Qi942.gif) While this demonstration is usually misleadingly explained using Bernoulli's principle (for instance look at this or other citation links from 62 to 66 here), it seems that it is explained using the Coandă effect. The Coandă effect is the tendency of a fluid jet to be attracted to a nearby surface. ![enter image description here](http://i.stack.imgur.com/SyXfE.jpg) I am looking for a simple(and humbly realistic) model which is capable of answering the following questions quantitatively: Assume the ball's mass is $m$ and its radius is $r$, the diameter of our pen is $d$, the volumetric flow rate of air through the pen body is $Q$ and the gravitational constant is $g$   1. What is the minimum of $Q$ which makes this demonstration possible?   2. Perhaps using this minimum rate $Q$ and reasonable numbers, how can I estimate the longest time that a normal person can levitate a ping pong ball in the air using his breath? I have noticed that the smaller $d$ is the longer I can levitate the ball. For a normal pen body, my personal experience is about 3 seconds.   3. An estimation of the maximum angle that the pen can make with vertical direction?   4. If we deflect the ball by a tiny amount, what will be the frequencies of motion along the vertical and horizontal direction?


title: For mesons, or baryons, do sea quarks contribute to the angular momentum of the bound state?

text: The total angular momentum of a bound state of quarks, such as a meson say, can be done by studying the spin and orbital angular momentum of the 2 valence quarks. What about the sea quarks why they do not contribute? or do they?


title: inductance value of a single turn thin circular coil

text: the question may seem simple but I haven't found any fitting formula yet. The problem is the following: consider a single-turn, circular coil made of reasonably thin wire (diameter of the wire much smaller than diameter of the coil). What is the inductance of this coil, based on the coil radius, and the wire radius? The answer could be either an analytical solution, a good approximation, or a formula with elliptic integrals (for instance) I found the analytical formula for the magnetic vector potential of an infinitely thin coil on wikipedia (here), but I do not have a good software to integrate the elliptic integrals properly; and moreover, this formula does not cover finite wire radius. To have some first trends, I checked the value of the magnetic flux of such a single-turn coil on a physics modeling software, and the value of the integral (with sufficiently resolved mesh) seems to depend strongly (enough) on the wire diameter (going from 4cm to 2cm diameter wire increased the magnetic flux by 20% for a 2m diameter loop, although we have 4cm<<2m!). And the other thing, that actually surprised me, is that, with a constant current value (1 amp), a bigger loop shows a smaller magnetic flux, although the area is much bigger. I think this last effect is linked to the fact that I used 130kHz frequency to calculate the flux integral, so this needs to be clarified further, I will update on that. EDIT 8 May: this is an error because the program handled badly the unit conversion from 4 cm to 1 m. As one might expect, the flux, and the inductance, is actually proportional to the radius. That's all I can tell. Does anyone have more insight on this? Thank you very much!


title: Why does the comb attract the pieces of papers if they're neutral?

text: When we rub our hairs with a comb, and then try to attract small pieces of paper, they're attracted by the comb. The pieces of the paper were not electrified before they were attracted. Then they might be neutral. Why does the comb attract the pieces of the paper if they're neutral, and have no influence of a specific charge on them (Pieces of paper).


title: Why does a supernova explode

text: This is really bugging me. When you look up some educational text about stars life, this is what you find out:   1. Gravity creates the temperature and pressure to start fusion reactions.   2. The fusion proceeds to heavier and heavier cores ending with iron, which remains in the centre of the star.   3. One moment, all light cores are depleted and the gravity wins over the power of the fusion reactions, now absent.   4. The core of the star collapses into high density object, which may vary depending on the star mass.   5. And the top layers of the star **explode**. And I just cannot find clear explanation why. According to what I imagine, the top layers of the star should just fall into the collapsing core. Is that because of the 3rd Newtons rule? Or do the stars have some need to end with a cool boom?


title: How do levers amplify forces?

text: This is really bothering me for a long time, because the math is easy to do, but it's still unintuitive for me. I understand the "law of the lever" and I can do the math and use the torques, or conservation of energy. or whatever... And I can see that a lever can amplify a force you apply to it if you apply a force on the longer side of the beam. If I were to look at the molecular lever and see what actually happens when I push on the lever, and I give acceleration to the molecules, how does it actually happen that more force is transmitted to the other side? Thank you all p.s I'm looking just for an explanation in terms of forces and acceleration, it's clear to me how to do this in terms of energy or torques


title: What are all the equations we use to calculate how bounces work?

text: I mean, what is the object's final displacement, or the function that describes the object's height over time (see **[1]** ) of an object thrown by a height $h$ with a speed of $\vec{v_0}$, a mass of $m$, a "bounce factor" of $\Lambda$ and the floor with a friction of $F_f$ and the air with a friction of $F_a$. **[1]** > ![Bounces](http://i.stack.imgur.com/e8ce1.png)


title: How can we be so sure the universe has 3 dimensions?

text: I'm just a layman here so please bear with me if I don't get all the words or theories by correct name or whatever. I hope it will be clear enough what my question is. Here goes: When I read articles or watch documentaries about Einsteins theories, M-theory, string-theory and so forth it always seems as it is common knowledge that we as objects in space stretches out in 3 dimensions: length, width and height. I get that this is very convenient on Earth when building book shelves and perhaps when calculating for space travel and such, that we plaster some sort of abstract grid onto space so we then can use geometry for calculating distances and forces and what not. What I DON'T get is how this is applicable when thinking about space in cosmology, to me it seems like the universe would need to stretch out ONTO this grid instead of BEING the grid for this way of thinking to apply. Discussion 1: If the universe would only be made up of two infinite small particles, it would be a 1 dimensional universe because we only have the distance between the two. If we were to add a particle it would be a 2 dimensional universe since each of the particles now have to relate to each of the other two. If we add another one we have a 3 dimensional universe, yes? If we add another particle it would be a 4 dimensional universe, and for each particle added a new relation to each of the other particles is needed, which means that we have as many dimensions as the number of particles minus 1. I guess this argument gets more complicated when adding waves and width of particles etc. But please correct me where I'm wrong. Discussion 2: Purely hypothetical now, if all the particles and everything that make up the universe would align on any one axis, they still have the POSSIBILITY to move away from that axis but just don't, doesn't that mean that the universe would be 2 dimensional? If not, just because of the possibility, wouldn't that mean that the above argument in discussion 1 is sound? This has been bugging me for a while that every where I look it seems like it is something we take for granted (just because 3 dimensions is handy on a blue print), and I never ever have seen or heard an explanation on why we think the universe has 3 dimensions. Then again, could be just me..


title: Intensity of Hawking radiation for different observers relative to a black hole

text: Consider three observers in different states of motion relative to a black hole: **Observer A** is far away from the black hole and stationary relative to it; **Observer B** is suspended some distance above the event horizon on a rope, so that her position remains constant with respect to the horizon; **Observer C** is the same distance from the horizon as B (from the perspective of A), but is freefalling into it. All of these observers should observe Hawking radiation in some form. I am interested in how the spectra and intensity of the three observations relate to one another. My previous understanding (which might be wrong, because I don't know how to do the calculation) was that if you calculate the radiation that B observes, and then calculate how much it would be red shifted as it leaves the gravity well, you arrive at the spectrum and intensity of the Hawking radiation observed by A. I want to understand how the radiation experienced by C relates to that observed by the other two. The radiation fields observed by B and C are presumably different. B is being accelerated by the tension in the rope, and is thus subject to something like the Unruh effect. C is in freefall and therefore shouldn't observe Unruh photons - but from C's point of view there is still a horizon ahead, so presumably she should still be able to detect Hawking radiation emanating from it. So I would guess that C observes thermal radiation at a lower intensity than B, and probably also at a lower temperature (but I'm not so sure about that). So my question is, am I correct in my understanding of how A and B's spectra relate to one another, and has anyone done (or would anyone be willing to do) the calculation that would tell us what C observes? References to papers that discuss this would be particularly helpful.


title: The relativistic mechanics of a battery that is being charged and accelerated at the same time

text: This might be an interesting question: Let's attach a battery into one end of an electric cable. Then we rotate the battery around, with accelerating speed, using 100 Watts of power, while the battery rotates around frictionlessly. (The battery moves in a circle whose radius is the length of the cable) At the same time we apply 200 Watts of electric power into the cable, this energy travels through the cable into the battery. We apply 100 Watts of mechanical energy into our end of the cable, and we apply 200 Watts of electric energy into our end of the cable, then the mechanical energy and the electric energy travel into the battery. The question: What is the speed of the battery after a very long time? ADDITION: Is this question too easy or too difficult? Naively I would think that after 1 Giga seconds the battery has 200 Giga Joules of electric energy, and 100 Giga Joules of kinetic energy. If an object has rest energy of 200 Giga Joules and kinetic energy of 100 Giga Joules, then the object can be said to be a non-relativistic object.


title: Do I understand measurement of dispersion relation in a solid correctly?

text: I'm currently doing an introduction to solid state physics course and have a quick question about measurement of the dispersion relation of phonons in a solid: The way I understood it is the following. One can look at the vibrations of the atoms in a solid in a quantum mechanical way and introduce phonons as a quasiparticle which is a boson. One can then derive the following laws of energy and momentum conservation for the interaction of photons with phonons: $\hbar \omega(q) = \hbar \omega_0 \pm \hbar \omega_{Ph} $ (Energy conservation) $\hbar q = \hbar q_0 \pm \hbar q_{Ph} + G$ (Momentum conservation) Where the subscript ph denotes the frequency / q-value of a phonon, and the subscript 0 denotes the frequency/ q-value of the photon after the photon and phonon interacted with each other. G is a reciprocal lattice vector. To sample the dispersion relation of the phonons in the solid one can now simply shoot photons at the solid and look at the out coming phonons. Solving the second equation for $q_{Ph}$ and the first one for $\omega_{Ph}$ (assuming we can properly measure the frequency and impulses of the out coming and incoming photons) then give us a relation between the q-values of the phonons and the frequency which is exactly the dispersion relation. Did I understand this correctly or are there any flaws in my reasoning? Cheers!


title: Angular momentum - maximum and minimum values for $m_{\ell}$

text: I want to work out the maximum and minimum values for $m_{\ell}$. I know that $\lambda \geq m_{\ell}$, therefore $m_{\ell}$ is bounded. In the lectures notes there is the following assumption: $$ \hat{L_{+}}|\lambda,m_{max}\rangle=|0\rangle \\\ \hat{L_{-}}|\lambda,m_{min}\rangle=|0\rangle $$ I think I understand this. Since the action of the ladder operators is the keep the value of $\lambda$ and raise (or lower) $m_{\ell}$, you cannot "go up" from $m_{max}$ or down from $m_{min}$. However, I do not understand why the result of the operation should be $|0\rangle$. It turns out we can write the produt of $\hat{L_{-}}\hat{L_{+}}$ as: $$ \hat{L_{-}}\hat{L_{+}}= \hat{L^2}-\hat{L_{z}^2}-\hbar\hat{L_{z}} $$ Then we we evalute the following expression: $$ \hat{L^2} |\lambda,m_{max}\rangle = (\hat{L_{-}}\hat{L_{+}}+\hat{L_{z}^2}+\hbar\hat{L_{z}})|\lambda,m_{max}\rangle $$ Since $\hat{L_{+}}|\lambda,m_{max}\rangle=|0\rangle $, then $\hat{L_{-}}\hat{L_{+}}|\lambda,m_{max}\rangle=\hat{L_{-}}|0\rangle =|0\rangle $. And $\hat{L_z}|\lambda,m_{max}\rangle = \hbar m_{\ell}|\lambda,m_{max}\rangle $. These two relations imply: $$ \hat{L^2} |\lambda,m_{max}\rangle =\hbar^2 m_{max}(m_{max}+1)|\lambda,m_{max}\rangle $$ Now I want to know how to compute $\hat{L^2} |\lambda,m_{min}\rangle $ since my lecture notes only state the result. My problem is that I will have $\hat{L_{-}}\hat{L_{+}}|\lambda,m_{min}\rangle$, but I can no longer say that $\hat{L_{+}}|\lambda,m_{min}\rangle=|0\rangle$. I tried to compute $\hat{L_{+}}\hat{L_{-}}$ and try to plug in the expression, but I had no success. How can I solve this? PS. This is not homework, I'm just trying to derive the expression stated in the lecture notes.


title: Why is it tension in surface tension?

text: Why is it called surface tension not surface compression?


title: Is there a way of measuring the spin along an arbitrary direction of a spin 1 particle?

text: I am familiar with the expression for spin 1/2 but haven't seen one for spin 1.


title: Why are two independent sources incoherent?

text: Coherent sources are produced from a single parent source. But, why are two independent sources always incoherent? Two sources can produce light of the same frequency. Then, I guess the problem is with phase. Are two independent sources always out of phase? why so?


title: Inertial Frames of Reference - Description of an Inertial Frame of Reference

text: An inertial frame of reference is described as being a frame of reference in which the first law of Newton (the law of inertia) holds. This means that all events as described with respect to this frame of reference must have a zero net force acting on it and therefore traces a straight line with a uniform non-translatory motion. But, I have read in some books, especially "Introduction to Special Relativity" by the well-known Robert Resnick, wherein his definition of an inertial frame of reference also refers to such a frame of reference as being an unaccelerated system. This is where I am confused.      How can we describe a frame of reference as being unaccelerated if we occupy the frame of reference itself? No mechanical experiment conducted solely confined to a single frame of reference can determine the absolute motion of the frame of reference relative to another frame of reference. All that can be understood is that there is a certain uniform relative motion between frames of reference and no more. Is Robert Resnick saying that the inertial frame of reference is unaccelerated with reference to another frame of reference?


title: Can a classical (or quantum) field, particularly the EMF, have a frame of reference?

text: I understand that a massless particle (such as a photon) cannot have a frame of reference. But the electromagnetic field does have mass; does it have a frame of reference? If so, I have a second question. If one takes a charge as "setting up" its own EMF, then the field's frame of reference is the charges, so the distinction doesn't matter. But if we take the EMF to pervade all of space, and charges merely to alter the value of the field at any given location, then it makes sense to ask about the frame of reference of the field per se. However the EMF varies from point to point; how could a volume of the EMF, which is not homogenous, have a single frame of reference?


title: Is there another way besides $V = k\int dq/r$ to find potential?

text: > Suppose that the charge distribution $q(r)$ in the radial $r$-direction in a > thin hollow disc of inner radius $a$ and outer radius $b$ is given by the > power-law equation: > > $$q(r) = Q_0 \left[1 + \mu\left(\frac{r-a}{r-b}\right)\right]^n$$ > > where $Q_0$ is magnitude of the charge at $r = a$ and is a constant, while > $\mu$ is called the charge-variation parameter which can either be positive > of negative; whereas $n$ is the power-law exponent. * For $n = 2$, what is > the electric potential $V (r)$ at any distance $r$ between the inner and > outer radii? Also, how is $V (r)$ affected by the variation of $q(r)$ as you > move from $r = a$ to $r = b$, related to the variation of $\mu$? What would > happen if $n$ is increased? Basically, all I know about relating $Q$ and $V$ is through: $$V(r) = k \int \frac{dq}{r}$$ But I'm not confident enough that this would be the right choice. Are there any other approaches other than what I wrote?


title: What do we mean exactly by the positive/negative charges in cathode and anode?

text: In batteries, what exactly do we mean by negative and positive charges? My understanding is that the negative charge of the anode is basically an atom with an extra electron in the last orbital and a minus electron in the last orbit of the positively charged atoms in the cathode? And so the electrons keep flowing and interchanging because the atom is trying to reach a stable state. Is this correct or am I wrong? If my understanding of positively/negatively charged atoms is correct, what do we mean by saying that the electron in itself has a negative charge?


title: How can I destroy earth with physics?

text: I want to destroy the whole earth using physics, I would like to learn some of the ways that can be used to achieve this. I tried using a nuclear bomb but it takes so long, and I can't wait that much: Why does it take so long to make a nuclear bomb? What are some physics experiments and theories to help me destroy the earth? Note that I have all the money and privileges I need.


title: current density in 1-d

text: I have a slight problem with the notion of the current density in one dimension. For example the probability current in 1-d given by: $J(x) = -\frac{1}{m} Im(-i\psi^*\partial_x \psi)$ calculation the total current $I$ from the density is achieved by integrating the density over some surface through which we would like to find the current. So what surface can i take in a 1-d system? or is it that in 1-d $J$ ~ $I$? thanks in advance


title: PhD or master in physics

text: I am 35 years old and I've just got a bachelor in Science. I will pursue higher education in Physics but I have my doubts if either a PhD or a master is more convenient for me. My areas of interest are Plasma-Fusion Physics, Condensed Physics, and Applied Physics. If I enter a PhD program, will take me around 6 years to complete it and probably I would be kind of old to take advantage of it...in the other hand If I choose the master (which is shorter in time) probably won't help 'cause Universities and Companies in these areas mostly look for doctors. These are my doubts and concerns. Please help with your expertise.


title: How to find different operator representations in QM?

text: I read that any observable operator may be represented as: $$\Omega = \sum_n \omega _n | \omega _n \rangle \langle \omega_n |$$ Where the little omegas are the eigenvectors/eigenvalues of the operator. Now, this definition is completely useless for doing any sort of calculation.   1. So suppose I want to find the position representation of the momentum operator. I have $$P = \int_{-\infty}^{\infty}dp \ p | p \rangle \langle p|.$$ In position representation, this becomes $$\langle x | P = \int_{-\infty}^{\infty}dp \ p \langle x| p \rangle \langle p|.$$ This is supposed to be equal to $-i \hbar \partial_x$. But how does one get there?    2. How to find out the overlap $\langle x| p \rangle$?    3. What additional input does one have to make?


title: resistance in a dc circuit

text: Herein is my question. On my gm vehicle, I replaced the dash gauges with aftermarket gauges. So, I replaced the battery light with a volt meter. I have burned up several alternators.. Have been told that by removing battery light, I also removed the resistance from the circuit going to the alternator and thereby causing the alternator to fail. Question,, how much resistance do I need to put in the circuit to make the circuit operational and not ruin the alternator? The vehicle is a 95 S10 with the CS130 AC Delco alternator. Will resistance also drop the voltage? I really need some help here and greatly appreciate the help that you can give me. I am really dumb when it comes to this circuitry. Thank you very much.


title: Are communications between computers faster by electrical signals via copper cables or electromagnetic signals?

text: Assume that there are two computers which are connected with a copper cable, e.g. Ethernet. Also, there is a radio connection between them, e.g. AM radio, in order to exchange data. When we try to send data from one to the other, which one would be faster? Assume they are the same distance away. As far as I know, the propagation speed of electromagnetic signals in the air is better than the copper. On the other hand, frequency of the AM signal is smaller. Is there any other factors which affects the speed?


title: Protons (as opposed to neutrons) to mediate nuclear fission?

text: I am just wondering why are protons (as opposed to neutrons) not used to mediate nuclear fission? Is it because it is charged, so we will have to input more unnecessary energy to overcome the Coulomb force?


title: Thermal propagator for a scalar field (KMS condition)

text: I'm having some troubles following the derivation of the scalar field thermal propagator. I'm following the article "Finite Temperature Quantum Field Theory in Minkwoski space" by Niemi and Semenoff (equations 2.24-2.28). I want to find $D(x)$ such that $(\square_c -m^2)D(x)=\delta_c (x)$ where the delta function and the square operator are defined on the Schwinger Keldysh contour (but I don't think this matters now). I have the ansatz $D(x)=\theta_c(t_x)D^>(x)+\theta_c(-t_x)D^<(x)$ and the KMS conditions $D^>(t-i \beta,x)=D^<(t,x)$ Now, the article just says that the solution for $D$ is (after Fourier transforming the spatial variables) $D(t,\omega_k)=\frac{-i}{2\omega_k}\frac{1}{1-e^{-\beta \omega_k}} \\{ [e^{- i \omega_k t}+e^{-\beta \omega_k+ i \omega_k t }] \theta_c(t) +[e^{i \omega_k t}+e^{-\beta \omega_k- i \omega_k t }] \theta_c(-t) \\}$ where $\omega_k=\sqrt{k^2+m^2}$ Now, I'm used to see this propagator with $T=0$ (which you get by Fourier transforming $D(x)$ and then closing the contour in the complex plane), and right now I don't understand how to get this. To be more specific, how do I put the KMS condition when I'm solving the equation for $D(x)$? Thanks


title: Path integral measure and symmetry

text: For a generic field theory the path integral measure is defined as, \begin{equation} \mathcal{D}\Phi = \prod_i d\Phi(x_i), \end{equation} where $\Phi$ is a generic field (i.e. it may be scalar, spinor, vector field). Now let us consider a coordinate transformation, \begin{equation} x \to x'. \end{equation} Now my question is: **In general, HOW to show that the measure remains invariant under the co- ordinate transformation if the co-ordinate transformation be a isometry of the underlying space-time?**


title: What is heat and how does it effect an atom?

text: What happens in the atom when it gets heated or cooled and turns into gas, liquid or solid? I just want to know how does heat affect an atom.


title: If the universe started again with exactly the same conditions... would it be the same?

text: Sorry if this is a basic question but I can't find anywhere to ask else to ask it: If the universe was created again and exactly the same conditions where in place at the beginning, would everything happen exactly the same as it has now. IE: Is there any true randomness?


title: Calculating two particle position and velocity with each other?

text: I've calculated the displacements and average velocities of both particles. $A$ average velocity $= 1.1180m/s$ displacement is $11.1803m$.   $B$ average velocity $= 0m/s$ displacement is $0m$.               When, approximately, are the particles at the same position?      I can't figure out how to find the when they are at the same position. I can look at the graph and tell it's around the $1.75s$ mark.               When, approximately, do the particles have the same velocity?       It's when both angles are the same, around $6s$. ![Graph of the particles.](http://i.stack.imgur.com/hTQS2.jpg)


title: Why no thermal expansion for camping tents?

text: I know that solid materials expand when heated, which is called thermal expansion. But what is happening to tent square or tent canvas of ( _plastic?_ ) camping tents? I have noticed, that during the day, when it’s hot, there is a lot of tension in the skin of a tent. But when it gets colder (when raining or during the night), the tension is less and the fabric gets sloppy. What is the reason?


title: Liquid oxygen how do they use it as fuel?

text: Rockets are said to be using liquid oxygen as fuel. How do they use liquid oxygen since it's just oxygen, it only helps in the combustion process. How can it be a fuel on its own?


title: What would happen if a piece of matter, in our atmosphere, were to suddenly vanish?

text: What would happen if an object were to suddenly disappear? I mean actually disappear, as in, the space occupied by that object were now completely empty- and a vacuum was left. If I was holding a basketball in my hand, outside my house, and it vanished. Would the atmosphere's pressure crash down in the one spot? Would there be an explosion? What would the result of such an event? EDIT: I heard something about matter becoming plasma at the center. Is this true? Would there be any physical damage to the surrounding environment?