title: Penrose's Zig-Zag Model and Conservation of Momentum

text: I was reading through Penrose's _Road to Reality_ when I saw his interesting description of the Dirac electron (Chapter 25, Section 2). He points out that in the two-spinor formalism, Dirac's one equation for a massive particle can be rewritten as two equations for two interacting massless particles, where the coupling constant of the interaction is the mass of the electron. In the Dirac formalism, we can write the electron field as $\psi = \psi_L + \psi_R$ where $\psi_L = \frac{1}{2}(1-\gamma_5)\psi$ and $\psi_R = \frac{1}{2}(1+\gamma_5)\psi$. Then the Lagrangian is: $$ \mathcal{L}=i\bar{\psi}\gamma^\mu\partial_\mu\psi - m\bar{\psi}\psi $$ $$ \mathcal{L}=i\bar{\psi}_L\gamma^\mu\partial_\mu\psi_L+i\bar{\psi}_R\gamma^\mu\partial_\mu\psi_R - m(\bar{\psi}_L\psi_R + \bar{\psi}_R\psi_L) $$ This is the Lagrangian of two massless fields, one left-handed and one right- handed, which interact with coupling constant $m$. He then pictorially explains his interpretation by drawing this interaction Feynman-diagram-style. The initial particle (L or R) travels at speed $c$ (with luminal momentum) until it "interacts" and transforms into the other particle (R or L) which also travels at speed $c$ **but in the opposite direction**. This "zig-zagging" of the particles causes the two-particle system to travel at a net velocity which is less than $c$, thus giving the electron a subluminal momentum, thereby granting it with a mass of $m$. He later states that this interaction is mediated by the Higgs boson, which we get if we replace the coupling constant $m$ by the Higgs field. I've tried to put this argument into a mathematical setting, deriving the massive propagator from the two massless propagators via perturbation methods, but what I can't seem to get around is the **conservation of momentum**. When a "zig" particle changes into a "zag", the direction changes and thus the new particle gains momentum out of "nowhere." If we involve Higgs, then the Higgs boson could carry away/grant the necessary momentum, but I want to know if the model can work without involving the Higgs. Is this possible?


title: How fast do large asteroids usually travel?

text: I have just watched this video made by discovery channel, and you can hear the narrator at 0:51 saying that : "even though it is moving at 720 THOUSANDS kilometers per hour..." I stopped once I heard this, because I never thought an asteroid would ever travel that fast. So, is it a mistake since in the beginning of the video someone said "an object moving 20 kilometers a second.." or it could be correct ? Also, if it is a mistake, then would a 20 km/s asteroid of that size cause the damage mentioned in the video ?


title: Compute an aerodynamic shape

text: I'm interested in building a Fairing for a recumbent bicycle. A fairing is just a rigid curved surface (usually plastic or fiberglass) that surrounds the bike in order to reduce wind drag. Apparently it can increase your speed dramatically (as much as 300% with zero wind relative to the ground). Questions of building the thing aside (which looks feasible actually), how can I compute the "most" aerodynamic surfaces to enclose a given space? In other words, if I have a given three dimensional "blob" (my body's range on motion + the bike itself) and a given direction of motion, what formulae and/or software can I use to give me some highly-aerodynamic shapes that are just slightly larger than the blob?


title: Singularities in Bianchi models in general relativity ( physical science)

text: what are the conditions to check point type singularity in a bianchi type model ? bianchi type model are of Type I,II,III,IX,IV or u can say we use different Bianchi type models having some specific metric or line element as i know .


title: Prof Lewandowski's paper about different space time for different particles

text: I read here http://www.sciencedaily.com/releases/2013/07/130709115344.htm that Prof. Lewandowski's work suggests that different particles may experience different spacetimes. His work is in the context of Loop Quantum Gravity. Sadly there is no reference to the original paper. Can anyone provide it?


title: Deriving the transformation under Weyl rescaling in Polchinski eq. (1.2.31)

text: I have another question in Polchinski's string theory book volume 1, namely how to derive Eq. (1.2.32)? $$(-\gamma')^{1/2} R'=(-\gamma)^{1/2} (R-2 \nabla^2 \omega) \tag{1.2.32}$$ I have awared his Eq. (1.2.21) for the transformation of world-line metric $$\gamma_{ab}'(\tau,\sigma)=\exp(2 \omega(\tau,\sigma )) \gamma_{ab} (\tau,\sigma)$$ , the definitions of Ricci scalar, $R:=R_{\mu \nu} g^{\mu \nu}$, Ricci, and Riemann tensors. It is still far from obvious how to construct the transformation of $R'$ from metric->Christoffel symbols->Riemann->Ricci-> (1.2.32) My question is, how to derive Eq. (1.2.32)?


title: Preservation of a scalar along geodesic trajectory

text: Let $u^\mu$ be the velocity of a particle , and $\xi^\mu$ be a killing vector. would taking a contravariant derivative of to scalar product $\xi_\mu u^\mu$ , and showing that it equals to 0 shows that this scalar is preserved ?


title: Stress tensor in a cube with shear forces

text: I want to calculate stress matrix in a cube with two faces parallel to x axis and perpendicular to z axis (sorry I don't know how can I put a picture in this post). There are two force uniform distributions (that we'll indicate with p) over this two surfaces: the superior is in the x direction, the inferior in the -x direction. So, I'll have only shear stress, and a shear-modulus $\mu$ dipendence. We suppose uniform stress tensor in the cube, because every infinitesimal dV of medium is in statical equilibrium with +pdS force with x direction by the superior infinitesimal dV and -pdS force by the inferior infintesimal dV for 3rd Newton's law. Remembering that any stress over a surface is $t_{ij}n{j}$, where $n_j$ is the normal versor to the surface, we must write: $T_{ij}n_1=0$ because we haven't any force on the surfaces perpendicular to the x-axis; so the first column is composed from three 0; $T_{ij}n_2=0$ because we haven't any force on the surfaces perpendicular to the y-axis;so the second column is composed from three 0 too; $T_{ij}n_3=p n_1$, because we have the force-distribution p over the surfaces that are perpendicular to z axis. $n_1$ is $\begin{pmatrix} 1 \\\ 0 \\\ 0 \end{pmatrix}$, $n_2$ is $\begin{pmatrix} 0 \\\ 1\\\ 0 \end{pmatrix}$, $n_3$ is $\begin{pmatrix} 0 \\\ 0 \\\ 1 \end{pmatrix}$. So the matrix $T_{ij}$ becomes: $$\begin{pmatrix} 0 & 0 & p\\\ 0 & 0 & 0 \\\ 0 & 0 & 0 \\\ \end{pmatrix} $$ But this haven't any sense, because stress-tensor must be simmetric for conservation of angular momentum. Where is the mistake?


title: How much CP violation does the weak force cause?

text: I'm looking for the most readable **_quantitative_** explanation, with the least amount of difficult mathematics. Rather than an answer with just a number, I'd like to see the derivation from the CKM matrix.


title: Why can't I slap an insect into unconsciousness?

text: So it is summer, which means I have to deal with insects. Today one has made it into my room and while I tried to work this insect was constantly in my face, which caused me waving my hands in the air. Now I have actually hit it 3 times but it kept coming back. Then I was wondering, why don't insects actually really mind when you just basically slap them with one hand? They don't fall to the ground, they surely don't die and they don't even seem injured in any way. If a giant as much bigger than me than I am compared to an insect were to just slap me, I would imagine I were at least severely injured. Is a human hand alone - not using a wall or another hand etc. to mash - really too slow for an insect to cause damage? Or does the insects "skin", which is made out of chitin, just give the insects this super power?


title: How does rolling resistance of rail wheel depend on diameter?

text: Freight train is more efficient than truck due to lower rolling resistance. And I wonder which one has lower rolling resistance<>, small diameter or larger one or it doesn't not depend on diameter at all? Both are steel wheel on steel rail.


title: Formalism to deal with discontinuous potentials in classical mechanics (hard wall, hard spheres)

text: It seems to me that Hamiltonian formalism does not suit well for problems involving instantaneous change of momentum, like particle collisions with hard wall or hard sphere gas model. At least I could not apply it straightforwardly to the simplest possible problem of 1D particle hitting a wall:                           │/ wall                 │/       particle  │/     ───o────────┼────────────> x                 │/                 │/                 │/      My attempt was quite direct. I took the Hamiltonian to be $$H = \frac{p^2}{2m} + U(x)$$ with potential $U$ defined as $$U(x) = \cases{ 0, \; x < 0, \\\\[.5em] K, \; x > 0, \\\\[.5em] E, \; x= 0.}$$ where $E$ is the particle energy and $K > E$. Hamiltonian equations should read as $$\cases{\dot x = \frac{p}{m}, \\\\[.6em] \dot p = - \frac{d U}{d x}.}$$ It is not hard to integrate the first equation, but my attempts to integrate the second one did not lead to any meaningful result (that's why I do not share them here, it was a complete failure). So I ask whether it is possible to obtain the solution to the problem by directly integrating Hamiltonian equations in the form above, without relying on general mechanical theorems/principles like energy conservation? Or is such an approach completely unsuitable for the task? If so, what is the general (and elegant) approach to such systems? There exist a related question on PSE "Hamiltonian function for classical hard-sphere elastic collision", but the setting is more cumbersome.


title: Do the laws of physics evolve?

text: Hubble's constant $a(t)$ appears to be changing over time. The fine stucture constant $\alpha$, like many others in QFT, is a running constant that varies, proportional to energy being used to measure it. Therefore, it could be agued that all running constants have 'evolved' over time as the Universe has expanded and cooled. Both the local and global curvature of the Universe changes over time implying that so too does the numerical value of $\pi$. All these things are however constants (well, let's say parameters since they are not really 'constant'.) In a discussion with astronomer Sir Fred Hoyle, Feynman said " _what today, do we not consider to be part of physics, that may ultimately become part of physics?_ " He then goes on to say " _..it's interesting that in many other sciences there's a historical question, like in geology - the question how did the Earth evolve into the present condition? In biology - how did the various species evolve to get to be the way they are? But the one field that hasn't admitted any evolutionary question - is physics._ " So have the laws of physics remained form-invariant over the liftetime of the Universe? Does the recent understanding of the aforementioned not-so-constant constants somehow filter into the actual form of the equations being used? Has advances in astronomical observations, enabling us to peer back in time as far back as the CMB, given us any evidence to suggest that the laws of nature have evolved? If Feynman thinks that " _It might turn out that they're not the same all the time and that there is a historical, evolutionary question._ " then this is surely a question worth asking. NB/ To be clear: this is a question concerning purely physics, whether the equations therein change as the Universe ages, and whether there is any observational evidence for this. It is not intended as an oportunity for a philosophical discussion.


title: NMR Analysis of Ethylene glycol

text: I have the spectrum obtained by an NMR analysis of Ethylene glycol : ![Structural formula of ethylene glycol](http://i.stack.imgur.com/CLm4Y.png) ![Ethylene glycol NMR spectrum ](http://i.stack.imgur.com/mZCdB.png) I would like to explain the number of peaks in this spectrum, and assign them to the proton spins present in the molecule. I would say that one of the picks is due to the OH groups, and the other one to the CH$_2$, but I don't understand very well the physics behind this. Could anyone try to explain me ? Thanks for your help (and sorry for my bad english)


title: Conversion of QCD cross section formula

text: I'm writing a program to calculate NLO cross sections for semi-inclusive high-$p_T$ pion production in proton-proton-collisions for my bachelor thesis. I've got a paper describing the production cross section as function of the differential cross sections of the partons $\frac{d\sigma}{dv}$ and $\frac{d\sigma}{dv dw}$ where $v = 1+\frac t s$ and $w = - \frac{u}{s+t}$, with $s$,$t$,$u$ being the conventional mandelstam variables. And I found a paper tabulating all the partonic cross sections. But unfortunately they use the formula $$d\sigma = \frac{1}{32 \pi^2 s} \frac{d^{n-1}p_3}{p_3^0} \frac{d^{n-1}p_4}{p_4^0} \sum M_{fi} \delta \left(p_1+p_2-p_3-p_4\right)$$ How do I calculate the conversion from $\frac{d\sigma}{d^{n-1}p_3 d^{n-1}p_4}$ to $\frac{d\sigma}{dv}$? I know in principle how to deal with differentials, and that the conversion should somehow begin with $\frac{d\sigma}{dv} \frac{dv}{d^{n-1}p_3 d^{n-1}p_4}$ but I don't have a good ansatz how to deal with the posterior part...


title: Why wet is dark?

text: When something gets wet, it usually appears darker. This can be observed with soil, sand, cloth, paper, concrete, bricks ... What is the reason for this? How does water soaking into the material change its optical properties?


title: Showing that the maximum possible uncertainty for any observable is half the difference between its maximum and minimum eigenvalues

text: Show that the maximum possible uncertainty for any observable is $\frac{1}{2}|x_2 - x_1|$ where $x_1$ and $x_2$ are the extreme eigenvalues of **X** (Maximize $\Sigma_i p_ix_i^2 - (\Sigma_i p_ix_i)^2$) with respect to $p_i$ subject to $\Sigma_i p_i = 1$ and each $p_i \geq0$ Using Lagrange multipliers $ f(p) = \Sigma_i p_ix_i^2 - (\Sigma_i p_ix_i)^2$ $ g(p) = \Sigma_i p_i$ -1 $$ d(f + \lambda g) = [\Sigma_i x_i^2 - 2(\Sigma_i p_ix_i)(\Sigma_i x_i)+ \lambda]dp_i = 0.$$ Choose $\lambda$ so that it is zero for the lowest eigenvalue. $\lambda = 2p_0x_0(\Sigma_i x_i) - x_0^2$ So for all i: $(x_i^2 - x_0^2) + 2(p_0 x_0 - p_ix_i)(\Sigma_i x_i) $ That's as far as I can see. I have made several attempts to factorise this , but I can't see it. I may have looked right past the solution, but here it is.


title: Equation for magnetic field due to a current in straight wire

text: How can I derive the expression for the magnetic field due to a long straight current carrying wire using the Biot-Savart law? The final expression should be: $$B=\frac{\mu_0I}{4πr(\sin(a)+\sin(b))}$$ Facing difficulty in finding how sine angles appear in the equation.Know only some of calculus


title: Time travel to the past in binary black hole system

text: I just saw this YouTube video (only 3 mins long) of Neil deGrasse Tyson explaining what happens when two black holes collide. He says that when the black holes are rotating around each other, there exists a path that you could follow that would cause you to travel back in time. Any chance that someone could explain this in an intuïtive way to me? I just finished my 1st bachelor physics, to give you an idea of my knowledge. I also understand the basics of special relativity.


title: Are temperature increases discrete?

text: Are temperature increases ever discrete in nature, or is it a continuous variable? If a discrete case exists, is there any material that exhibits particularly strange behavior?


title: Density matrix as a simple state

text: I computed eigenvalues and eigenvectors of a density matrix for state $a|0\rangle+b|1\rangle$. For eigenvalue $0$ for example, I obtain an eigenvector $(-b^*/a^*, 1)$ before normalization. Now I would like to normalize it. After finding the common denominator for normalization factor, I have $(a^*)^2 + (b^*)^2$ in it. The whole thing gives the correct answer only if I assume that $(a^*)^2 + (b^*)^2$ equals to $1$. However, I think this is generally not the case... Could you explain? ![ph](http://s29.postimg.org/6gbqgjxif/image.jpg)


title: What is the difference between a pulse and a wave?

text: I wanted to ask what is the difference between a pulse and a wave. According to the definitions of them , they are almost the same. In the websites I looked at , the difference between them was described by examples which not reflect the difference itself.


title: Is Dark Matter evenly spread out in the universe?

text: Is Dark Matter evenly spread out? If no, could we ever find a correlation between the amount of dark matter and matter in a specific place?


title: Most useful maths for theoretical and mathematical physics

text: I am going to apply for a programme of mathematical and theoretical physics for graduate studies and I'm currently studying maths. What is a good area to do a thesis (that is to say, considerable extra work and research) in? Here "good" means useful for my following studies and possibly making a good impression in the selection process for graduate studies.


title: What is making these subatomic honeycomb shapes and gaps in this graphene image?

text: I just read yet another graphene discovery and saw this STM image: ![enter image description here](http://i.stack.imgur.com/G7hsL.jpg) Nice photo, it begs two questions:   1. What is that secondary honeycomb structure, the one about 20x smaller than the atoms themselves?   2. What are those gaps in it? They're fewer than one per atom, and I assume any impurities would make the primary lattice be irregular.


title: The ridge at LHC

text: One of the results obtained by LHC is the following diagram for p-Pb collisions: ![enter image description here](http://i.stack.imgur.com/YFtle.jpg) I would like to understand what is actually depicted in the figure, what should we expect based on theoretical approach? what can explain the ridge?


title: Lax-Pair for principal chiral model

text: This question concerns Eq. (2.10) of the paper http://arxiv.org/pdf/hep- th/0305116v2.pdf by Bena, Polchinski and Roiban. In section 2.1 they are showing that the infinite number of conserved quantities for the principal chiral model \begin{equation} L = \frac{1}{2\alpha_0} \mathrm{Tr}(\partial_\mu g^{-1}\partial_\mu g) \end{equation} are given by the fixed-time Wilson lines $U(\infty,t;-\infty,t)$ where \begin{equation} U(x;x_0) = \mathrm{P}\, e^{-\int_{\mathcal{C}}a} \end{equation} and $a$ is a 1-parameter family of flat connections given by Eq. (2.3). My question is what becomes of the last two terms (i.e. $-a_0a_1 +a_1a_0$) in the second line of Eq. (2.10). Do they cancel? I don't see why the should because the $a$'s are non-commuting (Lie algebra-valued). Thanks


title: Enthalpy Change in Reversible, Isothermal Expansion of Ideal Gas

text: For the reversible isothermal expansion of an ideal gas: $${∆H}={∆U}=0 \tag1$$ This is obvious for the case of internal energy because $${∆U} = \frac {3}{2} n R {∆T} = 0 \tag2$$ and $${∆U} = -C_P n {∆T} = 0 \tag3$$ For the case of enthalpy it is easy to see that $${∆H} = -C_v n {∆T} = 0 \tag4$$ I've also seen $${∆H} = ∆U + ∆(PV) = ∆U + nR{∆T} = 0 \tag5$$ Now for the part I don't understand. $$dH = dU + PdV \tag6$$ $$dH = dU + nRT \frac {dV}{V} \tag7$$ $${∆H} = {∆U} + nRT \ln\frac {V_2}{V_1} \tag8$$ $${∆H} = 0 + nRT \ln\frac {V_2}{V_1} = \ln\frac {V_2}{V_1} ≠ 0\tag9$$ Clearly, it is incorrect to make the substitution $ P = nRT/V$ in going from $(6)$ to $(7)$. Why is that? I thought equation $(6)$ was always valid, and integrating such a substitution should account for any change in the variables throughout the process. Why does this not yield the same answer as $(4)$ and $(5)$?


title: Meaning of the chemical potential for a boson gas

text: My lecturer told me that the mu is the Chemical potential is zero or negative, in the following example, mathematically it acts as a Normalisation constant. But is there any Physical insight about why Boson gas can be zero or negative? I think it is due to the fact the Photon gas can crop up from nowhere (i.e. Vacuum fluctuation). ![enter image description here](http://i.stack.imgur.com/n1VVi.png)


title: Superselection sectors and $p+1$ form potentials in String Theory

text: we often come across the phrase superselection sector. I have couple of questions from that. First of all, I learned from some references about it (as a overview) and how is it related to superselection rules etc. There is of course this mathematical definition (from C* algebra, unfortunately I haven't gone through that), but I wanted to understand it physically. It seemed to me that basically if I construct two sectors (two hilbert spaces?) whose basis vectors don't cohere, then they belong to two different superselection sectors. My first question will be : Is that it or there are more into it than meets the eye! Please. Secondly, I encountered it in String theory. I will probably mention the place (I have an unrelated question here too!) where I encountered it in string theory. Its the argument using which we can guess the presence of D branes from SUGRA action. We see that in the low energy ST action, there can be a term (some p+1 form potential, properly coupled etc.) Now I saw this argument that "this term has finite tension and hence its not finite energy excitation above the vacuum" (why? and how did we guess about finite tension?- Is it because of the very fact that this term exists in the ST action?) and also its suggested that these objects (multi form potentials) are in different superselection sector. So again, please let me know if there is any general comment on superselection sector that you want to make in this context and also about the arguments.


title: superposition of two states with different number of particles

text: In an electron gas with no mechanism of production or destruction of electrons is it possible to prepare an state which is superposition of states with different number of electrons? $|\psi\rangle=|N=1\rangle+|N=2\rangle+\cdots$


title: Evaluation of QED amplitude with 1 external photon

text: I'm trying to compute the exact QED amplitude with one external photon. Suppose that the photon has 4-momentum $q$ and polarization $\varepsilon^\mu$. Peskin and Schroeder (p318) claim that ignoring the contribution from the external photon line we get the amplitude $$\langle\Omega |S|q\rangle = -ie\int d^4xe^{-iq.x}\langle\Omega|T\bar\psi\gamma_\mu\psi|\Omega\rangle$$ I have no idea how they derive this however. I've tried two ideas and would appreciate if somebody would review both. **Approach 1** The result would seem to follow directly from the definition of the $S$-matrix if we replaced the $\Omega$ on the RHS by the vacuum of the free theory, namely $0$. Have P&S just made a typo? **Approach 2** Using the LSZ formula I get something completely different, namely $$\frac{\sqrt{Z}}{q^2 + i\epsilon}\langle\Omega |S|q\rangle\epsilon^\mu = \int d^4xe^{-iq.x}\langle\Omega|TA^\mu|\Omega\rangle$$ It looks like I've got the VEV of completely the wrong field now! Have I applied LSZ wrongly? And if so what's my conceptual mistake? Many thanks in advance for your help!


title: Can parallel universes constitute the missing mass aka dark matter?

text: I was pondering about this question for a while now. And with my limited knowledge of physics I can answer myself - if parallel universe theory is true and if indeed there is gravitation leakage from 1 universe to the other, there would be infinite number of universes and thus unimaginable amount leakage would occur making life or formation of any matter impossible. Does that sound like a plausible answer? :)


title: How can I see this equation describes advection?

text: My notes say that this is the equation for the "advection of a quantity **A** at speed **v** ": $$ \frac{\partial \mathbf{A}}{\partial t} = \nabla\times(\mathbf{v}\times\mathbf{A}) .$$ Is this true? How can I see (mathematically) that this equation corresponds to advection?


title: How to measure electrical resistance?

text: Is it possible to theoretically measure the resistance of passing of electrical current through liquid electrolyte (distilled water + NaOH) ?


title: Ferrous object detection-ability dependent on size and depth?

text: The question came about when I was using a gradiometer to detect ferrous metal. From the datasheet they recommend a certain depth range for a larger ferrous metal buried in the ground. My question is, if I have an "infinite ferrous metal volume", would I be able to detect the metal no matter how deep it is buried? Or is there a finite volume which I would result in a finite depth detection-ability? From what I know, a larger volume ferrous metal would result in a bigger magnetic dipole moment. But is there a limit to this?


title: Relation for Dirac-spinors of different helicities

text: Assume that we have massless spin-1/2 particles. The Dirac-spinor is the solution of the Dirac equation: $$ p^\mu \gamma_\mu u_\pm(p) = 0, \quad p^2 = 0$$ The subscripts $\pm$ denote two different solutions, belonging to two different helicities. Is it possible to find a representation for the $\gamma_\mu$ so that the following relation holds true? $$ u_+ = \left( u_- \right)^* $$ I know this possible for (polarization) vectors, i.e. one may choose $\epsilon^\mu_+ = \left( \epsilon^\mu_- \right)^* $, but I guess it is not for spinors (in the weyl representation it obviously is not). So here is my **Question:** Is it possible to find a representation of the Dirac-Gamma- matrices so that the spinors of different helicities are related by complex conjugation?


title: Moon and Sun at the same time

text: Why do you see half of the moon when you can see both the sun and the moon from earth at the same time on some days?


title: protons and electrons

text: I am a novice in physics and a few things are not clear to me in electromagnetism: Consider the experiment of giving a piece of metal positive charge (which I assume consists of protons), by keeping a piece of rod with negative charge close to the metal and touching by your finger the farther side of it from the rod. Assuming that the electrons and protons are parts of the atoms, does this mean that the rod is actually taking the electrons away from the atoms? Is the magnetic repulsion of a rod enough to do so? If this is the case my second question is, if this happens then it means that the atoms of the metal do not have the same (say chemical) structure as before, since they have their electrons and protons taken away from them. but Apparently this has not happened to the metal. Can you explain this experiment with regards to actually what is happening to the structure of atoms of the metal?


title: How can one always be standing still when compared to the speed of light?

text: I was thinking if I built a device with 7 clocks, synchronized to each other, one in the middle, one up, down, left, right, behind and in front of me, say 1 meter away, and I fired a laser from the middle and split it into six pieces would the light always arrive at all six "corners" at the same time? If we accelerated (say to the right) up to half the speed of light and fired the laser again, wouldn't the laser have to travel farther to reach the right sensor (as it's moving at half the speed of light) and travel less distance (as it is approaching the light at half the speed of light)? So the time the light arrives at the right sensor should be greater than the time it takes to arrive at the left sensor. I'm assuming you will tell me that this is not true, the light will still arrive at all the sensors at the same time. Why would this be true? Isn't the whole apparatus moving at half the speed of light in one direction? Why would that make no difference? In that case, why couldn't I just accelerate to half the speed of light again, to the right? (Then I will be going at the speed of light - half the speed of light, twice, but in each instance I was considered to be standing still.)


title: Integral in regularization involving exp of a parameter in the denominator

text: Can someone please point me a way to compute $$\int_0^\infty\frac1{s+t}\exp\left(-\alpha t+\frac{t^2\beta}{s+t}\right)dt$$ ? How about the following one? $$\int_0^\infty ds\int_0^\infty dt\,\frac1{s+t}e^{-(s+t)}e^{-\frac{st}{s+t}}$$


title: Lorentz transformation matrix and its meaning in Minkowski diagram

text: I have sucessfully derived the Lorentz matrix for the boost in $x$ direction and its inverse. So I know how to get these two matrices: $$ \Lambda = \begin{bmatrix} \gamma & 0 & 0 & -\beta \gamma\\\ 0 & 1 & 0 & 0 \\\ 0 & 0 & 1 & 0 \\\ -\beta \gamma & 0 & 0 & \gamma \end{bmatrix} $$ $$ \Lambda^{-1} = \begin{bmatrix} \gamma & 0 & 0 & \beta \gamma\\\ 0 & 1 & 0 & 0 \\\ 0 & 0 & 1 & 0 \\\ \beta \gamma & 0 & 0 & \gamma \end{bmatrix} $$ What I lack is a geometrical meaning of those two. I have read, that it is some sort of a rotation, so I have already drawn the Minkowski diagram for steady _observer 1_ and one for the moving _observer 2_ as _observer 1_ sees it. Examples are drawn for $\beta = 0.5$. I have noticed that:   * _base cell_ is a square (picture 1) for a steady observer and a parallelogram (picture 2) for a moving observer   * light travels only diagonals of _base cells_   * axis $x'$ has coefficient $\beta$   * axis $ct'$ has coefficient $1/\beta$ **question 1:** Is there a way I can help myself with what I have figured out so far to show, how the Lorentz matrix deforms a square into a parallelogram and vice versa? **question 2:** I have heard of the invariant quantity $\Delta s$ in relativity for which I have stumbled upon an equation $\Delta s^2 = \Delta x^2 - c \Delta t^2$. Is it possible $\Delta s$ is an area of a base cell? **question 3:** Why is it so necessary for a scalar product to be defined with a minus sign for Minkowski spacetime? **picture 1:** ![Minkowski diagram for steady *observer 1*](http://i.stack.imgur.com/nmqKd.png) **picture 2:** ![enter image description here](http://i.stack.imgur.com/9KD3S.png) Thank you.


title: Question on acceleration of gravity

text: In pre-calculus last week my teacher explained a scenario in which a man throws a rock down a well and takes 4 seconds for you to hear the rock hit the bottom. How far downs the well? Last years I took regular physics and learned that distance equals Velocity x Time. So knowing that gravity pulls with 9.8m/s/s I assumed I would just multiply 4 by 9.8. Sadly I was wrong and my quick method of solving this problem was incorrect, we ended up using a very very large quadratic equation. My question Is why I was wrong in my answer? When I think about the problem assume I went wrong by using 9.8 as if it were a regular velocity when in fact it is a constant acceleration, so the speed is constantly increasing. If that is the case then can anyone explain mathematically how quickly gravity accelerates in a 4 second period and also what the easiest and most efficient way of solving this problem would be to your knowledge? Ps, this isn't homework I'm curious as to how this works


title: Comparing work done for two accelerations

text: Suppose at a time $t_{0}$ a car goes from 0 to 30mph at time $t_{1}$, and then maintains a speed of 30mph until $t_{2}$. After that minute it accelerates, reaching 60mph at time $t_{3}$. I want to compare the work done, particularly between $t_{0}$ and $t_{1}$, and between $t_{2}$ and $t_{3}$. The laws of work that I know are: $W = \int F \, dx = -\Delta U$. We don't know the force function and we don't know the potential energy function. We could use kinetic energy at each point and think that the only forms of energy are potential and kinetic so that they just trade off. In that case, it would seem that the kinetic energy goes from 0 to $\frac{1}{2}m(30^{2})$ so potential energy is the negative of that and work done is again $\frac{1}{2}m(30^{2})$. Then going from 30mph to 60mph you go from kinetic energy $\frac{1}{2}m(30^{2})$ to $\frac{1}{2}m(60^{2})$ which is a much higher jump in kinetic, meaning a much bigger change in potential, meaning a much larger work done. Is that basically right?


title: helium balloon tied to a car

text: A _helium_ balloon is tied to the seat of a car. The doors and windows of the car is closed. If the car now starts moving, in which direction will the balloon move- front or back?


Frictional force between solids operates even when they do not move with respect to each other. Do we have viscous force acting between two layers even if there is no relative motion?


title: Intuition of Impulse Formula $J = \sum F \Delta t$

text: I understand that $$\begin{align} J = \sum F \Delta t &= \Delta p \\\ \sum F &= \frac{\Delta p}{\Delta t} \\\ &= \frac{mv_2 - mv_1}{\Delta t} \\\ &= m \cdot \frac{\Delta v}{\Delta t} \\\ &= ma \end{align}$$ > $J$ is impulse, $p$ is momentum However, by just looking at the equation $$J = \sum F \Delta t$$ I seem to think that impluse is high if for same force, time is higher. But I think that's wrong? I think for a certain force, if I exert it in a short time theres a larger impluse? How can I then understand the above equation more intuitively without expanding it out?


title: Experimental foundations of the "otherwise galaxies would not hold together" argument for dark matter

text: On the question of motivation and evidence for dark matter, there are some illuminating (so to speak) answers on this site. I'd like to understand in detail the foundations of one of the lines of reasoning, as stated succinctly in this answer: > ...stars within galaxies are moving faster than the escape velocity of the > gravitational pull of the center of mass of the galaxy... If I were to attempt to unpack this, I might say something like the following:   1. There a formula for escape velocity, based on on the classical laws of physics, which should be valid for measuring the rotation of a galaxy (ie, general relativity correction would not change the conclusion we are after?). $$ v = \sqrt(2GM/r) $$   2. However, there are other techniques for measuring `M`, `v`, and `r`.     * `M` can be measured indirectly via the mass-luminosity relationship     * `v` (here the speed of the outermost stars in the galaxy) and `r` (the radius of the galaxy) can be inferred, via the Doppler Effect, by measuring the galaxy's red-shift (and some other measurement, I suppose?).    3. The velocity formula does not add up when the measurements are plugged in. Therefore, we can assume that either one of the measurements is incorrect, or the escape velocity formula is incorrect.   4. The most likely candidate is that the measurement for mass is incorrect, specifically because the mass-luminosity relationship is invalid. The theory being that some mass doesn't interact with light. Am I roughly correct that the above is the argument for the existence of dark matter? I'm looking for the building blocks of the argument, its foundation in terms of measurements and theories. (And I do understand that the case for dark matter lies in a convergence of several lines of evidence, but I'm attempting to wrap my arms around just this one for now.)


title: The relation between the action of tunneling and the energy

text: In the semi-classical physics, the probability of the penetration through a barrier is given by $$ p \sim \exp \left( - A_{0} (E) \right), $$ where $A_0$ is the imaginary part of the action and $E$ is the energy. Now, my problem is that I do not understand the following relation: $$ \frac{\partial A_0}{\partial E}= -2 \tau_0 $$ where $\tau_0$ is the imaginary time of the motion under the barrier between two points. I would appreciate if someone could explain why the last relation holds. Thanks,


title: Why does earth's magnetic field change its direction?

text: I know that the earth's magnetic field is due to molten metal and ions flowing in its core. It has been found that the Earth's magnetic field changes its direction and it has even flipped once! But, in order for this to happen the molten ions flowing in the core should also change its direction. What is causing this change in direction of the ion flow?


title: How to find the required inlet velocity/pressure to achieve 0.5 m/s in one part of the underground tunnel?

text: I have this problem here related with tunnel and ventilation. As you can see at the picture below, it is a top view of the tunnel. The tunnel has an inlet and an outlet. The air flows from the inlet on the left and exits at the outlet on the right. There is one branch/drift to this main tunnel road (it's with the label "0.5 m/s"). At the end of this 0.5 m/s drift, it is required for the airflow to achieve a velocity magnitude of **0.5 m/s at the end of the drift**. The pressure at the outlet surface is also **already defined as 120 Pa**. My question is, how to find the required inlet air pressure or velocity for achieving that 0.5 m/s at the end of the drift? My fluid dynamics knowledge is pretty bad so I need your help please. The dimensions and the figure of the tunnel is well illustrated in that picture. I will be grateful for any help that you could provide. ![Illustration of my problem](http://i.stack.imgur.com/vCAPp.png)


title: How to calculate the differential scattering cross section?

text: I am doing some simulations with the Meep code to study some properties of a metal nanoparticle. The situation is this: an incident electromagnetic wave on a metal nanoparticle. By now I know how to calculate the scattering, absorption and extinction cross sections and their efficiencies but I need the scattering angular distribution, that's why I need to calculate the differential scattering cross section and now I am a bit lost. Is there a way to relate the cross section to the differential cross section?


title: Consistent histories and Bohm mechanics, many worlds in disguise?

text: This was posted on here in someone's Phys.SE answer: _No, in the many worlds interpretation, every parallel universe is real, but in consistent histories, once you choose your projection operators, only one possibility is real, and all the others are imaginary. This makes consistent histories a lot more like Bohmian mechanics with the world the Bohmian particle sits in being more real than the rest. Why should one world be more real than the others? There is no reason. T[w]o copies of you living in a parallel world, they are more real than you are._ If as he says, each copy of you believes they are more real than the other, is he saying CH and BM are many worlds in disguise? or is there no copies of you in CH and BM?


title: Differences of behaviour of a particle in a box in quantum theory between that in classic physics

text: Can anyone help me enlist 3 major differences between the quantum and classical physics of the behaviour of a particle in a box? I would like some insight into the differences without solving PDEs because I am not a physics student. Thank you very much


title: Does cosmological horizon grow or decrease as it radiates?

text: Ron Maimon in many posts claimed that cosmological horizon is like a big black hole. Black holes decrease as they evaporate and their radius decreases as well. So what is with a cosmological horizon? If cosmological horizon is just a black hole centered at the opposite side of the universe, we should see the radius of the external space groving as the radius of the BH decreases. But if the radiation decreases the horizon entropy we should see the area of the horizon decreasing. What is the correct conclusion?


title: Capacitor related query

text: Why does a capacitor charge only upto the voltage of the source? I mean in an r.c. circuit if we have a resistor that is in series with the capacitor then because of the potential drop there will be less amount of current flowing to the capacitor. So shouldn't its final steady state voltage value be equal to the potential drop across the resistor? And on the capacitors its written 63 volts. So I guess it can withstand a voltage of that value. So if its steady state voltage is not equal to the potential drop across the resistor then why isnt it equal to the max value that is 63 volts. I know its a stupid doubt, but any help will be appreciated. The voltage of the source is 12 volts.


title: Which force is responsible for victory in tug of war?

text: I'm confused: In a tug of war game on rough ground who will win? The one who applies greater force on the rope or the one who applies greater force on ground?


title: Angular magnification of a two-lens system

text: Suppose an object is a distance $d_o$ from a first lens (which has a focal length $f_1$), where $d_o>f_1$. Behind the first lens is a second lens with a focal length $f_2$, such as to produce a virtual image a distance $d_i$ away from the second lens. How would I find the angular magnification of this system? (Let L be the separation of the two lenses).


title: Fractional-Order Moments

text: Commonly seen in physics(and statistics) are the concepts of moments of order zero(mass), one(center of mass), and two(moment of inertia). In statistics a third moment (referred to as skewness) also exists and is used. Actually, mathematically the moment can (of order n) be simplified to an operation in the form of: ![enter image description here](http://i.stack.imgur.com/TiCfp.png) Once I look at this equation it begs the question: what if n is a ratio. What could be some physical applications of this fractional-order moment?


title: Why does light in a room not form constructive and destructive interference patterns?

text: This is something that I have wondered for a long time. How come when I walk around why do I not see random black spots where light has collided destructively and bright spots where it has collided constructively?


title: The effects of nuclear warhead detonations on the Earth's environment and global warming

text: For years I have been hearing all sorts of conflicting information about global warming. Many say that scientists have proven that humans have been warming the Earth significantly with their actions. Specifically with the release of carbon into our atmosphere. Others say that scientists have found other causes. Sources such as some animals or plants. Natural wobbles in the Earth's orbit around the sun. Increased activity from the sun due to normal cycles in the Sun's output. It goes on and on and on. Recently I was watching a documentary on the creation and testing of nuclear bombs. These bombs seem to constantly increase in power and destructiveness. Between the United States and Russia, along with other countries, there have been hundreds of insanely powerful nuclear bombs that have been detonated for testing purposes (and a few other purposes). My question is whether or not anyone has studied what effect, if any, this could have had on global warming. I would think such detonations would have a much more significant impact than carbon being slowly released into the atmosphere. Given how intense and wide reaching such things were. I am not saying that this has caused all the problems. I am just curious what effect they could have had. I think that a lot of people always want to find the silver bullet. The one cause/cure of/for a problem. In the case of global warming we keep looking for one source. Those that want to believe mankind is causing it find a magic cause of it. Those that can't accept that possibility look for causes outside our control. I think the truth is somewhere in-between. Our planet is a very complex and dynamic thing that is affected my millions of different things.


title: Convert acceleration as a function of position to acceleration as a function of time?

text: Suppose I have acceleration defined as a function of position, $a(x)$. How to convert it into a function of time $a(t)$? Please give an example for the case $a(x)= \frac{x}{s^2}$.


title: LQG Demystified Book?

text: Does a similar easy to read introduction like David Mc Mohan's Demystified books exist for LQG? I'd like something easier to start with than a paper but deeper than a popular book.


title: Degeneracy and the Hamiltonian

text: How many linearly independent eigenfunctions can be associated with one degenerate eigenvalue of the Hamiltonian operator? (Is there a limit since it contains a 2nd order differential operator?) Thanks.


title: Basic question on bra-ket notation

text: Which of the following corresponds to a $ \psi(x)$, a wavefunction written in the position basis: $ x| \psi\rangle $ or $ \langle x| \psi\rangle $? If it is the second expression (which my textbook asserts), what is the meaning of the first expression?


title: Do superheating and supercooling only occur from liquid?

text: Superheating of a liquid past its boiling point (retarded boiling) is a well- known phenomenon. However, is there such a thing as supercooling of gas past its condensation point? Conversely, supercooling of a liquid (retarded freezing) is also well-known. Does the reverse ever occur: can I superheat a solid up past its melting point?


title: How do I determine resistivity from electron defects of high purity gold?

text: I am trying to create a plot for the electrical resistivity of high purity gold from 1 K to 1000 K. I found gold's resistivity at 300 K using the Wiedemann-Franz Law based on thermal conductivity data I was provided. I also have a table that has the resistivity of gold at 295 K By Mathiessen's rule, the resistivity is $$r_e=r_{e-ph} +r_{e-d}$$ and I can find $r_{e-ph}$ at each temperature using Bloch's formula $$r_{e-ph}=4 r_0 (\frac{T}{\Theta_D})^5 \int_0^{\frac{\theta_D}{T}}\frac{x^5e^x}{(e^x-1)^2} \,dx $$ I figured I could solve for $r_0$ using the resistivity value at 295 K from the table. The I could solve for $r_{e-ph}$ at all the temperatures using Bloch's formula. The only thing I don't know how to determine is the $r_{e-d}$. When I don't account for the defects, I get the shape I am supposed to get, but the values at the low temperature region are way too low because that is where the defects dominate. Any help would be appreciated. The only other data I have is the electron scattering rate at each temperature since I have the thermal conductivity at each temperature.


title: What is a single word that describes the idea of the second time derivative of energy?

text: I think about position, its time derivative speed, and its second time derivative, acceleration. I would like to identify a single word that can be used as a handle for the second time derivative of energy (i.e., the time derivative of power). If there is a widely used term, I'd prefer to use it. If not, I'd like to get your suggestions as to what a term might be. Any ideas?


title: Has the existence of anyons been experimentally verified?

text: I've been wondering whether there has been any experimental evidence for the existence of anyons or are they just objects of purely mathematical interest?


title: Fourier transform with periodicity at the harmonic frequency

text: Let's suppose I have a signal $F(t)$ that is periodic, with two periodicities $P_1$ and $P_2$, with $P_1 > P_2$. Suppose that I know the values of the two periodicities. Using the Fast Fourier transform I can show the two values as peaks in a power spectrum. Now, let's suppose the second periodicity $P_2$ (the faster one), has exactly the same value as the first harmonic of the fundamental value, or $P_2 = 2\times P_1$. This means that I will be not able to distinguish it by using the power spectrum, at least not by looking at the frequency of the peak. My question is: is there a way to separate the contributions in such a case? For example, is it possible to predict the power of the first harmonic, so that the difference between the predicted power and the observed power of the harmonic peak gives a result significant enough (i.e., greater than $3\sigma$) to say that the first harmonic also "contains" the contribution from a periodicity?


title: Protons and Electrons Occupying the Same Space

text: When it comes to atoms electrons can't fall into the nucleus, which besides the off hand uncertainty explanation, I'm not sure which force prevents them from falling into the nucleus. I thought I heard it was the strong nuclear force-but is there a similar force preventing protons and electrons, outside of atoms, from falling into each other? I figure there must be otherwise the force will become infinite… I'm lost.


title: Why are noble gases used for lights?

text: I know that neon is used in advert signs due to its inertness. However, I am not entirely sure how the inertness is exploited. I think it is because Ne being inert means that after electricity frees an electron we can count on it returning to the ion left behind to emit a photon, whereas for more active elements, the gas may just exist as a plasma?


How does nature prevent transient toroidal event horizons?.. and does it really need to? **Steps to construct a (transient) toroidal event horizon in a asymptotically flat Minkowski spacetime:**   1. take a circle of radius $R$   2. take $N$ equidistant points in the circle.   3. consider tangent lines on each equidistant point, label the infinite directions on each tangent as their clockwise or counterclockwise direction relative to the circle   4. pick an orientation (CW or CCW), and then throw black holes of radius $r \approx \frac{\pi R}{N - \pi}$ from each tangent line from the asymptotic infinite. Choose the tangential momenta which they are sent to be $p$ When all the black holes arrive at the circle in time $t_0$, their event horizons become connected. Even assuming that nature is abhorrent to this event horizon topology, it will take at least $t= \frac{R}{c}$ for the event horizon to reach the center of the circle. So there is "plenty" of time for causal curves to pass through the inner region and reach infinity How does the topology censorship theorem avoid this to happen?


title: Pion Decay and Fractional electric Charge disappeared, why?

text: ![Pion Plus Decay](http://hyperphysics.phy- astr.gsu.edu/hbase/particles/imgpar/pidec1.gif) Since the charged pions decay into two particles, a muon and a muon neutrino Fractional electric Charge disappeared, why? The decay proceeds by the weak interaction $W^{+}$ and can be visualized in terms of Feynman diagrams. my suggestion is that the Fractional electric Charge disappeared because it does not exist at all in the Quarks mechanism.


title: How 'Faster Than Light' Neutrino Was Product of Loose Cable at CERN?

text: A "faster than light" neutrino discovery was actually the result of a loose cable. A fiber-optic cable in a GPS receiver at the European Center for Particle Physics ( CERN) near Geneva Can someone show me the light here, What's Going On ? how long was fiber-optic cable?


title: Why exactly are images formed by lenses/mirrors?

text: I just don't get the concept behind why a lens or a mirror forms a reproduction of the object at present. Is it to do with the object blocking parts of the light source? I just don't understand why an image of that object is produced. The question is essentially trying to understand **what causes the image to be produced from the object, in the first place.**


title: Interaction between a Pair of Particles

text: We consider a particle, A receiving energy from a second one,particle B in a one dimensional collision. $$E^2=p^2+m_0^2$$   $$EdE=pdp$$ For particle A: $$E_AdE_A=p_Adp_A{\;\;\;\;\;\;}(1)$$ For Particle B: $$E_B dE_B=p_Bdp_B {\;\;\;\;\;\;}(2)$$ Now $$\mid dE_A\mid=\mid dE_B\mid{\;\;\;\;\;\;}(3)$$ [Since enegry lost by one particle is gained by the other] From Conservation of linear momentum we have: $$\vec{p_A}+\vec{p_B}=\vec{K}$$ where $\vec{k}$ is a constant vector. Now, $$d\vec{p_A}+d\vec{p_B}=0$$ Or, $$d\vec{p_B}=-d\vec{p_A}$$ Or, $$\mid d\vec{p_B}\mid=\mid d\vec{p_A} \mid {\;\;\;\;\;\;}(4)$$ Applying relations (3) and (4) to (1) and (2) we have: $$\frac{E_A}{E_B}=\frac{\mid \vec{p_A} \mid}{\mid \vec{p_B} \mid}{\;\;\;\;\;\;\;\;}(5)$$ A pair of particles cannot interact unless relation (5) is satisfied. Can we conclude that that relation (5)to be a restriction for 1D collisions? Now let's move to the general type of 3d collisions between a pair of particles A and B. A frame is chosen where the particle B is initially at rest in it. We may write: $$E_BdE_B=\vec{p_B}\cdot d\vec{p_B}{\;\;\;\;\;}(6)$$ If the particle B is initially at rest the RHS of (6)is zero. But the LHS cannot be zero unless $dE_B=0$. How does one get round this problem?


title: What are cosmological "firewalls"?

text: Reading the funny title of this talk, Black Holes and Firewalls, just made me LOL because I have no idea what it is about but a lively imagination :-P (Sorry Raphael Bousso but the title is just too excessively funny). Since this was a session talk targetted at the experts, I did not dare to watch the video, but still I am curious to know a little bit more about it. So can somebody explain to me what firewalls are used for in cosmology? What do they have to do with black holes and how are they related to complementarity? **Edit to include followup questions** From just reading the abstracts of the papers the first two links in Lumo's answer point to, I'm wondering if such firewalls would contradict the holographic principle? In one paper they say holographic spacetime has no firewalls but what about the converse line of thought? And what happens to the microscopic degrees of freedom of the black hole, would the firewall change the description of them?


title: Universal gravity at small distance

text: Could it be that there is simply a maximum gravitational force that two bodies of finite mass can exert on one another? This would occur at $r=0$, so maybe there is some really really really small $a$ in the universal gravity equation, making it $$F=G\frac{m_1m_2}{r^2+a}$$ If $a$ was small enough, it would only become apparent at extremely small distances. So as $r$ got smaller, the force would approach it's actual maximum which would be proportional to the two masses rather than just approaching infinity. Feel free to tell me in layman's terms why that idea is no good, but I haven't had calculus in like 15 years so I doubt I will understand anything too complex.


title: Exotic Matter -- What is it?

text: I have a conceptual understanding of some theoretical physics concepts, so without getting into the math, what is exotic matter? I have read some articles that say it is anything not normal, while others say it has a negative mass (is that antimatter?). I got interested when I saw the article on potential for 'warp drive' by bending space-time.


title: Is a particular force different in different frames

text: Can a particular real force have different magnitude in different frames?


title: Modified Arrhenius equation source? year proposed?

text: I was curious to know when was the first time (which year) the modified Arrhenius expression was proposed: $k=A(T^n)e^{\frac{-Ea}{RT}}$. I know that the simple Arrhenius equation $k=Ae^{\frac{-Ea}{RT}}$ was proposed by Svante Arrhenius in year 1889. Did he also propose the modified form in that year? Or someone else did? I searched but could not find any references.


title: Sunrise time across the globe?

text: This question is more astronomy related, I started thinking when I heard from one of my friends living in Jiamusi, that Sun rise at 02:00 and sets at 14:00, I know that this place is more near to north pole but why sunrise time so different there? I appreciate your time and answers.


title: From quantization under external classical gauge field to a fully quantized theory

text: Let me take QED for example to clarify my question: The textbook-approach(at least for Peskin&Schroeder) to quantize ED is to first quantize EM field and Dirac field as free fields respectively, and then couple them together perturbatively to represent the interaction, and we will have a fully quantized theory. Here by "fully quantized" I mean both EM field and Dirac field are quantized. On the other hand, we may quantize the Dirac field under an given external classical EM field: briefly speaking, one solves minimally coupled Dirac equation and take the solution space as 1-particle Hilbert space, and then second quantize the theory by building a Fock space based on this 1-particle space.(A more detailed description can be found in chap 10 of "the Dirac equation" by Thaller.B) Such a theory is defined non-perturbatively, and describes many-electron systems interacting with an external classical EM field but non-interacting among themselves, so in this sense it's a "better" theory than a quantized free Dirac field. However I cannot see a way to go from this theory to a fully quantized one, if there is, do we get exactly the same theory as given by the text-book approach, or possibly a better one? Any answer, comment or reference will be appreciated.


title: What is the difference of the two stable states of a Flip-flop?

text: > In electronics, a flip-flop or latch is a circuit that has two stable states > and can be used to store state information. [...] Flip-flops and latches are > used as data storage elements. Such data storage can be used for storage of > state, and such a circuit is described as sequential logic. Source: Flip-flop (electronics), Wikipedia **What are these states in Flip-flops?** I know that DRAM-storage makes use of capacitors. The two states are: Either the capacitor is charged or not. But I have no idea how to imagine the state of Flip-flops. What changes in the flip-flop if the state changes? * * * I am not a physicist, so please keep your answer as simple as possible.


Do protons in the beam on LHC interact with photons? I wonder how a beam would look from outside. Is there a photo of a proton beam?


title: How the Lagrangian of classical system can be derived from basic assumptions?

text: It is well known that the Lagrangian of a classical free particle equal to kinetic energy. This statement can be derived from some basic assumptions about the symmetries of the space-time. Is there any similar reasoning (eg. symmetry based or geometrical) why the Lagrangian of a classical system is equal kinetic energy minus the potential energy? Or it is just because we can compare the Newton's equations with the Euler-Lagrange equation and realize how they can match?


title: Entropy and the principle of least action

text: Is there any link between the law of maximum entropy and the principle of least action. Is it possible to derive one from the other ?


title: How is it possible for orbits to maintain stability?

text: According to $a = v^2/R$, the circular velocity and radial distance between two attracting objects (such as planets), must remain in perfect proportion in order for orbital motion to take place. How is it possible for objects in nature to achieve this proportion perfectly? Not only that, to maintain an orbit seems to be impossible. For example, assuming that the moon is orbiting the earth 'perfectly'. Let us say the moon is then hit by a series of meteorites. This would shift the balance slightly, and cause the moon's orbit to decay? Apparently not... How is it possible for the moon to remain in orbit for so long?


title: What is an "inclined magnetic field"?

text: What is meant by an "inclined magnetic field"? How is it different from the usual magnetic field?


title: What is the current radius of cosmological event horizon?

text: Doing some crude calculations (using the value of $H_0$ at this point of time only, since it is time dependent but not distance dependent thanks to Johannes answer) what is the radius of cosmological event horizon at this point of time? (not looking for the changes of CEH through time) From here we have for $H_0 $: $$H_0 = 73.8 \pm 2.4 (\frac{km}{s})\frac{1}{Mpc}\tag{I}$$ We are seeking the distance $L$ s.t. $H_0L = c = 3\times 10^6 \frac{km}{s}$ $$L=\frac {c}{H_0} = \frac {3\times 10^6}{73.8 \pm 2.4} Mpc \tag{II}$$ Where 1 pc = 3.26 light years ($ly$), $$ L=\frac {c}{H_0} = \frac {3\times 10^6\times10^6\times3.26}{73.8 \pm 2.4} ly \tag{III}$$ $$ L= \frac {9.78\times 10^{12}}{73.8 \pm 2.4} ly \tag{IV}$$ $$L=1.3\pm0.1\times 10^{11} ly \tag{V}$$ Is this calculation correct? Would the correct calculation make sense? (By making sense I mean it would seem in accordance with some observation and not in contradiction to some other observations? Or results like this are unconfirmable, just mere flights of fancy were they do not relate to anything physical? The only thing I could use to see it is not invalid (yes double negative, I cannot say it was valid) is the fact that observable universe is $45.7×10^9 ly$ but then again by that account $L=10^{123}ly$ would seem just as valid.


title: Measurement of topological spin

text: How do you measure the topological spin of an anyon? So how could an experimental setup look like? Is topological spin an observable at all?


title: Static Water Pressure vs Height

text: OK, so I have water at a known pressure, lets say 12psi. I have 2 pipes, installed vertically, one 2" diameter, the other 1/2" diameter. To what height (in feet) will the water raise to in each case? I said it would be the same, but I can't explain why. Remember this is a static pressure, as the water is not flowing.


title: Positron and other particles

text: > Positron is a particle which has the same mass as an electron . But it has > positive charge on it . When electron and positron combine, they annihilate > each other with the release of energy in the form of two very high energy > photons. > > if the energy released on annihilation of an electron and a positron is 0.5 > Mev(Million electron volts),find the frequency of the wave which is carrying > this photon.


title: Interpreting perturbation theory in general relativity

text: In quantum mechanics we start with a Hamiltonian $H_0$ for which we know the exact eigenstates and energy eigenvalues. We perturb it by a known term $H$, and then attempt to compute (approximately) the new eigenstates and eigenvalues. In general relativity, my understanding is we start with a metric $g_{\mu \nu}$, and perturb it by a known $h_{\mu \nu}.$ But in my lecture notes (http://arxiv.org/pdf/0804.2595.pdf), the lecturer shows how to compute $h_{\mu \nu}$. I thought we perturbed a system by a known quantity; can someone clarify the regular procedure of perturbation theory in general relativity, and what typical 'goals' are? The only alternative I see is that we perturb a known solution $g_{\mu \nu}$ by an unknown perturbation $h_{\mu \nu}$, state how we would like the stress- energy $T_{\mu \nu}$ to change, and then try and compute $h_{\mu \nu}$ such that it does. Could this be the correct interpretation?


title: Dirac, Weyl and Majorana Spinors

text: To get to the point - what's the defining differences between them? Alas, my current understanding of a spinor is limited. All I know is that they are used to describe fermions (?), but I'm not sure why? Although I should probably grasp the above first, what is the difference between Dirac, Weyl and Majorana spinors? I know that there are similarities (as in overlaps) and that the Dirac spinor is a solution to the Dirac equation etc. But what's their mathematical differences, their purpose and their importance? (It might be good to note that I'm coming from a string theory perspective. Plus I've exhausted Wikipedia here.)


title: How organic impurities in the air affects laser?

text: We have a Nd YAG laser pumped Optical Parameter Oscillator in our lab. A technician came to our lab (my professor called him for its maintenance) and while he was working on it, he told me that because of the organic impurities in the air, the crystals got contaminated. He showed me a layer formation on the crystal where the laser hits. He explained this is due to the organics in the air which interacts with laser and get deposited. I am not clear what kind of organics (aromatic or some other?) which creates this and how it is get deposited there? Can anyone please explain me in detail how the atmospheric impurities (especially organics) can cause damage on lasers?


title: Oscillations and Majorana Neutrino

text: In neutrino oscillations, neutrinos can convert from one flavor to another. This implies individual lepton number is not conserved. Doesn't it also imply that, if the neutrinos have mass, the mass should be of Majorana type? But I'm confused. Can I define lepton number (and the states like $\nu_e ,\nu_\mu$ and $\nu_\tau$) if neutrinos are Majorana particles? Or in other words, can Dirac neutrinos exhibit oscillations? Is the total lepton number conserved?


title: Balloon rubbing; where do the electrons go?

text: If you rubbed a balloon with a towel, where would the electrons go: the balloon or the towel? Why? I'm guessing the electrons would go to the object with a larger mass, but it's just a guess. :)


title: Diagonalize mass matrix term for fermions and "doubling trick" in m(atrix) theory

text: Can someone help me understand the "Doubling trick" at page 36 in http://inspirehep.net/record/887513/files/sis-2002-060.pdf (named "Scattering in Supersymmetric M(atrix) Models" by Robert Helling) or help me in some other way to get the mass for the fermions from the given Lagrangian, preferably without knowing the explicit form of the SO(9) gamma matrices?


title: If we change the radius of spherical surface does electric field or flux change?

text: > Suppose a point charge is located at the center of a spherical surface. The > electric field at the surface of the sphere and the total flux through the > sphere are determined. > > 1).What happens to the flux and the magnitude of The electric field if the > radius of the sphere is halved? Our teacher said the flux decreases and the filed increases. But here: > A spherical gaussian surface surrounds a point charge $q$. Describe what > happens to the: flux through the surface if > > 2) The radius of the sphere is doubled Our teacher said the electric flux will not change > 3) The shape of the surface is changed to that of a cube Also the electric flux will not change So why the electric flux changed in question 1 but in question 2 did not change? And when does the electric flux and electric field change?


title: Unitarity of PMNS matrix

text: Why should the neutrino mixing matrix (PMNS matrix) be unitary? Is the unitarity dictated by experiments or is it a theoretical demand?


title: Getting an equivalent integral equation from a given one

text: I'm reading a paper and don't understand some of the calculations. We are given an integral equation with asymptotic boundary conditions $\rho_+(u)=\frac{1}{2\pi} \int\limits_{|v|>\mu}^{}\mathrm{d}v\,\frac{2\hbar \rho_+(v)}{(u-v)^2+\hbar^2}$ $\rho_+(u)=\ln(|u|)-\frac{1}{2} \ln\left(\Xi\right)+O(u-1),\,u\rightarrow\infty$ $\rho_+$ is a density of zeros an $\ln(\Xi)$ can be seen as a density of particles. But the exact meaning is not important for further calculations. The first integral ranges over $\mathbb{R}$ exceot the interval $[-\mu,\mu]$, where we have $\rho_+=0$. Now a new function $\rho_-$ eis defined as $\rho_-(u)=\begin{cases} 0 &\mbox{if } |u|\geq\mu \\\ -\frac{1}{\pi \hbar}[\rho_+(u)-\frac{1}{\pi}\int\limits_{|v|>\mu}^{}\mathrm{d}v\,\frac{\hbar \rho_+(v)}{(u-v)^2+\hbar^2} & \mbox{if } |u|<\mu \end{cases}$ with which we can rewrite the first integral equation as $\pi\hbar\rho_- +(1-K_+)\rho_+=0$ with $K_+\rho(u)=\int\limits_{-\infty}^{\infty} k_+(u-v)\rho(v) \mathrm{d} v$ and $k_+(u)=\frac{1}{\pi} \frac{\hbar}{u^2+\hbar^2}$. So far everything is clear. Obviously a constant function is an eigenfunction fo $K_+$ with eigenvalue 1 and so $(1-K_+)$ ist invertible. It is the said, that an operator $K_-$ can be definied on functions vanishing at $\infty$ by on functions vanishing at $\infty$ $1-K_-=(1-K_+)^{-1}$ This is still quite clear although I don't really understand, why the functions must vanish at $\infty$. Then it ist said, that due to the degeneracy of the operator $K_+$, $K_-$ ist just defined up to a constant. What ist meant by this? Is it just due to the fact that every constant function is an eigenfunction? Now without any calculations it is said, that the kernel $k_-(u)$ is given by $k_-(u)=\frac{1}{\pi\hbar} \Psi(1+i\frac{u}{\hbar})+\Psi(1-i\frac{u}{\hbar})+\Delta$ where $\Psi(u)=\Gamma\,'(u)/\Gamma(u)$ is the digamma-function. It is only said, that this can easily obtaind by fourier transform. I don't have really much knowledge about integral-equations and do not see how this result can be found. Could please someone show me? I have some other questions concerning the following calculations, but maybe I stop here until this problem is solved.


What is physical meaning of $\kappa$ and $R$ in curved space? $$dl^2 = \frac{dr^2}{1 - \kappa\frac{r^2}{R^2}} \+ r^2d\theta^2 + r^2\sin^2\theta d\phi^2$$


title: Interpretation of $e|\psi|^2$ as electron density

text: In solid state physics the electron density is often equated to $e|\psi|^2$. However, the Sakurai says (Chapter 2.4, Interpretation of the Wave Function, p. 101) that adopting such a view leads "to some bizarre consequences", and that Born's statistical interpretation of $|\psi|^2$ as a probability density is more satisfactory. I am aware that a position measurement of an electron leads to (in the Copenhagen interpretation) a collapse of the wave function into a position eigenstate $x$ with the probability given by $|\psi(x)|^2$, and that it is known from some scattering experiments that the electron behaves as a point- like particle. However, the electron density is experimentally observable, e.g. by X-ray scattering. One could argue that X-ray scattering is done with a large ensemble of atoms, so that what we actually observe is the average over many position measurements. I am wondering if this the only "bizarre consequence", since this distinction seems to be a minor difference to me. My questions aims at clarifying the difference between the probabilistic interpretation of $|\psi(x)|^2$ and the interpretation as an electron density.


title: Conservation laws in mSUGRA model

text: Can somebody list all the quantum numbers (beside R-parity) that are conserved in vertex for SUSY particles in mSUGRA model?


title: Photon pumping in Laser

text: Let's consider a ring laser where the laser must pass through the gain material before it is sent toward a partially reflective surface $\ R=1-T $. The other mirrors are perfect reflectors with $\ R_1=R_2=1 $. Furthermore the output irradiance of the setup is give as $\ I_{out} = I_{sat}(\gamma - \gamma_{th})L$ where $I_{sat}$ is the saturation irradiance, $\gamma_{th} $ is the threshold gain coefficient, and $\gamma $ is the small-signal gain coefficient. Now I am trying to understand what needs to happen in order to have the scenario when every pump event results in an output photon. That is to say a theoretical one to one correspondence, every photon I put in I will get right back. (Neglect spontaneous emission). Does this happen when $\gamma > \gamma_{th}$ ? or when they are equal?


title: Are Hall edge currents truly dissipationless?

text: Integer quantum Hall states has integer number of chiral edge current channels flowing around like supercurrent in a superconductor. Are they truly dissipationless? If so, what is the mechanism that protects the coupling of different channels or phonon scattering? (e.g. edge current of 2D topological insulator is protected but time-reversal symmetry. But QHS has no symmetry except for particle number conservation and energy conservation.) Can we roughly estimate the temperature upper bound for a QHS to exist?


title: S. Weinberg, "The Quantum theory of fields: Foundations" (1995), Eq. 2.4.8

text: Unfortunately I'm struggling to understand how do we get eq. (2.4.8) from eq. (2.4.7), p. 60; namely how $(\Lambda \omega \Lambda^{-1} a)_\mu P^\mu$ is transformed into $\Lambda_\mu^{\;\rho}\Lambda_\nu^{\;\sigma}(-a^\mu P^\nu + a^\nu P^\mu)$. Why there are those two terms in the end, and not just $2a^\nu P^\mu$? I believe it must be connected with the antisymmetric nature of $\omega_{\mu\nu}$, but I'd be glad to see a rigorous derivation. Thank you!


title: force on two particles moving with same velocity

text: When two charged particles having same charge $q$ are at rest, only electric force acts upon it: a repulsive force of magnitude $\frac{kq^2}{r^2}.$ Now let them move with a constant velocity perpendicular to the line joining them. Then one of the particles will produce magnetic field on other and the second particle feels a magnetic force towards the first particle other then $\frac{kq^2}{r^2}$, so the net force on second particle will be the difference of two forces. The net force decreases after the motion starts. This decrease in force will be same in all frame of refrence. So what must be the frame of reference of their motion so that net force decreases.


title: How many mirrors placed perpendicular to each other shall make a light ray completely reverse its path?

text: The answer is 3. But according to me 2 shall also work in 2 Dimensions? Does anyone has a theoretical proof?


title: When can we assume that the wavefunction is separable

text: While working out the stationary states of a single particle in a 3d infinite potential box ($V=0$ inside a cuboid of known dimensions, $V=\infty$ everywhere else), I realized I had to assume the wavefunction was separable into a product of three functions, $\psi(x,y,z)=X(x)Y(y)Z(z)$, in order to find $\psi$. Why is this so, and under what conditions? What guarantees me that I can do this? The text I'm following isn't particularly clear on this.


title: Measuring human resistance with a DMM

text: When I attach a Digital Multimeter (DMM) to read the resistance between my right and left hands, the resistance starts off high and reduces over time. It appears almost like "RC-type" behavior. Can someone explain to me what it is that I am seeing on the DMM in some detail?


title: Could you theoretically map the internal distribution of mass in a black hole using hawking radiation?

text: Assuming you could measure the qualities of the radiation emanating from all around a black hole, could this be used to determine the internal geometry or makeup of the mass inside?


title: Fresnel transform

text: How is it possible to encrypt multiple images using fresnel transform and inverse the operation to de-multiplex those images?


I'm now learning quantum mechanics with Liboff. In the book it deals with "a compete set of mutually compatible observables" in order to make a state maximally informative. How can one find such set? It seems very hard unless commutation relation is an equivalence relation. Is commutation relation an equivalence relation? That is, if $A, B, C$ are Hermitian operators, then does $AB=BA, BC=CB$ imply $AC=CA$?


title: How to combine measurement error with statistic error

text: We have to measure a period of an oscillation. We are to take the time it takes for 50 oscillations multiple times. I know that I will have a $\Delta t = 0.1 \, \mathrm s$ because of my reaction time. If I now measure, say 40, 41 and 39 seconds in three runs, I will also have standard deviation of 1. What is the total error then? Do I add them up, like so? $$\sqrt{1^2 + 0.1^2}$$ Or is it just the 1 and I discard the (systematic?) error of my reaction time? I wonder if I measure a huge number of times, the standard deviation should become tiny compared to my reaction time. Is the lower bound 0 or is it my reaction time with 0.1?


title: How to estimate a final error when you have N experiments with N errors?

text: if we have Xo±e1, X1±e2, X2±e3, ..., Xn±en when we know these, what is going to be our estimation for X, and it's error? for example if we have 344±1, 350±2, 345±5, 338 ± 2 estimation? error of our estimation?


title: understanding time: Is time simply the rate change?

text: Is time simply the rate of change? If this is the case and time was created during the big bang would it be the case that the closer you get to the start of the big bang the "slower" things change until you essentially approach a static, unchanging entity at the beginning of creation? Also, to put this definition in relation to Einstein's conclusions that "observers in motion relative to one another will measure different elapsed times for the same event." : Wouldn't it be the case that saying the difference in elapsed time is the same as saying the difference in the rate of change. With this definition there is no point in describing the "flow" of time or the "direction" of time because time doesn't move forward but rather things simply change according to the laws of physics. Edit: Adding clarification based on @neil's comments: The beginning of the big bang would be very busy, but if time was then created if you go back to the very beginning it seems there is no time and there is only a static environment. So it seems to me that saying time has a direction makes no sense. There is no direction in which time flows. There is no time; unless time is defined as change. So we have our three dimensional objects: and then we have those objects interact. The interaction is what we experience as time. Is this correct or is time more complicated than this?


title: What defines the mass of elementary particle?

text: The electron is particle. The mass of electron is $9.10938215(45)\times 10^{−31}\, {\rm kg}$. But why is the mass exactly what it is? What in physics defines the mass of elementary particle?


title: Understanding difficulty with proving strong cosmic censorship theorems

text: I want to get a grip about the technical problems related to proving Penrose conjecture Intuitively, it seems it should not be too difficult; just start by classifying it in 0-dimensional (i.e: Schwarschild singularity) and 1-dimensional singularities (i.e: Kerr singularities). Assume we have a naked 0-dimensional singularity, take a neighbourhood small enough around it, how does the field look as the neighbourhood shrinks? can't i say that the asymptotic symmetry around the singularity is spherical? why can't i use this symmetry effectively to prove that there must be a schwarschild-like event horizon? Assume now we have a naked 1-dimensional singularity, a neighbourhood small enough should have cylindrical symmetry. Find appropiate solutions given from symmetry, repeat and rinse Where is currently stuck the programme for proving this conjecture? is there some specific step that is problematic, or this whole approach doesn't work for some deeper reasons?


title: Does the heat death of the universe really imply a maximum entropy state *all* of the time? Or most of the time?

text: Statistically speaking, you're going to still encounter deviations from equilibrium, even though the expected value is equilibrium. But these rare deviations from equilibrium - which are inevitable - might have the power to do work. So does the universe inevitably descend towards a maximum-entropy state? Or is it only probabilistically destined towards a maximum-entropy state - that is - it will be in that state more than any other state. After all, someone has even hypothesized a Poincare recurrence time, as described below: http://en.wikipedia.org/wiki/Timeline_of_the_far_future > Scale of an estimated Poincaré recurrence time for the quantum state of a > hypothetical box containing an isolated black hole of stellar mass.[47] This > time assumes a statistical model subject to Poincaré recurrence. A much > simplified way of thinking about this time is that in a model in which > history repeats itself arbitrarily many times due to properties of > statistical mechanics, this is the time scale when it will first be somewhat > similar (for a reasonable choice of "similar") to its current state again.


title: What Is the Noise of an MRI Machine

text: What is the source of the loud noise of an MRI medical scanning machine? Is it due to some mechanical moving parts such as pumps, or is it an effect, similar to magnetostriction of superconducting electrical magnets?


title: Frame of reference and big bang

text: I have read Brief History of Time in which he has wonderfully described the formation of universe. What is the frame of reference from which we are viewing the big bang? What is the frame of reference from which we are seeing the events associated with the big bang? Where is this frame situated? **Edit:** The frame of reference cannot be same as the one we have now, right? Because physical laws break when we are witnessing BB. So this frame should be different somehow, right?


title: Computer parsable table of nuclides

text: I am looking for an computer parsable (CSV, XML, JSON, since I didn't find anything, I am happy with almost everything) file (or collection of files) that contains the information you can find in a simple table of nuclides. There are several more or less usable "browsers" and of course Wikipedia's tables, but none of them is quite what I want. What I'd like to have is a list as simple as: numer of protons, number of neutrons, half life, decay mode(s) for each known isotope. But any other format that is parsable with reasonable effort to extract this data from a bigger set is also very welcome. I have already found this http://ie.lbl.gov/databases/databases.html, wich contains in a self-extracting .exe file (as a Linux user, I was happy to find out that 7zip can extract it just fine) one file for each element, and each file contains more than enough data: on the page there is also an 100-something pages long manual for the data format. I don't want to read this, so if anyone knows of a better (according to my needs) source, that'd be great!


title: Reciprocal Lattice of a non-bravais lattice

text: Is a reciprocal lattice defined for non-Bravais lattices? I'm trying to work out one for HCP structure and not figuring it out.


title: Example of the time-independent Schrödinger equation having a complex solution?

text: We know $\Psi(x,t)$ is complex, but can $\Psi(x)$ be complex? I have seen _particle in a box_ , _well_ and _harmonic oscillator_. All have real solutions for time-independent Schrödinger equation. Hence, I am curious to know examples where it is complex. This question says that it is possible, hence my request is for examples and references.


title: Kolmogorov-Smirnov test vs Chi-squared test

text: What is the difference between the Kolmogorov-Smirnov test and the Chi-squared test? When should we use one instead of the other? I was reading this article, and I got confused a lot. It is hard to get a physics-related answer in the Cross Validated SE section, so this is why I am asking here.


How do you calculate the energy output if you know the volts, amps and time?


title: How do you derive Lagrange's equation of motion from a Routhian?

text: Given a Routhian $R(r,\dot{r},\phi,p_{\phi})$, how do you derive Lagrange's equation for $r$? Do you just solve the following for $r$? $$\frac{d}{dt}\frac{∂R}{∂\dot{\phi}}-\frac{∂R}{∂\phi}=0$$ And as a related question, what is the motivation for using a Routhian?


title: Help with ideal gas law manipulation

text: Equation 1 represents an improvement over the intial assumption of constant mass velocity (i.e., $\rho v=\rho_o v_o$). We can now improve Eqn. (2) to get to Eqn. (3). My question is: what are the steps to obtain Eqn. (3)? Equation 1 is: $$\rho v = \rho_o v_o(1+\delta f[c,g]) \tag{1}$$ where $\rho$ is gas density, $v$ is gas velocity. The subscript $o$ denotes that the variable is evaluated at the inlet face of the core. $(1+\delta f[c,g])$ is the gas expansion corection factor. Equation 2 is: $$q(x,t)=\frac{q_o(t)P_o(t)}{P(x,t)}=\frac{-V_t}{P(x,t)}\left(\frac{dP_o}{dt}\right) \tag{2}$$ Equation 3 is: $$q(x,t)=\frac{-V_t (1+\delta f[c,g])}{P(x,t)}\left(\frac{dP_o}{dt}\right) \tag{3}$$ Some background information about these equations. The equations are being used to describe a test flowing gas through a rock. The test setup consists of a tank and pressure transducer that can be pressurized with nitrogen. A rock-core holder is attached to the tank, separated by a quick opening valve. To perform a run, the tank is charged with nitrogen to an initial pressure. If the valve at the bottom of the tank is opened, nitrogen will flow (axially) through the core (a right-cylindrical rock sample) and the pressure in the tank will decline -- rapidly at first, then more and more slowly. The volumetric rate of nitrogen flow at the inlet face of the core can be derived from the ideal gas law, since the compressibility factor (deviation factor) is unity for nitrogen at low pressure and room temperature. The volumetric flow rate at any position downstream from the inlet face of the core can be calculated with correction factors that account for variable mass flow rate with position at any instant in time. Hence the use of the correction factor used in Eqn. (1). My steps taken to obtain Eqn. (2): The volumetric isothermal flow rate of nitrogen as a function of time, while the nitrogen is assumed to behave as an ideal gas, from a storage tank at pressure $P_o$ is: $$ q_o(t)=\frac{M}{\rho_o(t)}\frac{-dn}{dt} \tag{a} $$ where: $q_o(t)=$ the volumetric flow rate at the inlet face of the core at time $t$ $M=$ molecular weight, g/mol $\rho_o(t)=$ gas density at the inlet face of the core at time $t$ $n=$ the number of moles of nitrogen in the reservoir tank $t=$ time The subscript $o$ in these equations refers to conditions just upstream of the inlet face of the sample. Since the only measuring instrument in place is a pressure transducer, we need to find the relationships between gas density and moles of gas versus pressure. Doing so we can re-write our volumetric flow rate equation in terms of pressure. The Ideal Gas Law for the gas in the tank can be written as: $$ P_oV_t=nRT \tag{b}$$ where: $P_o=$ the pressure of nitrogen gas in the reservoir tank/inlet core face $V_t=$ the volume of the nitrogen tank, which remains constant $R=$ universal gas law constant $T=$ absolute temperature Equation (b) can be rearranged to solve for the number of moles of nitrogen just upstream of the core as: $$ n=\frac{P_oV_t}{RT} \tag{c} $$ Equation (c) can be substituted into Eqn. (a): $$ q_o(t)=\frac{-MV_t}{\rho_o(t)RT}\left(\frac{dP_o}{dt}\right) \tag{d}$$ The density of gas just upstream of the core as a function of time is defined as: $$ \rho_o(t)=\frac{m(t)}{V_t} \tag{e} $$ where: $m(t)=$ mass of gas just upstream of the core as a function of time $V_t=$ tank volume From the Ideal Gas Law, Eqn. (b), we can relate the density of the gas just upstream of the core in Eqn. (d) to the pressure just upstream of the core as: $$P_oV_t=nRT$$ where $n=\frac{m(t)}{M}$. Substituting this relation in to Eqn. (b) and solving for pressure: $$P_oV_t=\frac{m(t)}{M}RT$$ $$ P_o=\frac{m(t)}{V_t}\frac{RT}{M}\tag{f}$$ substituting Eqn. (e) into Eqn. (f) and then rearranging for solve for density: $$ P_o=\rho_o \frac{RT}{M} \rightarrow \rho_o=\frac{MP_o}{RT}\tag{g}$$ Substitute Eqn. (g) into Eqn. (d) and solve for the volumetric flow rate of nitrogen based on the pressure just upstream of the core, as a function of time: $$ q_o(t)=-\frac{V_t}{P_o(t)}\left(\frac{dP_o}{dt}\right)\tag{h}$$ Assume for the moment that at any instant in time the mass velocity throughout the length of the core is constant. (This is not rigorously true, hence the correction factor in Eqn. (1).) As the nitrogen flows through the core it expands as a function of pressure change throughout the position of the core; the volumetric flow rate will be proportional to the ratio of inlet pressure and pressure along the core and as a function of time: $$ \Delta q \propto \frac{P_o(t)}{P(x,t)}\tag{i}$$ Therefore, the volumetric flowrate throughout the core as a function of position and time can be related by equations h and i: \begin{equation} q(x,t)=\frac{q_o(t)P_o(t)}{P(x,t)}=-\frac{V_t}{P(x,t)}\left(\frac{dP_o}{dt}\right)\end{equation} So this is how I got to Eqn. (2). Any help with how to get to Eqn. (3) using Eqn. (1) would be much appreciated.


title: How to compute observables from the boson field operator?

text: I think I understand that if given the two boson wavefunction of two different states \begin{align} \Psi(\boldsymbol{r}_1,\boldsymbol{r}_2) = \dfrac{ \psi_1(\boldsymbol{r}_1)\psi_2(\boldsymbol{r}_2) \+ \psi_1(\boldsymbol{r}_2)\psi_2(\boldsymbol{r}_1) } {\sqrt{2}} \end{align} the expectation value of an operator $\hat{Q}$ for $\Psi(\boldsymbol{r}_1,\boldsymbol{r}_2)$ is \begin{align} \langle \Psi | \hat{Q} | \Psi \rangle = \langle \psi_1(\boldsymbol{r}_1) | \hat{Q}_1 | \psi_1(\boldsymbol{r}_1) \rangle \+ \langle \psi_2(\boldsymbol{r}_2) | \hat{Q}_2 | \psi_2(\boldsymbol{r}_2) \rangle \end{align} If I add one more boson to state $\psi_1$, then the total wavefunction changes to \begin{align} \Psi(\boldsymbol{r}_1,\boldsymbol{r}_2,\boldsymbol{r}_3) = \dfrac{ \psi_1(\boldsymbol{r}_1)\psi_1(\boldsymbol{r}_2)\psi_2(\boldsymbol{r}_3) \+ \psi_1(\boldsymbol{r}_2)\psi_1(\boldsymbol{r}_3)\psi_2(\boldsymbol{r}_1) \+ \psi_1(\boldsymbol{r}_3)\psi_1(\boldsymbol{r}_1)\psi_2(\boldsymbol{r}_2) } {\sqrt{6}} \end{align} The expectation value of $\hat{Q}$ for $\Psi(\boldsymbol{r}_1,\boldsymbol{r}_2,\boldsymbol{r}_3)$ becomes \begin{align} \langle \Psi | \hat{Q} | \Psi \rangle = 2\langle \psi_1(\boldsymbol{r}_1) | \hat{Q}_1 | \psi_1(\boldsymbol{r}_1) \rangle \+ \langle \psi_2(\boldsymbol{r}_2) | \hat{Q}_2 | \psi_2(\boldsymbol{r}_2) \rangle \end{align} I am stuck with understanding how to compute the expectation values of $\hat{Q}$ in terms of the boson field operator \begin{align} \Psi(\boldsymbol{r})=\sum_{\nu} \psi_{\nu} \left( \boldsymbol{r} \right) b_{\nu} \end{align} where \begin{align} \left[\Psi(\boldsymbol{r}_1),\Psi^\dagger(\boldsymbol{r}_2)\right]=\delta (\boldsymbol{r}_1-\boldsymbol{r}_2) \\\ [b_\alpha^\dagger,b_\beta^\dagger]=[b_\alpha,b_\beta]=0,\quad [b_\alpha,b_\beta^\dagger]=\delta_{\alpha\beta} \end{align} I am assuming that \begin{align} \langle \psi_{\mu}^\dagger \left( \boldsymbol{r} \right) | \psi_{\nu} \left( \boldsymbol{r} \right) \rangle = \int \psi_{\mu}^\dagger \left( \boldsymbol{r} \right) \psi_{\nu} \left( \boldsymbol{r} \right) d^3\boldsymbol{r} = \delta_{\mu\nu} \end{align}


title: Quantum Energy Teleportation and Stress-Energy tensor divergence

text: This question is about a paper from last year about Quantum Energy Teleportation. If I understand the main assertion of what QET is supposed to involve, basically you are teleporting energy as information, but without sending the energy directly. So unless I'm very confused about what that means, this seems to mean the differential version of stress-energy momentum conservation: $$ D_{\nu} T^{\mu \nu} = 0 $$ where $D$ stands for covariant differentiation, must be replaced by some nonlocal version that looks like an average $$ \int_{\Sigma} D_{\nu} T^{\mu \nu} d^4\Sigma = 0 $$ Does this looks suspicious to you? or should I rather expect some kind of quantum inequality to apply a bound on the amount of energy that can be teleported nonlocally?


title: Regimes of Josephson junction

text: There are several formulas to describe critical current in Josepshon S-N-S junction mainly based on Eilenberger and Usadel equations for quasi-classical Green's functions. The starting point is the Nambu-Gorkov equation and then some simplifications. Lately, I encountered scattering matrix approach to this problem (Beenakker) and I am not quiet sure if this method can also handle the quasiclassical Green's function. I really don't understand in what regimes I am working. While searching the net I have found the statement that the Green's function method do not include the phase coherent Andreev reflection (I can't find it now). What kind of Josephson junction regimes exist: classical, semi-classical and mesoscopic ? Can we have long junction limit in the mesoscopic regime ? This question arises because I am working on Josephson supercurrent through topological insulators. I am not sure which formalism should I use to describe the critical current. I have seen both of them in literature, but I think it depends on some parameters (length scales etc.). Thank in advance for Your help.


title: Projectile Motion with air Resistance - Adjust only the Angle to hit a target

text: OK, let me preface this by saying i am not a physics student, nor have i ever been. I am a software engineer working on a development degree. So this is homework but the assignment was just to create an app, i chose to do this app.. and the teacher said he couldn't assist because this is a development class not a physics class. This is what i have done: First, I made an app to calculate the angle required for a projectile to hit a given target. (using bullets as the projectile). The first attempt was successful however the only force acting on the bullet wa gravity and i was able to calculate the correct angle. NOW, I am attempting to do the same with regards to adjusting the shooting angle to hit a required target WITH air resistance. I have already taken the initial velocity and broken it into the x & y components, calculated a cross sectional area, have the weight of the bullet in grains, and the drag coefficient. How do i calculate the angle required to hit a target? lets assume the following: i am firing a bullet from (0,0) to hit a target 1000 yards away, my bullet weighs 170 grains, ballisticCoefficient of the bullet is .25 . so i have this muzzleVelocity = 2000 fps velocityX = muzzleVelocity * cos(targetAngle) velocityY = muzzleVelocity * sin(targetAngle) airDensity = calculate elsewhere 1.168783 crossSectionalArea = calculate by using the bullet diameter ((1/2 Diameter)^2 * PI) = area of a circle Drag = (airDensity * ballisticCoefficient * crossSectionalArea) AccelrationX = -(drag/grain)*muzzleVelocity * velocityX) AccelrationY = -g -(drag/grain)*muzzleVelocity * velocityX)   1. What do i do now?   2. Do i need to convert the crossSectionalArea to the same unit of measurement as the velocity? The things i know are total velocity, bullet weight, bullet diameter, air density, distance and angle to the target. How can i determine the angle required to hit a target? I am loosely following here http://wps.aw.com/wps/media/objects/877/898586/topics/topic01.pdf however i am a computer programmer and this looks greek to me. Thanks Jeff AFTER RECIEVING A DOWNVOTE I ADDED THIS: (void)calculateTrajectory{               float velocityX = 0;          float velocityY = 0;          float accellerationX = 0;          float accellerationY = 0;          float drag = 0;          float timeInterval = .001;          caliber = [self.caliberTextField.text floatValue];          ballisticCoefficient = [self.ballisticCoefficientTextField.text floatValue];          muzzleVelocity = [self.muzzleVelocityTextField.text floatValue];          grain = [self.bulletGrainTextField.text floatValue];          velocityX = muzzleVelocity * (cosf(targetAngle / 180 * M_PI));          velocityY = muzzleVelocity * (sinf(targetAngle / 180 * M_PI));                    if(self.metricSwitch.on == YES){              crossSectionArea = pow(((caliber * 0.00109361)/2),2)* M_PI;          }else{              crossSectionArea = pow(((caliber / 36)/2),2)* M_PI; //pow(((caliber / 36)/2),2)* M_PI;          }          NSLog([NSString stringWithFormat:@"airdensity %f, BC %f, cross %f", airDensity, ballisticCoefficient, crossSectionArea]);               drag = (airDensity * ballisticCoefficient * crossSectionArea);          NSLog([NSString stringWithFormat:@"drag %f", drag]);          accellerationX = -(drag/grain)*muzzleVelocity*velocityX;          accellerationY = -g - (drag/grain)*muzzleVelocity*velocityY;          NSLog([NSString stringWithFormat:@"X %f, Y %f", accellerationX, accellerationY]);      /* LOOP THROUGH ANGLES??? NO THERE HAS TO BE A BETTER WAY               for (x=0; x<3; x+=.001) {              accellerationX = -(drag/grain)*muzzleVelocity*velocityX;              accellerationY = -g - (drag/grain)*muzzleVelocity*velocityY;              float nextx = (velocityX + );               }          */      EDIT FOR CLARIFICATION So when you said "You need to use the current velocity at any step of the calculation", i was beginning to do that (in a small loop for test purposes). I was going to test output using just three seconds in my for loop (x = seconds) for (x=0; x<3; x+=.001) {               accellerationX = -(drag/grain)*muzzleVelocity*velocityX;          accellerationY = -g - (drag/grain)*muzzleVelocity*velocityY;     float nextx = (velocityX + );      } I was planning on solving the NEW x,y locations based off of the negative acceleration in both the X,Y plane and then adjust the x & y velocity each iteration (every .001 seconds), and thats when it dawned on me that since I am down to thousandths of seconds and thousandths of a degree... This could be a VERY large computational process. Suffice to say i could probably do split halves to narrow it down between whole degrees and then go from there with the other calculation methods. You have provided great info in the links... i had no clue what is what called. I just know what i needed to do and it looks like i was correct that there is no hard fast formula to plug in. I will finish reading the links provided and see if i can grasp a better understanding of either of the methods. to recap: I will try (0,5,10,15,....90) to see what range the angle should fall between. Then use the formula to further narrow it down to my desired precision level by calculating the next x,y by adjusting the position and current velocities. an after thought, i suppose i could increase my time interval as well.


title: Why angular momentum about three independent axes?

text: The generic commutation relations for the angular momentum operator are $[J_x, J_y] = i \hbar J_z$, where the $J_i$, $i = x,y,z$ are the components of the angular momentum vector operator, $\mathbf J$. The spin components obey the same commutation relations. My question is, in all physical systems using a non-relativistic model for spin, the $i$ index above always runs from 1 to 3. What is special about three? I understand that a $3 \times 3$ rotation matrix can be represented as an element of $SO(3)$ and that spin obeys the same commutation relations as angular momentum (The reps of $SO(3)$ are objects that are appropriate for acting on the kets in the space). So, in a three dimensional system, it makes sense that there are three independent directions in which we can define a spin component. However, even for the spin 1/2 electron system, which is a _two_ dimensional Hilbert space, we still permit these _three_ linearly independent rotations. Why is that? Many thanks.


title: How long does it take plasma to radiate its heat?

text: Lets say we have 1 gram of plasma (Argon) at 1 million kelvin confined in a vacuum with electromagnets. If we keep the magnets on but shut down the device that heated the plasma, how long will it take for it to cool off?