title: Do small-angle coherent scattering experiments really see coherent effects over arbitrarily large distances?

text: # Short version After integrating over all possible outgoing angles, the total cross-section of coherent elastic scattering from a fixed target of characteristic length $L$ scales like $L^4$. Does this mean, given a beam of sufficiently small angular dispersion and detectors capable of sufficiently fine angular resolution, there are coherent effects over macroscopic distances arbitrarily longer than the wavelength of the beam? # Long Version Consider a target of $N$ identical atoms located at positions $\vec{x}_n$ (where $n = 1,\ldots, N$) which is bombarded by a flux of incoming particles, which we'll call neutrinos. Set $\hbar = 1$ and let $\vec{k} = 2\pi/\lambda$ be the momentum of incoming neutrino, where $\lambda$ is the de Broglie wavelength. Let $f(\theta)$ be the scattering amplitude for a neutrino on a single free atom so the differential cross section is \begin{align} \frac{d \sigma_0}{d\Omega} = \vert f(\theta) \vert^2 \end{align} for one atom. As is befitting neutrinos, assume that $f(\theta)$ is very tiny so that the Born approximation is applicable. In particular, we can ignore multiple scattering events. For many scattering processes the cross section of the target $\sigma_\mathrm{T}$is just $N$ times the cross section of each atom: $N \sigma_0$. However, for elastic scattering of extremely low energy neutrinos from the relatively heavy nuclei in atoms, the very long wavelength of the neutrinos means the various nuclei contribute _coherently_ to the cross section. When $\lambda \gg L$, where $L$ is the size of the target, one has $\sigma_\mathrm{T} = N^2 \sigma_0$ rather than $N \sigma_0$ [1]. For general $k$, the total cross section is [2] \begin{align} \frac{d \sigma_\mathrm{T}}{d\Omega} = \left\vert \sum_{n=1}^N f(\theta) \, e^{-i \vec{x}_n \cdot (\vec{k}-\vec{l})} \right\vert^2 \end{align} where $\vec{l}$ is the momentum of the outgoing neutrino [3] and $\theta$ is the angle between $\vec{k}$ and $\vec{l}$. Because it's elastic, $|\vec{l}|=|\vec{k}|=k$. A property of equation (1) is that for $L \gg \lambda$ there is a large suppression of scattering in most directions because the phase in the exponential tends to cancel for the different atoms in the sum. The exception is when $\vec{l}$ is very close to $\vec{k}$ (i.e. low momentum transfer, very slight scattering), because then the phase of the exponent varies very slowly from atom to atom. This means that for large targets, the vast majority of the scattering is in the forward direction. Now restrict to the case $A \lesssim \lambda \lesssim L$, where $A$ is the typical atomic spacing. What's initially confusing about this is that if we ask for the total cross section by integrating over $\hat{l}$, we find for large $L$ that [4] \begin{align} \sigma_\mathrm{T} = \int_\Omega d \hat{l} \frac{d \sigma_\mathrm{T}}{d\Omega} \sim \frac{N^2}{L^2 k^2} = \rho^{2/3} \lambda^2 N^{4/3} \end{align} where $\rho = N/L^3$ is the number density of the atoms. This means that, for fixed density and fixed neutrino momentum, the total cross section grows faster that than the number of atoms in the target ---even over distance scales much larger than the neutrino wavelength. In this sense the coherent scattering effects is non-local over large distances. The story I have been told is that this is resolved by incorporating the realities of a real-world detector. For any traditional experiment, there is always a minimum forward acceptance angle $\theta_0$ that it can detect. Particles which are scattered at smaller angles are indistinguishable from unscattered particles in the beam. Indeed, if we let $\tilde{\sigma}_\mathrm{T}$ be the detectable cross section scattered at angles greater than $\theta_0$ for _any_ fixed $\theta_0>0$, we find \begin{align} \tilde{\sigma}_\mathrm{T} = \int_{\theta > \theta_0} d \hat{l} \frac{d \sigma_\mathrm{T}}{d\Omega} \sim \frac{N^2}{L^4 k^4} = \rho^{4/3} \lambda^4 N^{2/3}. \end{align} This accords with our intuition. Growing like $N^{2/3}$ is the same as growing like $L^2$, i.e. there is complete cancellation in the bulk of the target, and the only significant scattering is from the surface (which scales like $L^2$). Is this all there is to say? Can we potentially see scattering of particles (with atomic-scale wavelength) which demonstrates coherent contributions from target atoms separated by meters? Are there any other limiting factors besides a finite mean free path of the incoming particle (which breaks the Born approximation) and the angular resolution of the detector? # Form of answer For positive answer, I would accept (a) anything that points to a reputable source (textbook or journal article) which explicitly discusses the possibility of coherent effects over arbitrarily large distances or (b) an argument which significantly improves on the one I have made above. For negative answers, any conclusive argument would suffice. # Bonus It has been argued to me that the Born approximation is invalid for small momentum transfer $\vec{k}-\vec{l}$, because the approximation requires the energy associated with this transfer to be much larger than the potential (which cannot be the case for $\vec{k}-\vec{l}=0$). This seems to explicitly conflict with textbooks on the Born approximation which state things like > For any potential $V$ there is a $\bar{\lambda} > 0$ such that the Born > series (9.5) converges to the correct on-shell $T$ matrix for all values > $\vec{p}$ and $\vec{p}'$, provided $|\lambda| < \bar{\lambda}$. > > ["Scattering Theory: The Quantum Theory of Nonrelativistic Collisions" > (1972) by John R. Taylor] (Here, $\lambda$ is the coefficient of the expansion.) Is there any validity to this objection? * * * [1] For macroscopic $N$, this can be a stupendous boost. This was responsible for some (misplaced) optimism in the '70s and '80s that relic neutrinos might be detectable. [2] This form also appears in many less exotic places than neutrino physics, like neutron and X-ray scattering. [3] $d\Omega$ is the differential over the possible directions $\hat{l}$. [4] This behavior is independent of the details of the geometry. The $1/3$'s come from the integrals over 3 spatial dimensions.


title: Does the fact that protons and neutrons have larger mass than electrons mean they're bigger in size?

text: and so if a proton is so larger than an electron doesn't that mean it has a shape? What would be the shape of a subatomic particle? are they spherical?


title: Mechanical pulse reflection

text: When we have a rope with one fixed end and we send a pulse through it, the reflected pulse is inverted. My question is as follows - is it correct to say that near the end (when the pulse hits the fixed end) the reflected-inverted pulse is superposing with the part of the pulse which didn't hit the end yet? In other words, is it right that the pulse pattern which we can observe when the incident pulse hits the fixed end is simply overlapping of two waves - the part which is incident and the part which is reflected? For example: ![enter image description here](http://i.stack.imgur.com/mPN6e.png)


title: How to formulate collapse in polarization subspace of a photon?

text: I am wondering how to describe the collapse of a photon state when it is measured in the polarization degree of freedom (say by a filter which let pass just one particular polarisation). Let the free photon be in the state $$a^{\dagger}_{f,H}+a^{\dagger}_{g,V}$$ (disregarding the normalisation) where $$a^{\dagger}_{f,H} = \int \text d k \; f(k) \, a^{\dagger}_{k,H} $$ creates a horizontally polarized photon with frequency distribution $f(k)$. Now it passes the filter which means a measurement in the polarization degree of freedom took place. For a filter where horizontally polarized survives the measurement is trivially described by $$a^{\dagger}_{f,H}+a^{\dagger}_{g,V} \;\longrightarrow\; a^{\dagger}_{f,H}$$ But this is just handwaving and not mathematics. Of course I know the usual rule in quantum mechanics where after the collapse the state is calculated by applying the respective projector. But in the formalism of second quantization I don't know a comparable rule and unfortunately I can't find an answer in the usual QM textbooks. I guess one just has to find the right projector which transforms the creation operators.


title: How to demonstrate total internal reflection?

text: How can we demonstrate total internal reflection practically to explain it clearly?


title: Huygen's principle in EM waves

text: In trying to understand diffraction, I keep coming across Huygen's principle as the why behind diffraction, and I think I understand the principle itself all right. However, I was hoping to find an explanation of why exactly Huygen's principle explains what happens for electromagnetic radiation. I can visualize it easily with sound - a particular particle is slammed forward into another particle or group of particles. They don't likely hit head-on so that the second particle or group of particles travels in the exact same direction that the first was travelling in, but rather I imagine it more like billiard balls hitting at different angles with the effect of the particles spreading out in all directions. Please correct me if I'm wrong, but that makes sense to me with what Huygen said. I can see how water waves would follow Huygen's principle in a similar way. But in EM, there isn't a medium and there aren't particles moving, so would the field somehow behave the same way that the particles do in sound or water waves? I'd hope to get an intuitive explanation, but if I'm completely wrong in my thinking and the only explanation is that of a pure-wave (as in, not thinking about particles at all and focusing on math) analysis then, I'd like to know. I have a suspicion that this question is related to my query, but it's somewhat above my head, though I am working on understanding it.


title: What is the physical meaning/concept behind Legendre polynomials?

text: In mathematical physics and other textbooks we find the Legendre polynomials are solutions of Legendre's differential equations. But I didn't understand where we encounter Legendre's differential equations (physical example). What is the basic physical concept behind the Legendre polynomials? How important are they in physics? Please explain simply and give a physical example.


title: Deriving Rabi rotation matrix

text: I want to understand where the matrix: $$ \left|\psi(t)\right> = \binom{a(t)}{b(t)} = \begin{bmatrix} cos(\Omega t/2)&-ie^{i\phi_L t}sin(\Omega t/2) \\\ -ie^{-i\phi_L t}sin(\Omega t/2) & cos(\Omega t/2) \end{bmatrix}\binom{a(0)}{b(0)}$$ comes from. I've already derived $$\dot{a} = \frac{\Omega}{2}e^{i\phi_L t} b$$ and its counterpart $$\dot{b} = \frac{\Omega}{2}e^{-i\phi_L t} a$$ where $\Omega = \frac{1}{\hbar} \left<1\right|-d\cdot\varepsilon\left|0\right>$ and, I think, $\phi_L$ is the frequency of the incident radiation. But how can I get from the above two relations to the matrix?


Silly question ... An antenna is necessary to receive a transmitted signal from atmosphere, or otherwise. The transmitted signal weakens as it travels through the atmosphere. When the signal encounters an antenna (subject to antenna gain) a corresponding signal is induced in the antenna. Is the inducing signal weakened to the extent it induces a signal in the receiving antenna?


title: What does y with a line over it represent?

text: I've been asked to complete this chart and have never come across this symbol before, nor can I find anything about it on google: http://postimage.org/image/oe7hb9cy3/ What does the y with the line over it represent? And also, does the the t^2 column just mean square all of the time values for each row?


title: What is the difference between dispersion and differential mode group delay?

text: I want to know what exactly is the differential mode group delay in optical fibers, is it the same thing as dispersion?


title: How could a solar sail be powered 'Upwind' by craft's own (laser) light source?

text: I've just watched a Discovery Science program - Alien Encounters featuring SETI luminaries such as Jill Tarter, Frank Drake and Nick Sagan. A serious discussion on how to interpret ET messages coupled with a dramatised depiction of the social impact on Earth of an impending visit from ETs. My major point of confusion is that, within our solar system, the visitors head in a sunward direction to Earth powering a craft with a solar sail, powered from behind using very powerful lasers. I have viewed some topics on Phys.SE on the powering of solar sails, and including 'how massless photons can have momentum'. My problem with the described system is; 1) This sounds like 'pulling yourself up by your bootstraps' - If the photons hitting the sail have sufficient momentum to push the sail along, would there not be an equal and opposite momentum of the photons leaving the laser - and thus cancelling out the thrust? 2) If the solar sail craft is heading sunwards, you are already fighting a 'reverse thrust' from the sun. The program postulated tapping energy sources such as dark energy to power the lasers (OK that's sci-fi at the moment, I admit). So if problem 1. above is wrong, the power source for the lasers would have to be immense as it would to exert more than double the sun's photonics pressure to move forward. I thought the only sunward movement of a solar sail was possible only by an analog of land-sailing's 'tacking'. I know the TV program was speculative, but I hope this is not received as science fiction, as solar powered space probes are a reality. To clarify, as I'm getting tangential responses to this. Is the issue I called problem 1. a complete bust? That is, powering a solar sail from behind with your own (laser) light source a non-starter, because of the 'bootstrap' scenario or another more scientific principle? Attempting to load a screenshot of the craft as requested- ![Solar Sail craft](http://i.stack.imgur.com/qezFI.jpg)


title: Occupation probability in grand canonical ensemble

text: I have a system with energy spectrum which has two groups of $N$ degenerate levels (the gap between them is $E$). There are $N$ non-interacting fermions in the system. What is the occupation factor at $T\ll E$ for upper level in grand canonical? I can compute the factor in canonical ensemble but in grand canonical the $\mu.N/ T$ is problematic. grand canonical partition function: $$Z=\sum\limits_{n}{N.e^{-(E_n-\mu.N)/(kT))}}$$ and the probability is: $$p_n=\frac{N.e^{-(E_n-\mu.N)/(kT)}}{Z}$$ for two energy levels of $E_1$ and $E_2$ ($E_2-E_1=E$): $$\frac{p_2}{p_1}=e^{(-(E_2-E_1)+\mu(N_2-N1))/(kT)}$$ so $$\frac{p_2}{p_1}=e^{(-E+\mu(N_2-N1))/(kT)}=e^{-E/(kT)}.e^{\mu(N_2-N1)/(kT)}$$ Now for $T<<E$ the first part goes to zero (same result as canonical), but what happens to the second part. Can we say the whole thing goes to zero?? (I hope this shows enough effort)


title: Differences between hard-core boson and fermion

text: Hard boson has strong repulsion with each other just like fermion. What is the differences between hard-cord boson and fermion. Which materials are hard-core bosons?


title: Mirror Transportation

text: Why do people transport mirrors in inclined position? I frequently find people carrying mirrors in slant position on mini trucks.I guess it has some advantage of carrying them without breaking,and I am now so curious to know the reason behind it ![Either sides of the wooden piece are aligned by mirrors](http://i.stack.imgur.com/J1S63.jpg)


title: Quantum Electrodynamics

text: I was wondering if anyone could give a simple explanation of how light interacts with matter. From what I have read in QED, electrons will repel each other because of their ability to emit and absorb photons. However, I do not observe any _visible_ light between objects around me, repelling each other when I place them on a desk. They also don't appear to heat up, or emit IR light, from just being placed beside each other either. What type of light is being emitted by this matter constantly? Extending this idea, what happens to a single electron when placed in a box which allows no light to be input, but removes all light emitted? Does it simply lose energy in the form of light until the electron no longer exists? In addition, does anyone know of any good visualizations of photon waves interacting with atoms? I have heard of an analogy of an atom being similar to a cork in a pond, while the waves are like photons coming to excite the material. Does this idea sound correct, and if so, does it fit well with the idea of visualizing oscillating probability densities of electrons and how they interact with photons? It would be nice to be able to visualize these processes and understand how they fit with tight binding / Hubbard models, etc.


title: Is it easier to determine the number of states with raising/lowering operators or using scattering?

text: > A particle is bound by > > $$V(x) = \begin{cases}\infty,& x <0 \\\ \frac{-32\hbar^2}{ma}, & x\le a \\\ > 0, & x \le a\end{cases}$$ > > a) how many states are there? i'm attempting to solve this via raising/lowering operators, by taking $E_n = \langle \Psi|\hat{H}|\Psi\rangle$ then substituting the identity in for the Hamiltonian, but am confused on what to put for the wave function or states. Am I approaching this wrong, would it be easier to solve this as a scattering problem, and solving for coefficients A, B, & C and plugging it into the general solution form?


title: Measurement in Quantum mechanics

text: I have got a quantum conservative system whose Hamiltonian is $H$. I consider an selfandjoint operator $O$ whose eigenvalues and eigenvectors are: $$O|\psi _{n}\rangle = \lambda _{n}|\psi _{n}\rangle$$ At initial time, the system is at the state $|\psi_{i}\rangle$. At time $t= \delta t$, I measure with the operator $O$. The approximation of the measured state to second order in $\delta t$ is   $$|\psi(\delta t)\rangle=U(\delta t, 0)|\psi(0)\rangle \;,$$ with $$U(\delta t, 0) = e^{-\frac{i}{\hbar}\delta t H} \approx 1 - \frac{i}{\hbar}\delta t H - \frac{1}{2}\left ( \frac{1}{\hbar^2}(\delta t)^2 H^2 \right ) \;;$$ the probability of obtaining $ \lambda _{i}$ in the second order on $\delta t$ is $$P(\lambda_{i}) \approx 1-\left ( \frac{\delta t}{\hbar}\Delta H_{\psi_{i}} \right )^2 \;,$$ where I denote $$\Bigl(\Delta H_{\psi_{i}}\Bigr)^2= \langle\psi_{i}|H^2|\psi_{i}\rangle-\langle\psi_{i}|H|\psi_{i}\rangle^2$$ **Question.** In what regime the quadratic approximation on delta $\delta t$ is good?


title: How long does it take a warm object to cool in air?

text: This is a work-related question. A warm steel torus of a given diameter & thickness is left in a room held at a controlled temperature, how long does it take to reach equilibrium? Assume the air is circulating so that the torus doesn't create a warm spot. Of course it never _quite_ gets there but this is a practical problem so _almost_ is fine. For example, the torus is 250mm diameter and 50mm thick (so that it has a 150mm hole). When taken from processing it is 22°C and the room is 20°C, how long to reach 20.01°C? It is resting on a measuring instrument supported by 2 steel dowels so assume it is just in mid-air. (Such small differences in temperature are important here because we are taking very close measurements. For something this size each 1°C changes the diameter of the hole about 0.0007 mm which to us is a big deal.


title: Earliest terrestrial planet?

text: If I've understood correctly, the heavier elements needed for terrestrial planets such as iron can only form in supernova. If that is indeed true, how long since the beginning of the universe would it take to create the first non gaseous planets? Is earth in anyway an "early" terrestrial planet?


Where is the way on a rotating black hole to another Universe? Where and how should it be entered by to get away from here?


title: Commutation Relations for Creation & Annihilation Opertors of Two Different Scalar Fields

text: Let us consider two different scalar fields $\phi$ and $\chi$. The commutation relations for the creation and annihilation operators of the scalar field $\phi$ are given by $$ [a(\textbf{k}), a(\textbf{k}') ]= 0 $$ $$ [a^\dagger(\textbf{k}), a^\dagger(\textbf{k}') ]= 0 $$ $$ [a(\textbf{k}), a^\dagger(\textbf{k}') ]= (2\pi)^3 2\omega \, \delta^3(\textbf{k} - \textbf{k}'). $$ For $\chi$ similarly we have $$ [b(\textbf{k}), b(\textbf{k}') ]= 0 $$ $$ [b^\dagger(\textbf{k}), b^\dagger(\textbf{k}') ]= 0 $$ $$ [b(\textbf{k}), b^\dagger(\textbf{k}') ]= (2\pi)^3 2\omega \, \delta^3(\textbf{k} - \textbf{k}'). $$ Are there any commutation relations among the operators of the two different fields upon any condition?


title: Feynman rules of a theory in non-standard form

text: I am currently studying lecture notes by Akhmedov on interacting scalar field theory in de Sitter space. In these notes, he considers a scalar field theory of the form $$\mathcal{L}=\partial_\mu\phi_1\partial^\mu\phi_1+m^2\phi_1^2+\frac{\lambda}{3!}\phi^3_1-\partial_\mu\phi_2\partial^\mu\phi_2-m^2\phi_2^2-\frac{\lambda}{3!}\phi_2^3$$ This Lagrangian arises from the fact that we are in a non-equilibrium situation so that we have to 'close the time contour' and consider fields on the forward as well as the backward time 'branches' (this is where the minuses come from). There is a matrix of four propagators, which can be labeled by a combination of $1$'s and $2$'s: $$\hat{G}=\begin{pmatrix} G^{11} & G^{12}\\\ G^{21} & G^{22}\\\ \end{pmatrix} $$ They satisfy the identity $G^{11}+G^{22}=G^{12}+G^{21}$. * * * In order to exploit this identity, he now performs the change of coordinates known as the Keldysh rotation: $$\phi_a=\frac{1}{2}(\phi_1+\phi_2),\ \phi_b=\phi_1-\phi_2 $$ In terms of these new variables, the Lagrangian is $$\mathcal{L}=\partial_\mu\phi_a\partial^\mu\phi_b+m^2\phi_a\phi_b+\frac{\lambda}{3!}\biggl(3\phi_a^2\phi_b+\frac{1}{4}\phi_b^3\biggr) $$ And the matrix of propagators becomes $$\hat{D}=\begin{pmatrix} D^{K}&D^{R}\\\ D^{A}&0\\\ \end{pmatrix} $$ We now only have three propagators left. They are related to the 'old' propagators by $$D^K=G^{12}+G^{21},\; D^R=G^{11}-G^{12},\; D^A=G^{11}-G^{21} $$ My question now is: How do I derive the Feynman rules for this theory in terms of the new variables, which doesn't look much like anything I've encountered before (e.g. a 'mixed' kinetic term?!)? Akhmedov himself just provides an image which is supposed to summarize them, which I have reproduced below. Note that my $\phi_a$ and $\phi_b$ correspond to his $\phi_\text{cl}$ and $\phi_\text{q}$ respectively. However, I cannot make much sense out of it (I also don't understand where he got his vertex factors, they seem to be wrong to me. The ratio should at least be 12, right?) ![enter image description here](http://i.stack.imgur.com/H3xOV.png)


title: Which experiment gave scientists reason to believe nuclear fission/fusion produced energy?

text: Every piece of knowledge in science has a beginning lying in someone's experiment. I would like to know which experiment gave scientists the reason to believe nuclear fission/fusion existed and was instrumental in the development of the field of nuclear energy. I would also accept a thought experiment as an acceptable answer, as long as it answers the question.


title: What is the most distant object from the Earth that a spacecraft has visited to date?

text: What is the most distant object from the earth that a spacecraft visited has visited so far? What was the mission and when did it happen?


title: For observing planets what kind of telescope offers the best view?

text: I've heard that using a refractor is better than a reflector when it comes to planets to best reproduce their colors. But perhaps other factors can weight in too? For example, do you want a slow or fast f-stop to get the best view of a planet? What about eye-pieces, what style of eyepiece construction will give the best views?


title: Christoffel symbols and affine connection

text: What is the difference between the "affine connection" (S. Weinberg, _Cosmology_ ) and "Christoffel symbols?"


title: 2 protons collision (both with different kinetic energies) - I don't know what to put in for $p^2c^2$

text: **The problem statement:** > Two protons with kinetic energies $W_{k1}=4GeV$ and $W_{k2}=2GeV$ colide and > form new particles. What is the mass of newly born particles? There are as > many as possible new particles. Relevant equations here are: \begin{align} E_{before}&=E_{after}\\\ p_{before}&=p_{after}\\\ E^2 &= {E_0}^2 + p^2c^2 \longleftarrow \substack{\text{Lorentz invariant}} \end{align} First I wrote the energy conservation law: \begin{align} E_{before} &= E_{after}\\\ E_{k1} + E_{k2} + 2E_{0p} &= \sqrt{{E_{0~after}}^2 +p^2c^2}\longleftarrow\substack{\text{Here the $E_{0~after}$ is a full}\\\\\text{rest energy after colision}}\\\ E_{k1} + E_{k2} + 2E_{0p} &= \sqrt{\smash{\underbrace{\left( 2E_{0p} + E_{0m} \right)^2}_{\substack{\text{after collision we have}\\\ \text{2 $p^+$ and new mass $m$}}}} +p^2c^2} \\\ \\\ \\\ \\\ \\\ \end{align} At this point I am not sure what to do with the last part $p^2c^2$. The only thing I came up with was to set: \begin{align} pc &= \sqrt{{E_{k~before}}^2 - 2E_{k~before}E_{0~before}}\\\ pc &= \sqrt{\left(E_{k1} + E_{k2}\right)^2 - 2\left(E_{k1}+E_{k2}\right)2E_{0p}}\\\ \end{align} If I continued with this calculation I got $W_{0m}=1.87GeV$ while in the solutions it is said to be $W_{0m}=5.634GeV$. Where did I go wrong?


title: Radioactive decay - What mechanism decides when an unstable nucleus decays?

text: My first question on Stackexchange (if it is formatted wrong or something please tell me so I know in future) - here it is: **Given an unstable nucleus (exactly which nucleus is not particularly pertinent) - what decides _precisely_ when it will decay?** I am somewhat familiar with the concept- a nucleus becomes 'unstable' when the mutual repulsion between the protons exceeds the binding force of the strong nuclear force. That being said, if we were to theoretically isolate a single unstable atom, there are a few possibilities (correct me if I am wrong):   1. atom immediately decays   2. atom decays x seconds later   3. atom doesn't decay There is an unpredictable nature (which I assume arises from something to do with quantum uncertainty). But getting back to my question, what is it that suddenly makes an 'unstable nucleus' decay? Any further reading would be appreciated, although nothing too complicated (university level is my limit I think). UPDATE Thanks all, my first time using this forum and I was not let down :) With regards to the question itself, I'm a bit disappointed that there is no precise mechanism, but I guess Einstein was wrong when he quoted: "God does not play dice"


title: What is the difference between Linear and Non-Linear Magneto-optical effects?

text: Principle difference between Linear and Non-Linear Magneto-optical effects and how they affect the property of light passing through a Vapor medium.


title: When to pull out a negative sign from a variable

text: I get confused about when there should be a negative sign in certain equations or not. I will give three short examples (that I will make long with explanation) that show my confusion. Example 1: Rockets. Consider a rocket of mass $m$ traveling in the positive x direction with velocity $v$. The ejected fuel has a velocity relative to the rocket $v_{ex}$. At time $t$, the rocket's momentum is $P(t) = mv$. In a short time later, $t + dt$, the rocket's mass is $m + dm$ (where $dm$) is negative and momentum $P(t + dt) = (m + dm)(v + dv)$. The fuel ejected in a time $dt$ has mass (-$dm$) and velocity $v - v_{ex}$ relative to ground. The total momentum at time $t + dt$ is $$P(t+dt) = (m+dm)(v+dv) - dm(v-v_{ex})$$ $$ = mv + mdv + dmv_{ex}$$ 1a) Why is the negative contained in the $dm$ of the rocket mass while the negative is pulled out of the $dm$ for the fuel? To reconcile this, I thought of something like, if ejecting mass is a positive, then use positive $dm$ as they did in the text from which I'm reading out of. So ejecting means positive, while the fuel gains mass in a way. So I can define gaining mass with a $-dm$ as they did. However, if I was doing this problem, I would have made ejecting mass negative and gaining mass positive. But if I change the sign on all of the $dm$'s I don't get the same result in the text. Example 2: I'm not reading this one out of a text but I can explain the question. Often times you'll see something like $$\frac{d(Q)}{dx} = ax$$ where $Q$ is some function. The book or whatever will then say, but as $x$ increases $Q$ decreases so the expression should really be $$\frac{d(Q)}{dx} = - ax$$ 2a) Why can't the negative sign be contained in the constant of proportionality? Does it change the result? All it does by putting a minus sign there is force $a$ to be positive. Example 3: Projectile motion with quadratic drag. The equation of motion for this is given by $$m\ddot{\vec{r}} = m\vec{g} - cv^2\hat{v} = m\vec{g} - cv\vec{v}$$ where $v$ is the magnitude of the velocity of the projectile. If we only look in the vertical direction, with y measured vertically upwards, my text gives $$m\ddot{v_y} = -mg - c\sqrt{v^2_x + v^2_y}v_y$$ 3a) So $\vec{g} = -g\hat y$ and they pulled negative out of the $g$. So it makes sense that its $-mg$ since y is pointed upwards. Why isn't it $+ c\sqrt{...}$? Is the reason because $v_y$ is negative, so a negative times a negative is a positive? If so, why did the negative come out of the $g$ while the negative was kept inside the $v_y$?


title: How to calculate the refracted light path when refraction index continuously increasing?

text: Suppose an incident light from vacuum ($n_1=1.0$) into some media ($n_2=n_1+\mu\; x^2$) as in the figure below. How to calculate the refracted light path curve in closed form? ![enter image description here](http://i.stack.imgur.com/YLMUt.png) **Update:** Try to set up ordinary differential equation for the refracted light path per Snell's law. Suppose the curve is $y=y(x)$; Since $n_i \sin\theta_i=\text{constant}=n_1\sin\alpha=\sin\alpha$. For any point $P:(x_0,y(x_0))$ on the path $y(x)$, we have: $$\tan(\theta_P)=\dfrac{\sin\theta_P}{\cos\theta_P}=y'(x)=\dfrac{\rm{d}y}{\rm{d}x},\quad \text{where }\theta_P \text{ is incident / refracted angle}$$ Since $\theta_P$ is always an acute angle, we have: $$\dfrac{\sin^2\theta_P}{{1-\sin^2\theta_P}}=y'(x)^2\Rightarrow \sin\theta_P=\dfrac{\pm y'(x)}{\sqrt{1+y'(x)^2}}$$ Clearly $n_P\sin\theta_P=\sin\alpha$, where $n_P=1+\mu x^2$, then we have: $$\left(1+\mu x^2\right)\dfrac{\pm y'(x)}{\sqrt{1+y'(x)^2}}=\sin\alpha\quad\text{with: y(0)=5} ||y'(0)=\tan\alpha$$ Then it becomes how to solve the ODE with a boundary condition. Can the ODE be solved in closed form?


title: Is it possible to create matter from space?

text: Like could maybe fluctuate the space some how and make the virtual paricles turn into normal matter


title: Resonance and the driving frequency?

text: Why does resonance occur in a mass-spring system when the natural frequency = the driving frequency. I think it is because the driving force is always contributing to the kinetic energy of the system rather then acting against it i.e. the driving force is in phase with the velocity of the system, or is there another explanation?


title: Refractive index inside a fibre

text: The refractive index $n$ of a fibre is calculated as; $$ n = c_0/c_m $$ Where $c_0\approx 300000km/s$ is the speed of light in a vacuum, and $c_m$ is the speed of light in the fibre in question. How can I calculate the speed of light in a fibre? I am a complete physics novice so answers need to be in laymen's terms; Is there a simple equation that can be used to calculate $c_m$ given a set of properties of a fibre's manufacturing perhaps? **UPDATE** The reason I am asking this (after reading the answers I am getting) is because from the refractive index I can then go on to calculate other things like the Numerical Aperture. Perhaps I should be going around this another way?


title: Electron and photon interaction

text: Assuming I shoot a beam of light to an electron, the electron will take the energy and reemit EMR at all direction randomly? If it will, is it happening at the same time?


title: What is dimensional units/quantity and dimensional state

text: First, I am not a native English-speaking student so I am not good at physics definitions in English. I participated in the MIT e-learning course on classical physics. The 1st lesson is about 3 fundamental physical quantities (time, length and mass). > It mentions dimensional units/quantity and dimensional state. I couldn't > find the meaning of those in a dictionary. Can someone give me an specific > explanation? > > Also, in the lesson, why are the quantities always associated with some > power of unknown value like $\alpha,\beta$, etc when we predict the > equation? And how do we know what quantities should be added into the > equation of a model?


title: How to choose a location for rocket launching?

text: What's the relation of earth's magnetic equator with satellite launching? How does the nearness of a rocket launching station to the earth's magnetic equator make it an ideal location for scientists to conduct atmospheric research? But not all stations are near the magnetic equator. why? Are there any special requirements(in terms of geometry) that are taken into consideration before setting a rocket station?


title: Superluminal travel without time travel

text: Can superluminal travel or communication be possible without leading to the possibility of closed timelike curves (CTCs) and causality violations? I have seen conflicting opinions regarding this issue, and whether all possible methods of superluminal travel (e.g. wormholes, Kraskinov tubes, and warp drives like the Alcubierre drive) necessarily open the door to causality violations. I understand that wormholes may lead to CTCs in certain conditions (i.e. through time dilation or Roman rings, as well as Kraskinov tubes). My question is basically this: Are there possible (hypothetical) methods of superluminal travel that do not open Pandora's box of causality violation?


title: Can an electromagnet repel a strong magnet?

text: When the direction of current is reversed, the polarity of an electromagnet should be reversed. I am trying to understand why this did not happen in an experiment I conducted today. An electromagnet was connected to a 12v battery and I verified that it was working. I then touched the electromagnet to a strong regular magnet (rated to hold over 50 kg). There was a strong force between the magnets. I then disconnected the electromagnet from power, reversed the wires, and turned it back on. I expected to feel the electromagnet repel the other magnet, but instead they two magnets got stuck together again. It seems like reversing the wires had no affect on the polarity of the electromagnet. Here is a picture of the electromagnet I used for reference: ![enter image description here](http://i.stack.imgur.com/9mea3.jpg) And here is the strong regular magnet.


title: Is the stability of the nucleus due to pions or gluons?

text: > **Possible Duplicate:**   >  Is it pions or gluons that mediate the strong force between nucleons? (I'm currently studying particle physics in my physics classes at high school so simple language would be appreciated.) We have learnt that the proton is the only stable hadron and the neutron has a mean lifetime of approximately 20 minutes. This leads to the question of how nuclei are stable. As I currently understand it, pions are exchanged between the neutrons and protons which turn the proton into a neutron and the neutron into a proton. My question is how gluons are involved in this process and what simplifications have been made in my understanding? My initial guess is something to do with quark color.


title: Optics Brewster's Angle Reflection Intensity

text: An incident unpolarised light beam of intensity $I_{0}$ strikes glass plate B at Brewster's Angle. The reflected light travels vertically and strikes a second glass plate A, again at Brewster's Angle. (We ignore the light transmitted by the glass plates.) Plate A is then rotated about the z-axis as shown. Briefly explain how the intensity of the light reflected by the apparatus varies with the angle of plate A. Illustrate with a qualitative sketch of intensity vs. angle of plate A. ![Diagram](http://i.imgur.com/2AZstU1.png?1) We have an unpolarised beam incident on plate B at Brewster's Angle which upon reflection should become s-polarised or polarised perpendicular to the incident plane. My understanding of Brewster's Angle then says that upon the second reflection at plate A if the plate is oriented such that the incident plane is the same as for plate B there will be no reflection. Otherwise as it rotates there will be light reflected, is this correct? Also does the intensity of the light reflected off of A increase gradually to a maximum when the beam is parallel to the incident plane at A? Any help would be much appreciated.


title: Limits of Poisson's ratio in isotropic solid

text: For an isotropic solid, Poisson's ratio can be expressed in terms of stiffness constants as: $$\sigma = \frac{c_{11} - 2c_{44}}{2c_{11} - 2c_{44}}$$ Alternatively we may express Poisson's ratio in terms of the Lamé constants $\lambda$, $\mu$ where $\lambda = c_{12}$ and $\mu = c_{44}$. For an isotropic solid, we have $c_{12} = c_{11} - 2c_{44}$. When we solve this algebraically we get: $$\sigma = \frac{\lambda}{2(\lambda + \mu)}$$ I am supposed to show that $\sigma$ must lie between $-1$ and $+\frac{1}{2}$. I figure that by setting $\mu=0$, it follows from the last expression of $\sigma$ that we get the value $\frac{1}{2}$. However, I can't see how to show that the lower bound must be $-1$. I've tried fidgeting around with the expressions and letting them go towards $0$ or $\infty$, but I still don't get the $-1$ value. If anyone can help me out here, I would greatly appreciate it!


title: Electric potential and movement of charges

text: If electric field is not present does that mean that charges achieve condition of electrostatics I.e, charges are stationary ? But we say that stationary charge can produce electric field. I read this statement one a page (link given below) "Now if there's any $\vec E$ inside the conductor , we can surely say then the charges must move ." link: Charge inside conductor


title: Speed of different em radiation in a medium

text: Does all frequency of light travel with the same speed in a particular material like glass? Since by cauchy's relation refractive index change with wavelength which implies the speed also changes. But as derived from maxwells equation, the speed of light should only depent on medium through which it travel and not on its frequency or wavelength. V=1/sqrt(epsilon*mu) Does different colours have different speed in a particular medium?


title: Why doesn't Fermi's golden rule distinguish attraction from repulsion?

text: Let's say I have two distinguishable charged particles interacting electrostatically. In Fermi's golden rule, the two particles can change state at a rate proportional to: $$|\langle \psi_f | H_{int} | \psi_i \rangle|^2 = \left| \int d^3 r_1 d^3 r_2 \; \psi_{1f}^*(r_1) \psi_{2f}^*(r_2) \frac{q_1q_2}{|r_1 - r_2|} \psi_{1i}(r_1) \psi_{2i}(r_2)\right|^2$$ The weird thing is: **It doesn't matter whether they are attracting or repelling!** The rate with $q_1 = q_2 = e$ is the same as the rate with $q_1 = -q_2 = e$. I'm shocked that there can be an electrostatic calculation that turns out the same regardless of whether the particles are attracting or repelling each other. How do I make sense of that?


title: Domain walls intersection

text: I was reading this article( _On domain shapes and processes in supersymmetric theories_ ). In the paragraph about domain walls intersection (paragraph $4$, page $7$) the authors say: > In a one-field theory it is known that an intersection of domain walls is > unstable They explain it as follows: > Indeed the configuration shown in Fig. 2a has translational zero modes, > whose shape is given by the gradient of the field. Since there are > directions in the plane, where the field approaches the same values at both > innities, the component of the gradient in such direction necessarily has > zero. Thus the zero mode cannot be the lowest in the spectrum and a negative > mode exists, leading to a separation of the walls. So, i have the theory of one scalar field in $2+1$ with mexican hat potential: $V=\frac{\lambda}{4}\left( \phi^{2}-v^{2}\right)^{2} $. I consider configuration (domain walls intersection) with boundary conditions: $$ \phi_{d}(+\infty, +\infty)=-v $$ $$ \phi_{d}(+\infty, -\infty)=v $$ $$ \phi_{d}(-\infty, +\infty)=v $$ $$ \phi_{d}(-\infty, -\infty)=-v $$ Here $\phi_{d}$ satisfies the stationary equation of motion (recall that $\dot{\phi}=0$): $$ \Delta \phi_{d}-\frac{\partial V}{\partial \phi}(\phi_{d})=0. $$ I consider the small excitation over $\phi_d$: $\tilde{\phi}=\phi_{d}+\phi$. After linearizing, I obtain the following equation ($\phi=e^{i\omega t}f_{\omega}(x_1,x_2)$): $$\left[-\Delta + \frac{\partial^{2}V}{\partial \phi^{2}}(\phi_{d})\right]f=\omega^{2}f \tag{1}$$. I have found **zero modes** : its directional derivative : $f_{0}(x_{1},x_{2})=(\nabla\phi, \mathbf{n})$. It satisfies the equation $(1)$ with $\omega=0$. Now I can consider the directions along "diagonals", at their ends the field takes the same value $v$ or $-v$ respectively (see figure 2 at page 10 in the file). So at this diagonals there is a point where $f_{0}=0$, as authors said. But I cant understand the last sentence of their explanation: **How the existance of such point (where $f_{0}=0$) implies that we also have negative modes?** Can you explain me this? Thanks in advance. _PS. I will accept another approaches to show the existance of negative modes (and therefore instability of such configuration) too._


title: The value of one atomic mass unit

text: In my textbook, it is given: > 1 atomic mass unit equals $1/12$th mass of one carbon 12-atom. Since mass of > $6.02 * 10^{26}$ atoms of carbon-12 is $12\space\text{kg}$. Thus, > > $$1 \text{ a.m.u (or u)} = \frac{1}{12} * \frac{12}{6.02 * 10^{26}} \text{ > kg}$$ This makes sense to me. But what follows is that they make this value equal to: $$1.66 * 10^{-27} \text{ kg}$$ Which goes against the value that I had thought. I thought that the $12$s would cancel out resulting in only: $$(6.02*10^{26})^{-1} \text{ kg}$$ Please help my out in figuring where did I go wrong.


title: Can Bremsstrahlung radiation produced as a result of electron-electron interaction?

text: Can Bremsstrahlung radiation produced by the electron-electron interaction as e-e interaction can result acceleration due to coulomb repulsive force? If yes, how the energy of bremsstrahlung radiation produced by e-nucleus and e-e interaction can be compared?


title: $E=mc^2$ resembles kinetic energy formula?

text: The simplest equation expressing mass–energy equivalence is the famous $E=mc^2$ where $c$ represents the speed of light. Compare this with $E_K = \frac{1}{2}mv^2$. Since $E=mc^2$ can be applied to rest mass ($m_0$) and rest energy ($E_0$) to show their proportionality as $E_0=m_0c^2$, I ask whether this resemblance is just a coincidence created by the need of any equation to be homogeneous for units, or are the two equations fundamentally related? I know about Mass-Energy Equivalency but I could not find the answer I'm looking for there.


title: Equation of motion for a falling rod (with one end touching a frictionless surface)

text: I have a quick question about the equation of motion for a falling rod (with one end touching a frictionless surface). The end touching the surface is not fixed. I am given the moment of inertia about the center of mass. Only gravity is acting on the rod. The rod begins at an angle $\theta$ above the ground. I know that the rod will rotate about the center of mass, and the point touching the frictionless surface will slide along the surface, but I am having trouble calculating the torque. For reference, the left end of the rod is touching the surface, and the right end is in air. I have calculated the torque, $\tau$, from the right moment arm as $\tau=\frac{mg\cos{\theta}}{4I_G}$ because the right half of the rod contains half of the mass and half of the length. I don't really understand how to calculate the total net torque, though. Any hints would be appreciated.


title: Reference frequency

text: In my experiment, I have to obtain a relation between magnetic field of a solenoid and the applied Vp-p. The setup is as follows. * * * http://www.teachspin.com/instruments/faraday/ * * * Does my choice of reference frequency, in such an experiment where the slope of the obtained graph (B versus Vp-p) is dependent on the frequency, matter? I did it with three different values but then proceeded with the experiment with just one of them for no particular reason whatsoever. So I am not sure as to how to explain my choice of reference frequency (out of 800 Hz, 900 Hz and 1000 Hz, I chose 800 Hz for the subsequent parts of the experiment).


title: Huggins Displacement Theory and Retrocausality

text: I was looking at the Wikipedia entries on Time Travel and the Grandfather paradox and noticed a paragraph on the so-called Huggins Displacement Theory. I haven't been able to find the source although I assume it's originator was the Victorian astronomer Sir William Huggins. Does anyone know of the precise reference? Normally sending information into one's past can lead to contradictions. For example based on some condition now I could send a message into my past that prevents that very condition from occurring in my present. Thus sending information into the past, and therefore arbitrarily changing it, seems to be forbidden physically. However the Huggins Displacement Theory says that there is a loop-hole in this argument (or more accurately a "corner case"!) I could send information along my past lightcone so that if I influence something a year in my past then it must be a light-year away from me. Thus no change in the past has a enough time to travel back to me to cause a contradiction in my present. For example imagine that I see a stock price going up today. The Huggins Theory says that I can beam knowledge of this stock price rise back in time provided that the message is also displaced by a comparable distance. Thus if I send it back a year in time then it must also travel away from me by a light-year in distance. If my twin gets the message then he might try to profit from the information by sending back a message to me to buy the stock. Unfortunately the message will only get back to me at the instant that the price rises so that neither myself nor my twin will be able to profit from this situation. Maybe physics does allow information to be sent back in time along a past lightcone so that this effect could be measured. My hunch is that perhaps if one charged up an electrode then electrostatic influences from it might travel back along its lightcone to cause measureable repulsion or attraction on distant electrodes. I don't think electromagnetic radiation itself could go backwards in time as in principle it could be reflected back onto an observer's past worldline and therefore trigger contradictions in his/her present.


title: Time inversion for Euler equation in fluid dynamics

text: Consider Euler equation for continuum body: $$\frac{\partial u^i}{\partial t}+\mathbf u\cdot \nabla u^i=- \frac{1}{\rho} \frac{\partial p}{\partial x^i} $$ where $\rho$ is the mass density, $p$ is the pressure and $\mathbf u (t, \mathbf x)$ is the velocity field. Under the effect of a boost $(t=t',R=1)$: $$\mathbf u'(t,\mathbf x')= \mathbf u (t,\mathbf x' + \mathbf v t)- \mathbf v=\mathbf u (t,\mathbf x)- \mathbf v$$ (here $\mathbf v$ is the velocity of reference frame) we note that if we put in a natural way $\rho '(t,\mathbf x')=\rho (t, \mathbf x)$ and $p'(t,\mathbf x')=p (t, \mathbf x)$ we obtain that Euler equation is Galilean covariant BUT is not invariant for a time inversion (i.e. $t \rightarrow -t$). My question is: which transformations of $\rho$ and $p$ do we have to use in order to obtain Euler equation that is invariant for time-inversion, without losing Galilean covariance?


title: How would one compute the angle of deflection, in a relativistic collision - underspecified system?

text: Consider the simplistic case of two identical mass particles colliding elastically with the second particle initially stationary and the first particle travelling with energy $E$. By conservation of 4-momentum we have: $$p_{1}^{\mu}+p_{2}^{\mu}=p_{1}'^{\mu}+p_{2}'^{\mu}$$ Taking the inner product of this with itself: $$\left\langle p_{1}^{\mu} \middle| p_{1}^{\mu}\right\rangle + 2\left\langle p_{1}^{\mu} \middle| p_{2}^{\mu} \right\rangle+\left\langle p_{2}^{\mu} \middle| p_{2}^{\mu} \right\rangle = \left\langle p_{1}'^{\mu} \middle| p_{1}'^{\mu}\right\rangle + 2\left\langle p_{1}'^{\mu} \middle| p_{2}'^{\mu}\right\rangle +\left\langle p_{2}'^{\mu} \middle| p_{2}'^{\mu}\right\rangle $$ Using the fact that $\left\langle p_{i}^{\mu} \middle| p_{i}^{\mu}\right\rangle = m_{0}^{2}c^{2}$, we can simplify this: $$2m_{0}c^{2}+2m_{0}E= 2m_{0}c^{2}+2\left(\frac{E_{1}'E_{2}'}{c^{2}}-\vec{p}_{1}'\cdot\vec{p}_{2}'\right) \implies m_{0}E=\frac{E_{1}'E_{2}'}{c^{2}}-\vec{p}_{1}'\cdot\vec{p}_{2}'$$ We note that $\vec{p}_{1}'\cdot\vec{p}_{2}'=\|\vec{p}_{1}'\|\|\vec{p}_{2}'\|\cos(\theta)$, where $\theta$ is the inner angle between $\vec{p}_{1}'$ and $\vec{p}_{2}'$. However, this results in a system with $\|p_{1}'\|$,$\|p_{2}'\|$ and $E_{1,2}$ unspecified and I cannot see how we could thus extract $\theta$ from the initial conditions; what have I misunderstood or misapplied here?


title: What makes materials hard and strong?

text: This is something I have wondered for a long time. Why are some materials like steel, diamond, and even light materials like graphene stronger than others? Is it due to the strength of the electromagnetic/static fields that repel two binding atoms? Or is it something else?


title: Are gravitons bound by the event horizon?

text: I understand that photons, even when traveling at the speed of light, cannot escape the event horizon of a black hole. Are gravitons and other virtual particles traveling at the speed of light also confined by event horizons? If so, it seems that the gravitational field created by the black hole would result only from the mass of the black hole beyond the event horizon, where gravitons are capable of escaping. As a result, would there would be a disparity between the apparent mass of the black hole due to its gravitational field on other celestial bodies and the total amount of matter contained within the black hole? Also, I was reading this question: Nature of gravity: gravitons, curvature of space-time or both?, which suggests that gravitons and curved space may be indistinguishable. However, if gravitons are bound by the event horizon it seems that a black hole would act differently based on whether gravity results from gravitons or curved space-time. The existence of bound gravitons would negate the gravitational field of mass within the event horizon, resulting in a significantly lower gravitational field outside of the black hole. Would this occur, or am I neglecting some effect of relativity upon the gravitational field?


title: Derivation of boundary conditions in the context of lattice dynamics

text: **Context** : Solid state state physics, lattice dynamics. **Question** : I'd like to derive the condition for the wave vector $\boldsymbol{q}$ in order for the lattice to be periodic and infinite. My lecture notes state that the condition is $q_i = \frac{n_i}{N^{1/3}}$, where $n_i$ takes certain values so that $\boldsymbol{q}$ falls into the 1st Brillouin zone, but I'm unable to reproduce it. **Attempted solution** (The first relation is given by the lecture notes): \begin{align*} \newcommand{\vect}[1]{\boldsymbol{#1}} e^{i\left(\vect{q}\vect{r} - \omega t\right)} = e^{i\left[q\left(\vect{r} + N^{1/3}\left(\vect{a_1} + \vect{a_2} + \vect{a_3}\right) \right) -\omega t\right]} \Rightarrow \\\ \exp\left[ i N^{1/3} \vect{q}\cdot(\vect{a_1}+\vect{a_2}+\vect{a_3}) \right] = 1 = \exp(i2\pi n_i) \Rightarrow \\\ N^{1/3} \left( q_1\vect{g_1} + q_2\vect{g_2} + q_3\vect{g_3} \right) \cdot \left(\vect{a_1} + \vect{a_2} + \vect{a_3}\right) = 2\pi n_i\Rightarrow \end{align*} where $\vect{g_j}$ are the basis vectors of the reciprocal lattice: \begin{align*} \vect{g_1} = 2\pi\frac{\vect{a_2} \times \vect{a_3}}{\vect{a_1}\cdot(\vect{a_2}\times \vect{a_3})} \\\ \vect{g_2} = 2\pi\frac{\vect{a_3} \times \vect{a_1}}{\vect{a_2}\cdot(\vect{a_3}\times \vect{a_1})} \\\ \vect{g_3} = 2\pi\frac{\vect{a_1} \times \vect{a_2}}{\vect{a_3}\cdot(\vect{a_1}\times \vect{a_2})} \end{align*} We see that $\vect{g_1} \perp \vect{a_2}, \vect{g_1} \perp \vect{a_3}, \vect{g_1} \parallel \vect{a_1}$. Therefore: \begin{align*} q_1\vect{g_1}\vect{a_1} + q_2\vect{g_2}\vect{a_2} + q_3\vect{g_3}\vect{a_3} = \frac{2\pi n_i}{N^{1/3}} \\\ q_1 + q_2 + q_3 = \frac{n_i}{N^{1/3}} \end{align*}


title: Will an anti-neutron annihilate a regular neutron even though they have no charge

text: Since neutrons have no electric charge will an anti-neutron annihilate when it comes into contact with a regular neutron as protons and anti-protons do? if so what causes it to annihilate?


title: On the lattice structures of graphite

text: My question is the following: **What causes graphite to have either a cubic lattice structure or a hexagonal lattice structure? Does it depend on how it is grown? Or is it a random process?** I would also like to know if anyone has any good reading on the solid state physics of graphite. Thanks in advance!


title: Potential and kinetic energy on spherical surface

text: ![enter image description here](http://i.stack.imgur.com/vMfu0.png) > _A small particle of mass $m$ is atop of a semi-sphere as shown in the > figure. A little push was given to the particle. Prove that the particle > will leave the spherical surface at a height of (2/3)$R$._ Now, I've managed to get this result (i.e, (2/3)$R$) by equating the potential energy with the kinetic energy.But to be honest, it was a mere hunch. My question is why is that true?


title: Power as a Function of Distance

text: I am interested in trying to understand the meaning of the integral of power with respect to distance. In the case of force, we have the two formulae: $\int F \, \operatorname{d}\\!t = \Delta p$ and $\int F \, \operatorname{d}\\!x = \Delta E_k$. I understand that power is closely linked to time and work by the relationship $P = \frac{\operatorname{d}\\!W}{\operatorname{d}\\!t}$. However, imagine a body moving continuously from the left to the right. In such a case, we can express displacement as a one-to-one function of time, and time as a one-to-one function of displacement. We could place regular markers along the road side and take power readings as the body passes each of these markers. Assuming the distance between markers is $\Delta s$ and then letting $\Delta s \to 0$ gives power as a function of displacement. We may then ask: **What does the integral of this function mean?** $$\int_{s_1}^{s_2} P \, \operatorname{d}\\!x = \operatorname{F}(s_1,s_2)$$ **Addition:** I've tried to play with the calculus a bit. I'm not sure if this helps: $$P = \frac{\operatorname{d}\\!W}{\operatorname{d}\\!t} = \frac{\operatorname{d}\\!x}{\operatorname{d}\\!t} \frac{\operatorname{d}\\!W}{\operatorname{d}\\!x} = v\frac{\operatorname{d}\\!W}{\operatorname{d}\\!x}\implies \int P \, \operatorname{d}\\!x = \int v\frac{\operatorname{d}\\!W}{\operatorname{d}\\!x} \, \operatorname{d}\\!x = \int v \, \operatorname{d}\\!W$$ Has anyone seen a velocity-work graph? If so, what does the area underneath it represent?


title: Is there an easy way to get water at roughly 70°C in our kitchen?

text: Some green tea requires to pour water at 70°C. I have no thermal sensor or kettle with adjustable temperature with me. Do you know a way to get water at roughly 70°C like “boil water and wait for _x_ minutes” or “mix _x_ part of boiling water with 1- _x_ part of fresh water” ?


title: Electric circuit question

text: http://www.fisme.uu.nl/nno/opgaven/bestanden/Ronde1-Theorie-2009.pdf, question 5: ![enter image description here](http://i.stack.imgur.com/1kNED.png) > You have to calculate Id when the switch at S is opened, when its closed, > the Id = 0.5 A. All resistors are the identical in the circuit. So, the answer is B, Id = 0.67 A. But why? I don't get it, maybe its elementary, but I don't get it. Could you explain to me, in steps, how to solve this problem? Thanks.


title: Spinor and Scalar Bose-Einstein condensate

text: I read about an order paramater that describes a Bose-Einstein condensate. But I don't understand, the classification into "scalar" condensate and "spinor" one. Is it linked with spin of atoms that take part in condensation? Or is it a property of the condensate? Does anyone have a good explanation?


title: Locality in the scattering amplitude

text: Early in this talk by Nima Arkani-Hamed, he describes what locality means for scattering amplitudes. "Locality tells you that the only poles in the scattering amplitude occur when the sum of a subset of the momenta of the particles goes on shell." The talk then goes on to describe his recent research, but right now I'm not worrying about understanding that. I'm just trying to understand the statement quoted above. I know what locality looks like in the Lagrangian density (the terms of the Lagrangian density contain field operators which are all functions of the same spacetime coordinate), but it's not at all obvious to me how that is equivalent to the above statement about scattering amplitudes. Can you explain it (preferably at about the level of a first graduate course in QFT)?


title: S-matrix and it's exponential form

text: By using Dyson series for the representation of the $S$-matrix, it's expression can be written in a form $$ \hat {S}(\infty , -\infty) = \sum_{n = 0}^{\infty}\frac{(-i)^{n}}{n!}\int \limits_{-\infty}^{\infty}d^{4}x_{1}...\int \limits_{-\infty}^{\infty}d^{4}x_{n}\hat {P}(\hat {\Gamma }_{i}(x_{1}))...\hat {\Gamma}_{i}(x_{n})), \qquad (.1) $$ where $$ \hat {N}(\hat {H}_{k}(t_{i}), \hat {H}_{k}(t_{j})) = \left\\{ \begin{matrix} \hat {H}_{k}(t_{i}) \hat {H}_{k}(t_{j}), t_{i} > t_{j} \\\ \hat {H}_{k}(t_{j}) \hat {H}_{k}(t_{i}), t_{j} > t_{i} \end{matrix}\right. . $$ How to rewrite $(.1)$ into exponential form?


title: Tsunami : power of destruction

text: I'm not sure this is the good place to ask those questions. Sorry if it isn't. Are there studies that evaluated the resistance of buildings against tsunami ? What kind of buildings could manage a tsunami with waves of 30 meters ? or a mega tsunami with waves of 100 meters ? Are underground buildings safe in those cases ? Is the potential of destruction directly linked to the height of waves on the shore ?


title: A box with cooler and heater on opposite faces

text: Suppose there's a box with one face cold, and the opposite face hot. So when the air molecules hit the cooler face, it will transfer its momentum and energy to the wall, bouncing back with less momentum. And when molecules hit the hotter face, it will bounce back faster. And the hotter side will be hit more frequently by air molecules. So will the box feel any force due to the temperature difference?


title: is gravitational force dependent on objects radius?

text: I was told that objects with the same mass but different sizes (or densities) will have a different gravitational pull. Is this true? If so, why?


title: Static,total and velocity pressure at a point

text: I have a past exam question that I am stuck on. > $200$m$^3/s$ of air is entering a shaft of $5m$ diameter, the density is > $1.18$kg/m$^3$, and I have shock loss of factor $X=.5$ I want static, total and velocity pressure at the starting point. Now I know that Shock loss is $\dfrac{X\rho v_1^2}{2}=\dfrac{.5\times1.18\times v_1^2}{2}$ and $v_1=\dfrac{200}{2.5^2 \pi}=10.186$ So Shock loss is $ \dfrac{.5\times1.18\times 10.186^2}{2}=30.608$ and Velocity pressure $=\dfrac{\rho v_1^2}{2}=61.215$ So I have VP and I want Total and static pressure, but I have no idea how to do this, any ideas would be lovely!


title: How is voltage increased from a battery?

text: I have recently watched a video about a powerful Octa-copter that uses 30,000 Volt brushless motors to propell, the batteries however look quite small about a 19v laptop battery. How is the voltage increased to suffice the voltage requirements? where does the extra electrical energy come from or what is the tradeoff for stepping up the voltage? and if the batteries are at most about 19V how does it obey ohms law?


title: Simple Question: Speed of Electromagnetic Waves in a Medium

text: If the speed of an electromagnetic wave in a particular medium is such that $v = c$, the speed of light, does this mean that the permeability $\mu = \mu_0$, i.e. that of a vacuum and the index of refraction is also $n = \frac{c}{v} = 1$? Also $\epsilon = \epsilon_0$?


title: Why and how will we be able to image extraterrestrial planets in the next two decades?

text: According to a recently published article, a well-established Cambridge astrophysicist stated that: > the “origin of life, where it exists, and whether aliens exist, is going to > be crucial over the next four decades”. He added: “We know now that stars > are orbited by retinues of planets just as our sun is. We have learned this > in the last decade, essentially. “ _Within 10 or 20 years we will be able to > image other planets like the Earth_ , orbiting other stars." My question is, what technological limitation are we dealing with regarding making images of extraterrestrial planets, and how can this scientist be so certain that this limitation will disappear only 10 or 20 years from now?


title: The Unruh effect for temporarily accelerated particles

text: Do temporarily accelerated particles experience the Unruh effect? I think, they don't, because they don't see an apparent event horizon. On the other hand, if the duration of the acceleration is long enough, this case should be approximated during this phase by the eternal accelerated particle. And this particle sees the Unruh effect.


title: Anybody have example of two-qubit non-Pauli and non-Clifford quantum gate?

text: A lot of known quantum gates are in the Pauli group (I,X,Z,Y) or in the Clifford group (H,P,Cnot). I need examples of the quantum gates that aren't in this groups. Also, are there are matlab functions to check if a quantum gate (2x2 or 4x4) is in any of these groups? Or maybe there are matlab routines that generate quantum gates outside of this groups? Thanks..


title: Dirac's quantization rule

text: I first recall the Dirac's quantization rule, derived under the hypothesis that there would exit somewhere a magnetic charge: $\frac{gq}{4\pi} = \frac{n\hbar}{2} $ with $n$ natural. I am wondering how the quantization of electric charge can be deduced from it. The quantization of the product $gq$ is certainly not enough; what else is demanded?


title: Can anybody provide a simple example of a quantum computer algorithm?

text: Does anybody give a good textbook description of a quantum computer algorithm and how its different from an ordinary algorithm?


title: Speed of light that is traveling away from the observer

text: The second postulate of Special Relativity states: > The speed of light in a vacuum is the same for all observers, regardless of > their motion relative to the source. Now imagine the observer that is looking at the light which is traveling away from him in a direction he is looking. In this case, the light should look somewhat like a "dot" to such observer - not really moving at any direction for him. However according to postulate above the light should move at the speed of c (if we are in vacuum to be precise). What is wrong with this reasoning? Or is it just some kind of exception from the rule?


title: Laptop self charge cable?

text: Ok this is a really silly question, but I was really curious as to why this wouldnt work. I'm just starting my electricity and magnetism course, so I really dont know much about electricity. I know this is not possible, but why? ![alt text](http://i.stack.imgur.com/k2HD5.jpg)


title: What would the electromagnetic field of a massless electron look like

text: The Standard Model gives non-zero mass to the electron via the coupling to the Higgs field. Issues of renormalizability aside, this is fundamentally unrelated to the fact that the electron couples to the EM field. However, if the Higgs mechanism did not operate - that is, if there were no vacuum symmetry breaking, the electron field would have no effective mass term. In QFT perturbation theory, this model offers no special difficulty. My question is, what is the classical limit of this theory, if it has one? Does the electron acquire a purely EM mass? If the low energy renormalized mass is set to zero (is there an obstacle to doing that?) what do the classical field configurations look like? My puzzlement is related to the classical description of the EM field of a relativistic charged particle - namely, that it becomes "flattened" in the direction of motion, just like a Lorentz-contracted sphere. That is, the field is weaker than for a motionless particle in the longitudinal direction, and stronger in the transverse directions. The limiting case is that the field becomes concentrated on a 2-d surface transverse to the particle location, where it has infinite strength - obviously an unphysical situation. So what actually happens?


title: Two 60W-lightbulbs are connected on 220 V AC-voltage - how much electric power is spent by each lightbulb?

text: > Two 60W-lightbulbs are connected on 220 V AC-voltage - how much electric > power is spent by each lightbulb? There are two scenarios: In one, these two lighbulbs are connected serially, and in the other parallelly. The trouble is, no resistance value is given, so I don't know how electric power can be calculated for each bulb. No amperage given either. How do I proceed? Edit: Thanks to the answer I did figure out the power consumption for each connection. **Serially:** > $P=\dfrac{U^2}{R} => R=\dfrac{U^2}{P}$ At 220 Volt and 60 W the resistance is $806 \dfrac{2}{3}$ ohm for each bulb. In a serial connection the resistances add up, meaning, the two bulbs together have $1613 \dfrac{1}{3}$ ohm. Repeating the above formula, we get a total power consumption of $30$ watt. So each bulb consumes $15$ watt. **Parallelly:** The total resistance in a parallel connection can be calculated as > $R_T=\dfrac{R_1*R_2}{R_1+R_2}$ which is $403 \dfrac{1}{3}$ ohm. Thus, again repeating the formula for power consumption $\big(P=\dfrac{U^2}{R}\big)$, the total power consumption is $120$ watt. Which is $60$ watt for each bulb.


title: Alpha particle deflection by 180 degree in Rutherford's gold foil experiment

text: Did some of the aplha particles back trace their path after hitting the gold foil ? (Turn by 180 degrees.) If so, how were they detected ?


title: How can a star emit light if it is in Plasma state?

text: 1. I understand that star is in Plasma state (all nucleus and electrons are not bound to each other and moving around freely)   2. Photon is emitted when an excited electron moves back to lower orbit.   3. So in a star if electrons are not in any orbit then how can photons be produced? I am sure some of my understanding above is incorrect :) please help me understand. Thanks


title: Final theory in Physics: a mathematical existence proof?

text: Some time ago, I read something like this about the issue of "a final theory" in Physics: "Concerning the physical laws, we have several positions as scientists   1. There are no fundamental physical laws. At the most elementary level, the Universe/Multiverse is essentially chaotic and anarchical. There are no such laws.    2. There are a continuous sequence of more and more precise theories, but there is no a final theory. Physics will be always evolving from one approximate theory to another bigger and more accurate. In the end, we will be also able to find a better theory and additional levels of complexity or reality.    3. There is a final theory explaining everything, and we will find if and only if:  i) We are clever enough to find such a theory. ii) We make good and sophisticated enough mathematics. iii) We guess the right axioms/principles/ideas. iv) We interpretate data correctly and test the putative final theory with suitable instruments/experiments. " Supposing 3) is the right approach... Question: How could we prove the mere mathematical existence of such a theory? Wouldn't it evade the Gödel's incompleteness theorem somehow since, as a Theory of Everything, it would be explain "all" and though it should be mathematically self-consistent? How could a Theory of Everything be a counterexample of Gödel's theorem if it is so, or not? Note: The alledged unification of couplings in supersymmetric theories is a hint of "unification" of forces, but I am not sure if it counts as sufficient condition to the existence of a final theory. Complementary: Is it true that Hawking has changed his view about this question? SUMMARY: 1') Does a final theory of physics exist? The issue of existence should be tied to some of its remarkable properties (likely). 2') How could we prove its existence or disproof it and hence prove that the only path in Physics is an infinite sequence of more and more precise theories or that the Polyverse is random and/or chaotic at the most fundamental level? 3') How 1') and 2') affect to Gödel's theorems? I have always believed, since Physmatics=Physics+Mathematics (E.Zaslow, Clay Institute) is larger than the mere sum that the challenge of the final theory should likely offer so hint about how to "evade" some of the Gödel's theorems. Of course, this last idea is highly controversial and speculative at this point.


title: Asking the vector form of the equation about velocity

text: In the undergraduate text of physics, the equation for velocity and displacement at constant acceleration are given in scalar form. For example, my text reads $$v^2 = v_0^2 + 2a(x-x_0)$$ But today, I am reading another text which gives the vector form for displacement. I am trying to write above equation in vector form also but how do I deal with that square in vector? I am tackling that square with dot product of two vector such that $v^2 \to \vec{v}\cdot\vec{v}$, is that correct? So what about the multiplication of acceleration and the displacement? dot product again? This equation looks a bit confusing to me. Let's say I throw a stone upright with initial speed $v_0$ and it reaches the highest point sometimes later (so $v=0$), and the coordinate is given with y axis down and x axis horizontal. Note that the displacement is negative since the coordinate's yaxis is downward, also the gravity is along the positive y axis so the equation becomes $$ 0 = v_0^2 + 2g(-x) $$ which gives $$ x = \frac{v_0^2}{2g} $$ is that right?


title: Fourier transformation

text: I have recently studied Fourier and Laplace transformation in maths. I wanted to understand the utility in physics with some examples that requires this change in dimension and the reason why.


I would like to explore some ideas with different geometries exchanging energy mainly by radiation. This means solving Laplace equation for the steady state. However the boundary conditions are always Neumann's because the only constraints are fluxes. After many questions and attempts, I realized that Mathematica cannot (yet ?) treat PDEs with Neumann boundary conditions. To my great surprise I have also found no literature describing the treatment of the heat equation with the derivative at boundary proportional to $T^4$ even if I thought that similar problems would play a paramount role in engineering. What are some good books/papers to treat the Heat equation with Neumann boundary conditions and more specifically when the flux at boundary is proportional to $T^4$ ?


title: Equations of motion for bob-on-a-string -- am I missing some terms?

text: The dynamics of a type of physical system I am currently working on are modeled in most of the literature by replacing the moving parts of that system with an equivalent set of pendulums. Parameters and initial conditions for this approximate pendulum model are chosen such that the total mass and the center of mass of these pendulums describe those of the real system as accurately as possible. For one particular system however, I need to take this one step further because that system is losing mass _rapidly_. (Also, it is spinning about an axis that lies outside the system, but let's leave that out of consideration here). Therefore, I ended up modeling this particular system with a series of "specialized" pendulums:   * the pendulum masses are free to move in any direction (i.e., not constrained to move in any single plane)    * the point masses are attached to the pendulum hinge points by means of massless strings of infinite strength but finite elasticity (spring constant $k$)   * The pendulum mass $m$, the height of the hinge point $h$, the restoring force $m\mathbf{g}$ and the nominal string length $L$ are all functions of time. ![enter image description here](http://i.stack.imgur.com/RJv7C.png) In this figure, $h(t) = \left|\mathbf{h}(t)\right|$ and $\left|\mathbf{s}(t)\right| = L(t)$ when the string is _just_ stretched. $\dot{\mathbf{h}}$ and $\ddot{\mathbf{h}}$ are constrained to only the pure vertical direction. Leaving the fictitious forces out of consideration, I have come up with the following equation of motion for this kind of pendulum: $$ m\ddot{\mathbf{r}} = -m\mathbf{g} + \begin{cases} 0 & \text{if } |\mathbf{s}| < L\\\ m\left(\frac{\mathbf{s}\times\mathbf{\dot{r}}}{|\mathbf{s}|^2}\times\mathbf{\dot{r}}\right) - \gamma_1\left(\frac{\mathbf{s}\times\mathbf{\dot{r}}}{|\mathbf{s}|^2}\times\mathbf{r}\right) & \\\ \ \ \ \ \ \ \ \ \ - k\left(|\mathbf{s}|-L\right)\frac{\mathbf{s}}{|\mathbf{s}|} - \gamma_2\left(\frac{\mathbf{s}}{|\mathbf{s}|}\cdot\mathbf{\dot{r}} + \dot{L}\right)\frac{\mathbf{s}}{|\mathbf{s}|}& \text{if } |\mathbf{s}| \geq L \end{cases} $$ where $\gamma_1$ is the torsional damping ratio, $k$ the string constant of the (flexible) string and $\gamma_2$ its linear damping ratio. The first two terms of the second case lie in the current plane of rotation of the mass $m$ about the hinge point. They describe the centripetal force towards the hinge point, and torsional damping, respectively. The last two terms are in the radial direction, and describe the stretching and damping of the stretched string, respectively. Numerical results of this equation are encouraging. However, this is all a bit messy (especially with the terms for the ficticious forces included), so I'd really appreciate if someone else took a look at it. Why I ask this is because I can't figure out why the $\dot{\mathrm{h}}$, $\ddot{\mathrm{h}}$, $\ddot{L}$ and $\dot{m}$ do not show up in this equation... Am I missing any terms anywhere?


title: Integral related to particle diffusion

text: In the context of particle diffusion, I am trying to understand the equations that describe Brownian motion as a macroscopic process. Assume $N(x,t)$ is a number concentration and $D$ is a diffusion constant, then we can write: $$\frac{\partial N(x,t)}{\partial t} = D \frac{\partial^2}{\partial x^2} N(x,t)$$ next, we can multiply by the mean square displacement $x^2$ and integrate over $x$ from $-\infty$ to $\infty$: $$\int x^2 \frac{\partial N(x,t)}{\partial t} dx = \int x^2 D \frac{\partial^2}{\partial x^2} N(x,t) dx $$ Now the script that I'm reading writes that the left-hand side can be written as: $$\int x^2 \frac{\partial N(x,t)}{\partial t} dx = N_0 \frac{\partial \langle x^2 \rangle}{\partial t} $$ where I assume that $N_0$ is the initial number concentration of particles and $\langle \rangle$ is the mean square displacement. I do not understand the last equation. Could somebody explain why I can express the integral as the partial derivative wrt to time of the mean square displacement (multiplied by $N_0$). **EDIT: $\langle x^2 \rangle$ is the mean square displacement (contrarily to what I have written earlier)**


title: A rock connected to one end of string in circular motion gets released.. and what happens?

text: I know this is a basic question, but question: A rock is connected to one end of string and is in circular motion with the center being the other end of the string. Now the string gets released at some point. How do we know the direction in which the rock is going to fly?


title: Determining Maximum Velocity of an object traveling horizontally

text: I'm in the process of working on a physics related game. I'm looking to find the maximum velocity of an object given it's mass and the force acting on it when it is traveling horizontally. I believe there must be a method of calculating this but I'm unaware of it. Is there a formula for this scenario? I'm unaware if this is the same calculation as terminal velocity but I don't believe it is. Thanks for any help!


title: Obtain oxidation numbers from looking at electron configuration?

text: Suppose I know that Selenium has electron configuration $[Ar] 4s^2 3d^{10} 4p^4$. How can I use this information to then come to the conclusion that its possible oxidation states are 6, 4, 2, 1 and -2? I understand the 6: Give up all s electrons to reach $Ar$ configuration I also understand the -2: Obtian two electrons to complete the 4th shell. I can guess why 4 is plausible: Give up all your 4p electrons But 2 and 1? A guess for 2 would be to give up the 4s electrons, but why should it? And for the 1 I have no idea. Maybe give up one electron in the 4th shell and have it replaced by one of the 3d electrons?


title: Can nowadays spin be described using path integrals?

text: In Feynmans book, "Quantum mechanics and Path Integrals" he writes in the conclusions (chapter 12-10) > With regards to quantum mechanics, path integrals suffer most grievously > from a serious defect. They do not permit a discussion of spin operators or > other such operators in a simple and lucid way. ... It is a serious > limitation that the half-integral spin of the electron does not find a > simple and ready representation. This has been written in 1965. Has there been any progress on this problem? For example, it is possible nowadays to not only derive the Schrödinger equation, but also the Pauli equation from the path integral formulation of QM?


title: Static friction

text: Just a quick question, When a force on an object is equal to its static coefficient * normal force , would the object move or would it stay stationary? Or does the force always need to be greater than this value for the object to move? My guess is that it need to be greater but im not completely certain.


title: Question on static electricity & electron transfer

text: Static electricity is caused by the transfer of electrons between substances right? For example, take a balloon and your hair. Both are stable and electrically neutral. So why would electrons jump from the balloon to your hair?


title: Stress-Energy Tensor

text: As of recent, I've been doing a bit of self education in GR, equipped with a working knowledge of the key elements of the differential geometry in GR, and in looking at the Einstein-Rosen bridge, ![Einstein-Rosen Bridge](http://i.stack.imgur.com/bMT6L.jpg) I see that geometrically it is a hyperboloid of one sheet. Now when using this manifold to calculate things like curvature and geodesics, we need a metric, which we can somewhat easily derive from the equation of the aforementioned surface. Now per $ R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R = \frac{8\pi G}{c^{4}} T_{\mu\nu} $ ( setting $ \Lambda = 0 $ for the simple case ), one should be able to solve for the stress-energy tensor, albeit the grotesque mathematics. By this logic, based on the predetermined shape of the local spacetime, we should be able to calculate the mass and geometry of the object required to create such a metric. My questions then are: To what extent is this a valid statement if at all? Given the fact that there are going to be initial value problems embedded in the PDE's with conditions that require the input of some properties of the geometry of the aforementioned object: How is it that one goes about finding physically viable objects that satisfy the EFE under the constraints that we've applied if any exists. How does one test to see if there are even any solutions that work? For an attempt at the second questions, I was thinking that if we have to apply constraints to get rid of the unknown constants that resulted from the PDE, we just put constraints on what the solution can be, to ensure that it fits with what we want. Ie I want to make sure my sphere of matter is of this size, but can't exceed this mass. Am I at least on the right track with this?


title: The Maxwell and the Boltzmann distributions

text: I am trying to understand where the Boltzmann distribution comes from. I recently learned some interesting things of which my interpretation follows below. Did I interpret correctly? If so, is this all there is to it, or is this only part of the story? When sampling coordinates in a high-dimensional Euclidean space as independent and identically distributed Gaussians, you are essentially uniformly sampling on a sphere in that space. Maxwell and Boltzmann were well aware of this. The equiprobability principle loosely asserts that configurations at the same energy occur with equal probability. In an $N$ particle system in 3-space at constant total kinetic energy $E_\text{kin}$, it can be derived that the equiprobability principle is consistent in the thermodynamic limit $N\to\infty$ with each velocity component of each constituent particle being independently distributed with probability distribution $$\pi(v_i^\alpha) \propto e^{-\frac12{m(v_i^\alpha)^2\over k_BT}}= e^{-{E_\text{kin, i}^\alpha\over k_BT}}$$ where $i$ indexes the constituent particles and $\alpha$ the coordinates; namely, this leads to a distribution of $r = \sqrt{\sum\|v_i\|^2}$ of the form $$\pi(r) \propto r^{3N-1}e^{-\frac12 m{r^2\over k_BT}} = r^{3N-1}e^{-{E_\text{kin}\over k_BT}}$$ ($3N$ is the total number of velocity degrees of freedom). For large $N$, this distribution peaks sharply around $r = \sqrt{(3N - 1){k_BT\over m}} \approx \sqrt{3N{k_BT\over m}}$ and since the average kinetic energy per particle is $E_\text{kin,i} = {3\over2}k_BT$, this is equal to $r\approx\sqrt{{2\over m}E_\text{kin}}$, at which value indeed the kinetic energy is $E_\text{kin}$. Boltzmann postulated that this still holds when the total energy is non- constant, and not only for components of individual particles, but for entire subsystems, so that the probability of any configuration of energy $E$ becomes proportional to $$\pi(E) \propto e^{-{E\over k_BT}}$$


title: Irreversible work

text: I'm having some trouble trying to find the irreversible work done by a perfect gas when expanding. I know that the reversible work is greater than the irreversible one, but how can I find the irreversible work for an isothermal process? The exercise asks me to find the irreversible work for the expansion of a perfect gas from 10000 to 2000 bar at 315 K with $T_{cte}$. I also found that the irreversible work is simply $P\Delta V$, and even though the volume does vary, the exercise also tells me the pressure does too. I found the volumes for each state and got $V_1=2.619*10^{-3}L$ and $V_2=0.013095 L$. But now I'm not sure what pressure I should use? Or why should I just use one pressure. Thanks in advance. Also, is the irreversible work for a gas the same independently of the process? Is it the same for an isothermal, an adiabatic and isobaric process? Or just like the reversible work, does it change?


title: Windshield wipers on rear windows

text: Cars, like sedans, with slanted rear windows never seem to have windshield wipers on them. On the other hand cars, like mini-vans, with vertical rear windows generally do have windshield wipers on them. Yet it seems to me that the vertical rear windows would need them less than the slanted windows. Is my intuition about vertical windows needing them less incorrect?


title: What is the "associated scalar equation" of equations of motion?

text: In an essay I am reading on celestial mechanics the equations of motion for a 2 body problem is given as: $$\mathbf{r}''=\nabla(\frac{\mu}{r})=-\frac{\mu \mathbf{r}}{r^3}$$ Fine. Then it says the "associated scalar equation" is: $$r''=-\frac{\mu}{r^2}+\frac{c^2}{r^3}$$ I've never heard of such a thing. Can someone please explain what the "associated scalar equation" of an equation of motion is. If it is just the equation of motion in scalar form, then why does that extra term $\frac{c^2}{r^3}$ appear? Oh, $\mu$ is the mass constant. It's not clear from the essay what $c^2$ is. It might be the speed of light squared, or perhaps a constant of integration. EDIT: The essay in question can be found here. The equations in question are found on page 5.


title: Effect of temperature on optical rotation of sugar solution

text: How does temperature affect the specific optical rotation of sugar solution at constant concentration and why? If, for example, the temperature is increased, will the optical rotation for a given path length and concentration increase or decrease, and why?


title: Definition of a non-pertubative Quantum field theory

text: How do you define a non-perturbative Quantum field theory. What does it mean? I was just digging around some math about the meaning of $Z_1$, and such in terms of probabilities. It turns out these and associated counter terms are derived via perturbation theory. So the question is , if you associate meaning to things like probability of self interaction and such, how do you make these things make sense if you are working non-perturbatively. What is a self interaction outside of perturbation theory? I mean considering some probable sketch, for some arbitrary calculations , something looking like the lines below might appear, then some cleaver physics is said. $1 + \Sigma_a$ $\Gamma^{n} \rightarrow Z_{b} \bar{\Gamma^{m}}$ ok so what is a self interaction in non-pertubative models like. Is a non-pertubative model real? Can it make sense. . . . I don't know .. . . . .


title: taking the trace

text: could anyone show me the first couple of steps in taking the trace of something like this, im not sure how to start. 'Tr[$\gamma($$\gamma k + $$\gamma p + $$\gamma q + m) $$\gamma ($$\gamma k + $$\gamma p+ $m)]'


title: Normalization of the path integral

text: When one defines the path integral propagator, there is the need to normalize the propagator (since it would give you a probability density). There are two formulas which are used. 1) **Original (v1+v2)** : The first formula (which I can intuitively agree with) says that: $$\tag{1} \int_{Dx_b}dx_b\left|K(x_bt_b|x_at_a)\right|^2=1$$ for all values of $x_a$ on fixed values of $t_a, t_b$ and where $Dx_b$ means the domain of $x_b$. 1') **Update (v3+v4)** : I changed my mind (to get more into agreement with the Born-rules). The first formula (which I can intuitively agree with) says that: $$\tag{1'} \left|\int_{Dx_b}dx_bK(x_bt_b|x_at_a)\right|^2=1$$ for all values of $x_a$ on fixed values of $t_a, t_b$ and where $Dx_b$ means the domain of $x_b$. 2) The second formula (which is actually also very intuïtive) says that: $$\tag{2}\lim\limits_{t_b\rightarrow t_a}K(x_bt_b|x_a,t_a) = \delta(x_b- x_a).$$ Now these are usually treated as equivalent, but I can't directly see how this can be the case. Isn't the second formula less restrictive ?


title: Resistor and LED - together and separated

text: Something very basic which I don't get no matter how much I read.   * circuit A: only a resistor and power supply - the resistor will burn.   * circuit B: only a LED and power supply - the LED will damaged. As far as I understand these are the facts.   * in circuit B LED will draw as much current as we give it (we should limit it to about 20mA)   * in circuit A resistor will burn because it cannot dissipate the energy since the current is very high  If the LED is not limiting the resistor current, how come the mix of both is working, and the resistor does not burn? Let me try to clarify. Let's take an example: in a series circuit of LED and resistor,   * $V_\text{in} = 12\text{ V}$   * LED $V_f = 2\text{ V}$   * LED $V_i = 20\text{ mA}$   * $R = \frac{10\text{ V}}{0.02} = 500\ \Omega$   * $P = 0.2\text{ W}$ The voltage across the resistor is $10\text{ V}$. If you hook the same resistor to a $10\text{ V}$ power supply, wouldn't it burn?


title: Question regarding waves squeezed

text: in one MOOC I am taking, the professor had a slide stating "when squeezed into a narrow space wave is amplified". I am not sure I understand what he meant by that, and I am trying to think into terms of wave equations. Could anyone explain it to me? Thank you.


title: Question about "different" equations of motion in dependence of indices

text: Let's have the action $$ S = \int (\partial_{\mu}h^{\mu \sigma}\partial^{\nu}h_{\nu \sigma} - \Lambda h^{\mu \nu}T_{\mu \nu}) d^{4}x. $$ For definiteness, $$ h_{\mu \nu} = h_{\nu \mu} , \quad T_{\mu \nu} = T_{\nu \mu} \neq f(h_{\mu \nu}), \quad h^{\mu \nu} = \eta^{\mu \alpha}\eta^{\nu \beta}h_{\alpha \beta}, \quad \eta^{\mu \nu} = diag (1, -1, -1, -1). $$ Let's take the variation of $S$ by $h_{\mu \nu}$ and set the surface terms to zero: $$ \delta S = \int \partial_{\mu} \delta h^{\mu \sigma} \partial^{\nu}h_{\nu \sigma} d^{4}x + \int \partial_{\mu}h^{\mu \sigma}\partial^{\nu}\delta h_{\nu \sigma} d^{4}x - \Lambda \int \delta h^{\mu \nu} T_{\mu \nu}d^{4}x = $$ $$ = -\int (\partial_{\mu}\partial^{\nu}h_{\nu \sigma}\delta h^{\mu \sigma} + \partial_{\mu}\partial^{\nu}h^{\mu \sigma} \delta h_{\nu \sigma} - \Lambda \delta h^{\mu \sigma} T_{\mu \sigma})d^{4}x. $$ Then it's time to my stupid question. The expressions under the integral are scalar form and we may rename the indices. But when we take out the variation $\delta h^{\mu \sigma}$, all is changed. I can do that by the two "ways": $$ \delta S = \int \delta h^{\mu \sigma}(\partial_{\mu}\partial^{\nu}h_{\nu \sigma} + \partial^{\alpha}\partial_{\mu}h_{\alpha \sigma} - \Lambda T_{\mu \sigma})d^{4}x = \int \delta h^{\mu \sigma}(2\partial_{\mu}\partial^{\nu}h_{\nu \sigma} - \Lambda T_{\mu \sigma})d^{4}x \qquad (1) $$ and $$ \delta S = \int \delta h^{\mu \sigma}(\partial_{\mu}\partial^{\nu}h_{\nu \sigma} + \partial^{\alpha}\partial_{\sigma}h_{\alpha \mu} - \Lambda T_{\mu \sigma})d^{4}x . \qquad (2) $$ By setting variation to zero I obtain different equations of motion $(1), (2)$. Where did I make the mistake?


title: Analyse astronomical data

text: Recently I was told of a job offer to analyse astronomical data. The job offer states that they want somebody with knowledge of astronomical data analysis software and it will be a plus somebody who has experience with programming languages, giving as examples: IDL and Fortran. My first question is:   1. Is Matlab a good programming language? I mean, to treat astronomical data and any other things that they might want? From what I see IDL is comparable to Matlab, both have pros and cons but basically they both do the job.   2. For my second question, and I'm sorry if I should post this on a different post but are very closely related, the selection for this job will be made by analysing data from some surveys on the optic, millimeter and infrared frequencies from the Hubble, Spitzer, Herschel and Apex telescopes. My other question is: what kind of data is this and what kind of treatment I may have to do? If somebody could indicate me somewhere online where free data is available I would be deeply thankful.  My experience on this kind of job is, clearly, limited, but I believe I have all the theoretical knowledge to tackle the problem, I just need to know what to study and train.


title: Capacitor's voltage

text: Imagine I have two plates that form a capacitor, so the magnitude of the charge in each plate is, let's say, $Q$. So, the force, in respect to the distance $d$, is gonna be: $$F(d) = k\frac{Q^2}{d^2}$$ So, in order to calculate the **energy** stored in the capacitor, since it's equal the work needed to pull a plate next to the other, I can calculate the energy $E$ by: $$E = F(d)\cdot d$$ But since $F$ changes with respect to $d$, then I have to integrate $F$, and then multiply by $d$, so: $$E = \int F(d)dd = \int k\frac{Q^2}{d^2} = kQ^2\int\frac{1}{d^2}dd = -kQ^2\frac{1}{d}$$ So: $$E = -kQ^2\frac{1}{d} =-k\frac{Q^2}{d}$$ I also know that **`voltage`** is the same as **`energy per charge`** , so: $$V = \frac{E}{Q} = \frac{-k\frac{Q^2}{d}}{Q} = -k\frac{Q}{d}$$ Is it right? I don't know, because the capacitance $C$ is $$C = \frac{Q}{V}$$ or $$Q = CV$$ And $E = QV$ Then if I substitute $Q$ inside my **energy** formula, I get: $$E = -k\frac{CV^2}{d}$$ but the right should be: $$E = \frac{CV^2}{2}$$ Can somebody help me?


title: Casmir effect creating positive energies in vacumn

text: The Casmir force is explained by wave exclusion: Two parallel plates create a reduction in the possible modes of vacuum oscillations between them, reducing the energy of the vacuum. Since we define vacuum far from any plates as "zero" energy, a reduction of energy creates a negative energy. However, these sources also state that sometimes the regions in the vacuum gets a positive energy (depending on the geometry of the conductors). If so, how would "exclusion" create extra energy in a region?


title: Are silicone polymers transparent to infrared light?

text: I know that silicon and even black plastic bags are transparent to infrared light. What about light-colored silicone polymers (take white or green colors)? Are they transparent to infrared light too? And why?


title: How does the Feynman's $i\epsilon$-prescription make the Feynman propagator causal?

text: The Feynman propagator is non-vanishing outside the light cone, but still manages to be in accord with causality. How is this achieved? What does the $i\epsilon$-prescription have to do with this?


title: How was the formula for kinetic energy found, and who found it?

text: My questions mostly concern the history of physics. Who found the formula for kinetic energy $E_k =\frac{1}{2}mv^{2}$ and how was this formula actually discovered? I've recently watched Leonard Susskind's lecture where he proves that if you define kinetic and potential energy in this way, then you can show that the total energy is conserved. But that makes me wonder how anyone came to define kinetic energy in that way. My guess is that someone thought along the following lines: > Energy is conserved, in the sense that when you lift something up you've > done work, but when you let it go back down you're basically back where you > started. So it seems that my work and the work of gravity just traded off. > But how do I make the concept mathematically rigorous? I suppose I need > functions $U$ and $V$, so that the total energy is their sum $E=U+V$, and > the time derivative is always zero, $\frac{dE}{dt}=0$. But where do I go from here? How do I leap to either   * a) $U=\frac{1}{2}mv^{2}$   * b) $F=-\frac{dV}{dt}$?  It seems to me that if you could get to either (a) or (b), then the rest is just algebra, but I do not see how to get to either of these without being told by a physics professor.


title: Why is momentum conserved in an inelastic collision and kinetic energy is not conserved?

text: We know that in an inelastic collision that total momentum of the system before collision equals the total momentum after collision. But total kinetic energy before collision is not equal to total kinetic energy after collision. How is possible given that the formula of momentum is $mv$ and the formula of kinetic energy is $\frac{1}{2} mv^2$; both are dependent on mass and velocity.


title: Is gravity not a force?

text: When I jump out of a plane and begin to fall and accelerate, I feel no force whatsoever. When I stand on the earth, I feel only the force of the Earth pushing me outwards. I contend the accelerating condition is a natural state brought about by a local curvature of space towards the center of mass of the Earth. We explain it by Newton's law, but I can't see how two separated bodies can exert a force on each other unless the space between is curved.


title: How long would it take for a galaxy to collapse without dark matter?

text: I am trying to understand the effects of gravity in the Cosmos without complications of Dark Matter/Dark energy issues. So my question is, assuming that a galaxy (for example, the Milky Way) does not contain any dark matter, does anyone know the equation that would explain how long it would take for the galaxy to collapse into the massive black hole standing in the center? If we were to extend the question to the Universe, would the equations be different and why? (again no dark energy or $\Lambda$ or a Hubble Constant would be involved so that I can easily get my head around the question of gravity at a very large scale). I've been trying to get information on this issue for a while, so any help with equations will be welcome. I will also appreciate some good references especially a good book on the subject.


title: Length of the missing part

text: I have got a very interesting question. A car is driving this way on the picture down. In the looping is a missing part. The radius of the looping is r = 40cm. The car starts at the height h = 3r. If the car jumps from the interruption of the looping it flys and lands an the other side of the interruption and goes ahead to drive. Now I have to find out, how long the missing part of the looping is. phi has no value. I don't have to calculate the friction. ![enter image description here](http://i.stack.imgur.com/mAVxR.png) The car has got a velocity of 17,5 km/h in B, right?


title: A beam of light, traveling around a planet

text: Some time ago, I happened to have found one problem: > The refractive index of the planet's atmosphere decreases with height above > the surface under the following law: > > $$n=n_0-\alpha h$$ > > with condition $\frac{\alpha h}{n_0}\ll1$ > > Find the altitude $H$ above the surface of the planet, where a ray, emitted > horizontally, will bypass the planet, remaining at this elevation. Radius of > the planet is $R$. Since optics is not my field I decided to ask here. What would be the height $H$ for the Earth? (approximately)