question stringlengths 37 38.8k | group_id stringlengths 2 6 | sentence_embeddings listlengths 768 768 |
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<p>I am an undergrad interested in Condensed Matter Theory. Particularly topological phases and systems exhibiting topological order. A potential research advisor doing a lot of work in Symmetry Protected Topological (<a href="http://en.wikipedia.org/wiki/Symmetry_protected_topological_order" rel="nofollow">SPT</a>) Phases pointed me to a review paper by Senthil (<a href="http://arxiv.org/abs/1405.4015" rel="nofollow">http://arxiv.org/abs/1405.4015</a>) but I have found that despite having taken some of the CMT grad courses at my institution (many-body quantum field theory, cond-mat topics) there are quite a few tools I still lack practice in. Chern-Simons Theory being a prime example. Anyone know some good review papers or perhaps a good path through the literature for someone interested in doing this kind of work?</p> | g13880 | [
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<p>(Reformulation of part 1 of <a href="http://physics.stackexchange.com/questions/741/electromagnetic-field-as-a-connection-in-a-vector-bundle">Electromagnetic Field as a Connection in a Vector Bundle</a>)</p>
<p>I am looking for a good notation for sections of vector bundles that is both invariant <em>and</em> references bundle coordinates. Is there a standard notation for this?</p>
<p><strong>Background:</strong></p>
<p>In quantum mechanics, the wave function $\psi(x,t)$ of an electron is usually introduced as a function $\psi : M \to \mathbb{C}$ where $M$ is the space-time, usually $M=\mathbb{R}^3\times\mathbb{R}$.</p>
<p>However, when modeling the electron in an electromagnetic field, it is best to think of $\psi(x,t)$ as a section in a $U(1)$-vector bundle $\pi : P \to M$. Actually, $\psi(x,t)$ *itself* is not a section, it's just the image of a section in one particular local trivialization $\pi^{-1}(U) \cong U\times\mathbb{C}$ of the vector bundle. In a different local trivialization (= a different gauge), the image will be $e^{i\chi(x,t)}\psi(x,t)$ with a different phase factor.</p>
<p>Unfortunately, I feel uncomfortable with this notation. Namely, I would prefer an invariant notation, like for the tangent bundle. For a section $\vec v$ of the tangent bundle (= a vector field), I can write $\vec v = v^\mu \frac{\partial}{\partial x^\mu}$. This expression mentions the <em>coordinates</em> $v^\mu$ in a particular coordinate system, but it is also <em>invariant</em>, because I also write down the basis vector $\frac{\partial}{\partial x^\mu}$ of the coordinate system.</p>
<p>The great benefit of the vector notation is that it automatically deals with coordinate changes: $\frac{\partial}{\partial x^\mu} = \frac{\partial}{\partial y^\nu}\frac{\partial y^\nu}{\partial x^\mu}$.</p>
<p><strong>My question:</strong></p>
<p>Is there a notation for sections of vector bundles that is similar to the notation $\vec v = v^\mu \frac{\partial}{\partial x^\mu}$ for the tangent bundle? What does it look like for our particular example $\psi$?</p>
<p>If no, what are the usual/standard notations for this? How do they keep track of the bundle coordinates?</p> | g13881 | [
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<p>The title says it all. I think this should be a pretty simple question but I just couldn't find the answer.</p>
<p>Ok -- I'll give a bit more context to my question. I'm encountering this in the context of general relativity, when we're trying to solve for the Schwarzschild solution. We write down a metric that respects spherical symmetry. According to the textbook (Carroll), "we argue that a spherically symmetric space-time can be foliated by two-spheres - in other words, that (almost) every point lies on a unique sphere that is left invariant by the generators of spherical symmetry." Some time later he mentions the Killing vectors of $S^2$, which are</p>
<p>$$R=\partial_\phi$$</p>
<p>$$S=\cos \phi \partial_\theta - \cot \theta \sin \phi \partial_\phi$$</p>
<p>$$T=-\sin\phi\partial_\theta - \cot\theta \cos\phi\partial_\phi$$</p>
<p>and he observes that these vectors obey the following algebra:</p>
<p>$$[R,S]=T\quad [S,T]=R\quad [T,R]=S.$$</p>
<p>I'm not sure how the Killing vectors are exactly linked to spherical symmetry, and why such an algebra between the Killing vectors is important (specifically with regard to spherical symmetry). And I also don't know what he means when he says "generators of spherical symmetry", hence the original question.</p> | g13882 | [
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<p>I have 2 equation describing the alternating amperage $I_1$ and $I_2$. I need to get amount of these amperages.</p>
<p>My equations:
$$I_1=10\sin(\omega t+30)$$
$$I_2=20\sin(\omega t-50)$$</p>
<p>How can i make it? I forgot. </p>
<p>Thank you.</p> | g13883 | [
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<p>In this excerpt from Griffith's quantum mechanics book:</p>
<blockquote>
<h1>THE FREE PARTICLE</h1>
<p>We turn next to what should have been the simplest case of all: the free particle [$V(x) = 0$ everywhere]. As you'll see in a momentum, the free particle is in fact a surprisingly subtle and tricky example. The time-independent Schrödinger equation reads</p>
<p>$$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} = E\psi\tag{2.74}$$</p>
<p>or</p>
<p>$$\frac{d^2\psi}{dx^2} = -k^2\psi,\quad\text{where } k \equiv \frac{\sqrt{2mE}}{\hbar}.\tag{2.75}$$</p>
<p>So far, it's the same as inside the infinite square well (Equation 2.17), where the potential is also zero; this time, however, I prefer to write the general solution in exponential form (instead of sines and cosines) for reasons that will appear in due course:</p>
<p>$$\psi(x) = Ae^{ikx} + Be^{-ikx}\tag{2.76}$$</p>
</blockquote>
<p>Equation 2.76 is the solution for equation 2.74.</p>
<p>I don't understand why Griffiths says the first term in equation 2.76 represents a wave traveling to the right and the second to the left.
Some website says when we apply momentum operator to first term then we get positive value in the positive x direction, but why do we separate two terms? If I apply momentum operator on equation 2.76, I can't get the wavefunction back.
Since nobody know the physical meaning of wavefunction, then why do we simply separate it?</p> | g13884 | [
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<p>I was thinking about Einstein's train and platform experiment and was wondering if a beam of light experiences throw? Let me explain, if I take a water hose and point it straight out and then swing and stop my arm, the water will for a moment go past the end of the hose in the direction I was swinging my arm... </p>
<p>Does light exhibit the same behavior? If I took a laser and started from the origin of a Cartesian system and accelerated it to let's say 75% of the speed of light along the x axis with it pointing in the y direction. When said laser pointer hits a brick wall at x0 will the laser beam actual cross to x0+delta?</p> | g13885 | [
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<p>Are valence electrons in non-polar molecules localized or delocalized?</p>
<p>I'm quite confused about the whole electron localization and delocalization business.
I'd love a simplified explanation that a high-schooler can understand.</p> | g13886 | [
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<p>I was thinking about this today...</p>
<p>Is it possible for a planet to be made almost entirely of water (H20)? I'm sure that other elements would be in the mix, but I mean almost entirely made of water. Even the core. No rocky interior covered by water.</p>
<p>If so, what would the core be like? I imagine the pressure would create something different than liquid water.</p> | g434 | [
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<p>In quantum field theories (to be relativistic, (non-)relativistic statistical or whatever), we have the powerful diagrammatic approach at our disposal. Most of the time we can not sum up all the diagrams, only a sub-class of them. I was thinking of the Hartree-Fock ones to give an example, but they are several classes we can re-sum exactly (skeleton, ladder, ...). Somehow this re-summation of diagrams is <em>exact</em>. I will keep italics to refer to this <em>exactness</em>.</p>
<p>Clearly the solution we find that way is not the full solution of the problem we considered at the beginning. By "problem" I obviously mean the eigenstates/eigenvalues problem of a secular equation given by a Hamiltonian/Lagrangian.</p>
<p>My question is the following: <strong>do we know the problem we <em>exactly</em> solved summing only partial class(es) of diagram ?</strong> I mean: can we write out an effective Hamiltonian/Lagrangian that our <em>exact</em> summation is the <em>exact</em> solution of ? Can we at least know the <em>exact</em> sub-space (eigenstates and/or eigenvalues), the solutions we found <em>exactly</em> belong to ? </p> | g13887 | [
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<p>In special relativity one assume that spacetime
can be locally described by 4 coordinates, so it makes
sense to model it as a 4-dimensional manifold.</p>
<p>I had the impression that it is assumed that there is an atlas of
special charts called inertial reference frames.
Their transition maps are restrictions of affine maps
on $\mathbb{R}^4 \rightarrow \mathbb{R}^4$. The subgroup of invertable affine maps
which are allowed as transition maps is called the
kinematical group.</p>
<p>Einstein shows in his paper, I guess, that one can choose
the kinematical group to be the Poincare group
and for example [Bacry Levy-Leblond]
showed that there would only be a few possible choices for these kinematical groups.</p>
<p>In the books, I am aware of, comes now a huge step and one
immediately assume one is in Minkowski space.</p>
<p>So my questions are the following:
Given a manifold $M^4$ with an atlas such that the transition maps
are all restrictions of the affine maps of the Poincare group.</p>
<ol>
<li><p>How can one construct a Lorentz metric (a (3,1)-semi-Riemannian metric)
on $M$.</p></li>
<li><p>Does anyone know a reference for the fact that
a manifold $M^4$ with affine transition maps is covered by
an affine space (i.e. A simple connected one is topologically $\mathbb{R}^4$)?</p></li>
</ol>
<p>One might be tempted to consider the above as something belonging to general relativity,
as one uses charts etc. But note affine transition maps are a huge restriction and
at the end (maybe after taking a covering map) one talking about Minkowski space anyways. </p> | g13888 | [
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<p>My understanding of orbital mechanics is very limited, but as I understand geostationary satellite, they stay in place by having an orbital speed corresponding to the spot they're orbiting over.</p>
<p>So my basic intuition tells me that it's not possible to have a geostationary satellite over the poles, since they're effectively standing still, and to my knowledge, a satellite must keep a certain velocity to keep from getting pulled down by gravity.</p>
<p>Am I missing something?</p> | g13889 | [
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<p>If $ϕ^† ϕ$ is invariant under $SU(2) \times U(1)$, a Phys.SE <a href="http://physics.stackexchange.com/q/71481/2451">question</a> I recently posted, then does that mean $ϕ^† ϕ$ is invariant under rotations and translations?</p> | g13890 | [
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<p>I know that in principle, the Fock state is not the same as a product state because if it were, the photons would be uncorrelated. What I don't understand is, can it be expanded in product states? </p>
<p>For example, if two photons share the same mode, is the state $|2,0\rangle$ the same as the product $|1,0\rangle|1,0\rangle$? </p>
<p>If they are in different modes, is the Fock state $|1,1\rangle$ the same as superposition of two Fock states, $\frac{1}{\sqrt{2}}\left(|1_a,1_b\rangle+|1_b,1_a\rangle\right)$ (with $a$ and $b$ labeling the photons)?
If yes, then is this superposition, in turn, the same as $\frac{1}{\sqrt{2}}\left(|1_a,1_b\rangle+|1_b,1_a\rangle\right)$ ?</p> | g13891 | [
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<p>If we have a linear molecule with a dipole moment $\mu$ in a static electric field $E$, the potential is given by $V = - \langle \mu,E \rangle$. What is the appropriate equation for the potential if we have a polarizability $\epsilon_1$ along the direction of the dipole moment?</p>
<p>If anything is unclear, please let me know.</p> | g13892 | [
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<p>I'm trying to understand mathematically that for the free expansion of an ideal gas the internal energy $E$ just depends on temperature $T$ and not volume $V$.</p>
<p>In the free expansion process the change in internal energy $\Delta Q = \Delta W = 0$, therefore by the first law $\Delta E=0$. Energy is a function of state so $E=E(V,T)=E(p,T)=E(p,V)$, using $V$ and $T$ as the state variables we have $E=E(V,T)$.</p>
<p>I have the following written mathematically in my notes but do not understand how it is derived.</p>
<p>If</p>
<p>$E=E(V,T)$</p>
<p>then</p>
<p>$dE=\left(\frac{\delta E}{\delta V}\right)_T dV+\left(\frac{\delta E}{\delta T}\right)_V dT$</p>
<p>since we know experimentally that $dT=0$ and $dE=0$ from the first law we can write:</p>
<p>$\left(\frac{\delta E}{\delta V}\right)_T=0$, as required.</p>
<p>I have two main problems with this, firstly I do not understand where </p>
<p>$dE=\left(\frac{\delta E}{\delta V}\right)_T dV+\left(\frac{\delta E}{\delta T}\right)_V dT$</p>
<p>comes from. My guess is that it is the differential of $E(V,T)$ but I do not fully understand the notation of what $E(V,T)$ represents that is, I understand that $E$ is a function of temperature and volume but I do not know how this can be written mathematically other than $E(V,T)$ so I can't differentiate it myself to check this. Again if I were to guess I would get an expression for $V$ and $T$ using the ideal gas equation and try to differentiate that?</p>
<p>Secondly even with the results I do not understand why the end result is</p>
<p>$\left(\frac{\delta E}{\delta V}\right)_T=0$ and not $\left(\frac{\delta E}{\delta V}\right)_T dV=0$.</p>
<p>Has $dV$ simply been omitted because multiplying it by 0 in turn makes it 0?</p>
<p>This question has arisen from me looking back over my lecture notes so it is possible I have written something down incorrectly which is now confusing me. If you could help to clarify any of the points above that would be great. Please let me know if more information is needed.</p>
<p>Edit:</p>
<p>I should also add I am not clear on the significance of the subscripted values $T$ and $V$ on either of the fractions represent.</p>
<p>Update:</p>
<p>I found the suffixes $T$ and $V$ mean that the temperature and volume is constant in the process.</p> | g13893 | [
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<p>From what I understand acceleration does not cause a force, but rather forces cause acceleration, so if I have a ball moving with CONSTANT velocity, that hits a wall, then the wall must apply a force to deflect the ball (i'am assuming no force by the ball since it is moving with constant velocity), but where does the force by the wall come from? Also if the wall does not accelerate to cause the force, how would you calculate it?</p> | g13894 | [
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<p>This is kind of an elementary question, but since i cannot find a reliable source on this i wanted to ask what is the formula for energy conversion efficiency? I know efficiency is calculated as useful energy out/energy in, but i've been reading some papers on Q-switching, and some people have used Energy(pump)/Energy(Q-switch) which is the same as Energy(in)/Energy(out). Could anyone please clarify this? If it isn't as i said, can the energy conversion efficiency be more than 1? Also, i was wondering what is the max efficiency that Q-switched (Nd:YAG) lasers get? I ended up getting around 58%</p> | g13895 | [
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<p>Some author(in the paper of <a href="http://dx.doi.org/10.1119/1.15862" rel="nofollow">Evolution of the modern photon</a>) mentioned that photon cannot account for the zero intensity through two crossed polarizators by 90 degree. Why so? </p> | g13896 | [
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<p>The question can be formulated as following:</p>
<p>Suppose
$$\delta \int_{t_1}^{t_2}{[p\cdot \dot{q} - H(p,q,t) ]dt} = 0$$
$$\delta \int_{t_1}^{t_2}{[P\cdot \dot{Q} - K(P,Q,t) ]dt} = 0$$</p>
<p>in which $$P = P(p,q,t), Q = Q(p,q,t)$$ is an invertible transformation.</p>
<p>Can we prove that there must exist a $\lambda$ and function $G(p,q,t)$ (or $G(p,Q,t)$, $G(P,Q,t)$, $G(P,q,t)$), such that
$$\lambda[p\cdot \dot{q} - H(p,q,t) ] = [P\cdot \dot{Q} - K(P,Q,t) ] + \frac{dG}{dt}~?$$</p> | g13897 | [
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<p>In the book by Arthur Beiser, <a href="http://www.goodreads.com/book/show/1722720" rel="nofollow"><em>Concepts of modern physics</em></a>, in the chapter LS coupling there is this image: </p>
<p><img src="http://i.stack.imgur.com/BTWyz.png" alt="enter image description here"></p>
<p><strong>QUESTION:</strong></p>
<p>How do we get total orbital angular momentum $L=3$ (<em>image (a)</em>) out of quantum numbers $\ell_1=1 \Rightarrow m_{\ell 1}=-1,0,1$ and $\ell_2=2 \Longrightarrow m_{\ell 2} = -2,-1,0,1,2$? What about $L=2$ and $L=1$?</p>
<p><strong>My attempt:</strong></p>
<p>I tried to combine the $z$ components using equations $L_z=m_\ell \hbar$ and got these results for $\boxed{L\equiv L_z}$: </p>
<p>\begin{align}
L_{1z}=&
\left\{
\begin{aligned}
-1&\hbar\\
0&\\
1&\hbar
\end{aligned}
\right.\\
L_{2z}=&
\left\{
\begin{aligned}
-2&\hbar\\
-1&\hbar\\
0&\\
1&\hbar\\
2&\hbar
\end{aligned}
\right.
\end{align}</p>
<p>If I sum these up one of the results is $-3\hbar$, which is weird and isn't listed as one of the possible values in the picture. Also the picture states that one of the possible total orbital angular momenta is $L=3$ and not $L=3\hbar$. What is this a mistake or what?</p> | g13898 | [
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0.05151820182800293,
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0.0... |
<p>In my investigation, tires will quickly get bald if we ride a motorcycle at high speed. Is it correct? How to explain it in physics? Does the friction increase at high speed rotation?</p>
<p>Update: The word "quickly" means that the tire gets bald with less distance traveled. </p> | g13899 | [
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<p>According to this and other similar papers, <a href="http://www.nature.com/nature/journal/v352/n6337/abs/352699a0.html" rel="nofollow">http://www.nature.com/nature/journal/v352/n6337/abs/352699a0.html</a>, adding a current-carrying path can increase the voltage drop across a circuit. What is the simplest example of a circuit (preferably consisting solely of two-terminal devices) that exhibits such behavior?</p> | g13900 | [
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0.01876666396856308,
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0.05... |
<p>I am imagining an unusual experiment which will require intense beams of either protons or neutrons. The experiment would work better with neutrons, but neutron sources are much <strike>weaker</strike> messier so I am considering both. I am interested in beams of almost any non-relativistic energy. I am not sure yet about how focused I will need the beam to be.</p>
<p>I know this is a beginner question, but what are the basic options here for the strongest source (as measured by neucleons per second)?</p>
<hr>
<p>Neutron apparently come from two sources: reactors and accelerators. (<a href="http://vimeo.com/19475801" rel="nofollow">As Sir Patrick Stewert can attest</a>.) In the latter case ("spallation"), a proton beam is just slammed into a heavy-metal target and the neutrons that fly out are caught. It appears that the highest intensity protons beams are often used to produce the highest intensity neutron beams in this manner.</p>
<p><a href="http://accelconf.web.cern.ch/accelconf/e00/PAPERS/THOAF202.pdf" rel="nofollow">Saclay High Intensity Proton Injector Project</a> (IPHI): $6 \times 10^{17}$ protons/second.</p>
<p><a href="http://arxiv.org/abs/physics/0008097" rel="nofollow">Low-Energy Demonstration Accelerator</a> (LEDA): $6 \times 10^{17}$ protons/second.</p>
<p>(These agree with Andre's chart below showing 100 mA. Note also that the LHC current is about 500 mA, although the protons are grouped into bunches. And as Andre pointed out, the LHC circulates the same protons and so could not provide a continuous 500 mA beam into a target/experiment that consumed them.)</p>
<p>Neutron sources, being the result of either fission or spallation, are messy. I still don't have a good handle on what kind of neutron intensities are possible for what momentum ranges. The Lujan Neutron Scattering Center at <a href="http://en.wikipedia.org/wiki/Los_Alamos_Neutron_Science_Center" rel="nofollow">LANCE</a> is a neutron source for many experiments.</p>
<hr>
<p>In spallation, the number of neutrons produced per incident proton is usually on the order of 40, but the phase space distribution is much wider than the incident protons. (Protons beams can be directly focused using magnets.) According to a <a href="http://accelconf.web.cern.ch/accelconf/SRF93/papers/srf93e02.pdf" rel="nofollow">CERN report</a> by Lengeler, reactors typically can produce thermal neutrons (~0.025 eV) fluxes of order $10^{14}$ neutrons/cm${}^2$s, whereas spallation sources can exceed $10^{17}$/cm${}^2$s. </p>
<hr>
<p>It is probably best, then, to think of continuous neutron sources as phase-space-density limited, with the above numbers giving a rough estimate of the currently achievable densities.</p>
<p>Apparently, spallation sources are limited by the need to dissipate the heat from the incident proton beam. If high fluxes are needed only briefly, laser pulses <a href="https://www.lanl.gov/newsroom/news-releases/2012/July/07.10-world-record-neutron-beam-at-lanl.php" rel="nofollow">can produce</a> $4 \times 10^{9}$/cm${}^2$ over a nanosecond, i.e. an instantaneous flux of $4 \times 10^{18}$/cm${}^2$s.</p>
<p>As for protons, it's worth noting that the high-intensity 50 mA beam at the <a href="http://toddsatogata.net/Papers/PAC11-FROAN1.pdf" rel="nofollow">European Spallation Source</a> is limited by "space charge effects at low energy, by the power that can be delivered to the beam in each cavity at medium and high energies, and by beam losses."</p> | g13901 | [
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... |
<p>I was researching about the heat-death of the universe, and I was wondering if my understanding of it so far is factual to the theory?</p>
<p>However, the 3rd Law of Thermodynamics prevents the universe’s entropy from reversing because that law states that the temperature cannot equal 0 Kelvin. The heat of the universe does not equal 0 Kelvin and the only way to get it to 0 Kelvin would be to have an infinite amount of objects that are at 0 Kelvin to bring the universe’s temperature down to near 0 K.</p> | g13902 | [
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<p>I was given with the position state of hydrogen atom:</p>
<p>$$
R_{21} =\left(\sqrt{\frac{1}{3}}Y^0_1 + \sqrt{\frac{2}{3}}Y^1_1\right)
$$</p>
<p>I am getting confused about getting the expectation value of energy, I know from the form that:</p>
<p>$$ R_{21} \Rightarrow n=2, l=1$$</p>
<p>according to <a href="http://quantummechanics.ucsd.edu/ph130a/130_notes/node239.html" rel="nofollow">this website</a>.</p>
<p>I should just:</p>
<p>$$ \langle E_n\rangle = - \frac{1}{2} \alpha^2\mu c^2 \frac{\frac{1}{3} \cdot \frac{1}{2^2} + \frac{2}{3} \cdot \frac{1}{2^2}}{2^2}
\\= - \frac{1}{2} \alpha^2\mu c^2 \frac{\frac{1}{12} + \frac{2}{12}}{4}
\\= - \frac{1}{2} \alpha^2\mu c^2 \frac{\frac{3}{12}}{4}
\\= - \frac{1}{2} \alpha^2\mu c^2\frac{1}{16}$$</p>
<p>Am I right?</p> | g13903 | [
-0.008914562873542309,
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-0... |
<blockquote>
<p>Say you have a uniformly charged straight wire of length $2L$ going from $-L$ to $+L$ along the $z$ axis with its midpoint located at $z=0$. What is the electrostatic potential, in cylindrical coordinates, for points in the midplane of the wire ($x,y$ plane)? From this electric potential, calculate the electric field in the $x,y$ plane.</p>
</blockquote>
<p>My intuition is telling me that, because of the symmetry, the only direction the electric can be directed is in the xy-plane. So we have $$dq=\lambda dz$$ Thus, since $$dV=\frac{kdq}{r}$$ We have $$V=2\int_0^L \frac{kdq}{r}$$ Also $$r=\sqrt{x^2+y^2+z^2}$$ The integral is then $$2k\int_0^L \frac{\lambda dz}{\sqrt{x^2+y^2+z^2}}$$ Evaluating this I get $$V = 2 k \lambda \ln ( \sqrt{(x^2 + y^2 + z^2)} + z)$$ I know it's not in spherical coordinates, but can someone tell me if this is right so far? If so, the electric field is then given by the negative gradient, correct?</p> | g13904 | [
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0.030453987419605255,
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0.02652786485850811,
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0.016523... |
<p>When detecting radio waves in space, we use very large telescopes or arrays of telescopes. But according to QM, aren't photons point particles when measured? Does a photon with a large wavelength have a wider probability distribution with regard to where it can be expected to be detected? Is that why photons with long wavelengths require large telescopes to be detected relative to shorter wavelength photons?</p> | g13905 | [
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0.03876109793782234,
0.0... |
<p>It is better for you to have studied "Feynman lectures on Physics Vol.3", because I cannot distinguish whether the words or expressions are what Feynman uses only or not and in order to summarize my questions here, I have to just quote the contents of the book.</p>
<p>However, one thing I notice is that "base state" that Feynman explains seems to be "basic orthonomal state vector"...</p>
<p>With a pair of hamiltonian matrix equation</p>
<p>$i\hbar \frac{d{C}_{1}}{dt} = {H}_{11}{C}_{1} + {H}_{12}{C}_{2}$</p>
<p>$i\hbar \frac{d{C}_{2}}{dt} = {H}_{21}{C}_{1} + {H}_{22}{C}_{2}$</p>
<p>where ${C}_{x} = <x|\psi>$ , $\psi =$ arbitrary state, the book set the states 1 and 2 as "base states". There are only two base states for some particle. Base states have a condition - $<i|j> = {\delta}_{ij}$.</p>
<p>I think the "kronecker delta" means that once the particle is in the state of j, we will not be able to find the state i, so if we suppose all the components of hamiltonian are constant, we can say ${H}_{12}$ and ${H}_{21}$ should be zero. ..............(1)</p>
<p>However the book says that states 1 and 2 are base states and ${H}_{12}$ and ${H}_{21}$ can be nonzero at the same time (if you have the book, refer equ. (9.2) and (9.3) and page 9-3.). There can be probability to transform from state 1 to state 2 and vice versa.....</p>
<p>Then, the relationship that I think like (1) between the "Kronecker delta" and the components of hamiltonian is not correct at all?? Or, is my thought that ${H}_{12}$ and ${H}_{21}$ should be zero true?</p> | g13906 | [
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... |
<p>Pair production is where an energetic photon on its interaction with strong electric field surrounding a nucleus produces electron-positron pair. Annihilation of matter is its converse where an electron-positron pair produces two photons on interaction with nucleus.</p>
<p>But how do these happen? I mean, Why do they require strong electric field and nuclei..?</p>
<p>(Does this process requires Strong Nuclear force?)</p> | g13907 | [
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<p>I was reading that a new type of refrigerator might reach a coefficient of performance (COP) of 10. This seems quite the achievement and the authors state that their approach might achieve a Carnot efficiency of 75%. (I could not get hold of the original source as it is a part of an expensive book series "Zimm C, Jastrab A, Sternberg A, Pecharsky V K, Gschneidner K Jr, Osborne M and Anderson I 1998 Adv.
Cryog. Eng. 43 1759".)</p>
<p>Now I am a bit confused. <a href="http://en.wikipedia.org/wiki/Coefficient_of_performance#Derivation" rel="nofollow">Wikipedia</a> states that the COP should be the inverse of the Carnot efficiency but than these numbers do not fit together or am I missing something?</p> | g13908 | [
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0.010305176489055157,
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0.017725098878145218,
... |
<p>If I am given a system, which I might have to describe using a generalized entropy, like the "q-deformed" Tsallis entropy, do I have to fit q from experiment or might I know it beforehand?
How do I know the parameter q and/or how can I possibly obtain the degree of non-extensitivity via experiment?
How can I measure the entropy of a part of the system, if the system is non-extensive?</p>
<p><a href="http://en.wikipedia.org/wiki/Tsallis_entropy" rel="nofollow">http://en.wikipedia.org/wiki/Tsallis_entropy</a></p>
<p>Edit:</p>
<p>After some browsing I think the answer might be related to the fact, that for q-deformed entropy, the most probable distribution is not the Gaussian, but seems to be the q-deformed Gaussian:</p>
<p><a href="http://en.wikipedia.org/wiki/Q-Gaussian" rel="nofollow">http://en.wikipedia.org/wiki/Q-Gaussian</a></p>
<p>Then I played around a bit:</p>
<p><a href="http://img835.imageshack.us/img835/5841/qgaussian.png" rel="nofollow">http://img835.imageshack.us/img835/5841/qgaussian.png</a></p>
<p>Maybe one applies such an entropy concept if one comes across a distribution of such type, but that's only a guess. And I don't see why one would/could conclude non-extensitivity from a distribution?!</p> | g13909 | [
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0.03... |
<p>I have looked around, and I will admit that I'm a bit lost on the definitions.</p>
<p><a href="http://en.wikipedia.org/wiki/Pressure" rel="nofollow">Wikipedia</a>'s opening reads:</p>
<blockquote>
<p>Pressure (symbol: P or p) is the ratio of force to the area over which that force is distributed.</p>
</blockquote>
<p>That's a bit hard to understand at first, I'll have to say. Is vaguely saying "pressure" relative to "air resistance", or drag? Because they seem to be mixed around often.</p>
<p><a href="http://en.wikipedia.org/wiki/Drag_%28physics%29" rel="nofollow">Drag</a> </p>
<blockquote>
<p>refers to forces acting opposite to the relative motion of any substance moving in a fluid.</p>
</blockquote>
<p>Sounds unrelated to pressure, so let's move on.</p>
<p>If pressure is the ratio of force to an area which the force is distributed, wouldn't anything be force then?
Wind's force is covered over an area that it's distributed from. A punch would count too, based on how I understand pressure. I have always envisioned pressure as a uniform body of gaseous force or energy acting versatilely upon everything, such as below illustrates:</p>
<p><img src="http://i.stack.imgur.com/5kOg0.jpg" alt="enter image description here"></p>
<p>Higher pressure of the atmosphere is like a tighter "squeeze" on everything within it.</p>
<p>People often say pressure in exchange for force, such as when someone shakes someone's hand very hard (with a strong grip), one will say that the force is pressure. Is this correct? If so, why, and how do force and pressure coincide?</p> | g13910 | [
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-0.00... |
<p>Whether is it our solar system or a whole galaxy, there is usually a massive object (star or black hole) at the centre with gas and objects rotating around it. </p>
<p>The gravitational effect of the star/black hole extends uniformly (more or less) in every direction in 3d. Why does matter tend towards a single plane?</p>
<p>Furthermore, what happens to matter that approaches after the "disc" is formed when it is pulled in from anywhere off the plane, why does it join the plane rather than forming another plane?</p>
<p>I suspect angular momentum has something to do with it, but would appreciate a "pop science" explanation.</p>
<p>Many thanks</p>
<p>Andrew</p> | g13911 | [
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0.052885204553604126,
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0.04569724202156067,
0.0112138... |
<p>The stress applied on a rod is linearly proportional to its strain. But shouldn't the opposite be true? I mean if you pull particles further apart doesn't the force they apply on each other decrease because the distance between them increases? Kinda like gravity?</p> | g13912 | [
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<p>Naively, I would assume that a topological invariant remains invariant under continuous deformations of whatever space the invariant belongs to. In the case of topological insulators, this space is the space of parameters characterizing the material and its environment. If we consider the magnitude of a magnetic field incident onto a material supporting metallic edge states a parameter, then why does the Chern number (TKNN invariant) change when $|\textbf{B}|$ is increased smoothly and slowly? Since the Hall conductivity experiences sharp increases due to slow increase of magnetic field, the Chern number must be increasing sharply (by integer jumps), signifying sharp changes in the topology of the parameter space. What parameter is changing so drastically in response to the increasing magnetic field? </p> | g13913 | [
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<blockquote>
<p>A rotating charge such as the electron classically orbiting around the nucleus, will constantly lose energy in
form of electromagnetic radiation. </p>
</blockquote>
<p>I asked my teacher about how this radiation is created from the charge. My teacher said that energy is usually lost by the mechanisms such as <em>Bremsstranhlung</em>, <em>Dipole radiation</em>, etc. But, I don't have these mechanisms in my text book and I need to know how these mechanisms are carried out, so that I would understand the concepts better. So, I thought Physics Stack Exchange will help me. </p> | g13914 | [
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<p>Suppose that particles of two different species, A and B, can be chosen with
probability $p_A$ and $p_B$, respectively. </p>
<p>What would be the probability (and distribution) $p(N_A;N)$ that $N_A$ out of $N$ particles are of type A? </p>
<p>I'm trying to apply the Binomial distribution here but am bothered by the fact that it applies for N trials (whereas here we only do 1).</p> | g13915 | [
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<p>The cosmic microwave background that we observe uniformly around us is usually explained by assuming that our universe is the surface of a four dimensional sphere. That way the uniformity makes sense since there is no center. My question is if this is true then what is the explanation that describes the fact that the farther we look into space, the further we look back in time. I can't perfectly picture this and see how it would coexist. Help me out.</p> | g13916 | [
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0.043585628271102905,
0.07551182061433792,
0.024046115577220917,
0.0034529014956206083,
-0.06305250525474548,
0.008989779278635979,
0.016684819012880325,
0.03711375221610069,
0.055... |
<p>I understand a problem like this has already been asked, but I have not found an answer that makes it clear. A force is applied to a box on a table(lets ignore friction), and the box moves with some constant velocity. In this case, acceleration is equal to zero. The net force does not. How can we explain this?</p>
<p>$$
\sum F = F_{app} = m a
$$</p>
<p>The acceleration $a = 0$, but $F_{app} > 0$.</p>
<p>Can someone clarify this for me? </p> | g13917 | [
0.07439898699522018,
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-0.0845382884144783,
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0.0016566384583711624,
0.021920... |
<p>While studying thermal physics at school, I have been taught that solids simply have more potential energy than the liquids and gases. Note that it was said that this potential energy is due to the intermolecular bonds between the atoms. However, my intuition makes me doubt this, why would there be more potential energy in a solid??? Lets see: the internal energy is the sum of the kinetic and potential energies of a body. Well liquids and gases have more kinetic - pretty clear. As for, our original point, the potential: well, if the body has potential to do work, via lets say, chemical reactions, then its solid form will have a higher activation energy, as the intermolecular bonds have to be broken first - so the energy then released from combustion or what not, is less than that of its gaseous equivalent! Then how is it, that solids are somehow said to have more "potential" energy than the more energetic states. I, upon questioning this, have been told that this is due to the forces of the molecular bonds, but really I think, these forces signalize a lack of potential energy!</p> | g13918 | [
0.0512344092130661,
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0.01007724367082119,
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0.02129... |
<p>I am having trouble with the meaning of the $k$ vectors in an energy diagram.
If I want to populate some band, let say using a laser, what will be the significance of $k$?
Does it correspond to the polarization angle with respect to particular planes? </p> | g13919 | [
-0.045215677469968796,
0.00004785677811014466,
-0.002774042310193181,
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0.03966866061091423,
0.04051811993122101,
0.0021820631809532642,
0.05593487247824669,... |
<p>The free energy, $F$ of a thermodynamic system at a given temperature $T$, is defined as,
\begin{equation}
e^{-\beta F} = \mathcal{Z} = \sum_{\{configuration\}} e^{-\beta E(configuration)
}
\end{equation}
where $\beta = 1/k_BT$ and $E$ is the energy of a certain configuration.</p>
<p>Now my question is:</p>
<p><strong>Why the free energy is called 'free'? I mean it is free in which sense?</strong> </p> | g13920 | [
0.05125126987695694,
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... |
<p>Find your average speed if you go 15m/s for 1000 m and then 30m/s for 1000 m</p>
<p>I understand that average speed is total distance / total time. The distance for this problem would be 2000m travelled. How would I figure out the total time?</p> | g13921 | [
0.05364079773426056,
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<p>consider you are in a train that moves with a speed of $c/4$. A bulb is hanged from the ceiling. A light beam leaves the bulb downward. Exactly under the bulb on the floor is a detector. According to the principle of relativity the light will not strike the detector in the reference frame of a non-moving observer (standing out of train), because the <strong>velocity</strong> of light must be constant (light should continue it's motion vertically). But obviously the observer on the train should see that the beam strikes the detector.</p>
<p>But both can not be correct. (The beam is whether detected by the detector or not)</p>
<p>Whats wrong?</p> | g13922 | [
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0.0004410076653584838,
-0.00... |
<p>Some planetary orbits occasionally can appear to move backwards to an observer on the Earth? </p>
<p>Does anyone know concise, clear, web-based visualizations, animations or tutorials that clearly show how this might come about?</p>
<p>I have difficulties imagining it "in my head". A link to an animated solar system simulation would be fine. </p>
<p>Here is a good static picture,but does anyone know even better ones? </p>
<p>And yes, I know that Wikipedia.com has some links, too.</p>
<p><img src="http://i.stack.imgur.com/cuX4u.png" alt="Retrograde Motion"> <a href="http://wiki.astro.com/astrowiki/en/Retrograde_Motion" rel="nofollow">http://wiki.astro.com/astrowiki/en/Retrograde_Motion</a></p> | g13923 | [
0.0013756838161498308,
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-0.014083650894463062,
0.024020913988351822,
0.03915340453386307,
0.10035207122564316,
0... |
<p>I can't understand something about emissivity. Emissivity is defined as the ratio of the radiant energy of an object, to radiant energy of blackbody at a temperature $T$. So, The vegetation has approximate emissivity of $0.99$ and concrete smaller than this like $0.95$. But in thermal images vegetation appears darker than concrete. If its emittance is much more; so why is it dark? If the vegetation emits more energy why does it look dark in thermal imagery?</p> | g13924 | [
0.00938013568520546,
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-0.0095... |
<p>I am writing a lab report for class on PNMR experiment that we did. <em>How come in this experiment we don't worry about the electron spins in our sample?</em> Aren't the electrons affected by the PNMR machine as much as the protons? Isn't there electron proton spin interactions? For some reason, the lab only talks about the proton spins and does not mention the electron spins at all. </p>
<p>Since we did this at room temperature, does this mean that the electrons have so much energy that their spins doesn't even align with the permanent magnetic field, or are they all paired up with other electrons and therefore do not contribute much? I am very confused. Please help!!!</p> | g13925 | [
0.014156868681311607,
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0.00... |
<p>I am not comparing <em>passive</em> gravitational mass with rest inertial mass. Is there an evidence in Standard Model which says that <em>active</em> gravitational mass is essentially mass assigned by Higgs mechanism.</p> | g13926 | [
0.035815853625535965,
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<p>With the discovery of the Higgs Boson, some have been calling it the <a href="http://www.theatlantic.com/technology/archive/2012/07/why-the-higgs-boson-discovery-is-disappointing-according-to-the-smartest-man-in-the-world/259468/" rel="nofollow">end of experimental particle physics for our generation</a>, due to the fact that all of the particles predicted by the standard model have been found (the W and Z Bosons in 1983 and the Top Quark in 1995) and no new physics is expected at the energy ranges we can probe. I have also generally heard from my friends who work in experimental particle physics (or, at the very least, accelerator physics) a similar story, that there isn't too much left to be discovered. </p>
<p>Is this an accurate assessment? </p>
<p>What about dark matter particle models, like the axion/axino? Can the LHC (or SLAC, etc) probe enough of a range to perhaps shed light on this cosmological problem? </p>
<p>Can we hope for any more insights into beyond the standard model theories from current accelerators?</p> | g13927 | [
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0.02591293677687645,
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0... |
<p>Maybe some of my assumptions here are basically wrong, but isn't it true that</p>
<ul>
<li>pion is the "mediator" for the strong force field. </li>
<li>the quantum field theory basically says that there are no fields, instead all forces are caused by interchanging of mediator particles all the time.</li>
</ul>
<p>So to me this look like two particles in a nucleus, which are hold together by the strong force, are "sending" these pion particles all the time. But all references say that these particles have mass! Why doesn't this mean that mass is created/destroyed all the time? Is mass of the whole nuclei affected in some way by the constant stream of pions? If the question is basically irrelevant, what are the most important parts that I am missing here?</p> | g13928 | [
0.04848974198102951,
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0.006315282080322504,
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0.12310691922903061,
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0.017685... |
<p>We know that in high temperature, resistivity in metals goes linearly with temperature. As temperature is lowered, resistivity goes first as $T^5$ due to "electron-phonon" interaction, and then goes as $T^2$ as temperature is further lowered due to "electron-electron" interactions. </p>
<p>My question is, is there an <strong>intuitive</strong> way of understanding the above power-law dependence at each temperature range <strong>physically</strong>?</p> | g13929 | [
0.05344710499048233,
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0.... |
<p>Let's have generating functional in path integral form for gauge $SU(n)$ theory with interaction:
$$
\tag 1 Z[J] = \int DB D\bar{\Psi}D\Psi D\bar{c}Dc e^{iS}.
$$
Here
$$
S = S_{YM}(B, \partial B) + S_{M}(\Psi , D\Psi) + S_{ghosts} + \int d^{4}x \left[-\frac{1}{2\alpha}(\partial_{\mu}B^{\mu}_{a})^{2} + \sum_{\varphi}(J_{\varphi} \cdot \varphi ) \right],
$$<br>
$$
L_{ghosts} = \bar{c}^{a}(\delta_{ab}\partial^{2} + gc_{acb}B_{\mu}^{c}\partial^{\mu})c^{b}.
$$</p>
<p>I want to get Slavnov-Taylor identities for nonabelian case by using the fact that $(1)$ doesn't change under translation substitution $\varphi \to \varphi^{\omega} = \varphi + \delta_{\omega}\varphi $. In abelian case identity which acts role of Slavnov-Taylor identity (it shows that full propagator of EM field has transverse structure) can be derived by using the result of invariance of $\left(\frac{\delta}{\delta J_{\mu}^{A}(x)} Z[J]\right)_{J =0}$ under sets of transformations
$$
\Psi \to \Psi^{\omega} \approx \Psi (1 + i\omega ), \quad \bar{\Psi} \to \bar{\Psi}^{\omega}\approx \bar{\Psi}(1 - i\omega ), \quad A_{\mu} \to A_{\mu}^{\omega} \approx A_{\mu} + \partial_{\mu}\omega .
$$
The result is
$$
\left(\frac{\delta}{\delta \omega (z)}\frac{\delta}{J_{\mu}^{A}(x)} Z[J]\right)_{J, \omega =0} = \int \prod_{\varphi}D \varphi \left(\frac{\partial }{\partial x_{\mu}}\delta (x - z) - \frac{i}{\alpha}A_{\mu}(x)\partial^{2}\partial^{\nu}A_{\nu}(z)\right) e^{iS} = 0
$$
(you can read about it, for example, on page 99 <a href="http://www.itp.phys.ethz.ch/research/qftstrings/archive/11FSQFT2/script.pdf" rel="nofollow">here</a>).</p>
<p>But when I have tried to get Slavnov-Taylor identity (I mean <a href="http://www.scholarpedia.org/article/Slavnov-Taylor_identities" rel="nofollow">this one</a>; I need to variate it with respect on $J_{\mu}^{b}$ and then set it to zero) bu using the invariance of $(1)$ under transformations
$$
B_{\mu}^{a} \to (B_{\mu}^{a})^{\omega} = B_{\mu}^{a} + g\omega^{c}B_{\mu}^{b}g_{cba} + \partial_{\mu}\omega_{a}, \quad \Psi \to ... ,
$$
but I haven't got it. Is this way incorrect?</p> | g13930 | [
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0.003369663143530488,
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0... |
<p>If a thin sheet of some material in which a slit is cut is immersed in a liquid and removed, a liquid film (bridge) stays in the slit. Is it possible to calculate the shape of the film if the properties of the liquid and the dimensions of the slit are known?</p> | g13931 | [
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<p>What is wrong with this form of the <a href="http://en.wikipedia.org/wiki/Electromagnetic_induction" rel="nofollow">Maxwell-Faraday equation</a>?</p>
<p>$$\oint \vec{E}\ \partial \vec l= \bigcirc \hspace{-1.4em} \int \hspace{-.6em} \int \frac{\partial \vec B}{\partial t}$$</p>
<p>"Line integral of the electric field is equal to the double integral of partial derivative of magnetic field with respect to time".</p>
<p>So far as I remember the correct form used to be "Line integral of electric field is always (negative) surface <strong>integral of partially derived magnetic field...</strong>"</p> | g13932 | [
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<p>I have three questions about electromagnetic waves and was wondering whether anybody here could comment on these things:</p>
<p>Wikipedia says that there are no longitudinal EM waves, although TM and TE waves exist. How can I see that an EM wave can never be purely longitudinal?</p>
<p>Does it always hold that the E-field is perpendicular to the B-Field in an EM wave? </p>
<p>Wave packets have a group velocity that can be different from the phase velocity. Is it correct that due to $\omega= vk$, the phase and group velocity for a plane wave are always the same?</p> | g13933 | [
0.00092981947818771,
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0.0648434... |
<p>A $Z_2$ gauge theory with Ising matter field on a 2-dimensional square lattice has the Hamiltonian</p>
<p>\begin{equation}
H=-t\sum_{\vec r,j}\sigma_j^x(\vec r)-g\sum_{\vec r}\sigma^z_1(\vec r)\sigma^z_2(\vec r)\sigma^z_3(\vec r)\sigma^z_4(\vec r)-\lambda\sum_{\vec r}\tau^x(\vec r)-\mu\sum_{\vec r,j}\tau^z(\vec r)\sigma_j^z(\vec r)\tau^z(\vec r+\hat e_j)
\end{equation}
where $\sigma$'s are Pauli operators for the gauge field living on the links and $\tau$'s are the Pauli operators for the matter field living on the sites. $\vec r$ denotes the position of a site and $j$ denotes the link attached to the site which is in the $\hat e_j$ direction. The local gauge transformation is induced by
\begin{equation}
\tau^x(\vec r)\sigma^x_{\hat e_x}(\vec r)\sigma^x_{-\hat e_x}(\vec r)\sigma^x_{\hat e_y}(\vec r)\sigma^x_{-\hat e_y}(\vec r)
\end{equation}</p>
<p>In chapter 9.10 of the book <em>Field theories of condensed matter physics</em> by Fradkin, he said we can choose a unitary gauge defined by
\begin{equation}
\tau^z(\vec r)=1 \quad\forall\ \vec r
\end{equation}
so that the last term becomes
\begin{equation}
-\mu\sum_{\vec r,j}\sigma_j^z(\vec r)
\end{equation}</p>
<p>On the other hand, gauge invariance implies the Hilbert space we are considering is the one that consists of vectors which are invariant under the local gauge transformation, so the third term becomes
\begin{equation}
-\lambda\sum_{\vec r}\sigma^x_{\hat e_x}(\vec r)\sigma^x_{-\hat e_x}(\vec r)\sigma^x_{\hat e_y}(\vec r)\sigma^x_{-\hat e_y}(\vec r)
\end{equation}
so the Hamiltonian becomes
\begin{equation}
H=-t\sum_{\vec r,j}\sigma_j^x(\vec r)-g\sum_{\vec r}\sigma^z_1(\vec r)\sigma^z_2(\vec r)\sigma^z_3(\vec r)\sigma^z_4(\vec r)-\lambda\sum_{\vec r}\sigma^x_{\hat e_x}(\vec r)\sigma^x_{-\hat e_x}(\vec r)\sigma^x_{\hat e_y}(\vec r)\sigma^x_{-\hat e_y}(\vec r)-\mu\sum_{\vec r,j}\sigma_j^z(\vec r)
\end{equation}</p>
<p>My question here is: after choosing a gauge, why do we still have gauge invariance so that the gauge transformation generators act trivially? In this example, the question is that after choosing the unitary gauge $\tau^z(\vec r)=1$, why do we still have $\tau^x(\vec r)\sigma^x_{\hat e_x}(\vec r)\sigma^x_{-\hat e_x}(\vec r)\sigma^x_{\hat e_y}(\vec r)\sigma^x_{-\hat e_y}(\vec r)=1$?</p> | g13934 | [
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0.0... |
<p>Context is 1D Ising model. Metropolis algorithm is used for simulate that model. Among all possible spins configurations (states) that algorithm generates only states with the desired Boltzmann probability. </p>
<p>Algorithm chooses spin at random and makes a trial flip. If trial satisfies certain conditions related to Boltzmann probability, flip is accepted. Otherwise flip is rejected and system is unchanged.</p>
<p>Define "acceptance ratio" as a percentage of accepted trials. Simulation shows that acceptance ratio is higher on higher temperature (behave as increasing function of temperature).</p>
<p>Questions: </p>
<ol>
<li><p>Why Metropolis algorithm is not efficient at low temperatures? </p></li>
<li><p>Is efficiency of algorithm related to acceptance ratio?</p></li>
</ol> | g13935 | [
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0.06636153161525726,
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0.0689040943980217,
0.0522700659930706,
0.00979... |
<p>I have been asked to find a way to calculate the amount of time that a flight takes during night time.</p>
<p>So far, I have the departure latitude and longitude and the time of takeoff, the arrival latitude and longitude and the time of landing. I can easily calculate the sunset and sunrise time at the departure location, and at the arrival location. But I don't know how to proceed in order to find at which moment of the flight the sunrise and sunset will happen at the location where the aircraft is at that precise moment.</p>
<p>I'm just asking for some guidance on how to resolve this problem.</p>
<p><strong>PS:</strong> The calculation does not have to be very precise, and estimation is more than enough. We are just looking for the simplest solution to obtain an acceptable result.</p> | g13936 | [
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0.04199635609984398,
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0.0075722807087004185,
0.0009489707299508154,
0.04... |
<p>Why do we have so many dualities in string theory? Is there a reason for that?</p> | g13937 | [
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-0.028452565893530846,
-0.048074498772621155,
-0.006474032066762447,
0.02513197623193264,
0.05... |
<p>We suppose for semplicity to have a 1D oscillator, but this is a question abaout the general CCR algebra in oscillators, second quantization, quantum field theory etc.
We know coherent states are a non orthonormal base and are a overcomplete base.
We know them are given from $A |\alpha>=\alpha |\alpha>$. where $|\alpha>=e^{-\frac{|\alpha|^2}{2}} e^{\alpha A^+} \Psi_0$</p>
<p>How can we calculate generical scalar product between two different coherent states with eigenvalue $\alpha$ and $\beta$?</p>
<p>$$
e^{-\frac{|\alpha|^2+|\beta|^2}{2}}\Psi_0 e^{\beta A} e^{\alpha A^+} \Psi_0
$$</p> | g13938 | [
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... |
<p>Well, it does according to <a href="http://arxiv.org/abs/1102.3913" rel="nofollow">this preprint</a> for certain scales.</p>
<p>What would be a simple way to explain MOND to a layman?</p>
<p>Does it ignore mainstream physics? How much?</p> | g13939 | [
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0.0... |
<p>Why are so many condensed matter phenomena so sensitive to impurities? In fact, quite a number of them depend upon impurities for their very existence!</p> | g13940 | [
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0.02... |
<p>Why, if the Schwarzschild metric is a vacuum solution ($T_{\mu\nu}=0$) , do textbooks state that $T=\rho c^{2}$ when approximating Poisson's Equation from the Einstein Field Equations? </p>
<p>Thank you.</p> | g13941 | [
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<p>Just what the title states; and this isn't an original question ...</p>
<p>On full-moon, moonlight is almost as bright as mid-dawn ; yet there is hardly any colour visible. This is not the case with sunlight. </p>
<p>I'm curious to know why this is the case</p> | g13942 | [
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<p>I'm having a problem starting and solving the following problem:</p>
<p><img src="http://i.stack.imgur.com/QX76Y.png" alt="Fluid Mechanics"></p>
<p>My attempt at the solution was to realize you need to balance the upward forces with those of the downward forces in which case you have a buoyant force, for the upward force. And downward is the weight of the ball and the pressure of the ball at that height it's at. So then to solve it I assume you equate both and then you can find the specific gravity?</p>
<p>Thanks,</p> | g13943 | [
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<p>The S-matrix of superstring theory will have finitely many massive particles which decouple in the asymptotic future/past and massless particles like gravitons, photons and dilatons. Typically, there will be an infinite number of massless "soft quanta". In the far future/past, all of these dressed particles will decouple and essentially travel along straight worldlines. So, we can draw a spatial boundary enclosing all the massive dressed particles but none of the massless particles. In the far future/past, that is. According to the holographic principle of quantum gravity, the massive particles would then be completely determined by the massless particles. Everything about them can be deduced in principle from the massless particles. This isn't as implausible as it may seem because we have infinitely many soft quanta.</p>
<p>''Does this mean the entire concept of an S-matrix has to be overthrown in superstring theory?''</p> | g13944 | [
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... |
<p>Exactly what paper by W. Pauli introduced the idea for the existence of a neutrino and how was its existence confirmed experimentally (who did that and in what paper)?</p> | g13945 | [
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<p>I was told by my professor that the current in a purely LR Circuit with at initial state (JUST after closing the switch) is zero. Can you please help me understand why it that so? </p> | g13946 | [
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<p>I am designing a LED dimmer using software-controlled Pulse Width Modulation, and want to know the minimum PWM frequency that I must reach to make that LED dimming method indistinguishable from variable DC current.</p>
<p>To dim to X% power, the led will be fully on for X% of the PWM period, and fully off for the rest, with X from 0 to 100. I am assuming that the observer can compare with nearby LEDs driven by DC current at various power (but not exactly X% power), and can move the eye (in particular look at the LEDs in peripheral vision, which, I'm told, is the most sensitive to flicker). I disregard purposefully moving one's head relative to the LEDs, which (I guess) could be an effective distinguishing method; and use of a rotating slit disc or similar technique.</p>
<p>I read movie projectors use 48 or 72 Hz shutter frequency (twice to thrice the frame rate) to reduce flicker to something tolerable (but quite noticeable in peripheral vision, I guess).
I also read that the 50 or 60 Hz AC frequency has been chosen to avoid flicker, suggesting that that 100 or 120 Hz flicker is perceptible (but the limit could be much higher, because thermal inertia of the filament must damp variations).
Another reference point is that CRT with 100 Hz scan rate have been designed, presumably with some benefit (but again that is not an indication of the upper limit).</p>
<p>Edit: Found this relevant post, <a href="http://physics.stackexchange.com/q/15390/6954">Effects of high frequency lighting on human vision</a></p>
<p>Edit: My LED is "white", that is really blue with a luminescence converter consisting of
an inorganic phosphor material.</p> | g13947 | [
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... |
<p>I shine laser line in front of one side of multimode optical fibre.Light coming out of other side is projected into wall.Whether the intensity pattern seen on wall is same as the mode of propagation of light pattern drawn on text books?</p> | g13948 | [
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<p>I have a few basic questions about the Pauli-Lubanski spin 4-vector S.</p>
<ol>
<li><p>I've used it in quantum mechanical calculations as an operator, that is to say each of the components of S is a matrix operator that operates on an eigenvector or eigenspinor. But my question is about the utility of S in a classical sense, that is to say it represents the physical spin angular momentum. For example, in an electron's rest frame, is the spin 4-vector for the case spin-up along the z-axis given by S = (0, 0, 0, h/2) and for spin-down along x we have S = (0, -h/2, 0, 0) etc?</p></li>
<li><p>I know that in the particle's rest frame S = (0, Sx, Sy, Sz) where the spatial components are the spin angular momentum 3-vector components. However, when we Lorentz boost S, the time component is no longer zero. In this boosted case, do the 3 spatial components still give the spin angular momentum 3-vector (analogous to the case for 4-momentum where the 3 spatial components always give the 3-momentum), or do the spatial components now mean something else? The reason I'm not sure is that some 4-vectors, e.g. 4-velocity, have spatial components that do not represent 3-velocity at all since they may be superluminal, etc.</p></li>
</ol> | g13949 | [
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<p>This is my first question on the Physics portion of Stack Exchange. I was hoping to get some light on the topic of gravity. I don't have much background knowledge of physics so I might as well start here.</p>
<p>I have a few planet and gravity related questions:</p>
<p><strong>1).</strong> Does planetary gravity depend on size or density? Or both?</p>
<p><strong>2).</strong> Is it plausible for a large (very large), terrestrial planet to have tall mountains like Mount Everest or even like on Mars?</p>
<p><strong>3).</strong> Is there a way, given an average density per cubic meter along with the average radius of the planet, to calculate the pull of gravity?</p>
<p>I've been scouring the internet for these things with no luck so far. The idea behind this is to make a planet rendering program which takes all of this into effect. If this forum doesn't answer this sort of thing, feel free to point me in the right direction. Thanks!</p> | g13950 | [
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<p>According to Faraday's law, changing magnetic flux induces an electric field. Is that electric field always circular? </p> | g13951 | [
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<p>Look at <a href="http://www.modernrelativitysite.com/chap11.htm" rel="nofollow">equation 11.2.17 in this page</a>. The expression is:</p>
<p>$$ T = 10^{-5} \text{K m} \frac{\xi}{\frac{GM}{c^2} \lbrace \frac{GM}{c^2} + \xi \rbrace - e^2 }$$</p>
<p>where</p>
<p>$$ \xi = (r_s^2 - a^2 - e^2)^{1/2}$$</p>
<p>and the usual parameters</p>
<p>$$ r_s = \frac{GM}{c^2} $$</p>
<p>$$ a^2 = \frac{L^2}{M^2 c^2}$$</p>
<p>$$ e^2 = \frac{Q^2 G}{4 \pi \epsilon_0 c^4}$$</p>
<p>This formula is supposed to describe the temperature of a black hole with angular momentum $L$, charge $Q$ and mass $M$</p>
<blockquote>
<p><strong>Question:</strong> is the above formula correct?</p>
</blockquote>
<p>I'm trying to find the limit temperature for $a=0$ and $e=r_s$</p>
<p>The temperature expression could be simplified as</p>
<p>$$ T = 10^{-5} \text{K m} \frac{\xi}{\xi^2 + a^2 + \frac{GM}{c^2} \xi }$$</p>
<p>if $a=0$,</p>
<p>$$ T = 10^{-5} \text{K m} \frac{1}{\xi + \frac{GM}{c^2} } $$</p>
<p>so when the black hole has extremal charge, $\xi=0$ and the temperature looks like the normal black hole temperature for the Schwarzschild black hole, which looks very wrong</p>
<p>Any idea where is the mistake? I was expecting the temperature of the charged extremal black hole with $a=0$ to be infinite</p> | g13952 | [
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<p>I encountered a physics problem which inquired about a ball rolling inside a parabolic bowl (i.e. a bowl where any cross section through the vertex would make a parabolic shape given by $y = kx^2$). The ball was to be released from some initial height $z_0$ and allowed to roll back and forth inside the bowl.</p>
<p>At first, I thought that the motion would be simple harmonic for any displacement, since $U(x) = mgh = mgkx^2$, and simple harmonic motion is characterized by a potential energy which is proportional to the square of the distance from some equilibrium point. However, the answer key said that the motion was not truly simple harmonic (and the rolling ball only appeared to be so for small amplitudes of motion).</p>
<p>Why is motion with the a potential energy proportional to the square of displacement from equilibrium not simple harmonic in this case?</p>
<p>EDIT: To clarify, the "ball" should have really been a point mass.</p> | g13953 | [
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0.05950978025794029,
-0.... |
<p>I really need some assistance setting up this problem. any assistance would be a Godsend:</p>
<p>a uniform heavy chain of length <code>a</code> initially has length <code>b</code> hanging off of a table. The remaining part of chain a - b, is coiled on the table. Show that if the chain is released, the velocity of the chain when the last link leaves the table is $\sqrt{2g\frac{a^3 - b^3}{3a^2}}$</p>
<p>Okay, so this is a variable mass problem, so momentum is constantly changing:</p>
<p>$$F_{ext}=m(t)g=\frac{dP}{dt}= \frac{d(m(t)v(t))}{dt}=ma +v\dot{m}$$</p>
<p>$$mg = ma + v\dot{m}$$</p>
<p>Gravity acts on the mass hanging off of the table, mass can be written as a function of length, as can velocity (where $\lambda$ is the linear mass density)</p>
<p>$m(t) = \lambda l(t)$ <br />
$\dot{m(t)} = \lambda v(t)=\lambda \dot {l(t)}$<br />
$v(t) = \dot {l(t)}$ <br/>
$ a(t) = \ddot{l(t)}$</p>
<p>$$\lambda l(t)g=\lambda l(t) \ddot{l(t)} + \dot {l(t)} \lambda \dot {l(t)}$$</p>
<p>Assuming this all to be correct $\implies$ $0 =l(t)(g -\ddot{l(t)}) +\dot{l(t)}^2$</p>
<p>I've tried solved this DE, but I don't know many methods for non linear DEs. </p> | g13954 | [
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<p>I found recently that water fills a bottle quicker through only observation if I hold the tube away from the filling water than if I leave the tube inside the bottle. why does that happen? does potential energy have anything to do with this? <img src="http://i.stack.imgur.com/VUXJU.jpg" alt="enter image description here"></p> | g13955 | [
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<p>If I were on one side of the room and moved at the speed of light to the other side of the room, to an observer it would appear that I teleported. If time stops at that speed, it would be instantaneous but we know that light is not instantaneous (see light years).
So I'm curious what the difference between traveling at the speed of light and teleportation is. I think teleportation has to do with deconstructing objects and then reconstructing them which allows them to travel through things etc., but is there any other noticeable difference?</p> | g13956 | [
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<p>I am in undergraduate quantum mechanics, and the TA made an off-hand comment that currently no one knows how to describe fundamental particles with spin > 1 without supersymmetry. I was curious and tried to look up info on this, and wikipedia does make some comments about troubles with spin 3/2
<a href="http://en.wikipedia.org/wiki/Rarita%E2%80%93Schwinger_equation" rel="nofollow">http://en.wikipedia.org/wiki/Rarita%E2%80%93Schwinger_equation</a></p>
<p>So my questions are</p>
<ol>
<li>Is there an easy to understand reason why photons aren't a problem, but a hypothetical particle with spin 3/2 doesn't work?</li>
<li>Does this also mean there are troubles explaining 3/2 composite particles in a low energy regime where we can treat the composite as strongly bound / 'fundamental'?</li>
<li>How does supersymmetry help here?</li>
</ol> | g13957 | [
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<p>I know from Carroll that the integration in GR is basically a mapping from n-form to the real number. And it's given that </p>
<p>$$d^nx=dx^0\wedge\ldots\wedge dx^{n-1}=\frac{1}{n!}\epsilon_{\mu_1\ldots\mu_n}dx^{\mu_1}\ldots dx^{\mu_n}$$</p>
<p>Now, I have an expression that is given in spherical coordinate system, where I have </p>
<p>$$\int_\Sigma f(\theta,\phi) d\theta\wedge d\phi$$</p>
<p>when I want to integrate this (the epsilon part is already computed), do I just have $\int\int d\theta d\phi$ to integrate, or do I need to put the part from integration in spherical coordinate system $\int\int\sin\theta d\theta d\phi$?</p>
<p>I haven't done much integration that involved forms before, so any help is appreciated :)</p>
<p>EDIT:</p>
<p>From the article I have:</p>
<p>$$k_\xi[h,\bar{g}]=k^{[\nu\mu]}_\xi[h,\bar{g}](d^{n-2}x)_{\nu\mu}$$</p>
<p>$$(d^{n-p}x)_{\mu_1\ldots\mu_p}:=\frac{1}{p!(n-p)!}\epsilon_{\mu_1\ldots\mu_n}dx^{\mu_p+1}\ldots dx^{\mu_n}$$</p>
<p>$$k_\xi^{[\nu\mu]}[h,\bar{g}]=-\frac{\sqrt{-\bar{g}}}{16\pi}\ldots$$</p>
<p>where $\ldots$ is an expression.
Does this mean, since I have an $n-2$ form and I'm in 4 dimensional space, I need to include this $\sin\theta$ after all?</p>
<p>EDIT2: I need to add the $\sin\theta$. I got it. Thanks :D</p> | g13958 | [
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<p>I am reading <a href="http://www.ocr.org.uk/download/sm/ocr_13461_sm_gce_unit_g484_csa_m.pdf">a document</a> and in answer to the question <strong>State Newton’s second law of motion</strong> the candidate answers that <strong>The force acting on an object equals the rate of change of momentum of the object.</strong> While this is not a complete answer, the examiner picks on on a word <em>equal</em> and says that it is <em>proportional</em> instead.</p>
<p>Now I understand that $F=\frac{dp}{dt}$ where both $F$ and $p$ are vectors — what is "proportional" about it?</p>
<p>I checked my Good Old Ohanian (2nd edition) and it says explicitly "The rate of change of momentum equals force" in section 5.5 <em>The Momentum of a Particle</em>.</p>
<p>What is this examiner talking about?</p> | g13959 | [
0.07603802531957626,
-0.013372506946325302,
0.009853365831077099,
-0.03307436779141426,
0.06567512452602386,
0.022068867459893227,
0.015228701755404472,
0.03929983079433441,
-0.02348281815648079,
-0.020703179761767387,
0.0007158172084018588,
-0.016495512798428535,
0.037604156881570816,
-0.... |
<p>In many books the $\phi^4$ model can produce a topological soliton called kink. Are they right? In the case of sine-Gordon model you can have a topological soliton due to you can express the Lagrangian in terms of two scalar fields taking values in a circle. As the field is constrained on a manifold the model could have stability. I am not sure if $\phi^4$ model could be viewed as a field constrained on a manifold also.</p>
<p>So my questions are:</p>
<ul>
<li><p>Is it right to talk about topological solitons or kinks in $\phi^4$ model?</p></li>
<li><p>Has the $\phi^4$ model an alternative formulation as a constrained field with target space a circle or any other manifold?</p></li>
</ul> | g13960 | [
-0.006131629925221205,
0.015227673575282097,
-0.002602653345093131,
-0.019631998613476753,
0.08924480527639389,
0.021296996623277664,
0.017927492037415504,
0.009988360106945038,
-0.0014292116975411773,
-0.019680596888065338,
-0.037409707903862,
-0.002048443304374814,
0.036493174731731415,
... |
<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/8441/what-is-a-complete-book-for-quantum-field-theory">What is a complete book for quantum field theory?</a> </p>
</blockquote>
<p>I am looking for good books that deal with Quantum Field theory starting from basics. Please suggest something that deals with the physical interpretation of the subject, and not just the mathematical formalism. Books dealing with particle sources and particle detectors will be much appreciated. </p> | g282 | [
0.06999039649963379,
0.01539476029574871,
-0.0010284027084708214,
-0.01954580470919609,
0.010152248665690422,
0.000367906381143257,
-0.016759393736720085,
0.018284432590007782,
0.007886224426329136,
0.00858478806912899,
0.021167943254113197,
-0.01816859096288681,
0.0013658885145559907,
-0.... |
<p>I learned about the coronal discharge, and the common explanation is because the electric field is strong where radius of curvature is small. But I haven't found anything yet that explains why electrons like to crowd at the peaks, and and escape from the holes. </p>
<p>My intuition suggests electrons try to distribute on the surface as uniform as they can, but they don't. Why?</p> | g13961 | [
0.04533690959215164,
0.06806399673223495,
-0.020731978118419647,
0.022470396012067795,
0.03597762808203697,
0.05720827355980873,
0.051430076360702515,
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0.00276110228151083,
-0.012080689892172813,
-0.017883315682411194,
0.03811206296086311,
0.057741403579711914,
-0.001... |
<p>I have read <a href="http://en.wikipedia.org/wiki/Closure_phase" rel="nofollow">this page</a> on closure phase and could not understand the following:
\begin{align}
\psi_1 & = \phi_1 + e_B - e_C \\
\psi_2 & = \phi_2 - e_B \\
\psi_3 & = \phi_3 - e_C.
\end{align}
If the error in $\phi_3$ is $e_C$ and the error in $\phi_2$ is $e_B$, then why should the error in $\phi_1$ be connected with the errors of $\phi_2$ and $\phi_3$ in the way given?</p> | g13962 | [
0.037832800298929214,
-0.018089374527335167,
0.010846833698451519,
-0.014392663724720478,
0.05111953988671303,
0.015785275027155876,
0.05008590221405029,
0.03519907221198082,
-0.05307144671678543,
0.0020874659530818462,
0.02061169408261776,
0.04528730735182762,
0.002730415668338537,
0.0634... |
<p>What is the theoretical difference between the physical elementary interaction that causes an e+ to attract an e− when they exchange a virtual photon? Why is this exchange different from an e-e- scattering (which produces repulsion)?<br>
This question is related to but more specific than the recently posted <a href="http://physics.stackexchange.com/questions/2244/the-exchange-of-photons-gives-rise-to-the-electromagnetic-force">The exchange of photons gives rise to the electromagnetic force</a>. It has not been answered there. It would be good to see the answer somewhere.</p> | g13963 | [
0.041617460548877716,
0.0009211488650180399,
-0.008210602216422558,
-0.02650461159646511,
0.06307666003704071,
0.05704325810074806,
0.016580544412136078,
0.03363623842597008,
0.009121667593717575,
-0.008440718054771423,
0.03474096208810806,
0.028146222233772278,
0.03989819064736366,
-0.002... |
<p>Suppose we have a spacecraft just inside the event horizon of a black hole, struggling to escape, but slowly receding into it. Another (bigger) black hole expands until its event horizon includes the spacecraft as well. This pulls both the black hole and space-craft into itself. As the spacecraft is closer to the second black hole, it receives more force, perhaps just enough to pull it out.</p>
<p>Is it possible that the spacecraft, even momentarily, is pulled out of the event horizon of the original black hole (although it is now trapped inside the event horizon of the new one)?</p> | g444 | [
-0.010322658345103264,
0.02083318866789341,
0.060592036694288254,
0.028368929401040077,
-0.018162943422794342,
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-0.004785833414644003,
-0.004004874732345343,
0.0036803330294787884,
-0.033918000757694244,
... |
<p>I'm currently reading papers about the effect of repeated measurements, such as <a href="http://prl.aps.org/abstract/PRL/v90/i6/e060401" rel="nofollow">"Purification through Zeno-Like Measurements"</a>, <a href="http://arxiv.org/abs/quant-ph/0301026" rel="nofollow">arXiv:quant-ph/0301026</a> (DOI: 10.1103/PhysRevLett.90.060401). </p>
<p>It says </p>
<blockquote>
<p>After N such measurements have been done, the survival
probability of finding system A still in its initial
state is represented by
$$P^{(\tau)}(N)=\mathrm{Tr}\left[\left(Oe^{-iH\tau}O\right)^{N}\rho_{0}\left(Oe^{iH\tau}O\right)^{N}\right]=\mathrm{Tr}_{B}\left[\left(V_{\phi}(\tau)\right)^{N}\rho_{B}\left(V_{\phi}^{\dagger}(\tau)\right)^{N}\right] $$
Notice that the quantity $V_{\phi}(\tau)=\left\langle\phi\right|e^{-iH\tau}\left|\phi\right\rangle$ is an operator acting on the Hilbert space of system B. </p>
</blockquote>
<p>where $O=\left|\phi\right\rangle\left\langle\phi\right|\otimes I_B $ is the projection operator onto an eigenstate of subsystem A, and $\rho_{0}=\left|\phi\right\rangle \left\langle \phi\right|\otimes\rho_{B} $ is the initial density matrix.</p>
<p>My question is: how the trace is reduced to the partial trace with respect to system B?</p> | g13964 | [
0.02070268988609314,
-0.02570544183254242,
-0.01981169544160366,
-0.020296160131692886,
0.010218863375484943,
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0.05239071324467659,
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0.05308828875422478,
-0.04862566292285919,
0.0158272385597229,
-0.008849347941577435,
0.0510... |
<p>Consider a ring of radius $R$, and charge density $\rho$. What will be the potential energy of the ring in its self field?</p>
<p>The best I can do:
$$dq = \rho R \cdot \, d \alpha $$</p>
<p>$$E_p = 2 \pi R \cdot 2 \int_{0}^{\pi - \delta \alpha} \frac{\rho^2 R^2}{r(\alpha)} \, d\alpha$$</p>
<p>where</p>
<p>$$r(\alpha) = \sqrt{2} R \sqrt {1+ \cos (\alpha)}$$</p>
<p><strong>Edit</strong>:</p>
<p>This is wrong - $E_p$ doesn't even have the correct dimensions.</p> | g13965 | [
0.0362037755548954,
0.034387849271297455,
-0.014995027333498001,
-0.022083202376961708,
0.03413252905011177,
0.02739643305540085,
0.007853985764086246,
0.0108971381559968,
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0.06468656659126282,
0.008908991701900959,
0.026602450758218765,
-0.010047737509012222,
-0.03882... |
<p>Most of the literature says that for a quenched average over disorder, an average over the log of the partition function must be taken:</p>
<p>\begin{equation}
\langle \log Z \rangle,
\end{equation}</p>
<p>while for the annealed average, it's</p>
<p>\begin{equation}
\langle Z \rangle.
\end{equation}</p>
<p>But a while ago, I came across a book that said that the annealed average is not $\langle Z \rangle$, though I don't remember what it said should be calculated instead. </p>
<p>Does anyone know which book this is, or what they might want to calculate instead of $\langle Z \rangle$ for the annealed average?</p> | g13966 | [
0.016748404130339622,
-0.0451277419924736,
-0.020842716097831726,
-0.036390360444784164,
0.005227232817560434,
-0.02465943433344364,
0.011529930867254734,
0.05914141982793808,
0.006815649103373289,
0.025763018056750298,
0.015692340210080147,
0.04256280139088631,
0.04383203014731407,
-0.028... |
<p>To be clear, this example can't apply to the Solar System, since the barycenter is within the Sun, similarly the Earth/Moon system's barycenter is within the Earth.</p>
<p>But, given a system of gravitationally attacting masses revolving about a barycenter that is not contained within any of the masses, i.e. a barycenter in open space, would a photon passing near that point behave in the same manner as a photon passing a point mass of the same mass as the system itself?</p> | g13967 | [
-0.02666490338742733,
-0.01431173924356699,
0.027005914598703384,
0.01026646513491869,
-0.02430911362171173,
0.01053930539637804,
0.015877557918429375,
-0.029594307765364647,
-0.023248635232448578,
-0.03587133437395096,
0.058410827070474625,
0.06460344791412354,
0.004882406909018755,
0.017... |
<p>I want to know more about optimising an air gap between two surfaces, such as window panes or a building's walls.</p>
<p>I was taught that as a general rule, a 6cm air gap is the optimum. Any larger and convection counters the effectiveness. But something tells me this is hardly an accurate figure!</p>
<p>Can anyone improve on this rule of thumb? What are the most significant variables which would alter the optimum separation distance?</p>
<p>Examples of what I would want to apply this to are:</p>
<ul>
<li>A building's window panes of different sizes: glass, perspex or combinations</li>
<li>Oven / furnace surfaces of different sizes: between two ceramic refractory walls, or refractory walls and steel. (Temperature differentials and absolutes of perhaps up to 1000 degrees C)</li>
</ul>
<p>Perhaps there are some approximated formulas to calculate optimum separation distance? And if the effective thermal conductivity of the air changes, a formula for that too.</p>
<p>Thanks!</p>
<p>For interest to other readers, <a href="http://physics.stackexchange.com/questions/15622/what-are-typical-values-of-the-critical-thickness-of-insulation">this question</a> is related.</p> | g13968 | [
0.04721401631832123,
0.003929171711206436,
-0.0007756948471069336,
0.04055631533265114,
-0.04002960026264191,
-0.0014889203011989594,
0.025240780785679817,
0.05773426964879036,
-0.04393192008137703,
-0.046007346361875534,
0.04754188284277916,
0.029532859101891518,
-0.016627464443445206,
0.... |
<p>I recently studied Laplace's equation and numerical methods for solving it, and so I'm trying to simulate an electron gun.</p>
<p><img src="http://i.stack.imgur.com/Mwsk2.png" alt="Schematic of an electron gun"></p>
<p>(assuming cylindrical symmetry on the Wehnelt, anode, and beam.) The setup is simple. The filament releases electrons, the Wehnelt cylinder is the cathode and helps focus electrons, and the electrons accelerate towards the anode. This was enough for me to set everything up and solve for the electric potential, with potential=0 as distance goes to infinity. There's a problem though: There are an infinite number of ways to set the potential of the cathode/anode that leave the voltage difference the same. </p>
<p>For example, with a voltage of 1V between the plates...</p>
<p>Cathode set to -1V:</p>
<p><img src="http://i.stack.imgur.com/T7iav.png" alt="Cathode set to -1V"></p>
<p>Cathode set to 0V:</p>
<p><img src="http://i.stack.imgur.com/rcU0l.png" alt="Cathode set to 0V"></p>
<p>(the visualization is a slice through the potential. It is cylindrically symmetric. Space has been compactified so that the midpoint is at the origin but the edges represent infinity, according to the equation $x_{\text{world}}=x/(1-x^2)$, where $-1<x<1$ are the coordinates in the image and both the radial/axial directions behave the same.)</p>
<p>The gradients of the potentials are different, so these two potentials represent physically different situations. I realized that I don't know which one is more correct/reasonable. I understand that potentials are relative, but for example if a lab frame is taken to have a potential of 0... which picture is the true picture if I brought in a 1v battery? Are the terminals at $-0.5$V and $0.5$V? Why?</p> | g13969 | [
0.02141609787940979,
0.018529430031776428,
-0.019265318289399147,
-0.014176256023347378,
0.06999713182449341,
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0.027568774297833443,
-0.0007405646028928459,
-0.09217526763677597,
0.042203713208436966,
-0.016301879659295082,
0.07144052535295486,
0.007966127246618271,
0.... |
<p>When a shuttle is in orbit, it is essentially rotating around the "centre" of the Earth at a great speed. So why does there seem to be no centrifugal force sticking them to the 'ceiling' of the shuttle?</p>
<p>Is it because the shuttle is not actually being rotated around a point but rather continuously falling or something else?</p>
<p>Thanks</p> | g13970 | [
0.01697402074933052,
0.03964134305715561,
0.009186701849102974,
0.04396875575184822,
0.043338656425476074,
0.0733809620141983,
0.02392258122563362,
0.0032382162753492594,
-0.0370977409183979,
-0.06337500363588333,
0.07500550895929337,
-0.03842032328248024,
-0.000451297324616462,
0.04871724... |
<p>Can Heisenberg's Uncertainty principle be rewritten in terms of energy density</p>
<p>writing
$$\Delta E \Delta T \geqslant \hbar/2$$
in factors of energy density
$\Delta \sigma \text{ }= \frac{3\Delta E}{4\pi r^3}$</p>
<p>I get
$$\Delta \sigma \frac{3\text{$\Delta $T}}{4\pi r^3} \geqslant \hbar/2$$</p>
<p>does $$\frac{3\text{$\Delta $T}}{4\pi r^3}$$
represent something?</p>
<p>Or does it make more sense to write
$$\text{$\Delta $x} \frac{\text{$\Delta $E}}{c^2} \geqslant \frac{\hbar}{2}$$</p>
<p>EDIT after noting Willie's comment I've corrected the form to $\frac{\hbar}{2}$</p>
<p>However, now I see c is constant and can be moved to the right side - is it mathematical correct to write
$$\text{$\Delta $x} \text{$\Delta $E} \geqslant \frac{\hbar c^2}{2}$$
or does this violate the Cauchy–Schwarz inequality? This is the question I was trying to understand at the beginning.</p> | g13971 | [
0.013294768519699574,
0.03334431350231171,
-0.003556046402081847,
0.041415393352508545,
0.029377683997154236,
0.05915436893701553,
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0.028773058205842972,
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0.042065758258104324,
0.006671908311545849,
-0.003824715968221426,
-0.053208429366350174,
0.... |
<p>If you consider big bang. According to that What's left at the center of universe where the Big bang occured?</p> | g13972 | [
0.04616183042526245,
-0.03232688084244728,
0.014279733411967754,
-0.07252689450979233,
-0.02206851728260517,
0.04061305150389671,
0.04960839822888374,
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0.005047083832323551,
-0.07473830133676529,
0.023705314844846725,
-0.02991834655404091,
-0.04196587949991226,
0.012913... |
<p>I know that the relativistic formulae for energy and momentum are:</p>
<p>$E = \gamma mc^2$ and $\textbf{p} = \gamma m\textbf{v}$;</p>
<p>Can we derive these formulae?</p>
<p>If yes, where from?</p> | g13973 | [
0.0117154810577631,
-0.042753566056489944,
0.0019065424567088485,
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0.04720543324947357,
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0.025430187582969666,
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-0.01834920421242714,
-0.0009496381389908493,
0.... |
<p>Consider the following operator $\hat{x}=i\hbar \frac{\partial}{\partial p}$.</p>
<p>I am trying to show that the eigenfunctions of $\hat{x}$ are not square-normalizable. I am interested in doing so since theoretically, we notice that the eigenfunction of the momentum operator $\hat{p}=\frac{\hbar}{i}\frac{\partial}{\partial x}$ is not square-normalizable. How do we set up the equation for the $\hat{x}$ operator?</p> | g13974 | [
0.027310814708471298,
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0.04675687476992607,
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0.02988165244460106,
0.015283593907952309,
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0.008363153785467148,
-0.01359287928789854,
-0.03950558230280876,
-0.0028188778087496758,
0.... |
<p>When a single-tone continuous modulating signal modulates a sinusoidal carrier, isn't the modulated wave periodic? If so, can't we apply fourier series and determine the harmonic frequency components instead of finding the frequency components using fourier transform?</p> | g13975 | [
-0.06906897574663162,
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0.01021850761026144,
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0.028209855780005455,
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0.006014013662934303,
-0.038493797183036804,
-0.05163438245654106,
0.017527198418974876,
0.009408177807927132,
0.... |
<p>What are proven procedures for preparing a part that comes fresh out of the workshop for ultra-high vacuum (UHV) and extremely-high vacuum (XUV)? </p> | g13976 | [
0.014333303086459637,
0.00969858467578888,
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0.020856762304902077,
0.011050083674490452,
0.03393189609050751,
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0.05378655716776848,
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-0.012002003379166126,
-0.0011308486573398113,
0.017193952575325966,
0.006708681117743254,
-0.... |
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