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<p>I have a two-component field:</p>
<p>$$\phi(\vec{x}) = \left( \begin{array}{c} \phi_1(\vec{x}) \\ \phi_2(\vec{x}) \end{array} \right)$$</p>
<p>with $\phi^T = (\phi_1, \phi_2)$. And I am trying to evaluate:</p>
<p>$$(\vec{\nabla}\phi)^T(\vec{\nabla}\phi)$$</p>
<p>I'm pretty sure the result of that expression is supposed to be a scalar, but I can't figure out why. $\phi$ is a 2x1 matrix, which means $\vec{\nabla}\phi$ must be the result of a matrix multiplication between a Nx2 and 2x1 matrix. You would think N would be 3, since $\vec{\nabla} = \frac{\partial}{\partial x}\vec{x} + \frac{\partial}{\partial z}\vec{z} + \frac{\partial}{\partial z}\vec{z}$, but then why are there 2 columns? Not sure what to do here so any assistance would be quite helpful.</p>
<p>(I'll mention that the full thing I'm trying to evaluate is $\mathcal{L} = \frac{1}{2}\left(\dot{\phi}^T\dot{\phi}-m(\vec{\nabla}\phi)^T(\vec{\nabla}\phi)\right)$. $\mathcal{L}$ is a scalar, right?)</p> | 6,374 | [
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<p><strong>Observation:</strong></p>
<p>So, I know that all computer screens are able to project many different colors by varying how they display the RGB (Red, Green, Blue) pixels.</p>
<p><strong>Question</strong></p>
<p>What's the difference between say, yellow light (575 nm) from sunlight vs. the same yellow light that's displayed on the computer screen? Aren't they different? Is our brains mixing together the RGB lights together and its being interpreted as yellow light or is yellow light the same (575 nm) light that comes from sunlight actually hits our eyes?</p>
<p><strong>Relationship between Fourier Series and light?</strong></p>
<p>1) So I know that in Fourier Series, you have basis functions, sines and cosines that make up any other function depending on the coefficients that are in front of the sines and cosines terms.</p>
<p>2) I know that RGB colors on a screen are the "basis" for all the other colors that a computer screen makes up and the intensity of each of the RGB are like the "coefficients". So, am I making the correct relationship here?</p> | 308 | [
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<p>Optical coatings designed for reflection or anti-reflection are made of many thin layers which will expand when heated. What will the effect be on the wavelengths the coating will reflect when the coating is heated? </p>
<p>For a first guess, magnesium fluoride has a thermal expansion coefficient of about 10 µm/m/K , so the affected wavelengths could be red-shifted by 1E-5/K? But this doesn't take into account refractive index changes. Does anyone know of experimental data on actual coatings? </p> | 6,375 | [
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<p>Are there any other particle predictions by the <a href="http://en.wikipedia.org/wiki/Standard_Model" rel="nofollow">standard model</a>? </p> | 222 | [
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<p>Based on John Isaacks' question, "<a href="http://physics.stackexchange.com/questions/11940/if-you-view-the-earth-from-far-enough-away-can-you-observe-its-past">If you view the Earth from far enough away can you observe its past?</a>" and the responses, it appears that we could use mirrors to see into the past. Using Vintage's example, a single mirror on the moon would only allow an observer to see 0.1μs into the past. Would it therefore be theoretically possible to align a massive series of mirrors that sent a single beam of light back and forth over a larger span of time (e.g., months, years, decades)? If so, would it be possible to send messages to ourselves in the past via such an arrangement? Or would we always be looking into the past and only able to send messages to the future?</p> | 6,376 | [
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<p>I was reading through the Wikipedia article on <a href="http://en.wikipedia.org/wiki/Quasar" rel="nofollow">Quasars</a> and came across the fact that the most distant Quasar is 29 Billion Light years. This is what the <a href="http://en.wikipedia.org/wiki/Quasar#Overview" rel="nofollow">article</a> exactly says </p>
<blockquote>
<p><em>The highest redshift quasar known (as of June 2011[update]) is ULAS-J1120+0641, with a redshift of 7.085, which corresponds to a <strong>proper distance</strong> of approximately 29 billion light-years from Earth.</em></p>
</blockquote>
<p>Now I come to understand that the <a href="http://en.wikipedia.org/wiki/Big_Bang" rel="nofollow">Big Bang</a> singularity is believed to be around 13.8 Billion years ago.</p>
<p>So how is this possible? Does the presence of such a quasar negate the Big Bang Theory? </p>
<p>I'm not a student of Physics and was reading this out of (whimsical) curiosity. Is there something I'm missing here or the "proper distance" mentioned in the fact is a concept that explains this?</p>
<p>Edit: My Bad! Here's how..</p>
<p>A simple google search led me to this article (sciencedaily.com/releases/2011/06/110629132527.htm) which says the farthest quasar found is 12.9 billion LYs and not 29 billion.
So in the end we have just proven that wikipedia needs more moderation. Lol, what a waste of time!</p>
<p>Although I did find out a lot of new stuff from the answers, so thanks for all your responses!</p> | 10 | [
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<p>We are a group of rocket amatures who build rockets using LOX and ethanol as propellent. We normally buy the LOX from supplier. But lately we have been discussing if there is an easy way to produce it in house. </p>
<p>I have looked at several methods:
<a href="http://en.wikipedia.org/wiki/Gas_separation" rel="nofollow">http://en.wikipedia.org/wiki/Gas_separation</a></p>
<p>PSA seams to be the simplest:
<a href="http://en.wikipedia.org/wiki/Pressure_swing_adsorption" rel="nofollow">http://en.wikipedia.org/wiki/Pressure_swing_adsorption</a><br>
<a href="http://www.youtube.com/watch?v=LWM3ZImmVWQ" rel="nofollow">http://www.youtube.com/watch?v=LWM3ZImmVWQ</a></p>
<p>We do not need pure Oxygen, we think something like 80% will be OK. </p>
<p>Where can I find comparison of the pros and cons of the different techniques?</p>
<p>Does anyone know where I can find more information about which techniques has less technical/risk difficulty, while having a good cost balance between the fixed and variable costs of the production? </p>
<p>We typically like to produce 500 Kg. LOX per month.</p> | 6,377 | [
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<p>What exactly is a <a href="http://en.wikipedia.org/wiki/Frame_of_reference" rel="nofollow">frame of reference</a>? Does it have an objective existence and if so what is it? </p>
<p>What's the size of a reference frame? Is a reference frame the same size for a stationary frame of reference and one that's accelerating?</p>
<p>What's the smallest frame of reference? Does a photon or electron have a reference frame or does this just apply to classical objects?</p> | 6,378 | [
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<p>I have a couple of questions about how <a href="https://en.wikipedia.org/wiki/Railgun" rel="nofollow">railguns</a> actually work, and the mathematics behind them. I understand that the projectile is driven forward by the Lorentz force caused by high currents traveling through the projectile. Because of these high currents, when the projectile initially comes in contact with the rails it can be welded into place instead of pushed forward, and thus needs an initial velocity.
So...</p>
<ol>
<li><p>How do you calculate what the initial velocity needs to be in order to counteract the high current that wants to hold the projectile in place?</p></li>
<li><p>Would there be any difference in the Lorentz force in a pure-translational velocity versus a rotational and translational velocity?</p></li>
<li><p>How high do the currents need to be compared to the mass of the projectile? </p></li>
<li><p>Where would I find these kinds of formulas for these calculations? </p></li>
</ol>
<p>Thanks!</p> | 6,379 | [
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<p>I understand that the higher your velocity the slower light will move. But how does time itself slow down while you are moving faster?</p> | 6,380 | [
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<p>In <a href="http://arxiv.org/abs/quant-ph/0202122" rel="nofollow">quantum information theory</a>, one can adopt the basic formalism where every system is given by an operator algebra, state preparation procedures correspond to linear functionals on that algebra (macroscopic systems preparing quantum systems), yes-no measurements correspond to projections in that algebra (quantum system affecting a macroscopic measurement device), and finally evaluating a state at a given projection is interpreted as the relative frequency of obtaining `yes' for the given projection.</p>
<p>More general observables are given by projection-valued measures on a set of measurement outcomes, or equivalently, self-adjoint operators. The need for an operator algebra (in the notes above, for example) is apparently that the spectral projections of a self-adjoint operator must be found in the observable algebra.</p>
<p>The trouble I have here is that it is not clear where, in the above formalism, one is to multiply linear operators representing observables? If an observable corresponds to a measurement, hence giving a `reading' on a macroscopic device (which is stipulated in the definition of measurement), how is it possible to perform a second measurement once a first has been made? (I'm thinking of a detector, here...like in Stern-Gerlach or double-slit.)</p>
<p>Does the need for noncommutativity come from an observed statistical uncertainty relation? Does one prepare a system in a specific way, say one of the beams of a Stern-Gerlach apparatus, and then make a measurement of two different quantities on the single prepared beam, and then try to observe a statistical uncertainty relation on the measured quantities? (i.e. does one look for an uncertainty relation in the standard deviations of the resulting measurements?) Is this how we'd determine that the observables assigned to the two measurements taken do not commute?</p>
<p>Note: "composed" Stern-Gerlach apparati and composition of quantum operations are not what I'm interested in. I want to know about noncommutativity of observables (which are viewed as measurements).</p>
<p>EDIT: It should be noted that a positive answer to the above question essentially protects uncertainty principle from weak measurement objections...</p>
<p>Thanks for your help in advance!</p>
<p>Further Edit:</p>
<p>Please tell me if the following is flawed. In classical mechanics, every observable is given by a real-valued function on the phase space. Each such function has a Fourier decomposition. In the case of a classical radiating atom under Maxwell's equations, it is predicted that the Fourier transform of the algebra of functions (observables) yields (and is isomorphic to) the convolution algebra of an additive abelian subgroup of the real numbers. However experimentally, the Rydberg-Ritz combination formula holds. Heisenberg drew a parallel between the convolution algebra of the group and that of the "Rydberg-Ritz rules" which yields a matrix algebra. </p>
<p>My question originally asked, essentially, how one is to get a noncommutative algebra by looking only at measurement results. This seems to be precisely what Heisenberg did. </p>
<p>Regarding the observables of this algebra (hermitian elements): at first glance it seems strange that Heisenberg assigns a noncommutative algebra to rules derived from spacing of frequencies of spectral lines, when Quantum Mechanics tells us to use measurement outcomes to label the eigenspaces of a hermitian operator in an operator algebra. I was under the impression that the spectral projections of the Hermitian operator, should generate the observable algebra. This sometimes holds in the commutative (classical) case by coincidence...since all the operators commute, a hermitian operator can determine a common eigenbasis for all observables, therefore the spectral projections generate the algebra. What an observable seems to do in the general (perhaps noncommutative matrix algebra) case is select an orthonormal basis with respect to which the matrices of the algebra are represented. If the observable is not invertible (e.g. a projection) this requires the additional choice of a basis for the kernel. Of course, everywhere I'm restricting my attention to finite dimensions.</p>
<p>The latter role for the Hermitian operator is not incompatible with the view that "measurement is final". We never need to compose two observables, but can effectively look at every operator in the algebra representing the system in terms of this basis. The noncommutativity came before this fact. (von Neumann) measurement, as orthogonal projection into eigenspaces of the given observable seems to make more sense from this viewpoint, as well.</p>
<p>Would a genuine physicist tell me if I'm off the mark here?</p> | 6,381 | [
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<p>Expectation values of $(x,y,z)$ in the $| n\ell m\rangle$ state of hydrogen?</p>
<p>Does anyone know of a quick way of finding this (if there is even one)? Can I somehow use the relation that:</p>
<p>$$\langle r\rangle ~=~ \frac{a_0}{2}(3n^2 - \ell(\ell+1)),$$ </p>
<p>or do I just have to brute force and use properties of Laguerre polynomials and spherical harmonics and what not?</p> | 6,382 | [
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] |
<p>In the Standard Model, I understand that the mass of the electron is assume to arise from two effects:</p>
<ol>
<li><p>A bare mass given by Yukawa interaction with the Higgs field, and</p></li>
<li><p>A mass correction from mass renormalization effects</p></li>
</ol>
<p>In this framework, why do we need to assume 1? Could mass renormalization explain the mass of fermions in general?</p> | 6,383 | [
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<p>When we push something it moves due to the disturbance in it's molecular arrangement causing waves. How do I calculate the speed of push/waves?
<a href="http://www.youtube.com/watch?v=Dnv-Pm4ehFs" rel="nofollow">http://www.youtube.com/watch?v=Dnv-Pm4ehFs</a>
The push actually depends on the amount of the force applied, right? That is, the push is force dependent, so how does he state the magnitude of it?</p> | 6,384 | [
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<p>Is the <a href="http://en.wikipedia.org/wiki/Gravitational_constant" rel="nofollow">gravitational constant</a> $G$ a fundamental <a href="http://en.wikipedia.org/wiki/Physical_constant#Table_of_universal_constants" rel="nofollow">universal constant</a> like Planck constant $h$ and the speed of light $c$?</p> | 6,385 | [
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<p>Alpha Centauri Bb is an exoplanet orbiting Alpha Centauri B. It is asserted that given the close distance to the star the planet should be tidally locked. </p>
<p>The orbiting period of the planet is about 3.2 days.</p>
<p>If the planet has no atmosphere (which is very possible due to proximity to the star), its dark side should experience low temperatures like permanently-shadowed areas of Mercury do.</p>
<p>At the same time the planet is about 11 AU from the other star, Alpha Centauri A.</p>
<p>This possibly means that the planet should experience quasi-day/night cycle each 3.2 days.</p>
<p>My questions are:</p>
<ol>
<li><p>To what temperatures the dark surface of such planet could be heated? Is there possibility of liquid water?</p></li>
<li><p>Will the radiation of the second star be enough to provide normal day-like illumination and heating?</p></li>
<li><p>Will the calendar on such a planet differ sufficiently from a calendar of a planet that experiences day/night cycles from the nearest star?</p></li>
<li><p>Will such planet experience seasons and how they would be arranged?</p></li>
</ol> | 6,386 | [
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<p>Here is a question which frequently occurs on my school exam paper:</p>
<p>"Prove that Faraday's law of electromagnetic induction is consistent with the law of principle of conservation of energy." What does this actually mean? Any help would be better...</p>
<p>P.S. I apologize if the question is too elementary.</p> | 6,387 | [
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<p>What is a good resource to learn about higher degree <strong>degenerate</strong> <a href="http://en.wikipedia.org/wiki/Perturbation_theory" rel="nofollow">perturbation theory</a> - one that involves mathematics that isn't much more advanced than first order perturbation theory? I've looked around and I've only found Sakurai talk about it but he uses projections operators and other fancy mathematics. Also, does anyone have any examples of it being used?</p> | 6,388 | [
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<ol>
<li><p>Can some one please explain in simple words that what is <a href="http://www.google.com/search?hl=en&as_q=effective+refractive+index" rel="nofollow">effective refractive index</a>? </p></li>
<li><p>How it is different from the <a href="http://en.wikipedia.org/wiki/Refractive_index" rel="nofollow">refractive index</a>? </p></li>
<li><p>And how we can calculate the effective refractive index?</p></li>
</ol> | 6,389 | [
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<p>In relation to mass/gravitational/centrifugal force. Is the increased gravitational force due to the increase in mass of a planet (i.e. earth) from meteorites, etc.. directly proportional to the centrifugal force of the planetary body?</p> | 6,390 | [
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<p>When calculating in a rest frame, doesn't one assume both, definite velocity (zero) and position (origin)? Why is Heisenberg okay with that?</p>
<p>Edit: E.g. For a decay we can do calculations in which we say that the particle is truly at rest with 0 (three vector) momentum. Wouldn't that automatically mean that the particle could be literally everywhere in space? Isn't it overall a little bit troublesome, since nothing can actually really be truly at rest? So is that center of mass frame merely a good enough tool for calculation with no strict real meaning? </p> | 6,391 | [
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<p>As pressure is increased, do we require more energy to increase the temperature from given temperature for the same mass. For example, if we heat water at 1 atm to raise its temperature by 20 degree Celsius, will the heat required be the same when pressure is 3 atm?</p> | 6,392 | [
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<p>I'm not a physicist but I majored it at high school (a long time ago) and I study university math.</p>
<p>Me and my roommate discussed whether the performance of a Thermos bottle is influenced by how full it is. So if it is not full do the contents cool down faster, slower or equally?</p>
<p>Thanks.</p> | 6,393 | [
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<p>I'm trying to understand what is the <a href="http://en.wikipedia.org/wiki/S-matrix" rel="nofollow">S-matrix</a> in QFT. People say that it has to be a unitary matrix, but that I guess will change with a different normalization of the incoming and outgoing states. My question is, how do we choose the normalization of states when discussing the S-matrix? And why in general a state of two identical particles has a normalization with an additional factor of $1/\sqrt{2}$ in comparison to states with non-identical particles?</p> | 6,394 | [
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<p>Let's assume a close room with 1-2 people who only one of them smoking cigarette.
What is the equation describe the smoking spreading? is it diffusion? what are the parameters is so? </p>
<p>Is there a way/ device I can build or put that make the smoke go in some specific way, like a vacuum that will suck all the smoke even if it is far away? </p> | 6,395 | [
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<p>I understand that the objects acceleration is determined by the force exerted <em>on it</em>, and that the force exerted <em>on it</em> is <a href="http://en.wikipedia.org/wiki/Newton%27s_laws_of_motion" rel="nofollow">determined</a> by its acceleration.</p>
<p>But, does an object's (named A) acceleration (and mass) tell us anything about how much force the object will exert on another object (named B)?</p> | 6,396 | [
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<p>Two objects go in against each other, and then they collide, will object 1, exerting force 1, necessarily get on it a reaction equal in magnitude and opposite in direction?</p>
<p>EDIT:-
In my book it only says "for every action there is always opposed an equal reaction". But after reading on Newton's third law on Wikipedia, I read this "The action and the reaction are simultaneous"; it's not an intuitive statement (it sounds more like an object gets a reaction after an action), but anyway that solves my problem. If you care to explain how the two forces are always simultaneous please do.</p> | 6,397 | [
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<p>Suppose they have the same contact area with you.</p>
<p>The term "painful" is a bit ambiguous.
Let me ask in another way:</p>
<p>Two cars with same momentum but different masses are going to hit a wall. You must stand right in front of the wall. Which car will you choose to be hit by? </p> | 6,398 | [
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<p>If i have a small toy boat with a sail, and i attach a fan onto it, FACING THE SAIL, which runs by solar. Once the fan turns on will the boat move or will it remain at rest. Apparently it wont because according to Newtons third law the sail will give an equal force back on to the air pushed by the fan. But that doesnt really make sense, so how do real boats which rely on the wind work? If I turn the fan the other way round, the boat will move as the model is now like a planes engine but why not when the fan is facing the sail??</p> | 6,399 | [
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] |
<p>This is asserted by Trinh Xuan Thuan in his book "Chaos & Harmony" chapter String theory.</p> | 6,400 | [
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<p>This question follows from a <em>schooling</em> I received in <a href="http://physics.stackexchange.com/questions/45387/why-cant-missing-mass-be-photons/45389#45389">this thread.</a> </p>
<p>I figured that photons do not interact with gravity, except when they've spontaneously converted into a particle-antiparticle pair. So the equivalence $E=mc^2$ only means that a photon could transform into a massive particle.</p>
<p>Apparently I'm wrong; @JohnRennie succinctly writes:</p>
<blockquote>
<p>The source of the curvature is the <a href="http://en.wikipedia.org/wiki/Stress%E2%80%93energy_tensor" rel="nofollow">stress-energy tensor</a> and this makes no distinction between mass and energy. They are treated as related by the famous $E=mc^2$.</p>
</blockquote>
<p>This is cool because $E=mc^2$ can act as some sort of uncertainty relation; if you have a population of photons with energy $E$, they are engendered with a mass $\frac{E}{c^2}$, no matter what my intuition says. (Is that right?)</p>
<p>So my real question is, in any model of particle interactions, can a photon <em>directly</em> generate a gravitational field? For example, something like a Feynman diagram with a photon and a massive particle linked with a graviton. I suspect the answer is negative.</p> | 6,401 | [
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<p>Sorry if this is a naive question, not being even a part qualified physicist in any way shape or form.</p>
<p>I've read that the universe is expanding and the rate of expansion is increasing. The assumption being that it will continue expanding indefinitely.</p>
<p>However isn't there another possibility.</p>
<p>Let me illustrate my question prior to posing it.</p>
<p>If I throw a ball in the air.. it accelerates away from my hand and keeps accelerating, until momentum is lost and it starts to slow down, eventually falling back to my hand. </p>
<p>Is it possible that the expanding universe is also in the throws (excuse the pun) of an initial acceleration phase, prior to that acceleration slowing and eventually the universe compressing?</p>
<p>Again sorry if its a stupid observation!</p> | 6,402 | [
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<p>We wouldn't have computers if we didn't know about quantum physics. I understand understanding of general relativity is needed to make GPS work well. Has knowledge of quarks or string theory resulted in useful things? </p>
<p><strong>EDIT:</strong> I suspect quarks and string theory could be used by people who study traces of the Big Bang and similar topics. That aside have they resulted in technological advances that could be useful to people (besides improving our understanding of nature)? Perhaps advances in medical imaging have been possible. There might also have been improvements in nuclear fission, nuclear fusion. </p>
<p>Other than that I will be surprised to hear of applications. I suspect many applications are possible, but I expect it will take years to get there. </p> | 6,403 | [
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<p>I'm studying perturbation theory in the context of quantum mechanics. My lecture notes say that in order to calculate the first-order correction of eigenfunction $\psi_n$, that is $\psi_n^{(1)}$, I shall use the formula:</p>
<p>\begin{equation}
\psi_n^{(1)}=\sum_{m,m\ne n} \frac{V_{mn}}{E_n^{(0)} - E_m^{(0)}}\psi_m^{(0)}
\end{equation}</p>
<p>where $E_n^{(0)}$ is the zeroth-order correction (i.e., the unperturbed), and $V_{mn}$ are given by:</p>
<p>\begin{equation}
V_{mn} = \left(\psi_m^{(0)}, V\psi_n^{(0)}\right)
\end{equation}</p>
<p>and V is the "weak" potential that represents the physical disturbance to our system. $(x,y)$ represents inner product.</p>
<p><strong>My question is</strong>: What values does $m$ take in the above summation ? Say I'd like to calculate the first-order corrected $\psi_1$, that is to calculate $\psi_1^{(1)}$. What values would $m$ assume ? Infinite values except for $1$ ? Values $0,\ldots,n-1$ ? </p> | 6,404 | [
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<p>Suppose that a fluid is flowing parallel to and over a flat plate. Obviously, a boundary layer develops in which the velocity ranges from 0 to 99% of the upstream velocity U. Could somebody please show me how the streamlines would look in this boundary layer? What do they typically look like in boundary layers?</p> | 6,405 | [
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<p>I have seen it remarked in some problem sets that if you have an electromagnetic wave traveling in the $x$-direction with it's $y$-coordinate given as</p>
<p>$y(x,t)=y_0\sin (\omega t +kx)$</p>
<p>and you want a build an antenna to receive the wave, the antenna must be of length $2y_0$. I want to know how much truth there is to this statement.</p>
<p>The physical picture is quite nice; the electric field is "waving in space" so when it hits a conductor, the electrons accelerate along the conductor and you can measure the current to get the frequency. But "real" E/M waves don't have spatial position like a string; a solution to Maxwell's equations for traveling waves looks something like</p>
<p>$\vec{E}=E_0\sin (\omega t+kx)\hat{y}$</p>
<p>(taking the simplist possible conditions). The physical picture still works, since the electric field is "waving" along the conductor, but the length argument no longer makes any sense. And actually, don't dipole antennas work the $other$ way, by maximizing the voltage difference across the two ends by aligning themselves parallel to the direction of propagation (the $x$-axis here)?</p>
<p>So is the simple picture we paint for students completely incorrect, or is there any validity to it? </p>
<p>EDIT: As it turns out, the only examples of this misunderstanding I can easily find are those which I have some reservations about posting because of their relationship to graded problem sets. So, I will leave this up for a few days to see if I can attract anyone with a simple explanation of antenna design to answer it. If not, I will answer it myself.</p> | 6,406 | [
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<p>It makes sense, since gravity tends to push the surface of a body towards it's center. Unless I'm mistaken, everything with mass has it's own gravity, every atom and for instance, our own bodies should also have their own gravity. The question is: how strong is our own gravitational pull? I know it must be extremely weak, but is there actually anything at all that gets attracted to us, like maybe, bacteria or molecules?</p>
<p>And finally (this will sound ridiculous, but I'd really want to get an answer or at least a way of calculating it myself): What size would a human body have to reach in order for it to collapse into a sphere?</p> | 6,407 | [
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<p>As i understood , if you have a point charge in the center of a hollow conducting sphere then the electric field inside it, is zero because the charge distribution is spherically symmetric. </p>
<p>But what's going on if the point charge isn't in the center of the sphere?</p>
<p>Will then be an Electric field inside the sphere? And will the outside electric field change?</p> | 6,408 | [
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<p>I always see $\vec j\cdot \vec e$ as Joule's dissipation and I don't understand why. For example, if we have a uniform electric field $\vec e=e_o\vec u_x$ and we release an electron in it, it will start moving, accelerating in the direction of $-\vec u_x$, so the we will have a current $\vec j$, and this product $\vec j\cdot\vec e$ will exist, but I don't see that any energy is being dissipated in the form of heat there. What is going on?</p>
<p>I've seen this essentially in Optics texts, when introducing Poynting's theorem.</p> | 6,409 | [
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<p>I know that the sum of vacuum bubbles can be related to the Vacuum energy, but I'm trying to understand how this follows from the Gell-Mann Low theorem/equation. My question will use equations from Peskin and Schroeder's (1995) text.</p>
<p>I start from the equation just below (4.30) on page 87, reproduced here</p>
<p>$$1=\langle\Omega|\Omega\rangle=\big(\big|\langle 0|\Omega\rangle\big|^2 e^{-iE_0(2T)}\big)^{-1}\langle0|U(T,t_0)U(t_0,-T)|0\rangle\,,$$</p>
<p>where,</p>
<ol>
<li>$T$ is a large time, and $t_0$ is arbitrary reference time.</li>
<li>$|\Omega\rangle$ is ground state of interacting theory with energy $E_0$.</li>
<li>$|0\rangle$ is the ground state of free theory.</li>
</ol>
<p>I would now like to solve for $E_0$, the ground state energy. By first taking the Log of both sides. I get <em>almost</em> the desired result.</p>
<p>$$E_0=\frac{i}{2T}\ln \langle0|U(T,t_0)U(t_0,-T)|0\rangle-\frac{i}{2T}\ln\big|\langle 0|\Omega\rangle\big|^2\,.$$</p>
<p>The first term is what I'm looking for: this generates the vacuum bubbles. But then there's the extra second term which I don't know how to interpret. How do I understand this second term? or how can I justify dropping it?</p> | 6,410 | [
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<p>In Feynman's lecture notes, he said that it is not (at his time).</p>
<p>How is the situation today?</p>
<p>Can first-principle calculation accounts the ferromagnetism of iron quantitatively now?</p> | 6,411 | [
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<p>In 't Hooft's original paper:
<a href="http://igitur-archive.library.uu.nl/phys/2005-0622-152933/14055.pdf" rel="nofollow">http://igitur-archive.library.uu.nl/phys/2005-0622-152933/14055.pdf</a></p>
<p>he takes $N \rightarrow \infty $ while $ g^2 N$ is held fixed. Is this just a toy model? Or is there some reason to believe that $g^2$ goes like $\frac{1}{N}$ for large coupling? Thanks a lot. </p> | 6,412 | [
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<p>It's said in Chapter VI.4 of A. Zee's book <em>Quantum Field Theory in a Nutshell</em>, a theory defined as $L(U(x))=\frac{f^2}{4}Tr(\partial_{\mu}U^{\dagger}\cdot\partial^{\mu}U)$, can be write in the form of a non-linear $\sigma$ model (up to some order)</p>
<p>$L=\frac{1}{2}(\partial\vec{\pi})^2+\frac{1}{2f^2}(\vec{\pi}\cdot\partial\vec{\pi})^2+...$,</p>
<p>where $U(x)=e^{\frac{i}{f}\vec{\pi}\cdot\vec{\tau}}$ is a matrix-valued field belonging to $SU(2)$, $\vec{\pi}$ is a three components vector, $\vec{\tau}$ are Pauli matrices. Maybe it's not hard but I meet some problems to derive it.</p>
<p>I suppose the first step is the Taylor expansion of $U$, $U=1+\frac{i}{f}\vec{\pi}\cdot\vec{\tau}-\frac{1}{2f^2}(\vec{\pi}\cdot\vec{\tau})^2+...$, and then $\partial^{\mu}U=\frac{i}{f}\partial^{\mu}(\vec{\pi}\cdot\vec{\tau})-\frac{1}{f^2}(\vec{\pi}\cdot\vec{\tau})\partial^{\mu}(\vec{\pi}\cdot\vec{\tau})$, then</p>
<p>$(\partial_{\mu}U^{\dagger})(\partial^{\mu}U)=\frac{1}{f^2}[\partial(\vec{\pi}\cdot\vec{\tau})]^2+\frac{1}{f^4}.[(\vec{\pi}\cdot\vec{\tau})\partial(\vec{\pi}\cdot\vec{\tau})]^2$.</p>
<p>Now there are my questions,</p>
<p>(1) Can I write $\partial^{\mu}(\vec{\pi}\cdot\vec{\tau})=\partial^{\mu}\vec{\pi}\cdot\vec{\tau}$? Then by $\vec{\tau}^2=1$, I get </p>
<p>$L=\frac{1}{4}(\partial\vec{\pi})^2+\frac{1}{4f^2}(\vec{\pi}\cdot\partial\vec{\pi})^2$,</p>
<p>which is almost correct but differ to the wished answer by a pre-factor $\frac{1}{2}$.</p>
<p>(2) Suppose $\partial^{\mu}(\vec{\pi}\cdot\vec{\tau})=\partial^{\mu}\vec{\pi}\cdot\vec{\tau}$ is correct, however, if I do $\partial^{\mu}U=\frac{i}{f}U\partial^{\mu}(\vec{\pi}\cdot\vec{\tau})=\frac{i}{f}U\partial^{\mu}\vec{\pi}\cdot\vec{\tau}$ first, it seems $\partial^{\mu}U^{\dagger}\cdot\partial^{\mu}U=|\frac{i}{f}U\partial^{\mu}\vec{\pi}\cdot\vec{\tau}|^2=\frac{1}{f^2}(\partial\vec{\pi})^2$, say, only the first term of the wished answer.</p>
<p>I probably made something wrong somewhere, can anyone hit me? </p> | 6,413 | [
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<p>I've already faced this situation several times: given a statement (in area of thermodynamics) I used it to provide an example of some perpetual motion machine (of first or second kind). Therefore, I thought, proving the initial statement wrong. And sometimes I get a response that I haven't actually constructed a perpetual motion machine. Because the temperature reservoirs I use will eventually change their temperatures. Ergo, my construction is not perpetual.</p>
<p>I'm pretty sure that the argument is invalid (not to say -- silly). But I still can be wrong. </p>
<p>Whatever the case. I'd like to have a consistent way of rejecting it (or accepting and using it) starting from first principles and definitions of thermodynamics. References dealing with that argument are also welcome. </p> | 6,414 | [
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<p>In condensed matter physics, people always say <a href="http://en.wikipedia.org/wiki/Quantum_fluid" rel="nofollow">quantum liquid</a> or <a href="http://en.wikipedia.org/wiki/Spin_liquid" rel="nofollow">spin liquid</a>. What does liquid mean?</p> | 6,415 | [
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<p>Why is constant for the conversion of mass to energy square of the ligths speed? is it bedside it's the fastest real matter? </p> | 6,416 | [
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<p>i saw this Lagrangian in notes i have printed:</p>
<p>$$
L(x,dx/dt) = (m^2(dx/dt)^4)/12 + m(dx/dt)^2*V(x) -V^2(x)
$$</p>
<p>what is it? is it physical? it seems like it doesn't have the right units of energy,</p>
<p>thanks</p> | 6,417 | [
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<p>I'm not certain that this question will make sense, but here goes...</p>
<p>In most monte carlo generators, when Z events are produced, there is a lower mass cutoff on the Z pole. I've been told that this is because the interference term with the photon would otherwise cause a divergence if you have $\gamma \to ee$ with low invariant mass.</p>
<p>Can anybody explain the meaning of this statement?</p> | 6,418 | [
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<p>I've just found <a href="http://www.youtube.com/watch?v=tu57B1v0SzI" rel="nofollow">Dr Quantum</a> video sample where double split experiment is presented conducted out by researchers. Are there any papers published in peer reviewed journals on that experiment to read in detail?</p> | 6,419 | [
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<p>Won't it be correct to define a CFT as a QFT such that the beta-function of all the couplings vanish? </p>
<p>But couldn't it be possible that the beta-function of a dimensionful coupling vanishes but it does so at a non-zero value of it - then the scale invariance is not generated though the renormalization flow is stopped? Is this possible? </p>
<p>(..it is obviously true that a theory with no intrinsic scale or dimensionfull parameter can still not be a CFT - like a marginal deformation of a CFT may not keep it a CFT and then this deformation parameter has to flow to a fixed point for a new CFT to be produced at that fixed point value of the marginal coupling..) </p> | 6,420 | [
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<p>Looking at the relevant wikipedia page, one can read that the graviton should be massless. Is it 100 % certain that it is massless or is there room in any "nonstandard" models for a tiny non-zero mass (which could lead to a similar surprise as the detection of the neutrino oscillation) such that the graviton (if it exists ...) could maybe selfinteract and form something like "graviballs"?
(My present knowledge of GR and QFT is at the "Demystified-Level")</p> | 6,421 | [
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<p>Can there be some non-electromagnetic radiation which would be perceived by human eye as light? I mean, say ultra-sound or some particle rays etc.</p> | 6,422 | [
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<p>I have read that string theory predicts (or requires ?) the existence of gravitons.
So, would that make it a quantum theory of gravity ?</p>
<p>If so, I have also read that quantum gravity would allow us to understand what happens at the singularity inside a black hole. What does string theory say about this ?</p>
<p>If string theory can't make any predictions about this, what are we missing ?
Is the theory incomplete ? Or is it that we have multiple string theories - hence, multiple ways to quantize gravity - and don't know which is the right one ?</p>
<p>I'm not a physicist, so just looking for an explanation at the Brian Greene popular book level. :)</p> | 6,423 | [
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<p>Other than one having a positive invariant scalar product and the other a negative one, what are the actual physical differences between these vectors?</p> | 6,424 | [
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<p>I do have a question about an assumption made in the very interesting Hayden-Preskill paper of black holes as informational mirrors. Alice throws her top secret quantum diary which is $k$ qubits long into a black hole hoping to get rid of it. Unfortunately, the top forensic scientist Bob was already maximally entangled with the black hole, and in a sense, "knows" the full microstate of the black hole before Alice dumped her diary into it. If $c$ is the scrambling time, and Bob waits to collect $k+c$ qubits worth of Hawking radiation plus a few more to reduce the margin of error up to the desired level, then in principle, Alice's quantum diary can be read by Bob. This argument is purely quantum informational theoretic. What are the computational resources needed for Bob to decrypt Alice's diary? By assumption, the dynamics of black hole evolution are described by a reversible unitary transformation, and not by Krauss operators. So, if Bob is willing to wait until the black hole has completely evaporated away and reverse the computations, he can read Alice's diary in polynomial time. However, Bob is too impatient for that, and he wants to read it after only getting $k+c$ plus a few more qubits worth of information. If Bob is willing to wait exponentially long (and he's not), he can create exponentially many identical black holes one by one and go over all possible $k$ qubit combinations to dump into the identical black holes and wait until he finds one which gives Hawking radiation that matches the Hawking radiation he actually got from the original hole. Bob is also unable to reverse the data either because he doesn't know the microstate of the black hole after collecting $k+c$ qubits worth of Hawking radiation. And apparently, neither can Alice if she doesn't already have a copy of her diary.</p>
<p>There is an analogous situation in classical computation. A cryptographic hash function is a polynomially computable function which takes in as arguments a seed which can be publicly known, and some unknown string to be encrypted. It's a one-way function. It's not possible to compute to original string from the encrypted string using polynomial resources. However, if we are given another string and asked to test if it's identical to the original string, we can compute its hash function using the same seed and check if it matches. If it doesn't, they're not identical. If they do, there's a very high chance that they are indeed identical.</p>
<p>Are black holes quantum cryptographic hash functions? What is the best lower bound on the time it would take for Bob to read Alice's diary?</p> | 6,425 | [
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<p>Lets take a disc that is rolling without slipping which has moment of inertia $I=kmR^2$. It will have total kinetic energy $E=\frac{1}{2}mv^2+\frac{1}{2}I\omega^2=\frac{1}{2}mv^2(1+k)$.</p>
<p>Lets now use the parallel axis theorem to measure the moment of inertia around a different axis, lets say the point of contact with the ground. So $I=kmR^2+mR^2$ (since the distance from the center of mass is R). To my understanding, in this instance, we need to add another term to the equation, $m \vec{v}\cdot(\vec{\omega} \times \vec{R})=-m |\vec{\omega}|^2R^2$ where $\vec{R}$ is the distance from the point of calculating moment of inertial to the axis of rotation. </p>
<p>So $$\begin{align*}E &= \frac{1}{2}mv^2 + \frac{1}{2}(kmR^2+mR^2)\omega^2 + (-m |\vec{\omega}|^2R^2)\\
&= \frac{1}{2}mv^2 + \frac{1}{2}(k+1)mv^2 + (-m v^2)\\
&= \frac{1}{2}mv^2(1+k) - \frac{1}{2}mv^2\\
&= \frac{1}{2}kmv^2\end{align*}$$ which does not seem to equal $\frac{1}{2}mv^2(1+k)$ at all.</p>
<p>Is there something I did wrong?</p> | 6,426 | [
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<p>Is it correct to state that the elements of Spin(n) fulfill a Clifford algebra and that the Lie group generators of Spin(n) is given by the commutator of the elements?</p>
<p>If not, then what is the relation of the Spin(n) group, the matrices that fulfill the Clifford algebra (that can be used to construct SO(n) bilinears from spinors, like the Dirac gamma matrices) and the generators of Spin(n) as a Lie group?</p> | 6,427 | [
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<p>I've recently been to a converence on plasma physics were, to my surprise, a lot of presentations were concerned with <a href="http://en.wikipedia.org/wiki/Plasma_actuator" rel="nofollow">plasma actuator</a>. </p>
<p>Could someone, preferably in the field, tell me how long people have been reseaching this now and, more importantly, give an honest opinion on how far (in years) this technique is from regular industrial use?</p> | 6,428 | [
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<p>In a NPR News story from a few years back:</p>
<blockquote>
<p>"A gamma-ray burst from about 13
billion light years away has become
the most distant object in the known
universe."</p>
</blockquote>
<p>I'm a layman when it comes to physics, so cut me some slack if this is an ignorant question, but assuming the universe is around 14B years old, and has been expanding since the Big Bang, how is it that we can see events so far back in time?</p>
<p>I understand how it would work if you had a static universe and the GRB happened 13B years ago at 13B Light years away and the light just arrived. However, at the time of the burst wouldn't we (or at least the matter that we are made from) have been much closer to the source of that burst, and wouldn't the light have blown by us eons ago? How is it we are seeing it now? </p>
<p>If we were expanding away from it at close to light speeds it would seem to make sense for why it took so long for it to get here, except for that whole notion that light moves at the same speed relative the to the observer, which I think would blow that idea out of the water.</p>
<p>Perhaps gamma rays travel at sub-light speeds? But, I'd still think the math would require that they travel MUCH slower than light for this scenario to play out.</p>
<p>Another, possibility is that the light has wrapped around a finite universe a few times before reaching us. Of course if that were the leading theory, there wouldn't be any remaining controversy about the finite vs. infinite universe models.</p>
<p>What am I missing here? </p> | 6,429 | [
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<p>In D'Alembert principle forces are classified into constraint and applied forces ? Is this classification different from internal-external forces ?</p> | 6,430 | [
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] |
<p>In this particular example shown the image below (from engineering dynamics - Meriam), I do not figure out why the direction of friction is in direction of the translation of the car?
Or I just didn't get it right and the car is slipping down in that situation?</p>
<p>(see hint 2 and 3 on the right hand side of the page)
<a href="http://pasteboard.s3.amazonaws.com/images/jWg2h9U.jpg" rel="nofollow"><img src="http://pasteboard.s3.amazonaws.com/images/jWg2h9U.jpg" alt="Meriam - Engineering Dynamics"></a></p>
<p>if you've got problems with viewing the image:
<a href="http://oi47.tinypic.com/359z3f4.jpg" rel="nofollow">new link</a></p> | 6,431 | [
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<p>I'm reading page 488 of <a href="http://www.physics.usyd.edu.au/~fvon4093/introgr.pdf" rel="nofollow">Hobson, Efstathiou & Lasenby</a>, and I don't understand something they write... so I came here. </p>
<p>The concept they describe is in linearised general relativity. In particular, they are describing the averaging process over small spacetime regions that makes it possible to define the energy-momentum tensor of gravitatonal waves.</p>
<p>They say that since we are averaging of all directions at each point, first derivatives average to zero. That is, for any function of position $f(x)$, we have $\langle \partial_{\mu} f(x) \rangle=0$. Here $x = (x^1,x^2,x^3)$, which is an index-notation version of the 'standard' $(x,y,z)$.</p>
<hr>
<p><strong>Edit</strong></p>
<p>By 'average', I assume they mean an integral. So, if we take the simplest possible case for starters, we would have a 1D spacetime, say the $x$ lined. So, all functions of position, are just $f(x)$.</p>
<p>So, I would interpret
\begin{equation}
\langle f(x) \rangle = \int f(x) dx.
\end{equation}
Then,
\begin{equation}
\begin{split}
\langle \partial_{x} f(x) \rangle &= \partial_{x} \int f(x) dx \\
&=f(x).
\end{split}
\end{equation}
So, how can this be zero?</p> | 6,432 | [
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<p>Let us consider a Wightman field theory for the free scalar neutral field $\phi$, and let $O\mapsto\mathfrak F(O)$ be the net of local von Neumann field algebras. If we take a non-empty bounded open subset $O$ of $\mathbb R^4$, then according to the Reeh-Schlieder theorem the vacuum vector $\Omega_0$ of the Fock construction is cyclic. This implies that the set of vectors $\{W\Omega_0\ |\ W\in\mathfrak F(O)\}$ is total in the Fock Hilbert space $\Gamma(\mathscr H)$, i.e. it is dense.</p>
<p>Why is $\mathfrak F(O)$ not irreducible in this case? If we take $C$ in the commutant of $\mathfrak F(O)$, i.e. $C\in\mathfrak F(O)'$, then we can show that $\Omega_0$ is an eigenvector of $C$, i.e. $\exists\lambda_C\in\mathbb R$ s.t. $C\Omega_0=\lambda_C\Omega_0$. Moreover, using commutativity, we deduce that $(\psi,CW\Omega_0)=\lambda_C(\psi,W\Omega_0)$ for any $\psi\in\Gamma(\mathscr H)$ and $W\in\mathfrak F(O)$. But since the $W\Omega_0$s are elements of a dense subset of the Fock Hilbert space, then we must conclude that $C=\lambda_C\text{Id}$ for any $C\in\mathfrak F(O)'$, i.e. $\mathfrak F(O)$ is irreducible.</p>
<p>What's wrong with the above proof?</p> | 6,433 | [
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<p>I suppose to build a small dehumidifier, but I am narrowing the question down into its logics.</p>
<p>I have relatively powerful computer fans, water cooling radiator and some metal tubings. With the idea of compression and decompression I think air would do the job better than the original liquid coolant in the water cooling system.</p>
<p>The idea is to generate pressured air flow into narrowing tubes, and have it bursting out inside the radiator. Assuming ambient temperature around 25 degrees and above 60% relative humidity. (usually around 80% in my area but let's generalize it a bit)</p>
<p>My question is, could this lowers the air temperature below dew point? And how effective is it?</p> | 6,434 | [
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<p>I'm given with these data:<br><br></p>
<p>Lens 1 and lens 2 are convex lenses with different focal lengths<br><br></p>
<p>Distance of object from lens 1: 50 mm <br>
Distance between lens 1 and lens 2: 100 mm <br>
Distance of final image from lens 1: 300 mm<br><br></p>
<p>How to compute for the Distance of first image from lens 1? A ray diagram for this kind of set-up will be appreciated. Thank you!</p> | 6,435 | [
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<p>I have encountered those terms in various places. As I understand it, "soft wall" can correspond to a smooth cutoff of some spacetime, while "hard wall" can be a sharp one, which can be described in terms of D-branes. Could somebody please explain the terminology, and in which context it can occur?</p> | 6,436 | [
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<p>Data are coming in, and it seems that recent Higgs boson observation is eliminating many SUSY models. If so, what is happening to superstring theories, like M-theory?</p> | 6,437 | [
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<p>Usually, when asked whether the purple color exists rainbows, an answer similar to this is given:</p>
<p>The purple color is perceived by human eyes via the activation of both red-sensitive and blue-sensitive cone cells. It is known that purple isn't a physically monochromatic light (a light composed of a single wavelength).</p>
<p>However, on diffraction gratings, the spectrum repeats itself in such a way that part of a lower-order diffraction overlaps with that of a higher order. For example, it is conceivable that the red band of the second order diffraction overlaps with the blue-violet band of the first order diffraction, producing the purple color perception.</p>
<p>This would also occur in thin-film light interference, such as anti-reflective coatings of eye glasses, etc.</p>
<p>Could the same occur in rainbows, which is dispersion in water driplets, where red and blue-violet somehow "overlap"?</p>
<p>(As a side issue in answering this question, it is hard for a computer user to perceive true "monochromatic violet", since trichromatic displays are not capable of producing them. This issue extends to photographs captured by trichromatic devices. Not every one has the opportunity to see Argon-voliet in their own eyes - and then remember that perception.)</p> | 6,438 | [
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<p>When material of rest mass M falls from infinity onto a black hole accretion disk, it gets heated and then emits so much light that the energy radiated away can measure up to about 30% or so of M c^2. Let's say that ε is the fraction of rest mass energy radiated away. </p>
<p>My first question is, after this accreted material has crossed the event horizon, does the mass of the black hole (as an observer would measure by, say, examining keplerian orbits of nearby stars) grow by M or (1 - ε) M? I am quite certain that the answer is (1 - ε) M, but I am open to correction here.</p>
<p>My second question is, if the mass does indeed grow by (1 - ε)M, then if you wanted to compute, say, the gyroradius of electrons circling magnetic field lines in the accretion disk after it has cooled, would you use the regular electron mass or would you somehow include the mass defect?</p>
<p>The fine print: It would be helpful to agree on answers to the explicit questions before abstracting the answers. On the abstract side, I am comfortable with the idea that "the whole is not equal to the sum of the parts" when it comes to mass in bound systems. But that quote is often employed to compare the mass of the bound system with the masses of the unbound (free) components. I am trying to understand what the relation is between the masses of the components of the bound system to the total mass of the bound system, if an unambiguous relationship exists. </p> | 6,439 | [
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<p>From what I've read, the only remaining candidates appear to be either sterile neutrinos or MOND (MOdified Newtonian Dynamics -- it does seem to keep changing.)</p>
<p>Did I miss anything else plausible?</p> | 6,440 | [
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<blockquote>
<p>An insulated disk, uniform surface charge density $\sigma$, of radius $R$ is laid on the $x,y$ plane. Deduce the electric potential $V(z)$ along the z-axis. Next consider an off axis point $p'$, with distance $\rho$ from the center, Making an angle $\theta$ with the z-axis. Expand the potential at $p'$ in terms of Legendre polynomials $P_l(\cos\theta)$ for $\rho < R$ and $\rho > R$</p>
</blockquote>
<p>For the point on the z-axis, this is pretty easy. The differential Voltage from a differential ring of charge with radius $r$ is:</p>
<p>$$dV = \frac{1}{4 \pi \epsilon_o} \frac{dq}{ \mathscr{R}}$$</p>
<p>$$dq = \sigma dA = \sigma 2 \pi r dr$$ </p>
<p>$$\mathscr{R} = \sqrt{r^2 + z^2}$$</p>
<p>$$ \Delta V(z) = \frac{ \sigma}{2 \epsilon_o}\int_0^R \frac{ r dr}{\sqrt{r^2 + z^2}} = \frac{ \sigma}{2 \epsilon_o} \left( \sqrt{R^2 + z^2} - |z| \right)$$</p>
<p>Which is obtained by using a U substitution.</p>
<p>As for the second part, The only thing that changes is the distance from the differential of charge and the point of interest so I have:</p>
<p>$$dV = \frac{ \sigma}{2 \epsilon_o} \frac{r dr}{ \mathscr{R}}$$</p>
<p>But now using the law of cosines, I use the angle between $r$ and $\mathscr{R}$, Note: this is not the angle recommended in the problem. Thus
$\mathscr{R} = (r^2 + p^2 - 2rp\cos \phi)^{1/2} = r(1 - 2 \frac{p}{r}cos \phi + \frac{p^2}{r^2})^{1/2},$
using spherical polar coordinates, where $p$ is the distance from origin to the point of interest, $p'$.</p>
<p>This is the generating function of the Legendre polynomials,</p>
<p>$$\therefore \frac{1}{\mathscr{R}} = \frac1r G\left( \frac{p}{r}, \cos \phi\right)$$</p>
<p>$$dV = \frac{ \sigma}{2 \epsilon_o} G\left( \frac{p}{r}, \cos \phi\right) dr = \frac{ \sigma}{2 \epsilon_o} \sum_{l = 0} ^{\infty} p_l(\cos \phi) \left( \frac{p}{r} \right)^l dr$$</p>
<p>Okay, so my question is this, assuming all of this is correct (which I believe is not) How would possibly integrate this? Is it as simple as $\int_0^R \left( \frac{p}{r} \right)^l dr$? This creates an infinity. Any help would save me so very much.</p> | 6,441 | [
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<p>I've been asked to show that both the position-momentum uncertainty principle and the energy-time uncertainty principle have the same units.</p>
<p>I've never see a question of this type, so am I allowed to substitute the units into the expressions and then treat them as variables?</p>
<p>If so, here's my attempt. Forgive me if I've done something silly, as I'm no physicist.</p>
<p>Starting with the position-momentum uncertainty principle:</p>
<p>$$\Delta{}x\Delta{}p \geq h / 4\pi$$</p>
<p>Substituting the units into the expression (at this point, diving by $4\pi$ won't necessarily matter):</p>
<p>$$(m)\left(kg \cdot \frac{m}{s}\right) \geq J \cdot s$$</p>
<p>Combining $m$ and bringing $s$ to the other side:</p>
<p>$$\frac{kg \cdot m^2}{s^2} \geq J $$</p>
<p>Knowing that $J = kg \cdot m^2/s^2$:</p>
<p>$$J \geq J$$</p>
<p>Now, for the energy-time uncertainty principle:</p>
<p>$$\Delta{}E\Delta{}t \geq h / 4\pi$$</p>
<p>Substituting the units into the expression (again, diving by $4\pi$ won't necessarily matter):</p>
<p>$$J \cdot s \geq J \cdot s$$</p>
<p>Diving by $s$:</p>
<p>$$J \geq J$$</p>
<p>Is this valid? Or could I not be more wrong?</p> | 6,442 | [
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<p>Consider the following figures. </p>
<ul>
<li><p>The top one shows the construction of a spring where its left end is attached to the wall and its right end is stretched by a force.</p></li>
<li><p>The bottom one is supposed to be the free diagram of the spring. Is it correct?</p></li>
</ul>
<p><img src="http://i.stack.imgur.com/jyYcO.png" alt="enter image description here"></p> | 6,443 | [
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<p>From the perspective of <em>dimensional analysis</em>, in the Laplacian of Gaussian operator
$$LoG(x,y,\sigma)=\frac{\partial^2g}{\partial x^2} +\frac{\partial^2g}{\partial y^2}.$$ I think $x,y$ are variables with dimension $L$, $\sigma$ is a parameter with dimension $L$. But what about $g$? Since $g$ is a function of $x,y,\sigma$,
$$g(x,y,\sigma)=\frac{1}{2\pi \sigma^2}exp(-\frac{x^2+y^2}{2\sigma^2}),$$
and $x,y,\sigma$ are of the same dimension $L$, so I guess in $g$, the term $exp(-\frac{x^2+y^2}{2\sigma^2})$ is dimensionless, isn't it? And the term $\frac{1}{2\pi \sigma^2}$ is of dimension $L^{-2}$, right? So $g$ is actually of dimension $L^{-2}$, isn't it?</p> | 6,444 | [
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<p>the spectrum of <a href="http://en.wikipedia.org/wiki/Quarkonium#QCD_and_quarkonia" rel="nofollow">quarkonium</a> and the comparison with positronium potential for the strong force is:</p>
<p>$V(r) = - \frac{4}{3} \frac{\alpha_s(r) \hbar c}{r} + kr$</p>
<p>(where the constant $k$ determines the field energy per unit length and is called string tension. )</p>
<p>in other hand, one can talk about a <a href="http://en.wikipedia.org/wiki/Yukawa_potential" rel="nofollow">Yukawa potential</a> of the form</p>
<p>$$ V(r) = - \frac{g^2}{4 \pi c^2} \frac{e^{-mr}}{r} $$</p>
<p>where $m$ is roughly the pion mass and $g$ is an effective coupling constant.</p> | 6,445 | [
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<p>There is an art/science video clip on youtube, by Simon Faithful, called <a href="https://www.youtube.com/watch?v=_wnyp3Nrp0w" rel="nofollow">Escape Vehicle no.6 (chair in space)</a>. It shows a simple steel office chair, tied to a weather balloon, ascending into the upper reaches of the atmosphere. Apparently it reaches about 30km above sea level. It's quite a beautiful clip.</p>
<p>At the start, the chair rises quite violently, as it is buffetted by surface turbulence, as would be expected. After a while (2:07-2:33, I suppose once it gets above the planetary boundary layer), the ascent becomes fairly smooth - there is still a lot of movement, but it's not so violent. Then at 2:45, when the chair is near the top of it's ascent (~30km), suddenly things start getting really violent again, and the chair looses a leg, then half the back is torn off, and then the chair is apparently torn from the rope, and disappears.</p>
<p>What forces would cause such a violent disintegration of what is a reasonably strong object, in such a thin atmosphere? </p> | 6,446 | [
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<p>I'm undergraduate and I'm looking for a text about <a href="http://www.google.com/search?q=small+oscillations+matrix" rel="nofollow">Small Oscillations</a> in which matrices are used. Could you suggest me a book or a PDF file?</p> | 6,447 | [
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<p>The <a href="http://en.wikipedia.org/wiki/Shapiro_delay" rel="nofollow">Shapiro delay</a> was predicted in 1964 and observed by 1966, and is now a tool used to measure the mass of distant binary pulsars. The <a href="http://en.wikipedia.org/wiki/Terrell_rotation" rel="nofollow">Terrell-Penrose rotation</a> was published in 1959, but I can find no evidence for experimental confirmation. Wikipedia lists none, although it does reference a good animation. Everybody believes the math, but what physical proof is there? What would it take to observe something like the animation shows?</p> | 6,448 | [
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<p>My understanding is that in a double-slit experiment, quantum interference disappears if which-path information is <em>available</em>. How is <em>available</em> defined? Consider the following experiment:</p>
<p>SPDC is used to create an entangled pair of photons. The signal photon goes through a double-slit with a detector behind it. The idler photon hits the wall of the laboratory. Is which-path information available? After all, <em>theoretically</em> the information carried by the idler could be reconstructed from careful measurement of the wall's properties. In such a case is interference observed? How "available" must which-path information be?</p> | 6,449 | [
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<p>The state $$|\Psi \rangle = |0\rangle + \sum_j \int d\omega f_j(\omega)\hat{a}^\dagger_j (\omega) |0\rangle $$ is coming from a far field and incident on a double slit setup. Here j is the index of j-th atom of the detector and $\omega$ is the index of the $\omega$-th mode of the field.</p>
<p>On the other side there is a detector. The setup looks as follows (Prof. Shih's book).</p>
<p><img src="http://i.stack.imgur.com/hbU2J.jpg" alt="enter image description here"></p>
<p>So far, I have calculated the following. The field at the detector is described as follows.</p>
<p>The positive field arrived at the detector is:
$$E^{(+)}(x, z, t) = E^{(+)}_1(x, z, t) + E^{(+)}_2(x, z, t) $$</p>
<p>and the negative field arrived at the detector is:</p>
<p>$$E^{(-)}(x, z, t) = E^{(-)}_1(x, z, t) + E^{(-)}_2(x, z, t) $$</p>
<p>We can derive the terms separately (assuming single polarization).</p>
<p>The positive field at the detector coming from the $P_1$ slit is:
$$E^{(+)}_1(x, z, t) = \sum_{\overrightarrow{k_1}} g(\overrightarrow{k_1}, z-z_0, t-t_0) E^{(+)}_1(\overrightarrow{k_1}, x_0, z_0, t_0)$$</p>
<p>$$ = \sum_{\overrightarrow{k_1}} \frac{e^{-i[\omega_1(t-t_0)-k_1r_0(x)]}}{r_0(x)} \int^{\frac{b}{2}}_{-\frac{b}{2}} dx_0 e^{i(k_{1x_0} -k_x)x_0} \times i \mathcal{E}_{\overrightarrow{k_1}}\hat{a}_{\overrightarrow{k_1}} . e^{-i \overrightarrow{k_1} r_0}$$</p>
<p>, the positive field at the detector coming from the $P_2$ slit is:</p>
<p>$$E^{(+)}_2(x, z, t) = \sum_{\overrightarrow{k_2}} g(\overrightarrow{k_2}, z-z_0, t-t_0) E^{(+)}_2(\overrightarrow{k_2}, x_0, z_0, t_0)$$</p>
<p>$$ = \sum_{\overrightarrow{k_2}} \frac{e^{-i[\omega_2(t-t_0)-k_2r_0(x)]}}{r_0(x)} \int^{\frac{b}{2}}_{-\frac{b}{2}} dx_0 e^{i(k_{2x_0} -k_x)x_0} \times i \mathcal{E}_{\overrightarrow{k_2}}\hat{a}_{\overrightarrow{k_2}} . e^{-i \overrightarrow{k_2} r_0}$$</p>
<p>, the negative field at the detector coming from the $P_1$ slit is:</p>
<p>$$E^{(-)}_1(x, z, t) = \sum_{\overrightarrow{k_1}} g(\overrightarrow{k_1}, z-z_0, t-t_0) E^{(-)}(\overrightarrow{k_1}, x_0, z_0, t_0)$$</p>
<p>$$ = \sum_{\overrightarrow{k_1}} \frac{e^{-i[\omega_1(t-t_0)-k_1r_0(x)]}}{r_0(x)} \int^{\frac{b}{2}}_{-\frac{b}{2}} dx_0 e^{i(k_{1x_0} -k_x)x_0} \times (-i) \mathcal{E}_{\overrightarrow{k_1}}\hat{a}^\dagger_{\overrightarrow{k_1}} . e^{i \overrightarrow{k_1} r_0}$$</p>
<p>and the negative field at the detector coming from the $P_2$ slit is:</p>
<p>$$E^{(-)}_2(x, z, t) = \sum_{\overrightarrow{k_2}} g(\overrightarrow{k_2}, z-z_0, t-t_0) E^{(-)}(\overrightarrow{k_2}, x_0, z_0, t_0)$$</p>
<p>$$ = \sum_{\overrightarrow{k_2}} \frac{e^{-i[\omega_2(t-t_0)-k_2r_0(x)]}}{r_0(x)} \int^{\frac{b}{2}}_{-\frac{b}{2}} dx_0 e^{i(k_{2x_0} -k_x)x_0} \times (-i) \mathcal{E}_{\overrightarrow{k_2}}\hat{a}^\dagger_{\overrightarrow{k_2}} . e^{i \overrightarrow{k_2} r_0}$$</p>
<p>Now we plugin the values in original equations.</p>
<p>$$E^{(+)}(x, z, t) = E^{(+)}_1(x, z, t) + E^{(+)}_2(x, z, t) $$</p>
<p>$$= \sum_{\overrightarrow{k_1}} \frac{e^{-i[\omega_1(t-t_0)-k_1r_0(x)]}}{r_0(x)} \int^{\frac{b}{2}}_{-\frac{b}{2}} dx_0 e^{i(k_{1x_0} -k_x)x_0} \times i \mathcal{E}_{\overrightarrow{k_1}}\hat{a}_{\overrightarrow{k_1}} . e^{-i \overrightarrow{k_1} r_0} + \sum_{\overrightarrow{k_2}} \frac{e^{-i[\omega_2(t-t_0)-k_2r_0(x)]}}{r_0(x)} \int^{\frac{b}{2}}_{-\frac{b}{2}} dx_0 e^{i(k_{2x_0} -k_x)x_0} \times i \mathcal{E}_{\overrightarrow{k_2}}\hat{a}_{\overrightarrow{k_2}} . e^{-i \overrightarrow{k_2} r_0} $$</p>
<p>and</p>
<p>$$E^{(-)}(x, z, t) = E^{(-)}_1(x, z, t) + E^{(-)}_2(x, z, t) $$</p>
<p>$$ = \sum_{\overrightarrow{k_1}} \frac{e^{-i[\omega_1(t-t_0)-k_1r_0(x)]}}{r_0(x)} \int^{\frac{b}{2}}_{-\frac{b}{2}} dx_0 e^{i(k_{1x_0} -k_x)x_0} \times (-i) \mathcal{E}_{\overrightarrow{k_1}}\hat{a}^\dagger_{\overrightarrow{k_1}} . e^{i \overrightarrow{k_1} r_0} + \sum_{\overrightarrow{k_2}} \frac{e^{-i[\omega_2(t-t_0)-k_2r_0(x)]}}{r_0(x)} \int^{\frac{b}{2}}_{-\frac{b}{2}} dx_0 e^{i(k_{2x_0} -k_x)x_0} \times (-i) \mathcal{E}_{\overrightarrow{k_2}}\hat{a}^\dagger_{\overrightarrow{k_2}} . e^{i \overrightarrow{k_2} r_0} $$</p>
<p>So, the correlation fuction for state $|\Psi \rangle = |0\rangle + \sum_j \int d\omega f_j(\omega)\hat{a}^\dagger_j (\omega) |0\rangle $ is:</p>
<p>$$G^{(1)} = $\langle \Psi | E^{(-)}(x, z, t) E^{(+)}(x, z, t) | \Psi \rangle$$</p>
<p>$$ = (\langle0| + \sum_j \int d\omega f_j(\omega)\hat{a}_j (\omega) \langle 0 |)| \left[ \sum_{\overrightarrow{k_1}} \frac{e^{-i[\omega_1(t-t_0)-k_1r_0(x)]}}{r_0(x)} \int^{\frac{b}{2}}_{-\frac{b}{2}} dx_0 e^{i(k_{1x_0} -k_x)x_0} \times (-i) \mathcal{E}_{\overrightarrow{k_1}}\hat{a}^\dagger_{\overrightarrow{k_1}} . e^{i \overrightarrow{k_1} r_0} + \sum_{\overrightarrow{k_2}} \frac{e^{-i[\omega_2(t-t_0)-k_2r_0(x)]}}{r_0(x)} \int^{\frac{b}{2}}_{-\frac{b}{2}} dx_0 e^{i(k_{2x_0} -k_x)x_0} \times (-i) \mathcal{E}_{\overrightarrow{k_2}}\hat{a}^\dagger_{\overrightarrow{k_2}} . e^{i \overrightarrow{k_2} r_0} \right] \left[\sum_{\overrightarrow{k_1}} \frac{e^{-i[\omega_1(t-t_0)-k_1r_0(x)]}}{r_0(x)} \int^{\frac{b}{2}}_{-\frac{b}{2}} dx_0 e^{i(k_{1x_0} -k_x)x_0} \times i \mathcal{E}_{\overrightarrow{k_1}}\hat{a}_{\overrightarrow{k_1}} . e^{-i \overrightarrow{k_1} r_0} + \sum_{\overrightarrow{k_2}} \frac{e^{-i[\omega_2(t-t_0)-k_2r_0(x)]}}{r_0(x)} \int^{\frac{b}{2}}_{-\frac{b}{2}} dx_0 e^{i(k_{2x_0} -k_x)x_0} \times i \mathcal{E}_{\overrightarrow{k_2}}\hat{a}_{\overrightarrow{k_2}} . e^{-i \overrightarrow{k_2} r_0}\right] | (|0\rangle + \sum_j \int d\omega f_j(\omega)\hat{a}^\dagger_j (\omega) |0\rangle) $$</p>
<p>At this point I am confused how to solved this scary looking expression. </p> | 6,450 | [
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<p>Is it theoretically possible for humans to manipulate Atoms to the extent that we can recreate anything we want?</p>
<p>e.g.
'Computer, Milky Way Bar please [or other].' (out pops a Milky Way Bar after some Atom Manipulation). <br>
'thanks computer'.<br>
'no problem'.<br>
'this one's melted'<br>
'sorry i'll do you another one'<br></p> | 6,451 | [
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<p>A problem I am trying to work out is as follows: </p>
<p>A particle moves in a force field given by
$\vec F =\phi(r) \vec r$. Prove that the angular momentum of the particle about the origin is constant. </p>
<p>I set it up as follows: </p>
<p>$\vec F = m {d^2\vec r \over dt^2}$</p>
<p>$\vec v = \int {\frac {\vec F}{m} }\ dt = \int {\frac {\phi(r) \vec r}{m} }\ dt$ </p>
<p>which is equal to : </p>
<p>${\frac {\phi(r) t \vec r}{m} } + c$ </p>
<p>(I am not sure what I am doing at this point. Is my integrated expression correct?)</p>
<p>Assuming it is, we get:</p>
<p>Angular Momentum $L = m (\vec r \times \vec v) = \vec r \times (\phi(r) t \vec r + c)$</p>
<p>Now I don't know what to do with the constant term, but I do know that </p>
<p>$\vec r \times k\vec r = 0$</p>
<p>However, the problem states that we have to prove the result is a constant, so I think I'm wrong. Specific places where someone could help me out are: </p>
<p>(1) Is my integration correct? If not, how does one integrate a force (given in terms of position vector notation) w.r.t. time? </p>
<p>(2) What happens to the constant? Cross-product of a vector and a scalar doesn't make any sense. </p> | 6,452 | [
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<p>i have seen on internet the following, for $ E >> 1 $ the Eigenvalue Staircase can be approximated by $ N(E)= \frac{1}{\pi}argZ(1/2+i \sqrt E ) $</p>
<p><a href="http://books.google.es/books?id=hLEJkA34GMQC&pg=PA35&lpg=PA35&dq=quantum+chaos+bolte%27s+formula&source=bl&ots=jqMMu5KK-L&sig=L9ZlQ_Wq7XQLDvEHyf1nvIc1TdE&hl=es&sa=X&ei=oqHMT4zdEoO_8APlk4UM&ved=0CGUQ6AEwBzgK#v=onepage&q=quantum%20chaos%20bolte%27s%20formula&f=false" rel="nofollow">http://books.google.es/books?id=hLEJkA34GMQC&pg=PA35&lpg=PA35&dq=quantum+chaos+bolte%27s+formula&source=bl&ots=jqMMu5KK-L&sig=L9ZlQ_Wq7XQLDvEHyf1nvIc1TdE&hl=es&sa=X&ei=oqHMT4zdEoO_8APlk4UM&ved=0CGUQ6AEwBzgK#v=onepage&q=quantum%20chaos%20bolte%27s%20formula&f=false</a></p>
<p>PAGE 34 section 1.28 (of course you may use $ \lambda =1$ )</p>
<p>of course the situation is very familiar if we put $ Z(s)= \zeta (s) $ Riemann zeta, but how could i prove Bolte's Law ??</p>
<p>also from the reconstruction of the potential could we deduce that $ V^{-1} (x)=A \frac{d^{1/2}}{dx^{1/2}}argZ(1/2+i \sqrt E) $</p>
<p>where could i find some more info and a proof of Bolte's Law ?? thanks.</p> | 6,453 | [
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<p><img src="http://i.stack.imgur.com/5QlUP.png" alt="enter image description here"></p>
<p>I have to <strong>write the matrix element</strong> $i\mathcal{M}$ given the Feynman rules of QED for the $\gamma \gamma$ scattering as the example above. Suppose that time is the vertical. There are two times, $t, t'$ such that $t'>t$ with two nodes. I now more or less to write matrix elements with one node at each time, but not with more than one. How should I proceed?</p>
<p>Moreover, there is charge conservation at $t$ and $t'$?</p> | 6,454 | [
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<p>Consider the following Lagrangian for a massive vector field $A_{\mu}$ in Euclidean space time: $$\mathcal L = \frac{1}{4} F^{\alpha \beta}F_{\alpha \beta} + \frac{1}{2}m^2 A^{\alpha}A_{\alpha}$$ where $F_{\alpha \beta} = \partial_{\alpha}A_{\beta} - \partial_{\beta}A_{\alpha}$ which means $$\mathcal L = \frac{1}{4} (\partial^{\alpha}A^{\beta} - \partial^{\beta}A^{\alpha})(\partial_{\alpha}A_{\beta} - \partial_{\beta}A_{\alpha}) + \frac{1}{2}m^2A^{\alpha}_{\alpha} \tag{1}$$ The canonical energy-momentum tensor is supposed to be, using the relation </p>
<p>$$T^{\mu \nu}_c = -\eta^{\mu \nu} \mathcal L + \frac{\partial \mathcal L}{\partial (\partial_{\mu}\Phi)}\partial^{\nu}\Phi,\,\,\tag{2}$$ </p>
<p>$$T^{\mu \nu}_c = F^{\mu \alpha}\partial^{\nu}A_{\alpha} - \eta^{\mu \nu}\mathcal L$$ </p>
<p>Then from $T^{\mu \nu}_B = T^{\mu \nu}_c + \partial_{\rho}B^{\rho \mu \nu}$, it is found that $$B^{\alpha \mu \nu} = F^{\alpha \mu}A^{\nu}\tag{3}$$ using the formula $$B^{\mu \rho \nu} = \frac{1}{2}i \left\{\frac{\partial \mathcal L}{\partial (\partial_{\mu}A_{\gamma})} S^{\nu \rho}A_{\gamma} + \frac{\partial \mathcal L}{\partial (\partial_{\rho}A_{\gamma})} S^{\mu \nu}A_{\gamma} + \frac{\partial \mathcal L}{\partial (\partial_{\nu}A_{\gamma})} S^{\mu \rho}A_{\gamma}\right\}$$ My question is how is this equation obtained and how did they obtain $(3)$? Did they make use of the explicit form of the spin matrix for a vector field? (My question is from Di Francesco et al 'Conformal Field Theory' P.46-47).</p>
<p>Here is my attempt:
$$B^{\alpha \mu \nu} = \frac{i}{2}\left\{F^{\alpha \gamma}S^{\nu \mu}A_{\gamma} + F^{\mu \gamma}S^{\alpha \nu}A_{\gamma} + F^{\nu \gamma}S^{\alpha \mu}A_{\gamma}\right\}$$ from simplifying the above. Concentrate on the first term. Then $$F^{\alpha \gamma}S^{\nu \mu}A_{\gamma} = F^{\alpha \gamma}\eta_{\gamma c}\eta^{\nu a}\eta^{\mu b} (S_{ab})^c_dA^d$$ Inputting the form of $S$ for a vector field, I get $$F^{\alpha \gamma}\eta_{\gamma c}\eta^{\nu a}\eta^{\mu b}(\delta^c_a \eta_{bd} - \delta^c_b \eta_{da})A^d$$ But simplifying this and writing the other terms does not yield the result. Did I make a mistake upon insertion of the spin matrix?</p> | 6,455 | [
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<p>In practical transformers there is always a leakage reactance. If it is possible to construct such a transformer (ideally thinking) having zero leakage reactance will the transformer action work? or not.</p>
<p>Edited: Probably my approach was wrong for that question I actually wants to ask. Now I'm trying to say it clearly. If it is possible to construct the transformer in such a way so that no flux is coming out from the iron core. Means no flux is cutting either primary or secondary. Will the transformer action work? </p> | 6,456 | [
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<p>Consider a disk with a radius $R$ (I'll use $R=1$ at various points here) that has a constant surface charge density $\sigma$. Unlike the similar problem of the <a href="http://physics.stackexchange.com/questions/9830/tension-in-a-curved-charged-wire-electrostatic-force-does-wire-thickness-mat">field in the vicinity of a infinitely thin ring</a>, the field directly above the disk is very well behaved. Like the case of an infinite sheet of constant surface charge, the field limits to a constant as you get closer to the surface, and a point on the surface has a defined electrical potential.</p>
<p><a href="http://www.sciencedirect.com/science/article/pii/S0893965911002564" rel="nofollow">This recent paper</a> gives equations for the specific case of points on the plane of the disk. Since we have rotational symmetry, there is only one variable, $\rho$, the distance from the axis of rotational symmetry. Graph from the paper:</p>
<p><img src="http://i.stack.imgur.com/PvFoJ.jpg" alt="Potential"></p>
<p>As you can imagine, the point of the highest slope is the edge of the disk. Is the slope (field) at this point infinite? From what I can make out of their equations, signs seem to point to "yes". The Elliptic functions are tricky, however, and I don't trust my conclusions. Up to the point $\rho=R$, the form of the above function is:</p>
<p>$$ V(\rho) = V(0) \frac{2}{\pi} E(\rho)$$</p>
<p>Here, $E(\rho)$ is an elliptic function, and is different from the paper, because it uses the "Matlab" convention while I use the "<a href="http://en.wikipedia.org/wiki/Elliptic_integral#Notational_variants" rel="nofollow">Wikipedia</a>" convention. The slope of E() doesn't <em><a href="http://reference.wolfram.com/mathematica/ref/EllipticE.html" rel="nofollow">look</a></em> convincingly upright, but the derivative seems to indicate it should be. I'm also troubled by the fact that I can't seem to numerically observe a divergence at $\rho=0.9999999999$.</p>
<p>The physics at play are even more perplexing to me. I expected that a 2D surface, any 2D surface, would have a finite field on its face. Not so for a line, I understand that. But the ring that defines the edge of the disk only has a differential amount of charge on it.</p>
<p>Consider another thought experiment: start at $\rho=1$ and $z=0$, I'll denote it $(1,0)$. Now move upward, but keep $\rho$ the same. If you're at $(1,0.001)$ you should have a finite field. But what happens as $z\rightarrow 0$? Does it just jump from a defined finite value to infinity? What on Earth is going on with the electric field in the vicinity of this edge, or ANY edge of a 2D surface for that matter?</p> | 6,457 | [
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<p>According to neural sciences the brain process information and reacts in 1 teen of a seconds the numbers add up to be 10.596674 loest is 30 to 20 years the course of time in reality recorded by our time keeping devices. but whant to know if perhaps i estimated wrong or something or if it really is what it sound like and what i mean by this that like stars from far out galaxies we to are seeing what happen years. Back in the past </p> | 6,458 | [
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<p>RHIC experiment results of forming Quark-Gluon Plasma and a possible bubble was news few months ago. I cannot find any follow up news after that. Does anybody know anything new in this front? I am especially interested to hear about the bubble formed.</p>
<p>Edit: The following are some links about this story.</p>
<p><a href="http://www.sciencedaily.com/releases/2010/02/100215101018.htm" rel="nofollow">http://www.sciencedaily.com/releases/2010/02/100215101018.htm</a></p>
<p><a href="http://www.popsci.com/science/article/2010-02/rhic-collider-creates-72-trillion-degrees-fahrenheit-quark-gluon-plasma" rel="nofollow">http://www.popsci.com/science/article/2010-02/rhic-collider-creates-72-trillion-degrees-fahrenheit-quark-gluon-plasma</a></p>
<p>Did a bubble actually formed? What have they found about the bubble? I want more scientific details than one would expect from a newspaper article.</p> | 6,459 | [
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<p>Can you give details of a recent experiment of deflection of light by the Sun?</p>
<p>What is the distance from the surface of the Sun and what is the exact value of the angle of deflection?</p> | 6,460 | [
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<p>The book Nielsen & Chuang "Quantum Computation and Quantum Information" presents the concept of tensor products as follows. </p>
<p>Suppose we have the vectors $|v\rangle$ and $|w\rangle$ which exist in vector spaces $V$ and $W$ respectively. We also define the linear operators $A$ and $B$ which exists in same respective vector spaces. Then we can define the tensor product of these vectors and operators which behaves as follows</p>
<p>$$\left(A\otimes B\right)\left(|v\rangle\otimes|w\rangle\right) = A|v\rangle\otimes B|w\rangle \tag{1}$$</p>
<p>I can accept this as definition, but my question arises from an exercise where it asks to evaluate $$\langle\psi\,|E\otimes I|\,\psi\rangle\tag{2}$$ where $E$ is a positive operator and $|\psi\rangle$ is any of the four <em>Bell states</em>.
However the book does not describe the behaviour of the expression
$$(A\otimes B)(|v\rangle)\tag{3}$$ My assumption is that this is not a valid expression since $|v\rangle$ does not exist in the vector space $V\otimes W$ on which the operator is defined. Am I correct in thinking this? How does one expand the inner product in equation (2)? </p> | 6,461 | [
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<p>I've been reading several papers on Adiabatic Quantum Computation, and I am confused by the form in which the adiabatic approximation condition is presented. For example (<a href="http://arxiv.org/abs/quant-ph/0001106" rel="nofollow">quant-ph/0001106</a> eq 2.8) \begin{align*}
\frac{\mbox{max}_{0 \leq s \leq 1} |\langle 1;s| \frac{d\tilde{H}}{ds}|0,s\rangle|}{\left(\mbox{min}_{0 \leq s \leq 1} (E_1(s) - E_0(s))\right)^2} <<T
\end{align*}
where $T$ is the time-scale of the evolution, $s = t/T$, and $H(t) = \tilde{H}(t/T)$.</p>
<p>If you read <a href="http://en.wikipedia.org/wiki/Adiabatic_theorem#Proof_of_the_Adiabatic_theorem" rel="nofollow">Wikipedia</a> or Sakurai (2nd edition Pg 347), we find that we must have:
\begin{align*}
\bigg|\frac{\langle m;t | \frac{dH}{dt}|n;t\rangle}{E_n - E_m}\bigg| <<1
\end{align*}
and using $\langle\frac{dH}{dt}\rangle_{m,n} = \langle\frac{d\tilde{H}}{ds}\rangle_{m,n}\frac{ds}{dt} = \frac{1}{T}\langle\frac{d\tilde{H}}{ds}\rangle_{m,n}$,
\begin{align*}
\frac{|\langle m;s | \frac{d\tilde{H}}{ds}|n;s\rangle|}{E_n - E_m} <<T
\end{align*}
and thus, if we wish the entire process to be adiabatic, then:
\begin{align*}
\frac{\mbox{max}_{0 \leq s \leq 1}|\langle m;s | \frac{d\tilde{H}}{ds}|n;s\rangle|}{\mbox{min}_{0 \leq s \leq 1}(E_n - E_m)} <<T
\end{align*}
So my main conern: Why in this paper (and others) do we have the quadratic $(E_n-E_m)^2$ in the denominator? This can't be a typo because it turns out to be very important in determining run-times.</p> | 6,462 | [
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<p>I am reading paper <a href="http://jasss.soc.surrey.ac.uk/12/1/6/appendixB/Axelrod1997.html" rel="nofollow">Axelrod's model of dissemination of culture </a>, I am unable to understand the transition probabilities of time homogeneous Markov chain for this model. Can some one please explain it clearly how the expression </p>
<p>$$ p_{ij} = \frac{1}{L^2 h_{ij}} \sum_{r\in H_{ij}} \frac{n_{kr}}{f} \frac{1}{f-n_{kr}} \qquad (i \neq j)$$</p>
<p>is written in the paper.</p> | 6,463 | [
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<p>How would one most appropriately model the magnetic field around each bunch of protons in the LHC at maximum operating speed? Is the assumption of a wire of the same length of the beams (~30cm) with the beam current of 0.58A sufficient?</p> | 6,464 | [
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<p>In this video: <a href="https://www.youtube.com/watch?v=0jHsq36_NTU#t=55" rel="nofollow">https://www.youtube.com/watch?v=0jHsq36_NTU#t=55</a> at 0:55, it is claimed that "the sun is [...] dragging the planets in its wake".</p>
<p>Is this true? My understanding is that the sun and the planets are all moving together through the solar system at the same velocity, and there is no friction involved, so there's no dragging or indeed motor force required. </p>
<p>What's really going on? </p> | 6,465 | [
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<p>I'm asking this to get yet another lessson in the inability of QFT and GR to cohabit. Many people believe GR must yield to quantization. The question here is as to why the activity of the vacuum cannot be imagined as having some sort of baseline and the gravitational force having some origin in this baseline. And, of course then, that there could exist variations from this baseline on a more fundamental level than the wavelike disturbances of QFT. The variations I intend may have to do with types of virtual particle-antiparticle pair eruptions favored at a particular location, tendency toward shorter or longer lifespans of such events, etc.<br>
Is it just that it is impossible to say anything meaningful about gradings in the living vacuum, and that such hypothetical gradings, however dynamically conjured, could not ever come to minic aspects of a GR like picture?...due to subtle stuff like violation of basic physical principles and such. </p> | 6,466 | [
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<p>I'd like to build a simple desk; just a single plank of wood (or a few side-by-side) with solid supports on each end of the desk. What I'm trying to figure out is how thick a plank I want to use for the surface. Ideally, the desk will be able to support my weight (since sometimes I stand on my desk to change lights, etc), plus the weight of my computer stuff, plus some amount of safety margin.</p>
<p>So, supposing I have a desk surface that's eight feet wide, what equation do I need to use to figure out how thick it needs to be to support, e.g., 300lbs, placed in the middle? What properties of the wood type that I'm using do I need to determine? Is this an issue of tensile strength, or something entirely different?</p>
<p>This is obviously more a structural engineering problem than a physics one, but I didn't see a structuralengineering.stackexchange.com. Sorry if I posted to the wrong place.</p> | 6,467 | [
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<p>Is there some material state that can propagate light indefinitely without dissipation or absorption, like superconductors are able to transmit current indefinitely?</p>
<p>If not, then the question is, why not? would some fundamental principle being violated in such a material?</p> | 6,468 | [
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<p>I would like to know why $\mu^+$ muons can easily penetrate a solid metal such as Fe with negligible interactions while $\pi^+$ mesons lose their energy a lot faster when traveling through the metal. Is it because the interaction between electrons and muons is forbidden or should I somehow see it from the crossection?</p> | 6,469 | [
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<p>Is it possible to calculate the shortest time it takes a body to travel a certain distance if the only information provided is its maximum acceleration, its maximum retardation and the distance it travels (from rest to rest)? I've been trying to work it out but I was wondering if it is necessary to know either its maximum velocity or the distance at which the body switches from accelerating to decelerating.</p>
<p><strong>Example:</strong><br>
The maximum acceleration of a body is $9ms^{-2}$ and its maximum retardation is $3ms^{-2}$. <br>What is the shortest time in which the body can travel a distance of 7200 m from rest to rest?</p>
<p>Answer: $80m$ </p>
<p>I have no idea how this answer was reached, I've made velocity-time graphs and tried simulations equations etc. but I can't seem to reach this answer, best I could do was, if the object kept accelerating at $9ms^{-2}$ it would travel 7200 m in 40s. $7200=0t+\frac129t^2$.</p>
<p>This is my first question so I hope I asked it correctly, any help would be greatly appreciated as it is hurting my brain :/ </p> | 6,470 | [
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