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<p>In the compression stroke of a petrol engine, the mixture is compressed by the upward movement of the piston. From where does the piston get energy to compress the mixture?</p>
<p>Similarly, in the exhaust stroke, the piston again moves upward to expel the gases. Where does the piston get energy from?</p> | 2,628 | [
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<p>Say you have an electron departing from point A and reaching poing B after a time t.</p>
<p>According to some helping friend, the <em>Partition Function</em> for that electron going from point A to B can be written as</p>
<p>$$Z = \int_{A \to B} [\mathcal{D}x]~ e^{iS[x]}$$</p>
<p>where $\mathcal{D}x$ is the measure that sums up over all paths going from $A$ to $B$, and $e^{iS[x]}$ is the weight of each path, $S[x]$ is the action.</p>
<p>That friend states then "<em>From this partition function all desirable quantities can be obtained</em>."</p>
<p>Not having much idea about Feynman's Path Integral Formulation of Quantum Mechanics, I have looked around a bit, and I would like someone to confirm the following statement I make:</p>
<p>The amplitude for the electron to go from A to B in the time t can be found in terms of that Partition Function $Z$ given above, as</p>
<p>$$\langle B|e^{-iHt}|A\rangle = Z$$</p>
<p>Did I catch it right?</p> | 2,629 | [
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<p>I was reading <a href="http://www.weizmann.ac.il/home/kblum/IPC/tutorial%20notes/TutKfirSZ.pdf" rel="nofollow">this</a> writeup on the Kompaneets equation and the Sunyaev-Zel'dovich effect. </p>
<p>On page 3, section 2 the author states </p>
<blockquote>
<p>There is no way to increase the mean energy of a planckian distribution without changing the particle number.</p>
</blockquote>
<p>But as far as I understand, photon number is conserved during a scattering process, and therefore isn't it possible that for a given Planckian distribution all the photons gain the same energy through scattering and thus increase the mean energy of the Planckian distribution? Or am I missing something here?</p> | 2,630 | [
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<p>We start our question we a definition </p>
<p>A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
\</p>
<ol>
<li><p>$P$ is Lagrangian</p></li>
<li><p>P involutive</p></li>
<li><p>dim$P\cap\bar P \cap TM$ is constant
Now, Introduce an hermitian form on $P$ defined by </p></li>
</ol>
<p>$$b(X,Y)=i\omega(X,\bar Y)$$</p>
<p>. Note that when $P$ is real then $b$ is vanishes identically on $P$. Cinsequently, $b$ projects onto a non-degenerate form on the quotient $P/{(P\cap \bar P)}$ and we denote it by $\bar b$. $P$ is said to be of type $(r,s)$ if and only if $\bar b$ has signature $(r,s)$ i.e. its matrix is $$diag(\underbrace{1,1...,1}_{r},\underbrace{-1,-1...,-1}_{s} )$$ for $0\le r+s=n-dim_{\mathbf C }P\cap \bar P$
. Then, $P$ is said to be positive if $s=0$. In the case if $r=s=0$ then $P$ is real</p>
<p>Philosophically the main goal in defining polarizations in quantum mechanics is to find the wave functions that are covariantly constant along its directions. When the form $b$ is not positive show that there are no globally defined wave functions</p> | 2,631 | [
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<p>It seems that Planck's constant was made from fitting a curve for blackbody radiation, is it just experimental-further more his assumption that energy comes in quanta seems to have been a guess. Why would energy come in quanta? <a href="http://en.wikipedia.org/wiki/Planck%27s_law">Wikipedia</a> says that he didn't think about it much, but I don't know why harmonic oscillators would even be suggested to only have a quantized number of energy modes, moreover what an energy mode would be. With a set frequency, how does one vary the energy?
<em>The main question I have, is what the reasoning behind the quantization of oscillators?</em></p> | 2,632 | [
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<p>This question came up when I was talking about the atmosphere. Someone had mistakenly claimed that as temperature increases, the density of the atmosphere should increase as well. I reasoned from the ideal gas law that, as temperature ($T$) increases, then so should $V$, leading to lower density. But I realized that pressure could increase instead; the atmosphere is not in a sturdy laboratory container where volume is a constant. Under these circumstances, how can one predict whether pressure or volume (or both) will increase?</p> | 2,633 | [
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<p>What would the size of the universe be if it were physically possible to remove all of the empty space, leaving only matter?</p> | 2,634 | [
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<p>The heliopause is now estimated to be something around 100 AU (1 AU = Astronomical unit = about the earth sun distance). See the wikipedia article:<br>
<a href="http://en.wikipedia.org/wiki/Heliosphere">http://en.wikipedia.org/wiki/Heliosphere</a></p>
<p>From reading a book on NASA's Voyager mission, I learned that before the launch of these spacecraft, the expectation was that the heliopause was at around Jupiter or Saturn's orbit which is about 5 to 10 AU. The book says that as the spacecraft continued to move away from the sun, the space physicists kept increasing their estimate of the position of the heliopause. If it's about an energy or flux balance, then the estimate would be defined by a certain area of the sphere where the heliopause occurs, and since areas are proportional to the square of the radius they got the number wrong by as much as $(100/5)^2 = 400$. That's a huge underestimate of the sun's output or a huge overestimate of what goes on in interstellar space.</p>
<p>The book doesn't explain why it is that early estimates were wrong and I didn't see an explanation. Perhaps someone knows and will give a nice intuitive explanation for the estimates.</p> | 2,635 | [
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<p>How little mass can a black hole contain and still be a "stable" black hole? What would the diameter be, in terms of the event horizon?</p> | 624 | [
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<p>This question arose in a seminar today about the solar wind...</p>
<p>This is my vagueish understanding of the problem - please correct if you see errors!</p>
<p>The 'classical' picture of atmospheric electricity is that the Earth as a whole is neutral, but that thunderstorms maintain a voltage of around +300kV at the electrosphere with respect to the Earth's surface, with a current of around 1 kA slowly discharging around 500 kC of total charge separation. The solar wind is supposed to neutralise any net charge that might be there between the Earth as a whole and the solar wind.</p>
<p>However, positive and negative charges in the solar wind are differently trapped in the van Allen belts, from which they can then descend to the Earth's atmosphere, which implies that a net charge can be developed due to this differential leakage. This begs the question of whether there are any estimates of the total net charge. I've hunted in the literature but have found little useful material other than Dolezalek's 1988 paper: <a href="http://www.springerlink.com/content/u057683112l148x5/">http://www.springerlink.com/content/u057683112l148x5/</a></p>
<p>Can anyone offer an explanation, or point me to some more relevant papers?</p> | 2,636 | [
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<p>In classical (Newtonian) mechanics, every observer had the same past and the same future and if you had perfect knowledge about the current state of all particles in the universe, you could (theoretically) compute the future state of all particles in the universe.</p>
<p>With special (and general) relativity, we have the relativity of simultaneity. Therefore the best we can do is to say that for an event happening right now for any particular observer, we can theoretically predict the event if we know everything about the past light cone of the observer. However, it tachyons (that always travel faster than the speed of light) are allowed, then we cannot predict the future since a tachyon can come in from the space-like region for the observer and can cause an event that cannot be predicted by the past light cone. That is, I believe, why tachyons are incompatible with causality in relativity. Basically, the future cannot be predicted for any given observer so the universe is in general unpredictable - i.e. physics is impossible.</p>
<p>Now in quantum mechanics, perfect predictability is impossible in principle. Instead all we can predict is the probability of events happening. However, Schrodinger's equation allows the future wavefunction to be calculated given the current wavefunction. However, the wavefunction only allows for the predictions of probabilities of events happening. Quantum mechanics claims that this is the calculations of probabilities is the best that can be done by any physical theory.</p>
<p>So the question is: "Is the predictability of the future to whatever extent is possible (based on the present and the past) equivalent to the principle of causality?" Since prediction is the goal of physics and science in general, causality is necessary for physics and science to be possible.</p>
<p>I am really not asking for a philosophical discussion, I want to know if there are any practical results of the principle of causality other than this predictability of the future of the universe. Please don't immediately close this as being a subjective question, let's see if anyone can come up with additional implications for causality besides future predictability.</p> | 2,637 | [
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<p>I've watched this video on YouTube by Sixty Symbols entitled "<a href="http://www.youtube.com/watch?v=UV0wtX9AXq4" rel="nofollow">Currents and Magnets</a>". In the video, the professor demonstrates the expansion of a wire due to current heating it up and he also demonstrates how the current interacts with a magnet. I want to ask about the magnetic interaction.</p>
<p>At about 2:28 he shows how the orientation of the magnet relative to the wire/current determines whether the wire is repelled or attracted. It appears that, mathematically, whether the wire is pulled or pushed is determined by the cross product between the vector of the flow of the current, $\vec{c}$, and the vector of the magnetic field, $\vec{m}$.</p>
<p>However, the cross product is not commutative (order of the operands is important). What determines whether I should cross $\vec{c}$ with $\vec{m}$ or whether I cross them the other way around?</p> | 2,638 | [
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<p>I have come across two similar definitions of primary fields in conformal field theory. Depending on what I am doing each definition has its own usefulness. I expect both definitions to be compatible but I can't seem to be able to show it. By compatible I mean definition 1 $\iff$ definition 2. I will write both definitions in the two-dimensional case and restricting to holomorphic transformations.</p>
<p>Def #1 from Francesco CFT: A field $f(z)$ is primary if it transforms as $f(z) \rightarrow g(\omega)=\left( \frac{d\omega}{dz}\right)^{-h}f(z)$ under an infinitesimal conformal transformation $z \rightarrow \omega(z)$.</p>
<p>Def #2 from Blumenhagen Intro to CFT: A field $f(z)$ is primary if it transforms as $f(z) \rightarrow g(z)=\left( \frac{d\omega}{dz}\right)^{h}f(\omega)$ under an infinitesimal conformal transformation $z \rightarrow \omega(z)$.</p>
<p>Can someone show me how they are indeed the same?</p> | 2,639 | [
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<p>Could this device theoretically continue in motion forever? If not, why not? (click below for images):</p>
<p><a href="https://docs.google.com/open?id=0B1BhNZS4yw5pVGFqNWFvbk14UG8" rel="nofollow">Device description.</a></p>
<p><a href="https://docs.google.com/open?id=0B1BhNZS4yw5pYlQ1RWxqMjZyRDQ" rel="nofollow">Device process.</a></p>
<ol>
<li>The device is less dense than air, so it rises. The propeller spins slightly (maybe) charging the device battery.</li>
<li>After rising some distance X, the device compressor turns on to deflate the device.</li>
<li>The device becomes more dense than air and falls quickly spinning the propeller, charging the battery.</li>
<li>After falling X, the compressor releases and the device becomes less dense than air, going back to step 1.</li>
</ol>
<p>The logic here is that there must be some distance X that the device can rise than will generate more energy than what is needed by the compressor.</p>
<p>Here is the underlying math to help:</p>
<p>$$PE = mgh$$</p>
<ul>
<li>$m$ = mass of the device</li>
<li>$g$ = coefficient of gravity</li>
<li>$h$ = height device has traveled up </li>
<li>$PE$ = potential energy of the device as it travels up</li>
</ul>
<p>$$CW = nRT(\ln(V_b) - \ln(V_a))$$</p>
<ul>
<li>$n$ = number of moles of gas in balloon of device</li>
<li>$R$ = ideal gas constant</li>
<li>$T$ = temperature of the gas</li>
<li>$V_b$ = volume of the balloon after compression</li>
<li>$V_a$ = volume of the balloon before compression</li>
<li>$CW$ = work to compress the balloon</li>
</ul>
<p>As $h$ increases $PE$ increases but $CW$ stays the same resulting in energy gain of the system.</p> | 2,640 | [
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<p>When I searched on the Internet for the reason of formation of <a href="http://en.wikipedia.org/wiki/Rainbow">rainbows</a>, I got many explanations like <a href="http://www.wisegeek.com/why-do-rainbows-form.htm">this</a> one & <a href="http://www.scientificamerican.com/article.cfm?id=why-do-rainbows-form-inst">this</a>. All the explanations consider only one spherical water droplet (like <a href="http://www.rebeccapaton.net/rainbows/formatn.htm">this one</a>).</p>
<p>Why don't the refracted waves coming from the adjacent droplets interfere with each other? For example, why doesn't a red ray coincide with a violet ray coming from another droplet at a suitable position? </p>
<p>How are we able to see rainbows with distinct colors? Since there are thousands of raindrops in the sky when it is raining, shouldn't it be a mess due to overlapping of different wavelengths?</p>
<p>Please explain.</p> | 2,641 | [
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<p>If the range of energies of cosmic rays is not so far away respect to gamma's, why those are not found commonly in a nuclear reaction?</p> | 2,642 | [
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<p>It has been argued the exponential size of the wavefunction can be interpreted as many parallel worlds, and this explains how quantum computers can factor large integers and compute discrete logarithms (although on closer examination, Shor's algorithm doesn't work at all by trying out a superposition of all candidate solutions. It works by using number theory to find a <em>different</em> function with periodicity, and period finding.). </p>
<p>However, unless we can examine specific individual branches of our choice, why should we argue for the ontological reality of the branches? Measuring which branch we are in "collapses", i.e. picks out a branch sampled at random, and not a branch of our choice.</p>
<p>To be more specific, let's say we wish to invert a one-way function f with no periodicity properties, i.e. solve $f(x)=y$ for a specific y. We can easily prepare the state $C\sum_x |x\rangle\otimes |f(x)\rangle$ where C is an overall normalization factor. However, unless we can also postselect for the second register being y, how can we interpret this as many parallel worlds?</p>
<p>Doesn't this inability argue for a more "collectivist" interpretation where the individual branches don't have "individual" existence, but only the collective relative phases of all the branches taken together have any real existence?</p>
<p>To take a less quantum computing example, consider Schroedinger's cat. Suppose we have N cats in N boxes, and we perform independent cat experiments on each of them. Let's suppose I pass you a prespecified N bit string in advance, and I ask you to "subjectively" take me to the branch where the life/death status of the cat in the i<sup>th</sup> box matches the value of the i<sup>th</sup> bit. That, you can't do unless you perform this experiment over and over again for an order of $2^N$ times.</p>
<p><strong>PS</strong> Actually, there might be a way using quantum suicide. Unless the life/death status of each cat matches that of the string, kill me. However, this rests on the dubious assumption that I will still find myself alive after this experiment, which rests upon the dubious assumption of the continuity of consciousness over time, and that it can't ever end subjectively.</p> | 2,643 | [
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<p>Given a classical field theory can it be always quantized? Put in another way, Does there necessarily need to exist a particle excitation given a generic classical field theory? By generic I mean all the field theory variants, specially Higher Derivative QFT(particularly Lee-Wick field theories).<br>
I ask this question because, several times in QFT we come across non physical particles when we try to quantize some field theory for e.g. ghost fields and ghost particles. These fields have opposite sign in front of the kinetic energy term. Such terms are common in higher derivative field theories. Hence we have to ask should we talk about particles in such situations.<br>
Now if the answer is No! <strong>Then we have to ask what is more fundamental in nature, particles or fields?</strong><br>
<em>Schwinger disliked Feynman diagrams because he felt that they made the student focus on the particles and forget about local fields, which in his view inhibited understanding.</em> -Source Wikipedia</p> | 2,644 | [
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<p>Does quantum fingerprinting really argue for the exponential size of wavefunctions? Quantum fingerprinting is the idea that an exponentially long classical string can be encoded in a linear number of entangled qubits using quantum fingerprints. To an exponential degree of accuracy, but not exactly, the fingerprints of two different exponentially long classical strings will be nearly orthogonal.</p>
<p>However, the whole idea of quantum fingerprinting rests upon the ability to tell if two different pure quantum states are identical or orthogonal. Can such a comparator exist? Let's work with comparing qubits first. Our hypothetical comparator has the property that if two pure qubits are identical, it always outputs YES. If they are orthogonal, it always outputs NO. For other cases, it may output either. Then, it would definitely output NO for $|0\rangle|1\rangle$ and $| 1\rangle|0\rangle$. By the superposition principle, it would also have to definitely output NO for the linear superposition ${1\over\sqrt{2}} \left[ |0\rangle|1\rangle +|1\rangle|0\rangle\right]$. It would also have to output YES for both ${1\over 2}\left(|0\rangle + |1\rangle\right)\left(|0\rangle + |1\rangle\right)$ and ${1\over 2}\left(|0\rangle - |1\rangle\right)\left(|0\rangle - |1\rangle\right)$, and hence, also YES for their linear combination ${1\over\sqrt{2}} \left[ |0\rangle|1\rangle +|1\rangle|0\rangle\right]$. Contradiction.</p>
<p>If we can't compare whether or not two quantum fingerprints are identical or nearly orthogonal, how then are they supposed to work? Is their apparent exponentiality "fake"?</p> | 2,645 | [
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<p>How exactly does gravitational compression, or compression in general, increase temperature? It seems counter-intuitive seen as temperature usually increases from the solid to the gas phase. </p> | 2,646 | [
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<p>How many photons in one Planck volume would it take to form a tiny black hole?</p>
<p>A photon doesn't have mass but it does have energy, $1.0101 \times 10^{-37}$ Joule for red $650$ nm wavelength light if I'm correct. A photon is a point-like boson so an infinite number of photons can fit into any given area. </p>
<p>So more photons is equal to more energy which would bend space-time even more till they themselves wouldn't be able to escape from their own bend in space-time.</p> | 2,647 | [
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<p>Given a higher derivative classical/quantum field theory with say one scalar field, particularly the Lee-Wick standard model. It has been shown that such a field theory encompasses two kinds of fields, one normal and other a ghost field, in a particular limit, generally the high mass limit of the ghost field. We started with a single field but in ended up with a two particle field theory. </p>
<p>This makes me ask:<br>
<strong>Is there any general mathematical theorem by which we can show that a $n$th order derivative theory(scalar, spinor or vector) can be quantized into $n$ different kind of particles?</strong> Has it got to do something with the symmtries involved in the theory?<br>
And what is the physical importance of the constraint? And how exactly is the system behaving when that constraint is not satisfied?(It seems to be in some sort of a enmeshed state of the two particles.) </p>
<p>My question mainly relies on the paper in the following link [The Lee-Wick standard model]: <a href="http://arxiv.org/abs/0704.1845" rel="nofollow">http://arxiv.org/abs/0704.1845</a> . In section 2 (A Toy Model) he considers a higher derivative Lagrangian eq.(1) for a scalar field $\phi$. And then introduces an auxiliary field $\tilde{\phi}$. And then it is shown that given the condition $M > 2m$, the Lagrangian sort of gets decoupled into two different fields. One normal field and other a ghost field, both interacting with each other. </p> | 2,648 | [
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<p><em>I am aware that there are plenty of questions regarding book recommendations, however, I have not found one that fully matches what I intend to ask. I have provided a list of links to some similar questions under my question.</em></p>
<hr>
<p>I am 2nd year Undergrad doing physics. So far we have completed three "Mathematics for Scientisits" courses and they are all heavily based on matrices, vectors, orthogonality and linear algebra in general. However, the motivation behind the methods is not explained at all! All sorts of terminology is thrown around and I hit a brick wall when I try to connect it all together.</p>
<p>I can, for the most part, do the maths behind them but I am failing to fully grasp the pure and applied concepts of:</p>
<ul>
<li>Orthogonality</li>
<li>Eigenvalues and Eigenfunctions</li>
<li>Inner product, vector space</li>
<li><em>(Essentially, the range of topics in Linear Algebra)</em></li>
</ul>
<p>Mainly, I am curious as to why they are used so much in <strong>quantum mechanics</strong>. We have not been introduced to Hilbert Space yet (nor Dirac notation, which worries me); I think those things will be studied next year but might as well learn them now.</p>
<p>Having said all that, I am looking for books, both for mathematicians and/or physicists, which explain the <strong>meanings, concepts and uses</strong> of all this abstract mathematics well. It would be an added bonus if they have good practice questions too. <strong>I am looking for books that have <em>thorough, well written explanations.</em></strong></p>
<p>After doing a bit of research I am most intrigued by:</p>
<ul>
<li><em>Linear Algebra: Concepts and Methods</em> by M. Anthony and M. Harvey</li>
<li><em>Linear Algebra and Its Applications</em> by David C. Lay</li>
</ul>
<p>Any comments on those before I buy them?</p>
<hr>
<p>List of links of related questions:</p>
<p><a href="http://physics.stackexchange.com/questions/193/best-books-for-mathematical-background/10684#10684">Best Books for Mathematical Background</a></p>
<p><a href="http://physics.stackexchange.com/questions/39165/linear-algebra-for-quantum-physics">Linear Algebra for Quantum Physics</a></p>
<p><a href="http://physics.stackexchange.com/questions/884/books-that-every-physicist-should-read">Books Every Physicist Should Read</a></p>
<p><a href="http://physics.stackexchange.com/questions/33215/what-is-a-good-introductory-book-on-quantum-mechanics">What is a good introductory book on quantum mechanics?</a></p> | 53 | [
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<p>I am currently a senior in high school. I have spent the past four years participating in physics research at a local lab. Are there any journals in which I as a high schooler can publish my research?</p> | 2,649 | [
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<p>Assuming water volume ($V$), initial water temperature ($T_0$) and environment temperature ($T_e$) are known, what is the easiest way to calculate temperature of water in given time ($T$)?</p>
<p>For the sake of question, let's assume water container is so thin it doesn't matter. Also, water container is airtight (for example, soda can or sparkling wine bottle). Environment temperature is constant and air is not moving remarkably.</p>
<p>I already experimented with sparkling wine bottle (filled with water and temperature logger). Blue line is for fridge, red for freezer and greenish for ice water (without salt). But it's not very practical to repeat same experiments with different and different containers. </p>
<p><img src="http://i.stack.imgur.com/iJ2H6.png" alt="Temperature of sparkling wine bottle"></p>
<p>Why? I thought it would be nice to make a simple mobile application for giving alarms when your drinks are chilled to whatever temperature you want.</p> | 2,650 | [
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<p>How can the integral form of <a href="http://en.wikipedia.org/wiki/Gauss%27s_law" rel="nofollow">Gauss's law</a> for magnetism be described as a version of general <a href="http://en.wikipedia.org/wiki/Stokes%27_theorem" rel="nofollow">Stokes' theorem</a>? How does it follow?</p> | 2,651 | [
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<p>It seems that this question has not really been explored in the literature. Do isolated neutron stars (which do not accrete material) emit stellar wind? If yes, what composition would it have? If yes, what will be the rate of mass loss for the star?</p>
<p>One might also think up that the process of Hawking radiation might possibly be applicable to neutron stars, where negative energy particles get trapped in the nuclear star atmosphere (instead of crossing the horizon in case of black holes), which would also lead to some sort of evaporation and corresponding wind. However, in this question I am more interested in 'classical' winds.</p> | 2,652 | [
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<p>If Alice wants to send one <strong>bit</strong> of classical information she can use a <strong>qbit</strong>. Then Bob needs to know which axis to measure to get the information. This needs an extra agreement between Alice and Bob before starting communication. I can figure out why up or down agreement has no importance here since 0 and 1 are just different and need the same level of agreement (two choices available, one for 0 another for 1).</p>
<p>Using <strong>qbits</strong>, an extra agreement is needed before sending information. A lamp could be on or off without any need for more agreement (which axis in <code>qbit</code> is an extra agreement).</p>
<p>Has this extra contract any importance?</p> | 2,653 | [
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<p>According to some theory, currents in <a href="http://en.wikipedia.org/wiki/Quantum_Hall_effect#Integer_quantum_Hall_effect_.E2.80.93_Landau_levels" rel="nofollow">IQHE</a> are due to the edge states. The edge current on the upper edge flows from left to right, that on the lower edge flows from right to left.</p>
<p>Did anyone detect or try to detect the magnetic field associated with the edge currents?</p>
<p>Is this a way to test the theory? </p> | 2,654 | [
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<p>Angular momentum is defined from linear momentum via $\vec L = \vec r\times\vec p$, and is conserved in a closed system. Since energy is the time part of the linear four-momentum, is there a quantity defined from energy that's also conserved?</p> | 2,655 | [
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<p>The position center of gravity of a bicycle and its rider is known, and the distance from it to the point of contact of the front wheel with the ground, in terms of horizontal and vertical distance (x cm and y cm), are both known, as well as the total weight of the bike and the rider. The brakes are then applied, making the bike turn around the point of contact with the ground of the front wheel, and this force is also known.</p>
<p>Is there a way to calculate the amount of torque generated from this information?</p> | 2,656 | [
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<p>How can you have a negative <a href="http://en.wikipedia.org/wiki/Voltage" rel="nofollow">voltage</a>? I don't really understand the concept of negative voltage, how can it exist? </p> | 2,657 | [
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<p>When trying to solve the Schrödinger equation for hydrogen, one usually splits up the wave function into two parts:</p>
<p>$\psi(r,\phi,\theta)= R(r)Y_{l,m}(\phi,\theta)$</p>
<p>I understand that the radial part usually has a singularity for the 1s state at $r=0$ and this is why you remove it by writing:</p>
<p>$R(r) = \frac{U(r)}{r}$</p>
<p>But what is the physical meaning of $R(r=0) = \infty$. Wouldn't this mean that the electron cloud is only at the centre of the atomic nucleolus?</p>
<p>Thanks in advance!</p> | 2,658 | [
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<ol>
<li><p>What exactly are <a href="http://www.google.com/search?as_epq=crystal+plane" rel="nofollow">crystal planes</a> and how do they reflect x-rays?</p></li>
<li><p>Are crystal planes real physical planes or just an abstract concept? </p></li>
<li><p>What are these planes made of? </p></li>
<li><p>If they are an abstraction, what do the x-rays hit and get reflected by? Individual atoms? </p></li>
<li><p>Then where is the concept of a plane coming in? </p></li>
<li><p>And why do different crystal structures have different 'active' reflecting planes? </p></li>
<li><p>Could someone please clear up the concept of these planes and Bragg's Law for me?</p></li>
</ol> | 2,659 | [
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<p>Suppose that there are negative charges (e.g. electrons) only. There are more negative charges on left than on right. How would electric field be constructed? (So, What would be the direction?) And how would electrons flow (because of electric field)? The text says that it flows from left to right, but I am not getting this. </p> | 2,660 | [
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<p>If I have a Lagrangian of the form:</p>
<p>$$ \mathcal{L} = k \bar{\psi} \varepsilon^{\mu \nu} \lambda^a \phi G^a_{\mu \nu} + h.c. $$</p>
<p>[where $\phi, \psi$ are fermions, $\lambda^a$ are Gellmann matrices, $\varepsilon^{\mu \nu} $ is some antisymmetric tensor and $G^a_{\mu \nu}$ is the gluon field stength tensor.]</p>
<p>And I want to have the Feynman rule for the vertex and it's conjugate I would usually compute the following:</p>
<p>$$ \frac{\delta S}{\delta \psi \delta \phi \delta A_\mu^a} $$
[where $A_\mu$ is a gluon field].</p>
<p>Following that prescription, I have
$$ S = k \int d^4 x \bar{\psi} \varepsilon^{\mu \nu} \lambda^a \phi G^a_{\mu \nu} $$
ignoring color for a moment,
$$ \implies \frac{\delta S}{\delta \psi \delta \phi } = 2k \int d^4 x \gamma^0 \varepsilon^{\mu \nu} \left( \partial_\mu A_\nu + i g_s A_\mu A_\nu \right) $$</p>
<p>using the convention that $G_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + i g_s \left[ A_\mu, A_\nu \right] $</p>
<p>Taking the Fourier transform to do the last variation I get a leftover $A_\mu$ in the expression though, don't I? </p>
<p>Something seems off here.</p>
<p>With a nicer field strength tensor, like $F^{\mu \nu}$ I get a nice expression at the end only in terms of the momentum of the photon field...here I have a leftover term involving an $A_\mu$.</p>
<p>The initial variation shouldn't be with respect to the field strength tensor, should it?</p>
<p>I'm messing up something trivial, could someone point me in the right direction?</p> | 2,661 | [
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<p>I have a doubt about the electric potential of a body. Well, I know that given a continuous distribution of charge we can find the potential at a point $a$ using the following relation:</p>
<p>$$V(x,y,z)=\frac{1}{4\pi\epsilon_0}\iiint_{S} \rho(a') \frac{1}{|a-a'|}dV'$$</p>
<p>However, this gives a function that at each point tells me the electrostatic potential energy divided by charge. The difference of potential between two points can be found then calculating this function at those points and subtracting.</p>
<p>My problem is that I've found exercises that asked something like: "find the difference of potential between two concentric spheres with uniform charge density and radii $a$ and $b$". However, what does it means calculate the difference of potential between two spheres? How do we deal with problems like this? And last but not least, what it means the potential of a sphere? The potential of a charge as the difference of potential between infinity and the place of the charge I understand, but what should be the potential of a whole sphere?</p>
<p>Thanks in advance for the help.</p> | 2,662 | [
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] |
<p>In statistical physics, mean-field theory (MFT) is often introduced by working out the Ising model and it's properties. From a spin model point of view, the mean-field approximation is given by requiring that :</p>
<p>Eq.(1)$\hspace{75pt}$$\langle S_i S_j \rangle = \langle S_i \rangle \langle S_j \rangle $ for $i\neq j$ </p>
<p>Where $S_i$ is the local spin observable supported at site $i$ of a given lattice (in the classical Ising case, it is just $\pm1$). </p>
<p>I divide my questions/comments into two parts :</p>
<p>Part 1 (a): I know that there are more sophisticated ways of formulating much more rigorously mean-field theory in statistical physics, but is the above relation an equivalent definition for the particular case of a spin model ?</p>
<p>Part 1 (b) :
Given that the above relation is an equivalent definition of MFT for a spin model, is it true to say that :
"Mean-field theory is equivalent to taking out any spatial spin correlations of our system."
?
I think this follows from the Eq.(1). </p>
<p>Part 2 :
However, and here is what is confusing me :
Why can we define a correlation length $\xi$ and a corresponding critical exponent $\nu$ (c.g. $\nu=1/2$ for MFT applied to the Ising model) from the connected two-point correlation function ?</p>
<p>Eq.(2) $\hspace{75pt}\langle S_iS_j \rangle - \langle S_i \rangle \langle S_j\rangle\sim e^{-|i-j|/\xi}$</p>
<p>To me, Eq. (1) and Eq. (2) look contradictory for distances $|i-j|$ smaller than the correlation length, yet there are both derived from MFT... </p> | 2,663 | [
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<p>In a series or parallel circuit, if two bulbs have the same resistance, do they have the same voltage drops? The problem I am asking about is below. Do A, B, and C have the same voltage drops since they have the same resistance? If so, how do I determine the current?</p>
<p><img src="http://i.stack.imgur.com/2wyv1.png" alt="Problem"></p> | 2,664 | [
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<p>As explained for example in <a href="http://profmattstrassler.com/2012/08/15/from-string-theory-to-the-large-hadron-collider/">this</a> article by Prof. Strassler, modern <a href="http://physics.stackexchange.com/q/32491/2751">twistor methods</a> to calculate scattering amplitudes have already been proven immensely helpful to calculate the standard model background in searches for "new physics". </p>
<p>If I understand this correct, the "practical" power of these methods lies in their ability to greatly simplify the calculation of scattering processes, which are due to limited computer power for example, not feasable applying conventional Feynman diagrams.</p>
<p>Depending on the system considered, a renormalization group transformation involves the calculation or summation of complicated Feynman diagrams too, which usually has to be simplified to obtain renormalization group equations which are numerically solvable in a finite amount of time.</p>
<p>So my question is: Could the new twistor methods to calculate scattering amplitudes be applied to simplify investigations of the renormalization group flow, in particular investigations of the whole renormalization group flow field beyond a single fixed point, too? Are such things already going on at present?</p> | 2,665 | [
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<p>I was thinking of making a simple 2D model of the solar system, with planets moving along ellipses like</p>
<p>$$x(t) = k_x \sin(t + k_t) (\sin(k_\phi) + \cos(k_\phi))$$</p>
<p>$$y(t) = k_y \cos(t + k_t) (cos(k_\phi) - \sin(k_\phi))$$</p>
<p>and, for earth at least, a angle that some longitude (say the Greenwich Meridian) is facing in the $xy$ plane:</p>
<p>$$d(t) = k_dt+k_e$$</p>
<p>or something equally minimal.</p>
<p>Two questions:</p>
<ul>
<li>Where can I find the appropriate constants in the most cut-and-paste-able form?</li>
<li>How long will this kind of model be accurate for? (If I want to use it to vaguely look in the right part of the sky for a particular planet)</li>
</ul> | 2,666 | [
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<p>I have hit a bit of a roadblock in my simulation. In the equation I require the aperture size for the laser source.</p>
<p>$$I_a ≈ \frac{\lambda d}{A}$$</p>
<ul>
<li>$I_a$ = aperture atop the atmosphere ($m$)</li>
<li>$\lambda$ = wavelength of laser (in this case 1023nm)</li>
<li>$A$ = aperture of the laser transmitter (...? $m$)</li>
<li>$d$ = distance from SPS to atmosphere (39,000km)</li>
</ul>
<p>But all the articles I have read from NASA, JAXA, ESA, sciencedirect concepts, no one specifically talks about the details of laser transmission if used for SBSP. </p>
<p>I just need some reasonable value for the aperture size. So we are talking about GW scale laser transmission from a single main source, if anyone has knowledge of optics/photonics/lasers, could you recommend me a sensible size along with a citation. </p>
<p>Thanks</p> | 2,667 | [
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<p>Is it a myth that yelling to a coffee mug will heat it? I have been hearing my friend saying that screaming will heat coffee or water.</p> | 2,668 | [
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<p>Reading about nuclear models, nuclear physics and the mythical ``stability island'' I just wondered about the next question: </p>
<p><strong>How can the lifetime of any undiscovered superheavy element be calculated or estimated theoretically?</strong> </p>
<p>Moreover...Does it correspond to a numerical recipe or a theoretical closed approximated formula? Is the lifetime of superheavy atoms the same as that of the superheavy nuclei it has? Any good rereferences about this hard topic? For instance would be welcome! </p>
<p>Comment: I would also be interested in how could someone calculate (if possible) the lifetime of, e.g., Ubh-310 from first principles with the aid of PC or grid computing...</p> | 2,669 | [
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] |
<p>I was going through the concept of designing a Nuclear Reactor that uses Spent Nuclear Fuel(SNF) to generate power as proposed by Transatomic Power .</p>
<p><a href="http://transatomicpower.com/white_papers/TAP_White_Paper.pdf" rel="nofollow">http://transatomicpower.com/white_papers/TAP_White_Paper.pdf</a></p>
<p><strong>Is it possible to design such a Nuclear Reactor that uses spent nuclear fuel to generate power ?</strong></p> | 2,670 | [
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<p>Imagine that there is a cube box that has mirrors all 6 faces in . If we use a strong laser and enter in the box from a small hole on the box. The laser light travels in the box long time that we can detect the laser via a detector on other hole of the box. </p>
<p>1) Is it possible to simulate the long light travel in it (for example a day or week)?
2) Is it possible to proof that there is no ether via that box? If we move the box in a fixed velocity what we can observe about receive time on detector.</p>
<p>I ask a question first time in this website If it is asked question or not appropriate for your format sorry for that.</p> | 2,671 | [
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<blockquote>
<p>Show that in the WKB approximation, the mean kinetic energy $T_{n1}$ in a bound state $\psi_n$ in a potential $V(x)$ is given by</p>
<p>$\langle T_n \rangle = \frac{1}{2}\left(n+\frac{1}{2}\right) \frac{dE_n}{dn}$</p>
</blockquote>
<p>This is a homework problem and I'm having trouble understanding what it's asking.</p>
<ul>
<li>What does the notation $T_{n1}$ mean?</li>
<li>What is the meaning of the expression $\frac{dE_n}{dn}$? Bound states should have discrete spectra.</li>
</ul>
<p>Could someone provide a clearer statement of this problem?</p> | 2,672 | [
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<p>I have a solver for Poisson's equation and it works nicely. It uses finite differences. It works in the presence of multiple dielectrics.</p>
<p>It also solves the <a href="http://en.wikipedia.org/wiki/Poisson%E2%80%93Boltzmann_equation" rel="nofollow">Poisson Boltzmann equation</a>. That is, fixed charges with free moving charges, as in a molecule immersed in a solution with salt assuming that the molecule and liquid can be approximated as a continuum medium.</p>
<p>Now, what happens if there are currents? this violates the assumption of equilibrium required for Poisson Boltzmann. I'm looking for the equation that describes this situation. I guess it should have the form</p>
<p>$$
\nabla\cdot(\epsilon\nabla \phi) = -\rho_\text{fixed} + \text{<ions effect>} + \text{<current effect>}
$$</p>
<p>I'm pretty sure this has already been studied. Can anyone direct me towards where to look for more detail? is there an equation with a name (like Poisson Boltzmann) for this?</p> | 2,673 | [
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<p>On the plane z=0 there is a superficial charge distribution such that $\sigma$ is constant. </p>
<p>Near to the plane, there is a bar, charged uniform with total charge q. At the extremities the bar has two constraints, so it can't turn.</p>
<p>If I want to find the constraints force and the force momentum needed to block the bar, can I consider the charge q as a single point charge and put it in the midpoint of the bar? </p> | 2,674 | [
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<p>In Sun-Earth system, what transformation law allows me to find the Hamiltonian H in an arbitrary inertial system knowing the same in the center of mass?</p> | 2,675 | [
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<p>I can't seem to think of any way to envision electron spin. Can it be thought of as the uncertainty in angular momentum?</p> | 2,676 | [
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<p>Consider a single coil that is turning in a constant and uniform magnetic field thanks to a motor. The normal to the coil is given by:</p>
<p>$${\bf u}(t)=\sin (\omega t){\bf u_x}+\cos(\omega t){\bf u_z}$$</p>
<p>How can we obtain the energy that the motor has to spend in a period $T=\frac{2\pi}{\omega}$?</p>
<p>I have thought that it could be given by the integral of -$U_p$ ($U_p$ is the potential energy of the coil) from 0 to T (we should have to integrate because the normal vector to the coil depend on t, I have thought). But I'm almost sure that it is wrong.</p> | 2,677 | [
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<p>I don't mean <a href="http://en.wikipedia.org/wiki/Line_integral" rel="nofollow">line integrals</a>, I am talking about <a href="http://en.wikipedia.org/wiki/Path_integral_formulation" rel="nofollow">path integrals</a> or <a href="http://en.wikipedia.org/wiki/Functional_integration" rel="nofollow">functional integrals</a> like the ones that Feynman introduced to quantum mechanics. And what are the prerequisites to this field of study?</p> | 2,678 | [
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<p>When you measure the position of an electron that is in a pure energy state, what happens the energy becomes non-deterministic. That is future measurements of energy can only be predicted with respect to a probability distribution. </p>
<p>This seemingly violates the conservation of energy since future measurements of energy may give different results. </p>
<p>I assume here that the measurement process somehow adds some non-determinism or transfers energy to the measuring apparatus. But this is just a vague idea. How do you clearly explain this?</p> | 2,679 | [
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<p>These papers describe a phenomenon referred to as "atomic collapse" and "supercritical charge" in graphene: <a href="http://www.sciencemag.org/content/340/6133/734.abstract" rel="nofollow">Wang et al.</a>, <a href="http://prl.aps.org/abstract/PRL/v99/i16/e166802" rel="nofollow">Pereira et al.</a></p>
<p>"Atomic collapse" appears when you have a large enough Coulomb potential in a system described by the Dirac Equation. In the case of graphene, a localized state is embedded in the continuum below the Dirac Point, causing an electron to fall into the state, leaving behind a hole.</p>
<p>I was wondering whether something similar could happen for gravity. I know gravity is due to the curvature of spacetime rather than a Coulomb potential, but the phenomenon described by the papers cited above seem reminiscent of Hawking Radiation. Is "atomic collapse" the electromagnetic analog of Hawking Radiation?</p> | 2,680 | [
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<p><strong>Problem:</strong><br />
<em>A block B of 1,5 kg is attached to the right of a spring (not deformed, with its right side attach to a wall) with a constant of $k = 80 N/m$ and, at rest, the block enter in collision with another identical block A. The block A moves to the left with a velocity of 5 m/s before the collision. Consider successively a restitution coefficient between the two blocks of $e = 1$ and $e = 0$. There is no friction in this problem.<br />
a) the maximum deformation of the spring.<br />
b) the final velocity of the block A.</em></p>
<hr />
<p>I have some difficulty to start this problem. I have an example with a block falling on a spring, but I have a hard time to apply it on a horizontal collision. I started by drawing a diagram of the two blocks before the collision. Block B is not moving and block A is going to the left. I also draw a diagram of the two blocks after the collision. The block B is going to the left and pushing the spring, and the block A is going to the right. (I'm going to use the french notation)</p>
<p>Since the restitution coefficient is 1 (impact perfectly elastic), the two blocks will go in an opposite direction with the same velocity they had before the impact. So block A will go right at 5 m/s and B will stay at 0 m/s but the spring will be compress.</p>
<p>Later, the block B will be push to the right with the elongation of the spring and then the two blocks will stick together and continue to the right with a restitution coefficient of 0 (impact perfectly plastic).</p>
<p><strong>So, I understand what is happening, but I need some help for the steps to follow to resolve this problem. Can someone help me understand?</strong></p>
<hr />
<p><strong>Note:</strong> if you have some problem understanding the problem, it is probably because I translated it from a french book. We do not use the same exact term in french (for example, the exact
translation from french of "velocity" is "speed"), so just ask for more informations.</p> | 2,681 | [
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<p>Photons have no mass. Yet they interact gravitationally, as all energy does, with other energetic and massive particles. This means that if you put multiple photons in a system, you get something that appears to have mass, even though none of the constituent particles do have mass.</p>
<p>That makes me wonder:</p>
<p>Is mass really a fundamental part of reality? Couldn't it be that massive particles (protons/neutrons/electrons) are just composed of massless particles like photons knotted up, confined to a small area and whizzing around in very tight orbits? So everything is, in a sense, massless?</p>
<p>The search for and discovery of the Higgs Boson suggests to me, in my limited understanding, that scientists believe mass is a fundamental property that some particles have. And also that mass is fundamentally different than other types of energy (though conversion is possible). Does all of this preclude a system like I describe?</p> | 2,682 | [
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<p>Trying to work out some pesky flywheel dynamics for a project I'm working on, would love some for your assistance to better understand the underlying concepts. </p>
<p>For a given flywheel (thin-walled cylinder, assume a spoked bicycle wheel) rotating in the x-y plane, I'm trying to calculate the force generated in either direction along the z-axis. </p>
<p>It seems to me, in line with Newton's first law of motion extended to rotational dynamics, what forces are physically being generated that prevent a rolling wheel from falling over?</p> | 2,683 | [
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<p>Given particles A, B, C and D, where:</p>
<ul>
<li>A and B have an equivalent mass</li>
<li>C and D have an equivalent mass, both larger than A (or B)</li>
<li>D is the antiparticle of C.</li>
</ul>
<p>A and B start close to C, but with velocity, such that they have moved away from C, so their kinetic energy has been converted to GPE. If D then collides with C such that C and D annihilate, what is the fate of the GPE that A and B previously acquired?</p> | 2,684 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/5456/the-speed-of-gravity">The speed of gravity</a><br>
<a href="http://physics.stackexchange.com/questions/26742/does-gravity-travel-at-the-speed-of-light">Does gravity travel at the speed of light?</a> </p>
</blockquote>
<p>Imagine there is a large mass $m_1$ (e.g. a star) 1 light-year away from us. It is stable, stationary relative to us and has been in place for a long time, much more than a year. A small mass $m_2$ (e.g. a proton) has just been created locally, 1 light-year away from $m_1$.</p>
<p>How much time does it take for $m_2$ to feel the gravitational pull of $m_1$, and how can this be explained with the virtual-graviton theory of gravity?</p>
<p>Some possible answers I can imagine:</p>
<p>a) Immediately $m_2$ interacts with virtual-gravitons sent by $m_1$, a year ago.</p>
<p>b) 1 year. It takes this long for freshly launched virtual-gravitons from $m_1$ to reach $m_2$ and vice-versa before any effect is felt on either mass</p>
<p>c) 2 years. There needs to be an exchange of information / virtual-gravitons between $m_1$ and $m_2$ and this is the minimum time it could take.</p>
<p>d) None of the above</p> | 4 | [
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<p>Why do I keep getting a slightly different result from the following two ways of determining the center of mass of a rigid, geometrically simple object? </p>
<p>The object is a rectangular 5(x) by 7(y) sheet of uniform rigid material, with a 2(x) by 3(y) rectangle missing from the upper right corner.</p>
<p><strong>Method 1)</strong></p>
<p>The official formula in my textbook:</p>
<p>x center of mass = ( m1x1 + m2x2 ) / ( m1 + m2 )</p>
<p>y center of mass = ( m1y1 + m2y2 ) / ( m1 + m2 )</p>
<p>This gives results of x = <strong>2.18</strong> and y = <strong>3.09</strong>.</p>
<p><strong>Method 2)</strong></p>
<p>A torque calculation that would seem more intuitive to me: </p>
<p>The xy coordinate plane is parallel to floor and I watch it from above. Object is in the first quadrant (top right), touching the x and y axes.</p>
<p>I imagine balancing the object on a razor blade parallel to the y-axis. I solve for x by assuming that the torques on the left and right side must cancel out. </p>
<p>I rotate the razor blade so that it's parallel to the x-axis. I balance the object again. I solve for y by assuming again that the torques on each side must cancel out.</p>
<p>This gives results of x = <strong>2.07</strong> and y = <strong>2.9</strong>.</p>
<p>The difference between the results given by the two methods is small, but significant. What's going on? </p>
<p>I checked my math several times and even tried a different problem with a simpler geometry. Again the results differed by a small but significant value. I'm fine with having to learn the textbook method, but would like to know why the torque approach gives results that are 3-5 % off.</p> | 2,685 | [
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<p>I'm trying to calculate the amount of fluid that would flow through an area dependant on the amount of pressure that there is. I'd also like to know the rate at which it would flow.</p>
<p>Essentially I have a very basic model of a well drilling system. At the moment the variables / parameters for each of the objects in question are defined by the user but may be set to constants to enable testing.</p>
<p>The assumptions I'm working on are:</p>
<p>The viscosity of the oil / hydrocarbons is set to constant.
Once drilled the area of the hole will not change (there will be no collapse etc).
The well is never ending so constant pressure.</p>
<p>If someone could point me in the right direction I'd be very thankful. Some of the assumptions may be way off so guidance in general is also appreciated, many thanks.</p> | 2,686 | [
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<p>A uniform cylinder was placed on a frictionless bearing and set to rotate about its vertical axis. After a cylinder has reached a specific state of rotation it is heated without any mechanical support from temperature $T$ to $T+\Delta T$.
Angular momentum will be conserved but rotational kinetic energy will decrease as angular velocity decreases.
What I fail to understand is where does this energy go?</p> | 2,687 | [
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<p>I am a total beginner in the field of Quantum Mechanics. So, the question I am asking may be a silly one. So kindly give me possible answers or advice for modifications. </p>
<p>Recently I am learning the concept of qubit. The quantum theory tells that a $n$-qubit system is represented by a unit vector in $(\mathbb{C^2})^{\bigotimes{n}}$ with some basis set. Now, we can also express a $2^n$ dimensional complex signal as a vector at any point of time, and with suitable normalization it is nothing but a qubit. </p>
<p>Now, as far as I know, the evolution of a qubit is always dictated by a unitary operator, whereas, the evolution of a signal can be dictated by any arbitrary operator. So, is there any way, any sort of transformation that allows,given a signal and its evolution, to create its equivalent representation as an evolution of qubit, so that we can solve problems of signals using methods of Quantum Mechanics. </p> | 2,688 | [
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<p>Is there even a relativistic mass or just relativistic momentum? How does one reason to prefer one over another? What is the problem with saying a hot gas will have more mass/inertia to it? </p> | 54 | [
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<p>I am reading a quantum transport book, where they often mention: phase breaking length and Fermi wavelength. I have looked up and found that:</p>
<p><strong>Phase breaking length</strong>= length over which electron remains its phase.</p>
<p><strong>Fermi wavelength</strong>= Wavelength associated with the maximum energy of electron (Fermi energy). This is often equal to the distance between 2 electrons.</p>
<p>What I couldn't find, however (and in which I am interested) is:</p>
<p>1) <em>What is the difference between the Fermi wave length and the phase-breaking
length?</em> </p>
<p>2) <em>In which transport regimes do they play a role?</em></p> | 2,689 | [
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<p>I live in a very old house - build 1902 - in the 4th floor in the city of Karlsruhe (Germany). I have a shower and the gas-heater for the water is in it:</p>
<p><img src="http://i.stack.imgur.com/tpiBK.jpg" alt="enter image description here"></p>
<p>If I hold the shower head over some height (see image), the water pressure is suddenly reduced. Whats the reason for that? </p>
<p>(I guess the way the pump works might be the reason.)</p>
<p><img src="http://i.stack.imgur.com/rvn6k.jpg" alt="enter image description here"></p> | 2,690 | [
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] |
<p>Let's say that there is a parallel circuit with two identical resistors in parallel with each other. If a third resistor, identical to the other two, is added in parallel with the first two, the overall resistance decreases. </p>
<p>Why does this overall resistance decrease?</p> | 2,691 | [
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<p>In Peskin & Schroeder (and also Cheng which I have skimmed through) they motivate the <a href="http://en.wikipedia.org/wiki/Operator_product_expansion">Operator Product Expansion</a> with a lot of words. </p>
<p>Is there any way to motivate it mathematically, e.g. Taylor expansion or similar?</p> | 2,692 | [
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<p>I didn't have much luck getting a response to this question before so I have tried to reword and expand it a little:</p>
<p>In early 2010 I attended this inaugural lecture by string theorist- Prof. Mavromatos entitled 'MAGIC strings'. In it he proposes that some string theory models may violate Lorentz symmetry at the Planck scale resulting in a kind-of foamy spacetime that could be observed by differing arrival times of photons of different energies reaching us from distant astronomical sources. See <a href="http://www.kcl.ac.uk/news/events_details.php?year=2010&event_id=2178" rel="nofollow">http://www.kcl.ac.uk/news/events_details.php?year=2010&event_id=2178</a> or here for one of the papers: <a href="http://iopscience.iop.org/1742-6596/174/1/012016" rel="nofollow">http://iopscience.iop.org/1742-6596/174/1/012016</a></p>
<p>Furthermore, in 'Cycle's of Time' that I read recently, Prof. Sir Roger Penrose mentions (page 203) that Wheeler and others have strongly argued that if we could examine spacetime at the Planck scale we would see a turbulent chaotic situation (from vacuum fluctuations of the quantum fields I suppose) or perhaps a discrete granular one. Penrose goes on to list some other approaches that may suggest how this discrete structure may manifest itself. Loosely transcribed these are: <strong>spin foams, casual sets, non-commutative geometry, Machian theories, twistor theory, [EDIT] loop quantum gravity, or strings and membranes</strong> existing in some higher-dimensional geometry...</p>
<p>I have studied some QM, introductory QFT and the Standard Model as well as some basic GR but I have no formal experience of string theory. My questions are therefore:</p>
<ul>
<li><p>What's involved with each of the above approaches? I.e. in what way does the spacetime become discretised? (Particularly in string theory)</p></li>
<li><p>Are there any other popular(ish) approaches that should be added to the list?</p></li>
<li><p>Supplementary query, with GR being a background-independent theory, I fail to see how one can end up with discretised spacetime without it being a pre-defined background onto which a theory of the dynamics would have to be 'bolted-on'??</p></li>
</ul>
<p>Please forgive my ignorance if what I have said is misinformed, all comments and elucidations would be most welcome.</p> | 114 | [
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<p>Given a collection of point-particles, interacting through an attractive force $\sim \frac{1}{r^2}$.</p>
<p>Knowing only $m_1a=\sum_i \frac{Gm_1m_i}{r^2}$ and initial conditions we can deduce the motion of the system.</p>
<p>Consequently we can observe that three quantities remains constant
A) center of mass of the system
B) total energy
C) angular momentum</p>
<p>How can we derive these 3 facts directly from $m_1a=\frac{Gm_1m_2}{r^2}$ ?</p>
<p>Are these quantities conserved for any attractive force $\sim\frac{1}{r^n}$ ?</p>
<p>Given any monotonically decreasing force for $r$ in $(0,\infty)$, which are the conserved quantities?</p> | 2,693 | [
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<p>There are some things I encountered, studying the Bernouilly equation, that I don't understand. I was studying in the following book: <a href="http://www.unimasr.net/ums/upload/files/2012/Sep/UniMasr.com_919e27ecea47b46d74dd7e268097b653.pdf" rel="nofollow">http://www.unimasr.net/ums/upload/files/2012/Sep/UniMasr.com_919e27ecea47b46d74dd7e268097b653.pdf</a>. At page 72-73 they derive the Bernouilli equation for the first time, from energy considerations. They state that it can be applied if the flow is steady, incompressible, inviscid, when there is no change in internal energy and no heat transfer is done (p.72 at the bottom). I understand this derivation, my problems arise when they derive the equation again at page 110-111, this time from the Navier-Stokes equation. </p>
<p>I don't quite understand the derivation yet, but I am more confused about the outcome. It seems that the Bernouilli equation can now be applied under the four conditions they state (inviscid flow, steady flow, incompressible flow and the equation applies along a streamline) while it seems that there are no limitations to the internal energy of the flow and the heat transfer done. If I understand the last paragraph right, they state that the requirement that the flow is inviscid already comprises the claim that there is no change in internal energy ("the constant internal energy assumption and the inviscid flow assumption must be equivalent, as the other assumptions were the same", I don't really understand this reasoning, because the "inviscid flow assumption" was also already made in the previous derivation). </p>
<p>Furthermore, if I follow the reasoning (so if I assume that inviscid flow indeed implies that there is no change in internal energy or if I assume that the Bernouilli equation can also be applied, without the assumption of "no change in internal energy") then example 4 at page 74 seems strange to me. It shows that in this situation the Bernouilli equation can't be applied directly (because headloss has to be included in the equation). However I think that we can easily repeat the derivation, assuming that the flow is inviscid (and assuming that the other conditions are satisfied), but this example shows that the internal energy increases, so inviscid flow can't imply "no change in internal energy". And the example also shows that the Bernouilli equation can't be applied, because head loss has to be considered (so this example seems to contradict the result at page 110-111).</p>
<p>I hope someone can explain where my understanding is lacking, because this really confuses me about the conditions under which the Bernouilli equation can be applied.</p>
<p>It would also be helpful, if someone could explain the derivation at p.110: what I don't understand is the definition of "streamline coordinates". Do they just take a random point on the streamline, where they place a cartesian coordinate system?</p> | 2,694 | [
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] |
<p>When analysing powder diffraction patterns, the broadening of peaks can be used to estimate crystal sizes. Smaller crystal size gives larger broadening according to the <a href="http://en.wikipedia.org/wiki/Scherrer_equation" rel="nofollow">Scherrer equation</a>:</p>
<p>$$ \beta = {{K\cdot\lambda}\over{D\cdot\cos \theta}}.$$</p>
<p>What is the physical origin of this effect?</p>
<p>Edit:
Added homework tag as this is related to my thesis on soil studies using diffraction methods.</p> | 2,695 | [
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<p>I'm not sure if this is the appropriate forum for my question as I actually am studying this as part of electrical engineering and I don't actually study physics. Nonetheless, I shall ask and if need be, move my question to another venue.</p>
<p>My question is with regard to how complex permittivity is defined. According to my book
$$
\begin{align*}
\nabla \times \mathbf{\tilde{H}} &= \sigma \mathbf{\tilde{E}} + \jmath\omega\varepsilon \mathbf{\tilde{E}} \\
&= (\sigma + \jmath\omega\varepsilon)\mathbf{\tilde{E}} \\
&= \jmath\omega\underbrace{\left(\varepsilon - \jmath\frac{\sigma}{\omega}\right)}_{\varepsilon_c}\mathbf{\tilde{E}} \\
&= \jmath\omega\varepsilon_c\mathbf{\tilde{E}}
\end{align*}
$$
($\mathbf{\tilde{E}}$ and $\mathbf{\tilde{H}}$ are phasors.)</p>
<p>I really do not understand why $\varepsilon_c \equiv \varepsilon - \jmath\frac{\sigma}{\omega}$ and not $\varepsilon_c \equiv \sigma + \jmath\omega\varepsilon$. What is the sense in creating a complex value, $\varepsilon_c$, and then multiplying by $\jmath\omega$ when you could just modify the definition of $\varepsilon_c$ such that $\nabla \times \mathbf{\tilde{H}} = \varepsilon_c \mathbf{\tilde{E}}$?</p>
<p>I also have a conceptual question: From what I understand, $\epsilon$ determines the phase delay between the H and E fields. This phase delay, as far as I know, comes from the finite speed involved in 'rotating' the dipoles in the medium. When these dipoles are 'rotated' though, since they take a finite time to rotate, that implies to me that there are some sort of losses involved in rotating the dipoles. These losses, though, as far as I can tell, are not accounted for in $\varepsilon_c \equiv \varepsilon - \jmath\frac{\sigma}{\omega}$ (I figure the loss due to rotating the dipoles should be part of $\Im{\{\varepsilon_c\}}$ (from what I can tell, $\Im{\{\varepsilon_c\}}$ accounts for the loss and $\Re{\{\varepsilon_c\}}$ accounts for the phase delay); however, $\Im{\{\varepsilon_c\}}$ only seems to take into account frequency and loss from electorns crashing into atoms).</p>
<p>Similarily, I would've thought that the loss that comes from electrons crashing into atoms (which $\sigma$ looks after), would also have the affect of at least somewhat slowing down the wave and causing lag.</p>
<p>Basically, what I'm saying is, why aren't $\varepsilon$ and $\sigma$ also complex numbers? Or maybe they are...</p>
<p>Thank you.</p> | 2,696 | [
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<p>I am looking the derivation of the speed of sound in Goldstein's Classical Mechanics (sec. 11-3, pp. 356-358, 1st ed). In order to write down the Lagrangian, he needs the kinetic and potential energies.</p>
<p>He gets the kinetic energy very easily as the sum of the kinetic energies of the individual particles (the sum going over to an integral in the limit). Let $\eta_i, i=1,2,3$ be the components of the displacement vector (each $\eta_i = \eta_i(x,y,z)$ being a function of position). So the kinetic energy density is ${\cal T}=(\mu_0/2) (\dot{\eta}_1^2+\dot{\eta}_2^2+\dot{\eta}_3^2)$, where $\mu_0$ is the equilibrium mass density.</p>
<p>For the potential energy, he uses a thermodynamic argument, relying on the work done in a PV diagram, and using the equation $PV^\gamma = C$. His ultimate result, after several steps, is</p>
<p>${\cal V} = -P_0 \nabla\cdot\vec{\eta}+\frac{\gamma P_0}{2}(\nabla\cdot\vec{\eta})^2$</p>
<p>Here, $P_0$ is the equilibrium pressure, and $\gamma$ is the ratio of specific heats.</p>
<p>He later shows that the term $P_0 \nabla\cdot\vec{\eta}$ has no effect on the equations of motion, and so he drops it. So his final formula for the Lagrangian density is:</p>
<p>${\cal L} = (1/2)(\mu_0\dot{\vec{\eta}}^2 - \gamma P_0(\nabla\cdot\vec{\eta})^2$</p>
<p>and the Lagrangian of course is the integral of this over all space.</p>
<p>Now in the case of an ideal gas (or better yet, a perfect gas), my understanding is that the internal energy is <em>entirely kinetic</em>. Naively, the statistical model is a bunch of non-interacting point particles racing around, bouncing off the walls of the container. (For simplicity, ignore gravity.)</p>
<p>This seems contradictory. Shouldn't we get the same results from a microscopic and a macroscopic viewpoint?</p>
<p>To put it another way, this suggests that in a gas made up of non-interacting point particles, with no external forces except for the hard-wall forces, sound waves could not propagate (since the Lagrangian density would reduce to the kinetic part). That doesn't seem right.</p> | 2,697 | [
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<p>The following passage has been extracted from the book "Modern's abc of Chemistry": </p>
<blockquote>
<p>..Heisenberg in 1927, put forward a principle known as Heisenberg's uncertainty principle. It states that, it is not possible to measure simultaneously both the position and momentum (or velocity) of a microscopic particle, with absolute accuracy. </p>
</blockquote>
<p>Lets fill an isolated atom by subatomic "Rutherford projectiles"-alpha particles. I hope it is possible. This doesn't seem to be a limit of our technology. Isn't it? </p>
<p>If we are successful in filling the an atom with alpha particles, we are decreasing the space for the electron and confining them to a least distance, isn't it? </p>
<p>Doesn't this experiment, give us belief of measuring simultaneously both the position and momentum with little greater (or even complete) certainty than what predicted by Heisenberg's principle? This doesn't seem to allow us to fill an atom with alpha particles. </p>
<p>So, can we fill an atom with alpha particles?</p> | 2,698 | [
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<p>I am looking for patterns to efficiently disperse reflected ultrasound in the range of wavelengths 1mm to 4mm within the interior of a narrow tube (I do not want wall reflections). For various reasons absorption does not appear to be a solution. Thickness is also critical in that the surface must not be much more than 1mm deep. This is part of a setup for measuring absorption in gases using reflected pulses.</p> | 2,699 | [
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<p>In quantum mechanics, what is the difference between <a href="http://en.wikipedia.org/wiki/Group_velocity#Matter-wave_group_velocity" rel="nofollow">group velocity</a> and <a href="http://en.wikipedia.org/wiki/Phase_velocity#Matter_wave_phase" rel="nofollow">phase velocity</a> of matter wave? How can it also be that phase velocity of matter wave always exceeds the speed of light?</p> | 2 | [
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<p>I have read the following: <a href="http://www.feynmanlectures.caltech.edu/I_20.html#Ch20-S1" rel="nofollow">http://www.feynmanlectures.caltech.edu/I_20.html#Ch20-S1</a> </p>
<p>The formula for $\tau_{xy}$ is derived in this chapter: <a href="http://www.feynmanlectures.caltech.edu/I_18.html#Ch18-S2" rel="nofollow">http://www.feynmanlectures.caltech.edu/I_18.html#Ch18-S2</a>. In this derivation until equation 18.11 ($\Delta W=(xF_y-yF_x)\Delta\theta$) the term $xF_y-yF_x$ does not seem to be arbitrary. It would not make any sense the other way around like: $yF_x-xF_y$, because of the equations 18.6 ($\Delta x=-PQ\sin\theta=-r\,\Delta\theta\cdot(y/r)=-y\,\Delta\theta$) and 18.7 ($\Delta y=+x\,\Delta\theta$). </p>
<p>Then $\tau_{yz}$ and $\tau_{zx}$ are derived (in the first link) by symmetry.
$$\begin{alignedat}{6}
&\tau_{xy}~&&=x&&F_y&&-y&&F_x&&,\\[.5ex]
&\tau_{yz}~&&=y&&F_z&&-z&&F_y&&,\\[.5ex]
&\tau_{zx}~&&=z&&F_x&&-x&&F_z&&.
\end{alignedat}$$ </p>
<hr>
<p>I have some ideas and wonder whether or not they are true: I have got the feeling that they are wrong but I have no idea why they should be wrong.</p>
<ol>
<li><p><em>So, since the formula for the torque in the $xy$-plane ($\tau_{xy}$) is not arbitrarily-derived, there is to torque $\tau_{yx}$, since $yF_x-xF_y$ would not be correct in equation 18.11.</em></p></li>
<li><p>In my first link, below equation 20.1 there are 2 pictures which shall demonstrate how the letters $x$, $y$ and $z$ can be interchanged. <em>How would that make sense?</em> Since equations 18.6 and 18.7 are true, we "live" in a right-handed coordinate system and $\tau_{yx}$ would not make any sense.</p></li>
<li><p>Since $\tau_{yx}$ would not make any sense, the arguments in my first link down to equation 20.9 show that the right-hand rule is not arbitrary.</p></li>
</ol>
<p>What is wrong with my arguments? What does $\tau_{yx}$ mean?</p> | 2,700 | [
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<p>How can we define U(1) local gauge transformation for Dirac spinor field?, like scalar fields? </p> | 2,701 | [
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<p>I want to be able to modify the height of a quantum locked superconductor. My original plan was to levitate the superconductor over an electromagnet and modify the current to said magnet (increase the current for higher levitation height, decrease for lower). Further research has made it apparent to me that this won't work. Is there a way to change the levitation height of a superconductor short of mechanical interference within the system?</p> | 2,702 | [
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<p>In this problem set I have a passage that describes an experiment that looks at the changing temperature as an air filled balloon rises to the surface from the bottom of a water filled tank. The graph they provide shows that as the balloon rises to the surface the temperature of the air in the balloon decreases. (I'm paraphrasing a lot of the question because it's long and has a bunch of information that's not pertinent such as dimensions of the tank, dimensions of a valve that's not even used in any problem!!! etc.)</p>
<p>I'm struggling with the idea of the temperature changing at all. From the answers from the problems it seems that the gas is doing work by expanding thereby losing internal energy and thus temperature. This to me makes sense, but I keep looking at the PV=nRT and thinking well wait, wouldn't the temperature just be fixed and the volume and pressure change correspondingly? Help getting through these basic concepts would be great!</p>
<p>Also one of the problems I struggled with:</p>
<p>Which of the following items of information would NOT help in predicting the results [shown in the graph]?</p>
<p>A) The number of air molecules inside the balloon</p>
<p>B) The thermal conductivity of the rubber (of the balloon)</p>
<p>C) The variation with depth in the speed of the balloon</p>
<p>D) The total mass of the water in the tank</p>
<p>Answer is D</p>
<p>I'm confused what the test makers were hinting at with C; if C could be included in your explanation of this experiment that would be great (does this somehow give the amount of internal energy lost??). I can't find other examples of this kind of experiment, is it a specific type? I.e. Does it have practical applications (even a name of like a standardized experiment similar to "pendulum motion" etc) or is it simply a conceptual check?</p>
<p>Thanks!</p>
<p>Update: <a href="http://s24.postimg.org/yzvb1ci5x/balloon_gas_temp_experiment.jpg" rel="nofollow">http://s24.postimg.org/yzvb1ci5x/balloon_gas_temp_experiment.jpg</a></p> | 2,703 | [
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<p>This <a href="http://travel.stackexchange.com/a/21886/4868">travel stackexchange answer</a> has kinda got me wondering... how long do experiments involving the large hadron collider usually take? I'd expect you run it for a few seconds and bam - higgs boson detected or whatever. Maybe it'd take a few months to set the experiment up but once it's setup it doesn't seem like it'd take that much time at all to run the experiment?</p>
<p>I mean, maybe you'd want to run it a few times to verify your results but if each run takes just a few seconds it seems like you could still be done with your multiple runs even in a single day.</p>
<p>Any ideas?</p> | 2,704 | [
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<p>$$
\hat{\Omega}_j{(\tilde{q}_j)}=\Omega_j(\tilde{q}_j-\hat{q}_j)
$$</p>
<p>$$
[\hat{q}_j,\hat{q}_l]=ik_{jl}
$$</p>
<p>Implies</p>
<p>$$
[\hat{q}_j,\hat{\Omega}_l]= \frac{\partial\Omega_l(\tilde{q}_l-\hat{q}_l)}{\partial\hat{q}_l}.ik_{jl}
$$</p> | 2,705 | [
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<p>Consider some 1D Lattice of atoms with nth neighbor coupling of strength k_{n}. I'm looking for the dispersion relation for acoustical phonons under these conditions. </p>
<p>I start with the Lagrangian,
$$L = K- V$$
$$L = \sum^{\infty}_{n} \frac{1}{2}m \dot{x}_{n}^{2} - \sum^{\infty}_{p=1} \frac{1}{2}k_{p} \{(x_n-x_{n+p})^2 + (x_n - x_{n-p})^2\}$$</p>
<p>Mass is the same for each atom. The Lagrange equation should be</p>
<p>$$m \ddot{x}_{n}=\sum_{p=1} k_p(x_{n-p}+x_{x+p}-2x_n)$$</p>
<p>Now, if I use a travelling wave solution as an ansatz, I should get my dispersion relation as some infinite series. Is this correct? If so, help me out because I can't make it work. Thanks!</p> | 2,706 | [
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<p>This website <a href="http://www7b.biglobe.ne.jp/~kcy05t/" rel="nofollow">http://www7b.biglobe.ne.jp/~kcy05t/</a> appears to refute Quantum mechanics using some proof.</p>
<p>An important paper involved is this 'Calculation of Helium Ground State Energy by Bohr's Theory-Based Methods' <a href="http://arxiv.org/abs/0903.2546" rel="nofollow">http://arxiv.org/abs/0903.2546</a> (written by the website author)</p>
<p>How to disprove the author's claims, assuming his refutation of QM is unacceptable/false.</p>
<p>Note: I don't know if this question belongs here.</p>
<p>Edit:It may take considerable effort to refute or support his claims.</p> | 2,707 | [
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<p>For simplicity let us consider one-dimensional quantum-mechanical systems only. Given any state $\rho\in\mathcal{B}(\mathcal{H})$ and its Wigner function $W_\rho(x,p)$, there are several properties it always satisfies,</p>
<ul>
<li>it is real-valued and bounded in absolute value by $2/\hbar$ from above,</li>
<li>its marginals are well-behaved probability distributions,</li>
<li>its overlap $\int W_\rho(x,p) W_\sigma(x,p) \mathrm{d}x\mathrm{d}p$ with the Wigner function of any other state $\sigma$ is nonnegative.</li>
</ul>
<p>Given an arbitrary function $W(x,p)$, we can reconstruct a state $\rho$ it corresponds to iff all of these conditions are met using the Weyl transform. If some properties are not satisfied, the resulting operator will be unphysical—non-Hermitian, unnormalized or non-positive. The question is how to check the conditions prior to the calculation.</p>
<p>The first two are usually fairly easy to check but the latter involves going through all states of the system. Even if we restrict ourselves to pure states, this still iterates over all rays in an infinite-dimensional linear vector space and is as such usually done finding a proof tailored to the specific case. Is there a simpler (algorithmic) condition which can replace the third bullet above and implies its validity?</p> | 2,708 | [
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<p>Consider a Pure state of a two dimensional system $|\psi\rangle={1\over\sqrt{2}}(|e_1\rangle|e_1\rangle+|e_2\rangle|e_2\rangle)$ where $\{|e_i\rangle\}$ is an orthonormal basis.</p>
<p>Could any one just confirm me whether the corresponding density matrix $\rho$ is defined as $|\psi\rangle\otimes |\psi\rangle$? If I am misunderstood could anyone tell me what will be the density matrix?
also how to find the partial trace of $\rho$? </p>
<p>I am bit lost to find the partial trace from the formuale $\sum_{i=1,2} (I\otimes\langle e_i|)\rho(I\times\langle e_i|)$</p>
<p>is $I$ a $2\times 2$ Identity matrix?</p>
<p>Thanks for the help.
Source: Quantum Computing:From Linear Algebra to Physical Realizations, Page-45</p> | 2,709 | [
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] |
<p>What is the absorption cross section, how is it measured?</p>
<p>How to convert it to the absorption coefficient (measured in cm$^{-1}$)?</p> | 2,710 | [
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] |
<p>What is the relationship between earthquakes power and nuclear explosions?</p>
<p>How to compared power of earthquake with nuclear explosion?</p> | 2,711 | [
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<p>I was referring Electron microscopes and read that the electrons have wavelength way less than that of visible light. But, the question I can't find an answer was that, If gamma radiation has the smallest of wavelengths <em>of all</em>, why can't it be used to reach to even finer details in microscopy?</p> | 2,712 | [
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<p>Consider that space is uniformly charged everywhere, i.e., filled with a uniform charge distribution, $\rho$, everywhere.</p>
<p>By symmetry, the electric field is zero everywhere. (If I take any point in space and try to find the electric field at this point, there will always be equal contributions from volume charge elements around that point that will vectorially add up to zero).</p>
<p>Consequently, from Gauss' law in the differential form $$\nabla\cdot E = \frac{\rho}{\epsilon_0}$$</p>
<p>if $E$ is zero, the divergence is zero hence the charge density is zero.</p>
<p>What is going on here? is a nonzero uniform charge distribution that exists everywhere has no effect and is equivalent to no charge at all?</p> | 427 | [
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<p>"The flag of a ship that is moving northward with 10 km/h, points exactly southwestward. The windsock at the lighthouse points under 30 relative to the western direction southward. Calculate the velocity of the wind at land and on board of the ship".</p>
<p>I have done my homework and solved this problem, so this is not a homework question, but a conceptual one about flags. First, I have assumed that the ship velocity given in the problem is relative to the earth (that is the velocity as viewed by an observer on the earth). Second that the vector sum of the <em>the wind velocity</em> and <em>the ship velocity relative to the wind</em> equals to <em>the ship velocity relative to the earth</em>.</p>
<p>These assumptions all make sense to me, while I am a bit uncertain about the third assumption I needed to do in order to solve the problem: the indication given by the flag on the ship! My understanding is that when you have a flag on an object moving in the wind, this flag - as seen from the earth - waves in the direction <strong>opposite</strong> of that of the velocity of the object <strong>relative to the wind</strong> (that is the velocity produced by the ship's engines in this case).
My question: Could someone please explain if this correct and why it is so? </p>
<p>In this problem this means that this latter velocity makes an angle of 45 degrees northeastward, the same angle made by the flag southwestward (which is the opposite direction in words).</p>
<p>My results are: speed of the wind: 27.32 km/h relative to earth, but 33.46 km/h relative to the boat. The latter higher value I got makes sense, intuitively speaking, since the wind is a headwind, so it is stronger relative to the ship!</p> | 2,713 | [
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<p>I have the following situation in mind: </p>
<p>A big airtight bag of arbitrary shape with a person standing on it. The bag gets inflated with air to lift the person.
Assuming that the bag is much larger than the persons footprint, how do I find the minimal overpressure in the bag that I need to lift the person of the ground?</p>
<p>I was thinking of just dividing the normal force of the standing person by the footprint area, but I am not sure on that approach
$$F_n = 80×9.81 = 784\text{ N}$$ $$P_n = \frac{784}{0.2×0.3} = 13066\text{ Pa}$$</p>
<p>I have the feeling that the bag dimensions play a role as well, as intuitively I would say that to do this, a small bag would work better than a big bag, but again I'm not sure...</p> | 2,714 | [
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<p>My friend said this to me and just want to make sure this is right " when we connect the a battery to a LED and the 2 poles are connected, electrons flow from the (-) to the (+) but with very low velocity, but when you do the same with a capacitor , once the 2 plates are connected electrons flow incredibly fast that's why it's discharged fairly quickly" is that true?</p> | 2,715 | [
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<p>I have encountered a hard exercise which i cannot quite solve. Could anyone help me with it? This is the exercise:</p>
<blockquote>
<p>Lets say we have a photon whose energy $W_f$ is equal to the
relativistic energy of an electron $W_{e0}=m_e c^2$. What is the energy of
a scaterred photon $W_f'$ if after the "collision" the electron is
moving in a direction $\vartheta =40^\circ$ according to the original direction of a
photon?</p>
</blockquote>
<hr>
<p>What i did first was to draw the image <em>(It is in Slovenian language so dont bother about what is written)</em>: </p>
<p><img src="http://i.stack.imgur.com/5UCXT.png" alt="enter image description here"></p>
<p>Now i decided to first calculate the $\lambda$ of the incomming photon:</p>
<p>\begin{align}
W_f &= W_{0e}\\
\frac{hc}{\lambda} &= m_e c^2\\
\lambda &= \frac{hc}{m_e c^2}\\
\substack{\text{this looks similar to the}\\\text{1st part of the Compton's law}} \longrightarrow \lambda &= \frac{h}{m_e c}\\
\lambda &= \frac{6.626\times 10 ^{-34} Js}{9.109\times10^{-31}\cdot 2.99\times 10^8 \tfrac{m}{s}}\\
\lambda &\approx 2.45pm
\end{align}</p>
<p>Now if i try to get the energy of a scattered photon i write down the conservation of energy: </p>
<p>\begin{align}
W_{before} &= W_{after}\\
\frac{hc}{\lambda} + m_ec^2 &= \frac{hc}{\lambda'} + m_ec^2 + W_{ke}\\
\frac{hc}{\lambda} &= \frac{hc}{\lambda'} + W_{ke}\\
\end{align}</p>
<p>This is one equation with two variables ($\lambda'$ and $W_{ke}$) so i am kinda stuck here and i need a new equation which must be a Compton's law. If i implement it i get: </p>
<p>\begin{align}
\frac{hc}{\lambda} &= \frac{hc}{\lambda'} + W_{ke}\\
\frac{hc}{\lambda} &= \frac{hc}{\lambda + \Delta \lambda} + W_{ke}\\
\frac{hc}{\lambda} &= \frac{hc}{\lambda + \tfrac{h}{m_ec}(1-\cos \phi)} + W_{ke}\\
\end{align}</p>
<p>Still i notice that i have 2 variables (now they are $\phi$ and $W_{ke}$). At this point i know i need 1 more equation. I presume it is from the momentum conservation so i write down the momentum conservation for direction $y$ and $x$: </p>
<p>Direction $y$:</p>
<p>\begin{align}
p_{before} &= p_{after}\\
0 &= \frac{h}{\lambda'}\sin\phi - p_e \sin\vartheta\\
p_e &= \frac{h}{\lambda'}\frac{\sin\phi}{\sin\vartheta}
\end{align}</p>
<p>Direction $x$:
\begin{align}
p_{before} &= p_{after}\\
\frac{h}{\lambda} &= \frac{h}{\lambda'}\cos\phi + p_e \cos\vartheta\leftarrow \substack{\text{here i implement what i got from the }\\\text{conserv. of momentum in direction $y$}}\\
\frac{h}{\lambda} &= \frac{h}{\lambda'}\cos\phi + \frac{h}{\lambda'}\frac{\sin\phi}{\sin\vartheta} \cos\vartheta\\
\frac{1}{\lambda} &= \frac{1}{\lambda'} \left(\cos\phi + \frac{\sin\phi}{\tan\vartheta}\right)\\
\lambda' &= \lambda \left(\cos\phi + \frac{\sin\phi}{\tan\vartheta}\right)\leftarrow\substack{\text{It seems to me that i could solve}\\\text{this for $\phi$ if i used Compton's law}}\\
\lambda + \Delta \lambda &= \lambda \left(\cos\phi + \frac{\sin\phi}{\tan\vartheta}\right)\\
\lambda + \tfrac{h}{m_e c} (1 - \cos\phi) &= \lambda \left(\cos\phi + \frac{\sin\phi}{\tan\vartheta}\right) \leftarrow \substack{\text{I got 1 equation for 1 variable $\phi$ but}\\\text{it gets complicated as you will see...}}\\
1 + \tfrac{h}{\lambda m_e c} (1-\cos \phi) &= \frac{\cos\phi \tan\vartheta + \sin\phi}{\tan\vartheta}\\
\tan\vartheta + \tfrac{h}{\lambda m_e c}\tan\vartheta - \tfrac{h}{\lambda m_e c}\tan\vartheta \cos\phi &= \cos\phi \tan\vartheta + \sin \phi\\
\tan\vartheta \left(1 + \tfrac{h}{\lambda m_e c} \right) &= \cos\phi \tan\vartheta \left(1 + \tfrac{h}{\lambda m_e c}\right) + \sin\phi\\
\tan\vartheta \left(1 + \tfrac{h}{\lambda m_e c}\right) \left[1 - \cos\phi\right] &= \sin \phi\\
\tan^2\vartheta \left(1 + \tfrac{h}{\lambda m_e c}\right)^2 \left[1 - \cos\phi\right]^2 &= \sin^2 \phi\\
\tan^2\vartheta \left(1 + \tfrac{h}{\lambda m_e c}\right)^2 \left[1 - \cos\phi\right]^2 + \cos^2\phi&= \sin^2 \phi + \cos^2\phi\\
\underbrace{\tan^2\vartheta \left(1 + \tfrac{h}{\lambda m_e c}\right)^2}_{\equiv \mathcal{A}} \left[1 - \cos\phi\right]^2 + \cos^2\phi&= 1 \leftarrow \substack{\text{i define a new variable $\mathcal{A}$}\\\text{for easier calculations}}\\
\mathcal{A} \left[1 - 2\cos\phi + \cos^2\phi \right] + \cos^2 \phi - 1 &= 0\\
\mathcal{A} - 2\mathcal{A} \cos\phi + \mathcal{A}\cos^2\phi + \cos^2 \phi - 1 &= 0\\
(\mathcal{A}+1)\cos^2\phi - 2\mathcal{A} \cos\phi + (\mathcal{A} - 1) &= 0\leftarrow \substack{\text{in the end i get the quadratic equation}\\\text{which has a cosinus.}}
\end{align}</p>
<hr>
<p><strong>Question:</strong> Is it possible to continue by solving this quadratic equation as a regular quadratic equation using the "completing the square method"? </p>
<p>I mean like this: </p>
<p>\begin{align}
\underbrace{(\mathcal{A}+1)}_{\equiv A}\cos^2\phi + \underbrace{-2\mathcal{A}}_{\equiv B} \cos\phi + \underbrace{(\mathcal{A} - 1)}_{\equiv C} &= 0
\end{align}</p>
<p>and finally: </p>
<p>$$ \boxed{\cos \phi = \dfrac{-B \pm \sqrt{B^2 - 4AC}}{2A}}$$</p>
<p>Afterall if this is possible i get $\cos \phi$ and therefore $\phi$, $W_{ke}$ and finally $W_f'$. </p>
<hr>
<p>EDIT: </p>
<p>I did try to solve this using the quadratic equation and i got solution:</p>
<p>\begin{align}
\cos \phi &= \dfrac{-B \pm \sqrt{B^2 - 4AC}}{2A}\\
\cos \phi &= \frac{2\mathcal{A} \pm \sqrt{4\mathcal{A}^2 - 4(\mathcal{A}+1)(\mathcal{A} - 1)}}{2 (\mathcal{A}+1)}\\
\cos \phi &= \frac{2\mathcal{A} \pm \sqrt{4\mathcal{A}^2 - 4(\mathcal{A}^2-1)}}{2 (\mathcal{A}+1)}\\
\cos \phi &= \frac{2\mathcal{A} \pm \sqrt{4\mathcal{A}^2 - 4\mathcal{A}^2 + 4}}{2 (\mathcal{A}+1)}\\
\cos \phi &= \frac{2\mathcal{A} \pm \sqrt{4}}{2\mathcal{A}+2)}\\
\cos \phi &= \frac{2\mathcal{A} \pm \sqrt{2}}{2\mathcal{A}+2)}\\
\end{align}</p>
<p>So if i apply "+" i get $\cos \phi = 1$ which is impossible for a photon to hold its original direction! But if i apply "-" and insert a variable $\mathcal{A}$ i get:</p>
<p>\begin{align}
\cos \phi = \frac{2 \cdot \tan^240^\circ \left(1 + \tfrac{6.626\times10^{-34}Js}{2.45\times10^{-12}m\cdot 9.109\times10^{-31}kg\cdot2.99\times10^{8}m/s}\right)^2 - 2}{2 \cdot \tan^240^\circ \left(1 + \tfrac{6.626\times10^{-34}Js}{2.45\times10^{-12}m\cdot 9.109\times10^{-31}kg\cdot2.99\times10^{8}m/s}\right)^2 + 2} = \frac{5.59 - 2}{5.59 + 2} = 0.47
\end{align}</p>
<p>Now i can calculate:
\begin{align}
\phi&=\cos^{-1}0.47 = 61.74^\circ\\
\Delta \lambda &= \frac{h}{m_e c} (1-\cos\phi) = 1.28pm\\
\lambda' &= \lambda + \Delta \lambda = 3.73pm\\
W_f' &= \frac{hc}{\lambda'} = 5.31\times10^{-14}J = 331.55 keV.
\end{align}</p>
<p><strong>And the result is correct according by my book. But this way of solving a problem is very long and in a case if i get it in my test i cannot solve it in time i think. So how can i solve it faster? In the comments it was mentioned that i should use the momentum coordinate system? How do i do that?</strong></p> | 2,716 | [
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<p>There is a popular belief that wet skin burns or tans faster. However, I've never heard a believable explanation of why this happens.</p>
<p>The best explanation I've heard is that the water droplets on the skin act as a lens, focusing the sunlight onto your skin. I don't see how this would affect an overall burn, because the amount of sunlight reaching the skin is the same (ignoring reflection).</p>
<p>Is this 'fact' true, and if so, what causes it?</p> | 2,717 | [
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<p>So i have a force field $F(x,y)$ and i have to find out wether it is a potential or not.</p>
<p>My first idea was to calculate : $dU=Fdr$ (where $r$ is the radius vector) , to integrate on both sides and hence to see if $U$ is path dependent or not. That turned out to be right.</p>
<p>I was thinking about another approach: we know that , if a potential exists then $F=-\nabla U$
So if $F$ is a potential it must be possible to find such a $U$ , wich in my case wasn't possible, hence $F$ is not a potential.</p>
<p>My question:</p>
<p>(1) Is the second approach right?</p>
<p>(2) are there othere ways for solving this problem? </p>
<p>Thanks in advance.</p> | 2,718 | [
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<p>For a two-dimensional collision experiment, I must explain in one of the questions the problems in analyzing a non-ideal two-dimensional collision. As the question states:</p>
<blockquote>
<p>Suppose that the projectile marble and target marble do not collide with their centers of mass equidistant from the floor, as shown in figure 8.4. What problems in analyzing this experiment are cause by the non-ideal collision?</p>
</blockquote>
<p><img src="http://i.stack.imgur.com/1Pzya.jpg" alt="enter image description here"></p>
<p>I am not so sure that I have everything or if it is a valid awnser, but so far I have concluded the following:<strong>The Problems in analyzing non-ideal collisions or any other non-ideal conditions lies with either the decrease of precision of data, accuracy of data, or both.</strong></p>
<p>Note: For the sake of the quality of this post, I ask you to please feel free to make any edits so that people in the future can have this question answered. Also I thank everyone in advance for the contribution of this post.</p> | 2,719 | [
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<p>When a magnet passes through a copper coil and electricity is induced into the coil, is there a magnetic resistance on the magnet as it passes through the coil?</p> | 2,720 | [
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