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<blockquote>
<p>Perpetual motion describes hypothetical machines that operate or produce useful work indefinitely and, more generally, hypothetical machines that produce more work or energy than they consume, whether they might operate indefinitely or not.</p>
</blockquote>
<p>(Source:<a href="http://en.wikipedia.org/wiki/Perpetual_motion" rel="nofollow">Wikipedia</a>)</p>
<p>With this definition in mind, particularly the "operates indefinitely" (I don't care about producing work), won't quantum mechanics allow perpetual motion due to energy quantization?</p>
<p>For example, an electron in hydrogen can be thought of as perpetual motion. It's indefinite(I think so); unlike gravitational orbits (which slowly release energy). This is due to the quantization of energy. Without it, the electron would have fallen into the nucleus.</p>
<p>More generally, if we energy is quantized in a system, dissipative forces of lesser magnitude cannot act on it, due to quantization. </p>
<p>For example, if a block can have only an integer value of energy in Joules, then frictional forces of power $P<\frac{1 J}{\text{planck time}}$ cannot act. Or something like that.</p>
<p>So does quantum mechanics permit an infinitely advanced civilization to build a machine which operated indefinitely without doing work?</p>
<p>I'm not well versed in quantum mechanics, so I may be making a mistake here, or I may just be confused. Refer to equations if you want, but try not to use them too heavily unless the answer depends on it. It's OK if they're explained a bit.</p> | 5,097 | [
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<p>I think now is a good time to introduce my son to theoretical physics. He asks so many questions about the universe, black holes, gravity, atoms, molecules, light, etc. He's borderline obsessed with the idea of more than 3 space dimensions. And he tries to apply math to everything. So I'd love to find a great introductory book, lecture, online class or something of that nature to feed his curiosity.</p>
<p>The issue is he's only 6 years old (his birthday was only a few months ago). So his life experience is obviously very limited. His vocabulary and general knowledge is probably not a whole lot more than the average 1st grader. However, his math skills are extraordinary. He's working on calculus now, having mastered algebra and everything that's come before.</p>
<p>Is there any introductory physics material that is heavy on the math and light on the vocabulary and assumed general knowledge? This would also be beneficial to non-native English speakers.</p> | 98 | [
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<p>From Wikipedia <a href="http://en.wikipedia.org/wiki/Spin_triplet" rel="nofollow">here</a> and <a href="http://en.wikipedia.org/wiki/Triplet_oxygen" rel="nofollow">here</a>:</p>
<p><em>''Almost all molecules encountered in daily life exist in a singlet state, but molecular oxygen is an exception.''</em></p>
<p><em>''The unusual electron configuration prevents molecular oxygen from reacting directly with many other molecules, which are often in the singlet state. Triplet oxygen will, however, readily react with molecules in a doublet state, such as radicals, to form a new radical. ''</em></p>
<p><a href="http://en.wikipedia.org/wiki/Singlet_oxygen" rel="nofollow">This wiki page</a> is also relevant. <a href="http://upload.wikimedia.org/wikipedia/commons/c/c7/Molek%C3%BClorbital-Sauerstoff.png" rel="nofollow">Here</a> is a picture (which I can't read).</p>
<hr>
<p>How is this triplet state property quantitatively computed and why is it such an exceptional feature? </p>
<blockquote>
<p>How does the triplet state come about in oxygen? Don't more electrons in electron shell mean more complicated factoring into representations and therefore even more complicated states?</p>
</blockquote>
<p>How does that impact the thermodynamical properties of the element and where is the thermodynamical difference (besides the different energy) to the singlet state? How can the reaction features be understood? </p>
<blockquote>
<p>Where does the energy difference of the two state come from. Taking a look at the periodic table of elements, has $\text S$ or $\text {Se}$ similar properties? </p>
</blockquote>
<hr>
<p>Btw. I don't mind any math, but I'd probably need explanations for <a href="http://en.wikipedia.org/wiki/Molecular_term_symbol" rel="nofollow">expressions like</a> $\text O_2(b^1\Sigma_g^{+})$.</p> | 5,098 | [
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<p>How many percent of the whole visible light reaching the Earth are from other stars than the Sun?</p>
<p>Is it maybe 0,5 - 1% or is my guess already too much?</p>
<p>I am interested mainly in visible light, but if you have knowledge about other parts you can drop it too ;)</p> | 5,099 | [
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<p>ive been given a problem in physics, its to prove if a lump of metal we have been given is real gold or not. one way to to do it would be to work out its density, which is fairly easy, mass/volume. (its not real gold by the way, we just have to show it). But as this is the route that all the other groups are gonna go down, id like to do something more interesting, but im not sure if its possible with the equipment, which consists of, basic optics equipment like a ray table, lasers,(which would give us a known frequency of light), prisms, mirrors, rheostats, transistors, geiger counters, thermometers, diffraction gratings, and a few other things.
Maybe its also possible to measure some absorbtion(fraunhoffer) lines?
does any body have any ideas or know problems that i might come into?</p> | 5,100 | [
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<p>I'm still struggling a bit with some ideas around double slit experiments. One that keeps cropping up for me is the role of observers. </p>
<p>Imagine a classic double slit experiment with a hidden observer who has arraigned an apparatus to detect which slit the electron's are passing through. This person and their measurements are hidden to you and you have no interaction with them.</p>
<p>So the question is, do you see an interference pattern or not?</p>
<p>Additionally:
And if the answer is Not, then is the reason because they "disturbed" the electron (by say firing photons at them) or is it for another reason? And if it is because they "disturbed" the electron, then how is it that unobserved electron's are not disturbed since they certainly interact with other objects, for example other atoms in the matter around the slit(s) will feel a slight gravitational tug as it passes through.</p> | 5,101 | [
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<p>If my understanding is correct, the temperature of space (as defined by the temperature that a black-body will reach) has been decreasing since the big bang. It has never increased. Additionally, because of the existence of dark-energy, it seems most likely that the temperature never <em>will</em> increase.</p>
<p>Taking the large-scale universe as homogenous, how many times greater would the energy release rate of all cosmic processes have to be in order for the average temperature to hold constant against the cooling effect of dark-energy?</p> | 5,102 | [
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<p>In the end, are they not a condensed form of energy?</p>
<p>I want the 'Why' to go as far as it takes. Where it ends?</p>
<p>What are the constituents of the electron and proton, that make them attract each other?</p> | 176 | [
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<p>When we go to bed at home, we started to put a bowl of water on the radiator (the air gets a bit dry).</p>
<p>By instinct I put a soaked piece of paper (e.g. toilet paper) into the bowl and let it touch the radiator. The next morning all the water in the bowl was gone. My wife was not so sure that the paper actually had any effect. So I put two bowls of water.. one with the paper and one without. The next morning the bowl without the paper had all the water remaining and the ball with the paper was empty and the paper completely dry.</p>
<pre><code> paper
_____
/ |
\ -----/---/ |
\ ------ / \
\------/ \__
------------------------- radiator
</code></pre>
<p>What is the physical mechanism for this "paper pump"? By how much does the temperature difference between the water and the radiator overcome the gravity force?</p> | 5,103 | [
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<p>In a lab experiment, we connected a simple circuit: an AC voltage source, connected (in series) to a variable resistor and an inductor. We measured the current in the circuit, and the voltage that falls over the inductor.
We calculated the phase difference between the voltages and used it to calculate $V_L$, and used it to calculate $R_L$, the effective resistance of the inductor.
We got that $R_L(I)$ rises up to a maxima, and then decreases, but we couldn't understand why - as we understand, $R_L=2\pi fL$, so it should be constant...</p>
<p>What did we not understand?</p> | 5,104 | [
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<p>A neutron can decay into a proton, an electron, and neutrino. Could an antiproton, a positron, and a neutrino combine into a neutron? Could this be where much of the "missing" antimatter is?</p> | 5,105 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/6108/comprehensive-book-on-group-theory-for-physicists">Comprehensive book on group theory for physicists?</a> </p>
</blockquote>
<p>I'm having a hard time trying to get my head around the fundamentals of gauge theory. I've taken classes in QFT and Particle Symmetries recently, both of which touch on the topic, but neither explained it in detail. Browsing the Wikipedia entry on gauge theory gives me the same heuristic arguments I've read hundreds of times, together with some mathematical formalism that's totally impenetrable.</p>
<p>Does anyone know of an introductory book that will explain gauge symmetries, the gauge group and their applications to a grad school student? Ideally I'd like a book which starts slowly but gets far, and isn't too basic. However I'm not looking for a text where you need to have done a couple of courses in differential geometry (fibre bundles etc) to get started!</p>
<p>Many thanks in advance.</p> | 177 | [
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<p>I heard the term <em><a href="http://www.google.com/search?q=conformal+quantum+mechanics" rel="nofollow">Conformal Quantum Mechanics</a></em> used today.</p>
<ol>
<li><p>What exactly does this mean? </p></li>
<li><p>Why would one want to study this?</p></li>
</ol> | 5,106 | [
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<p>In the classic experimental physics text "<a href="http://rads.stackoverflow.com/amzn/click/B0000CKKGJ" rel="nofollow">Statistical Theory of Signal Detection" by Carl. W. Helstrom</a>, Chapter II, section 4 concerns Gaussian Stochastic Processes. Such a process is observed at times $t_1, t_2, t_3, ... t_n$ to obtain $n$ random variables $x_1, x_2, x_3, ... x_n$, then the probability density for the $n$ variables is of the form:<br>
$$p_n(x_1,t_1; x_2,t_2; ...; x_n,t_n) = M_n\;\exp(-0.5\Sigma_j\Sigma_k\mu_{jk}x_jx_k).$$<br>
where $\mu_{jk}$ is a positive definite matrix and $M_n$ is a normalization constant to give unit probability when you integrate over all possible values:<br>
$$M_n = (2\pi)^{-n/2}\;|\;\textrm{det}\;\mu|^{0.5},$$
where $\mu$ is the determinant of the matrix $\mu_{jk}$. We assume that the expected values are all zero: $E(x_k)=0$.</p>
<p>He notes that the expected value of the product of any odd number of $x_j$ is zero (which seems to follow from symmetry), and gives a formula for the expected value of a product of even numbers of variables. We define $\phi_{jk}$ as:<br>
$$\phi_{jk} = E(x_jx_k).$$
He then notes that:<br>
$$E(x_1x_2x_3x_4) = \phi_{12}\phi_{34} + \phi_{13}\phi_{24} + \phi_{14}\phi_{23}.$$</p>
<p>Is there a simple proof? And is there a simple proof that relates to the methods of quantum mechanics / quantum field theory?</p> | 5,107 | [
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] |
<p>I think most physicists mostly model physical systems as some kind of Hilbert space. </p>
<p>Hilbert spaces are a strict subset of <a href="http://mathworld.wolfram.com/BanachSpace.html" rel="nofollow">Banach spaces</a>.</p>
<p><strong>Questions:</strong></p>
<ul>
<li><p>Can physical systems really have non-compact topologies, as a Banach
space has?</p></li>
<li><p>Does anyone have an example of physics which requires a physical space
which is Banach and not Hilbert?</p></li>
</ul> | 5,108 | [
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<p>A Newtonian homogeneous density sphere has gravitational binding energy in Joules $U = -(3/5)(GM^2)/r$, G=Newton's constant, M=gravitational mass, r=radius, mks. The fraction of binding energy to gravitational mass equivalent, $U/Mc^2$, is then (-885.975 meters)(Ms/r), Ms = solar masses of body, c=lightspeed.</p>
<p>This gives ratios that are less than half that quoted for pulsars (neutron stars), presumably for density gradient surface to core and General Relativity effects (e.g., billion surface gees). <strong>Please post a more accurate (brief?) formula acounting for the real world effects.</strong></p>
<p>Examples: 1.74 solar-mass 465.1 Hz pulsar PSR J1903+0327, nominal radius 11,340 meters (AP4 model), calculates as 13.6% and is reported as 27%. A 2 sol neutron star calculates as 16.1% and is reported as 50%. There is an obvious nonlinearity.</p>
<p>Thank you.</p> | 5,109 | [
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<p>Is it true that the whole galaxy is actually revolving, and powered by a black hole?</p>
<p>Has it been proven, and if it is true, how can our solar systems actually keep up the momentum to withstand the pull?</p> | 5,110 | [
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<p>Why GPS/GLONASS/Galileo satellites are on low earth orbit?</p>
<p>Why geostationary orbit is so bad? Sattelites might be placed there 'statically' and more precise...</p>
<p>The only problem I can see is navigation close to poles, but they have this problem anyway.</p> | 5,111 | [
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<p>this is my mental picture for how they travel without a medium, how (like water waves) some can't stay still, why they have wave and particle properties, energy/mass equivalence, conservation, etc. it might capture uncertainty too -- i've heard that all waves have an uncertainty relation (say in their power spectrum), but i don't get why -- it seems like we can discuss waves with absolute precision.</p> | 5,112 | [
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<p>This is the problem I am dealing with:
"One of the functions of an automotive air bag is that it lengthens the collision time. Without an air bag, suppose the 100Ns impulse of a collision is taken up by a fairly rigid dashboard, requiring a time of 15 ms. Find the amount of force exerted on the passenger."</p>
<p>Up till this problem I have only dealt with problems using the equation F x Change in Time = Mass x Change in Velocity. This problem only gives me 2 of the 4 variables in that equation, so I am at a loss of how to figure it out. Please help!</p> | 5,113 | [
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<p>When an electric dipole of moment $\mathbf{P}$ is located in a non-uniform electric field $\mathbf{E}$, there is an net force exerted on it.</p>
<p>However, the formula of the force in some books is read $\mathbf{F}=\nabla(\mathbf{P}·\mathbf{E})$, while in other books, it is $\mathbf{F}=(\mathbf{P}·\nabla)\mathbf{E}$. Obviously, the two formula are not the same. So, which one is true?</p> | 5,114 | [
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<p>I'm reading a review about membranes properties and I have reach a section about fluid membranes. The section discuss the principal curvatures ($c_1, c_2$) and the spontaneous curvatures ($c_0$). After stating some properties of $c_0$, the following two sentences appear:</p>
<blockquote>
<p>The membrane also is assumed to be incompressible. Hence, all the contributions to the surface tension vanish.</p>
</blockquote>
<p>I'm not sure about the meaning of these two sentences, I think it means that by bending the membrane I can not generate surface tension, is that correct?</p> | 5,115 | [
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<p>So there appears to be quite a bit of misinformation on the web as to why people should stay in their cars during a thunderstorm. So I'd like to clear some things up. One such non-nonsensical answer is that cars have rubber tires which insulate you from the ground. I believe this contributes little to nothing to the actual reason.</p>
<p>The "correct" answer appears to be because the car acts like a Faraday cage. The metal in the car will shield you from any external electric fields and thus prevent the lightning from traveling within the car.</p>
<p>However, what happens if you have an imperfect Faraday cage around you? Say for example, you had a window open. I think the car would still protect you a) because it still acts as a Faraday cage, albeit a bit not perfect and b) because electrons will travel the path of least resistance which would be through the body of the car and not through you.</p>
<p>Now going with my b) reasoning, wouldn't you be just as safe standing next to a giant conductive pole (i.e. a lightning rod)? Wouldn't the lightning just go through the lightning rod and you'd be 100% safe?</p>
<p>Also a side question: lightning is essentially just an huge electric arc from the clouds to the ground, correct?</p> | 5,116 | [
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<p>I'm wondering why the weak interaction only affects left-handed particles (and right-handed antiparticles).</p>
<p>Before someone says "because thats just the way nature is" :-), let me explain what I find needs an explanation:</p>
<p>In the limit of massless fermions, chirality (handedness) becomes helicity $(\vec S \cdot \hat p)$. Now, helicity is a property of the state of motion of an object in space. It is pretty unobvious to me how the internal symmetry SU(2) × U(1) would "know" about it, and be able to distinguish the two different helicity states of motion.</p>
<p>On a more technical level, IIRC, left and right handed spinors are distinguished by their transformation properties under certain space-time transformations, and are defined independent of any internal symmetry. If we want to get the observed V-A / parity violating behavior, we have to plug in a factor of $(1 - \gamma^5)$ explicitly into the Lagrangian.</p>
<p>Is there any reason this has to be like this? Why is there no force coupling only to right handed particles? Why is there no $(1 + \gamma^5)$ term? Maybe it exists at a more fundamental level, but this symmetry is broken?</p> | 5,117 | [
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<p>The canonical momentum is a fundamental conserved quantity from Noether's theorem for translational invariance of the Lagrangian. Yet I'm finding it very difficult to see its derivation, or even a statement of what it is for something as fundamental as a relativistically moving charge in a static Coulomb electric field.</p>
<p>Can anyone state what it is, or even give a derivation if it's not too much trouble?</p> | 5,118 | [
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<p>In <a href="http://physics.stackexchange.com/a/18721/2451">this</a> Phys.SE answer
Ron Maimon stats: </p>
<blockquote>
<p><em>there is no relativistic particle formalism in which the particles have postive energies and casual propagation. You can either deal with fields in which case the particle notion is non local or you can deal with particles. But then they go back in time.</em></p>
</blockquote>
<p>Is this true that there can be no particle interpretation where particles only travel forward in time? I'm not asking about path intergrals but the particle interpretation in general, I'm also aware the field interpretation is the more popular.</p> | 5,119 | [
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<p>My question is in reference to Landau's Vol. 1 Classical Mechanics. On Page 6, the starting paragraph of Article no. 4, these lines are given:</p>
<blockquote>
<p>If an inertial frame $К$ is moving with an infinitesimal velocity $\mathbf\epsilon$ relative to another inertial frame $K'$, then $\mathbf v' = \mathbf v+\mathbf \epsilon$. Since the equations of motion must have the same form in every frame, the Lagrangian $L(v^2)$ must be converted by this transformation into a function $L'$ which differs from $L(v^2)$, if at all, only by the total time derivative of a function of co-ordinates and time.</p>
</blockquote>
<p>1) Doesn't this hold for same frame? Why is Landau changing the Lagrangian of frame $K$, $L$, to $L'$ with the change satisfying this condition? So, how can he assume that the action would be minimum for the same path in $K'$ as there was in $K$? In two frames the points $q_1$ and $q_2$ aren't same which are at $t_1$ and $t_2$.</p>
<p>2) How did he assume that this is the one and only way to change the Lagrangian without changing the path of least action? Can we prove this?</p>
<p>With respect to first question, I feel that there is something fundamentally amiss from my argument as the Lagrangian is dependent only on magnitude of velocity, so $q_1$ and $q_2$ won't matter. I have made an explanation myself that since the velocity is changed infinitesimally, it should essentially be the same path governed by the previous Lagrangian, the path it took with constant velocity $v$. But, still I am not convinced. The argument isn't concrete in my head. Please build upon this argument or please provide some alternative argument.</p>
<p>I know the question (1) and argument above are very poorly framed but I am reading Landau alone without any instructor and so have problems forming concrete ideas.</p> | 5,120 | [
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<p>The Liouville-von Neumann equation for the <a href="http://en.wikipedia.org/wiki/Density_matrix" rel="nofollow">density matrix</a> is:</p>
<p>$$ i\hbar\frac{\partial\rho}{\partial t}=[H,\rho],$$</p>
<p>while in the Heisenberg picture:</p>
<p>$$ \frac{d}{dt}A(t)=\frac{i}{\hbar}[H,A(t)] +\frac{\partial A(t)}{\partial t}$$</p>
<p>if we adopt some kind of conservation law (like in classical theory), i.e.</p>
<p>$$\frac{d}{dt}A(t)=0. $$ Then we have</p>
<p>$$\frac{i}{\hbar}[H,A(t)] +\frac{\partial A(t)}{\partial t}=0 $$</p>
<p>by replacing $\rho=A(t)$, we can directly reach the von Neumann equation.</p>
<p>Is this derivation correct? if it is, what is the physical meaning behind $\frac{d}{dt}A(t)=0$ then?</p> | 5,121 | [
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<p>In reading a lot of articles about HEP and what the LHC could detect or what it has excluded (like low-mass superpartners) it seems every author essentially assumes that things like low-mass superpartners and dark matter could be produced at the LHC.</p>
<p>How reasonable is it to assume that these particles are producible in principle? Could there be a new conservation law that makes it impossible for ordinary matter / anti-matter collisions to ever produce these particles? Would this conservation law contradict what we currently know about the Standard Model and laws of physics? Could there be low to medium mass dark matter particles or superpartners that we'll simply never be able to produce?</p>
<p>Or does the interaction with gravity ensure that these particles would always be producible, in principle?</p> | 5,122 | [
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<p>I have seen many questions on SE on the dual nature of electrons behaving in certain circumstances as particles and as waves in some other circumstance. There is one thing I couldn't get a clear answer on.</p>
<p>When making double slit experiment, we all agree that the electrons behave as waves. The same is true in atoms, where electron levels are described by Schrödinger equation. However, if we speak about a field like plasma physics (my field of work) and maybe beam physics, electrons are treated classically as particles with applying Newton's equation to describe their motion. The models built on particle treatment of electrons show an excellent agreement with experimental results.</p>
<p>From experimental results and testing, we know that electrons behave like waves (in double slit experiment) or as particles (gas discharge models). My question is, is experimenting the only way to decide which model (wave/particle) describes electrons better in particular circumstances? Isn't there any theoretical frame that decides whether electrons will behave as particles or wave in particular circumstance??</p>
<p>For the record, in plasma physics the strongest type of theoretical models is called Particle In Cell models (PIC). In those models Newton equation of motion is solved for a huge number of particles including electrons. Then the macroscopic properties are determined by averaging. This method although it treats electrons classically it is very successful in explaining what happens in experemints</p> | 5,123 | [
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<p>I read some material in this forum and realize that entanglement entropy does not correspond to long range entanglement. Then what quantity can be used to characterize the topological order in 1+1D topological superconductor that can be obtained numerically? </p> | 5,124 | [
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<p>Does it make sense to introduce Faddeev–Popov ghost fields for abelian gauge field theories? </p>
<p>Wikipedia says the coupling term in the Lagrangian "doesn't have any effect", but I don't really know what that means. If it doesn't work at all (probably because structure constants are zero?) then <em>why</em> doesn't it work, physically? I mean you still have to do things like getting rid of unphysical degrees of freedom/gauge fixing.</p>
<p>Since it isn't done in QED afaik, I guess its not a reasonable way of doing things like computation of self energy/computing corrections of propagators. I wondered, because the gauge field-ghost-ghost vertices for Yang-Mills basically look just like the gauge field-fermion-fermion vertex in electrodynamics.</p> | 5,125 | [
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<p>A particle of mass $m$ moves freely in the interval $[0,a]$ on the $x$ axis. Initially the wave function is:
$$f(x)=\frac{1}{\sqrt{3}}\operatorname{sin}\Big( \frac{\pi x}{a} \Big)\Big[1+2\operatorname{cos}\Big( \frac{\pi x}{a} \Big) \Big]$$</p>
<p>I let the normalised wave function in one dimension be:</p>
<p>$$\Phi_n(x,t)=\sqrt{\frac{2}{a}}\operatorname{sin}\Big( \frac{n\pi x}{a} \Big)\operatorname{exp} \Big( \frac{-in^2\pi^2 h t}{2ma^2} \Big)$$
Where $h$ is $2\pi$ Planck's constant.
So here I have said that as
$$f(x)=\frac{1}{\sqrt{2}}[\Phi_1(x,0)+\Phi_2(x,0)]$$
That for later $t$ the wave function of the particle is given by:
$$\Phi(x,t)=\frac{1}{\sqrt{2}}[\Phi_1(x,t)+\Phi_2(x,t)]$$
Could anyone help me go about normalizing this? (or showing it is already normalized, however I am guessing it is not). And help me find the probability that the particle is in the interval $[0,a/2]$.</p>
<p>A problem I am encountering is:</p>
<p>$$|\Phi|^2=\frac{1}{2}|\Phi_1+\Phi_2|^2=\frac{1}{2}(\Phi_1+\Phi_2)\overline{(\Phi_1+\Phi_2)}=$$
$$\Rightarrow |\Phi|^2=\frac{1}{2}(|\Phi_1|^2+|\Phi_2|^2+[\overline{(\Phi_1)}{(\Phi_1)}+\overline{(\Phi_1)}{(\Phi_1)}])$$
$$\Rightarrow \int_0^a|\Phi|^2=\frac{1}{2}\int_0^a|\Phi_1|^2+|\Phi_2|^2\operatorname{d}x+ \frac{1}{2}\int_0^a\overline{(\Phi_1)}{(\Phi_2)}+\overline{(\Phi_2)}{(\Phi_1)}$$
$$\Rightarrow \int_0^a|\Phi|^2=1+ \frac{1}{2}\int_0^a\overline{(\Phi_1)}{(\Phi_2)}+\overline{(\Phi_2)}{(\Phi_1)}$$</p>
<p>From here I can't see how to proceed to normalize the wave funtion.</p> | 5,126 | [
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<p>A hepa vacuum cleaner will pick up fine dust from the floor, filter it and send the clean air out through the exhaust. However with movement in the room fine dust will also be goinng up in the air and so the vacuum will not take it in and this fine dust will settle hours later.</p>
<p>As far as i can see, a vacuum cleaner is very similar to an air scrubber, takes air in, filters it and sends it out.
<p>1) is it not possible to close windows and leave the vacuum on in the middle of the room and expect it to filter fine dust in the air/room?<br>
<p>2) what if i maneuvered around and tried to vacuum the air aswell as the floor for several hours, would this do the job?
<p>3)is a air scrubber/filter necessary?</p>
<p>The room in question is about $18\,\mathrm{m}^2$ and the vacuum cleaner i intend to use is a sealed hepa unit which is about 500 air watts and says it can do $58\,\mathrm{L/s}$ which i think is litres per second. Please give general answers to the questions I have asked aswell as specific to the vacuum and room in question. Thanks.</p> | 5,127 | [
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] |
<p>To add to <a href="http://physics.stackexchange.com/questions/884/books-that-every-physicist-should-read">Books that every physicist should read</a>:<br>
A list of popular physics books for people who aren't necessarily interested in technical physics. </p>
<p>(see also <a href="http://physics.stackexchange.com/questions/12175/book-recommendations">Book recommendations</a>)</p> | 1,025 | [
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<p>The friction term in Navier-Stokes equation assumes that the viscosity coefficients are the same for the longitudinal and transverse directions. This doesn't seem intuitive, because the former is essentially a bulk modulus while the latter doesn't involve any compression of the fluid. How is the assumption justified?</p> | 5,128 | [
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<p>In Jackson's text he says that Faraday law is actually:
$$
\oint_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell} = -k\iint_{\Sigma} \frac{\partial \mathbf B}{\partial t} \cdot \mathrm{d}\mathbf{S}
$$
where $k$ is a constant to be determined.<strong>(page 210, third ed.)</strong>.He claims that $k$ is not an independent empirical constant that must be measured from experiment, but is an inherent constant which for each system of units can be determined by <strong>Galilean invariance</strong> and also <strong>Lorentz force law</strong>.He writes the Faraday's law in two frames, lab frame and a moving frame with velocity $\mathbf{v}$, and by writing the above law in each of two frames and assuming :</p>
<ul>
<li><p>electric field in one frame is $\mathbf{E}'$ and in the other is $\mathbf{E}$ (so they are different) , but magnetic field is $\mathbf{B}$ in both frames!</p></li>
<li><p>Galilean invariance needs :$$\iint_{\Sigma} \frac{\partial \mathbf B}{\partial t} \cdot \mathrm{d}\mathbf{S}
$$ be equal in two frames deduces that :</p></li>
<li><p>$k=1$ </p></li>
</ul>
<p>and also </p>
<ul>
<li>the electric field in the moving reference frame is $$\mathbf{E}' = \mathbf{E} + \mathbf{v} \times\mathbf{B}$$.</li>
</ul>
<p>I know that this electric field ($\mathbf{E}'$ ,in the moving frame ) is only an approximation and the real $\mathbf{E}'$ that can be obtained using Lorentz transformations. Now the question is that</p>
<ul>
<li><p>how Galilean transformations which are wrong (are approximately correct) give the correct answer for $k$ ?</p></li>
<li><p>Why we should assume that there are two electric fields ,one in the lab frame and one in the other , but just one magnetic field in both frames?</p></li>
</ul> | 5,129 | [
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] |
<p>The second Bianchi Identity is
$$
\nabla_{[a}R_{bc]de}=0
$$</p>
<p>As far as I know, the proof (say, <a href="http://mathworld.wolfram.com/BianchiIdentities.html" rel="nofollow">Walfram Mathword</a>) start by stating the representation of <a href="http://en.wikipedia.org/wiki/Riemann_curvature_tensor" rel="nofollow">Riemann tensor</a> in <a href="http://www.google.com/search?as_epq=local+inertial+coordinates" rel="nofollow">local inertial coordinates</a>
$$
R_{abcd}=\frac{1}{2}(\partial_a\partial_cg_{bd}-\partial_a\partial_dg_{bc}-\partial_b\partial_cg_{ad}+\partial_b\partial_dg_{ac}).
$$</p>
<p>Then we calculate
$$
\partial_aR_{bcde}
$$</p>
<p>accordingly. Then we say that it is true in a local inertial coordinate, and after changing partial derivative into covariant derivative, it is true in general.</p>
<p>My concern is, I think we cannot express the Riemann tensor and the covariant derivative into local frame one by one, but should simultaneously. Say
$$
\nabla_{a}R_{bcde}=\frac{1}{2}(\partial_a+\Gamma_1)(\partial_a\partial_cg_{bd}-\partial_a\partial_dg_{bc}-\partial_b\partial_cg_{ad}+\partial_b\partial_dg_{ac}+\Gamma_2)
$$</p>
<p>where $\Gamma_1$ and $\Gamma_2$ are some terms involving the Christoffel symbol. When we only concern $R_{bcde}$ in a local frame, $\Gamma_2$ vanishes. But now we get a new term
$$
\partial_a\Gamma_2
$$</p>
<p>which I cannot see to vanish because it involves derivative of the Christoffel symbol. So I think in a local frame $\nabla_aR_{bcde}$ is not $\partial_aR_{bcde}$.</p>
<p>Is there anything wrong?</p> | 5,130 | [
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<p>I'm refering to the Paper:
PHYSICAL REVIEW B 80, 195310 (2009)
"Möbius graphene strip as a topological insulator"
Z. L. Guo, Z. R. Gong, H. Dong, and C. P. Sun. </p>
<p>The paper is also available as a preprint version via:
<a href="http://arxiv.org/abs/0906.1634" rel="nofollow">http://arxiv.org/abs/0906.1634</a></p>
<p>When I'm refering to an equation, I'll also use the arxiv-reference (since it's freely available). </p>
<p>I'm refering to Section II: Edge States in Möbius Graphene Strip, Equations (12), (13). </p>
<p>I can show that these linear combinations do indeed satisfy the periodic boundary conditions. What I do not see is why you have to use y>0 and y<0 in the proof?
At which stage of the periodicity proof does one need to distinguish between y>0 and y<0? </p>
<p>It's true that is seems to be a natural distinction, because one can cut the moebius strip in the middle and obtain two cylinders as shown in Fig 3. </p>
<p>I'd be more than happy on some advice. </p>
<p>Best regards.</p> | 5,131 | [
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<p>The quaternion Lorentz boost $v'=hvh^*+ 1/2( (hhv)^*-(h^*h^*v)^*)$ where $h$ is $(\cosh(x),\sinh(x),0,0)$ was derived by substituting the hyperbolic sine and cosine for the sine and cosine in the usual unit quaternion rotation $v'=hvh^*$ and then subtracting out unwanted factors in <em><a href="http://visualphysics.org/preprints/qmn10091026" rel="nofollow">Lorentz boosts with Quaternions</a></em>.</p>
<p>Does this transformation form a group? Do two transformations of the form $v'=hvh^*+ 1/2( (hhv)^*-(h^*h^*v)^*)$ make another transformation of the form $v'=fvf^*+ 1/2( (ffv)^*-(f^*f^*v)^*)$? </p>
<p>Is this transformation useful?</p>
<p>For more information, see his blog, <em><a href="http://www.science20.com/standup_physicist" rel="nofollow">The Stand-Up Physicist</a></em>.</p> | 5,132 | [
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<p>In one of the documentaries hosted by Morgan Freeman, a reference was made that just like an ordinary three-dimensional object like a ruler has scratches and cracks, in the same way there might be minuscule loops in space-time. My question is this: If many bodies of extremely high densities are brought close together, will it deform the space-time fabric and hence create a permanent loop in space-time that might be accessible if ever achieved?</p> | 5,133 | [
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<p>I once tried to make a magnetic wheel using different shaped magnets and placing them in different orientations and of different shapes.But it did rotate making a small angle ie it didnt rotate even a full circle.I still think I can do this with strongest magnet with a magnet with required north south pole arrangement or with two opposite poles neutral and other two opposite poles working.Is this feasible?I require some suggestions please.</p> | 5,134 | [
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<p>I tried calculating this, but it gets too complicated.</p>
<p>Assume, we have a Moon orbit station and ISS on Earth orbit. We have a Moon base. We want to send a tourist for a week on the Moon and back. We need to launch only the oxygen/fuel cells and the fuel for trans-lunar and trans-earth injection.</p>
<p>The lunar lander lands and takes off in one piece (we don't need a pile of used stages on the lunar base). It can be lighter than 10 metric tons of Apollo LEM. Fuel cells (200 kg each) may go, because now we have solar panels.</p>
<p>So, how low can we get with newer technologies? And the main question, how small can the launch weight be? Is this doable with conventional Soyuz or Proton rocket?</p>
<p>Assume, we might have a coilgun to do trans-Earth injection right from the lunar surface. Delta-v of 2.7 km/s means 90 km of rail and accelerating @ 5g, if I remember my calculations correctly.</p>
<p>Some data from Wikipedia:</p>
<ul>
<li>Apollo Lunar Module
<ul>
<li>Ascent stage 4,547 kg of which 2,353 kg is propellant</li>
<li>Descent stage: 10,149 kg (8,200 kg of propellant for 2,500 m/s delta-v)</li>
</ul></li>
<li>Command and Service module
<ul>
<li>command module 5,809 kg</li>
<li>service module 24,523 kg</li>
</ul></li>
</ul>
<p>all together 46,980 kg</p>
<ul>
<li>Saturn V
<ul>
<li>1st stage with fuel: 2,300,000 kg</li>
<li>2nd stage: 480,000 kg</li>
<li>3rd stage: 120,800 kg</li>
</ul></li>
</ul>
<p>The third stage fired only partially (165 + 335 seconds) to get to LEO, and then was used for TLI.</p>
<p>So, 120 tons were sent to LEO, and only 47 tons left after TLI. How low can the latter get?</p>
<p>The lunar module will be orbiting the Moon and reused. This means several tonnes less for TLI and Lunar orbit injection. But the fuel for it has to fly from Earth.</p>
<p>If CM gets smaller, this can also make the vehicle lighter.</p>
<p>Service module can be reduced by using inflatable materials, and it can be reused and stored at ISS. So we save 25 tons x 9 km/s (launch from Earth), but add 3km/s of delta-v to park it after the way back.</p>
<p>So, all together, we need to launch</p>
<ul>
<li>the Earth landing module (CM analog)</li>
<li>fuel for
<ul>
<li>TLI</li>
<li>LOI</li>
<li>lunar landing and takeoff</li>
<li>TEI</li>
<li>fuel to park SM in LEO</li>
</ul></li>
</ul>
<p>If coilgun is used, we don't need takeoff and TEI fuel. (Hm... we need to launch the SM back too :)</p>
<p>How much does this weigh?</p> | 5,135 | [
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<p>During a lecture that I missed, I was trapped when the lecturer uses the relation</p>
<p>$$dp_x~ dp_y ~dp_z ~=~d^3\mathbf{p} ~=~ 4\pi p^2 dp.$$</p>
<p>Can I know how is this relation derived please?</p> | 5,136 | [
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<p>After reading <a href="http://physics.stackexchange.com/questions/16090/how-do-we-resolve-a-flat-spacetime-and-the-cosmological-principle/">How do we resolve a flat spacetime and the cosmological principle?</a> I still remain perplex.<br>
Please excuse my ignorance and try explaining to me : </p>
<p>I thought that basically, when we rewind back to the big bang, we get down to planck's dimension (something like 10exp-35) which is small and therefore (?) finite. (i acknowledge we have yet no theory beyond that).<br>
Since :<br>
<strong>big bang => small</strong><br>
<strong>small => finite</strong><br>
<strong>finite * whatever_expansion = finite</strong><br>
<strong>finite ~> curved</strong> (but see below point #2)<br>
I derive :<br>
<strong>big bang ~> should still be curved</strong></p>
<p>So, just like @adam asked (see link above), how can spacetime be said to be flat now ?<br>
May be my question simply gets down to clarify :<br>
When experts say "flat", do they mean : </p>
<ol>
<li><strong>strictly flat whatever the geometry</strong> (and then i am lost) </li>
<li><strong>strictly flat, but in the sens of specific geometry</strong> like <em>"Flat universe ... In three dimensions, there are 10 finite closed flat 3-manifolds, of which 6 are orientable and 4 are non-orientable"</em> as mentionned in [wikipedia Shape_of_the_Universe] (http://en.wikipedia.org/wiki/Shape_of_the_Universe)</li>
<li>or : <strong>nearly flat</strong> only, as we can observe, (but can't be strictly, because ... see above my reasoning). </li>
<li>other ? (please elaborate ...) </li>
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<p>I am reading a book about wave mechanics. There are two different cord (one light and one heavy) connected together, one person waving the lighter one, the wave transverse to the right from the lighter one to the heavy one. The frequency and the wavelength is given so the speed and the tension of the lighter cord are unknown. The textbook stated that when the wave traveled into the heavier cord, the tension and the frequency doesn't change. But I am wondering why the tension as well as the frequency stay unchanged? Why not the wavelength stay unchanged? What's the physical explanation of it?</p> | 5,138 | [
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<p>I am planning a series of science fiction novels that take place on an imaginary binary planet system. Both planets have a lower surface gravity than the Earth and one has slightly more mass than the other. </p>
<p>If average temperature is the same as earth or cooler and the radius of the larger planet is allowed to vary, what is the minimum amount of gravity required for the larger planet to sustain a 50:50 Nitrogen/Oxygen atmosphere with a total pressure of 100 mmHg at 2000 m above sea level?</p>
<p>What law or theorem would I need to solve this problem.</p>
<p><strong>Extra stuff about the planets, my motivations and other things I'm trying to figure out that you don't have to read if you don't want to:</strong></p>
<p>I want the larger planet to have an atmosphere that is thinner and whose pressure varies to a greater degree by altitude than the atmosphere of the Earth, but is still breathable by humans at sea level. Basically I want Denver to feel more like Mount Everest and Mount Everest should be pretty much vacuum. Mount Everest has an oxygen partial pressure of about 43 mmHg, and Denver is at an altitude of 1.6 km. Let's say we want 50 mmHg of oxygen at 2km. I figure if oxygen concentration is higher on the planet then on Earth, then atmosphere can be thinner and overall pressure can be lower. I picked a 50:50 mix not knowing whether it is really feasible, but if it is then pressure could be 100 mmHg at 2km. I don't know how to calculate how big the radius and mass of the planet has to be to satisfy these or what the pressure would be at sea level and at what altitude pressure will be 0.</p>
<p>The smaller planet should have an atmosphere that is too thin for humans but could conceivably host sentient life.</p>
<p>Both planets should have lower gravity than the earth. Each planets has a different species of sentient beings and I would like them to be able to lift stuff into space for much cheaper than here on earth. Basically by the time they have something like a Saturn V rocket, there should already be some trade in manufactured goods between the two. </p>
<p>Both planets have awesome magnetic fields that are way better than Earth's at doing the stuff that magnetic fields do. I also would like the binary planets to be in a binary star system and have one or two small moons each, but this is probably pushing it. Even one small ice moon would serve as a handy plot device but I don't want to make it too improbable and I do want to get as much of the Physics right as I can and figuring out the orbits, climate map and the seasons will be difficult enough as it is with two planets, especially considering how little I know.</p> | 5,139 | [
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<p>I'm trying to learn theoretical physics up to string theory. I know linear algebra, calculus 1+2, complex analysis. I know the basics of homology, homotopy, group theory and differential geometry. Now I'm starting to read a first introduction to QM, which is this PDF: <a href="http://arxiv.org/abs/1007.4184" rel="nofollow">http://arxiv.org/abs/1007.4184</a> . And I read a lot on physics exchange on these topics. My main goal is to get a good sense of string theory and maybe that I can solve some basic problems. Now I did read that the historical background of string theory is the S-Matrix(which I guess has to do with the matrix formulation of QM).</p>
<p>Should I first learn this approach to QM and then switch to S-Matrix and finally string theory? Or just completely skip these topics(S-Matrix and Matrix Mechanics) and learn string theory in an ordinary style?(like in the usual literature, which I assume is not taught with the S-Matrix model). </p>
<p>Edit/ I know usual mechanics, theoretical mechanics(Lagrangian and Hamiltonian too) and a bit electromagnetism, but didn't do much problems on these topics, but I know the concepts.</p> | 5,140 | [
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<p>At a certain instant in AC generator, when the normal of the plane (rectangular coil) makes an angle of 270 degrees with with the magnetic induction B, the value of emf is:</p>
<p>$E = -NAB\omega$</p>
<p>My teachers would usually say that this is the minimum value of emf that a generator produces. Does it really mean that? Or does the negative sign only mean that emf is at its peak value but the current is flowing in opposite direction?</p> | 5,141 | [
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<p>Consider the "half BHZ" Hamiltonian</p>
<p>$${\cal H}=\sum_{\mathbf{k}}\left(A\sin(k_{x})\sigma_{x}+A\sin(k_{y})\sigma_{y}+{\cal M}(\mathbf{k})\sigma_{z}\right)c_{\mathbf{k}}^{\dagger}c_{\mathbf{k}}$$ where $M(\mathbf{k})=M-2B\left(2-\cos(k_{x})-\cos(k_{y})\right)$</p>
<p>For simplicity, take A=1 and B > 0. M < 0 and M > 8B give a trivial phase, while 0 < M < 4B and 4B < M < 8B give topological phases with Chern number $\pm1$ and the corresponding edge states. The existence of the edge states can be easily verified by numerically diagonalizing the Hamiltonian for periodic boundary conditions in, say, the x direction and vanishing boundary conditions in the y direction. They can also be found analytically by expanding the Hamiltonian around the TRIM points $(k_x, k_y) = (0, 0), (0, \pi), (\pi, 0), (\pi, \pi)$ to second order and looking for solutions localized near a boundary. The trouble is, the expansion around $(0,0)$ seems to imply that the only condition for the existence of a localized state at $k_x=E=0$ is MB > 0. But we know that for M > 4B there's no such state, so what happens to it? I think it's supposed to somehow pair-annihilate with the $(0, \pi)$ edge state when M becomes bigger than 4B, but according to the expansion around $(0, \pi)$ the $(0, \pi)$ edge state vanishes on its own. Is there some other explanation for the disappearance of the $(0, 0)$ edge state? Or does it annihilate with the $(0, \pi)$ edge state after all, but there's no way to see it from the local Hamiltonians?</p> | 5,142 | [
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<p>I have a professor that is fond of saying that vorticity cannot be destroyed. I see how this is true for inviscid flows, but is this also true for viscous flow? The <a href="http://en.wikipedia.org/wiki/Vorticity_equation">vorticity equation</a> is shown below for reference. From this equation, it looks as if vorticity only convects and diffuses. <strong>This would suggest that it can't be destroyed.</strong></p>
<p>$$\frac{D\boldsymbol{\omega}}{Dt} = (\boldsymbol{\omega}\cdot\nabla)\boldsymbol{V} + \nu\nabla^2\boldsymbol{\omega}$$</p>
<p>However, consider this thought experiment:</p>
<p>Suppose we have a closed container filled with water with initial vorticity field $\boldsymbol{\omega}_0$ at time $t_0$. If the container is allowed to sit undisturbed, as $t\to\infty$ the water will become stationary ($\boldsymbol{V}\to 0$) with zero vorticity ($\boldsymbol{\omega}\to 0$).</p>
<p><strong>This suggests vorticity can be destroyed.</strong></p>
<p>My professor claims the boundary-layer vorticity at the sides of the container is equal and opposite in sign to the bulk vorticity. If this is the case, the vorticity cancels out after a long time resulting in the stationary fluid and vorticity is not destroy (just cancelled out).</p>
<p><strong>EDIT:</strong> I'm looking for either a proof that the boundary-layer vorticity is equal and opposite to the bulk vorticity <strong>or</strong> a counter-explanation or proof. (I'm using proof in a very loose hand-wavy sense)</p> | 5,143 | [
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<p>If I want to punch a person inflicting maximum damage, what do I need to care about?</p>
<p>My force of punching, i.e, do I need more acceleration? Or do I need momentum, i.e my velocity for punching?</p> | 5,144 | [
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<p>I am using a thermoelectric cooler from a pc's heatsink to produce electricity. Its size is 30mm by 30mm. I will cool it on one side at -10 degree Celsius and the other at 24 degree Celsius. Can anyone tell me how much electricity in watts will be produced. I'm using this heatsink cooler because it's the most easily available to me. Can anyone suggest an even better thermoelectric generator? Also how much electricity it would produce and how much it will cost?</p> | 5,145 | [
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<p>The <em>back motor effect</em> (see <a href="http://en.wikipedia.org/wiki/Counter-electromotive_force" rel="nofollow">Counter-electromotive force</a>) is the counter torque which opposes the rotational motion of the coils in a generator when the generator is under load.</p>
<p>The back motor effect depends upon the load connected to the generator. The more load, the more current will be produced which will increase the counter torque. But to produce more current, the normal torque will also increase. So, shouldn't this balance the effect? If it does, then why do small generators change sound (nearly shutting down) when load in increased on them?</p> | 5,146 | [
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<p>For a charged quantum particle, say, an electron or a quark, how in the particle's wavefunction is the electric charge represented? Is it truly possible to represent electric charge using the wave mechanics formulation of quantum mechanics, or is that something only matrix mechanics can satisfy? And does the same apply for other quantum properties, such as color charge or spin?</p> | 5,147 | [
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<p>Perhaps it is some free moving spinner attached to the wheel, but as opposed to this question: <a href="http://physics.stackexchange.com/questions/41006/why-does-the-wheel-of-a-car-appear-to-be-moving-in-opposite-direction">Why does the wheel of a car appear to be moving in opposite direction?</a>
I have seen car wheels that appear to spin backward both while speeding up, and moving at a constant rate.
Am I just confused, or is there a reason for this?</p> | 5,148 | [
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<p>Follow by the Michelson–Morley experiment, What happen if we use something that has a speed lower than light instead of light in Michelson–Morley experiment? How about the result?</p> | 5,149 | [
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<p>Currently, my experiment involves the three elements: <code>Au, Ga, Bi</code>, and I need <code>Au-Ga-Bi ternary phase diagram</code> to explore the underlying reason for my experimental phenomenon. Unfortunately, I can't find any information about this ternary phase diagram both in <code>Web of Knowledge</code> and <code>Google</code>. It should be impossible that nobody would think of this ternary phase diagram since the first phase diagram was published, because there are so many scientists. So I wonder whether Au-Ga-Bi ternary system has something special that impedes the calculation of its phase diagram. Otherwise, why no one does it?</p>
<p>I'd appreciate it greatly if anyone could provide some information.</p> | 5,150 | [
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<p>Einstein described his discovery of the equivalence principle as the "happiest thought of my life". Why? What, in broad conceptual terms, is the logical chain of reasoning that leads from the equivalence principle to general relativity?</p> | 5,151 | [
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<p>How can you calculate the amount of energy an object produces that falls from a particular height?
Or water.</p>
<p>In an eg i would like to know how can you calculate the amount of electricity water can produce at a certain height?</p> | 5,152 | [
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<p>In my acoustics books I see</p>
<p>$$c^2 = \frac{\mathrm{d}P}{\mathrm{d}\rho}$$</p>
<p>where $c$ is the speed of sound, $P$ is the pressure and $\rho$ is the density. Where does this equation come from? In my books it appears almost as a definition. Could you explain this, or at least point me to an article or a book that addresses this question? Thanks.</p> | 5,153 | [
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<p>I notice that the larger the truck the greater the magnitude of the lurch. Can anyone give a physical explanation to this?</p> | 5,154 | [
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<p>According to the Wikipedia article on <a href="http://en.wikipedia.org/wiki/Black_hole#Gravitational_collapse">black holes</a>:</p>
<blockquote>
<p><em>Even though the collapse takes a finite amount of time from the reference frame of infalling matter, a distant observer sees the infalling material slow and halt just above the event horizon, due to gravitational time dilation. Light from the collapsing material takes longer and longer to reach the observer, with the light emitted just before the event horizon forms delayed an infinite amount of time. Thus the external observer never sees the formation of the event horizon; instead, the collapsing material seems to become dimmer and increasingly red-shifted, eventually fading away.</em></p>
</blockquote>
<p>If this is the case, then why is there a problem about information loss in a black hole? To any observer not inside a black hole, black holes do not exist.
Thus there is no problem about information loss. In the same vein, isn't it misleading to say, for instance, that the engines of quasars are black holes? When we observe a quasar, all its mass is still observable in principle (though too red-shifted to see in practice) and no black hole exists.</p> | 178 | [
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<p>Could someone please explain the difference between a <a href="http://en.wikipedia.org/wiki/Wave_packet" rel="nofollow">wave packet</a> and a wave train? I have rummaged around online but have not been able to find a definitive definition. </p> | 5,155 | [
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<p>In <a href="http://www.tapir.caltech.edu/~chirata/ph236/2011-12/lec19.pdf" rel="nofollow">this PDF</a> [1], is made reference to specific energy and angular momentum of a particle. If the particle has no mass, like a photon, how should I define these terms in the equations further down for the path of the particles?</p>
<p>[1] Lecture XIX, Christopher M. Hirata, Caltech M/C 350-17</p> | 5,156 | [
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<p>I've been experimenting with radiative cooling lately and in my mind <a href="http://en.wikipedia.org/wiki/Convective_heat_transfer" rel="nofollow">convection</a> is always an enemy (especially forced convection since the modules are outdoors).</p>
<p>Looking at <a href="http://en.wikipedia.org/wiki/Lumped_capacitance_model" rel="nofollow">Newton's law of cooling</a>, it doesn't seem to me that it accounts for surfaces made of different materials losing/gaining heat (can be wrong though, not uncommon especially when working with formulas).</p>
<p>Does it matter (much), that surfaces are made from different materials when we talk about convective heat gains/losses?</p>
<p>I'm only saying this because metal cooled more under radiation than a type of plastic I used, and to me that doesn't make much sense as the convective heat gain should, in my mind, be greater in the metal panel and therefore it shouldn't cool more than the plastic one (both had similar emissivity because of a very thin paint coating).</p>
<p>My experiments tell me that I'm most likely wrong in assuming the convective heat gains/losses will be greater in metals, but I thought it's most likely better to check with those who actually know what they're talking about, it may be some weird fluke in the tests.</p> | 5,157 | [
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<p>As I understand it, gravity is inherent to mass and therefore even a small rock has its own gravitational pull. It seems entirely plausible then that a rock 1" in diameter could orbit a bigger rock, say 10" in diameter. Is this actually possible? Are there any practical limits on the size of objects that could have another object orbiting them?</p>
<p>What kind of environment would be necessary to facilitate something like this occurring? Could it happen in a high or low earth orbit? Would it have to be far away from any stars, planets, or other large bodies? It seems to me that the speed of the objects would have to be very, very slow.</p>
<p>I have no background in physics and don't know a ton of math so answers without a lot of complicated equations would be appreciated. Thanks.</p> | 5,158 | [
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<p>Systems of charged particles (such as atomic nuclei and electrons) can be described by nonrelativistic quantum mechanics with the Coloumb interaction potential. A fully relativistic description is given by quantum electrodynamics which is much more complex.</p>
<p>Is it possible to expand various quantities in QED as power series in 1/c i.e. around the nonrelativistic approximation? Examples of relevant quantities are:</p>
<ul>
<li>Ground state energy of a given set of charged particles</li>
<li>Excited state energies</li>
<li>Scattering cross sections of charged particles & their bound states (assuming we trace over the photons in the final state)</li>
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<p>Quantum entanglement links particles through time, according to this study that received some publicity last year:</p>
<p><ul>
<li><a href="http://www.technologyreview.com/blog/arxiv/26270/" rel="nofollow">New Type Of Entanglement Allows 'Teleportation in Time,' Say Physicists</a> at <a href="http://www.technologyreview.com/blog/arxiv/" rel="nofollow">The Physics arXiv Blog - Technology Review</a> </li>
<li>S.J. Olson, T.C. Ralph, <em>Extraction of timelike entanglement from the quantum vacuum</em>, <a href="http://arxiv.org/abs/1101.2565" rel="nofollow">arXiv:1101.2565</a></li>
<li>S.J. Olson, T.C. Ralph, <a href="http://dx.doi.org/10.1103/PhysRevLett.106.110404" rel="nofollow">Entanglement between the Future and the Past in the Quantum Vacuum</a>, Phys. Rev. Lett. 106, 110404 (2011)</p>
<blockquote>
<p>Timelike entanglement may be regarded as a non-classical resource in a manner analogous to the spacelike entanglement that is often studied in the Minkowski vacuum, since any quantum information theoretic protocol may utilize conversion of timelike entanglement to spacelike entanglement as a step in the protocol.</li>
</ul>
I have read all the reviews in the popular press, most of the "time-travel-discovered" type, but I am looking for more sober comments, i.e. material written by physicists who have studied the paper. What does this result <em>really</em> mean? I would appreciate pointers to reviews, comments etc.</p>
</blockquote> | 431 | [
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<p>Is it possible to disregard drag force of water with no viscosity that affects lightweight pop-up ball (its weight is assumed to be zero)? There is a discussion, on which I consider that although weight is small, the influence of drag force does not depend on ball but on water, therefore this force must be taken into account. What can you say?</p> | 5,160 | [
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<p>For a countable sequence of positive numbers $S=\{\lambda_i\}_{i\in N}$ is there a construction producing a Hamiltonian with spectrum $S$ (or at least having the same eigenvalues for $i\leq s$ for some $s$)?</p>
<p>Here by 'Hamiltonian' I understand a polynomial of $p_i$ and $q_i$ (or equivalently - $a_i$ and $a_i^\dagger$) of $k$ pairs of variables and of order $2n$. Both $k$ and $n$ can be functions of $S$ and $s$.</p>
<p>For example, for the spectrum
$$S =\{3,5,7,9,\ldots\}$$
one of Hamiltonians working for any $s$ is
$$H = 3+a^\dagger a.$$</p> | 5,161 | [
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<p>When an electron emits a photon from changing energy levels, the frequency of the photon depends on the difference between the energy levels.</p>
<p>But if someone is moving with respect to the atom, the frequency will be apparently red shifted or blue shifted.</p>
<p>Does this mean that the energy levels of the orbits of the atom look to have different values if you are moving with respect to them? But, the apparent energy difference can be different depending on whether the photon is emitted towards you or away from you.</p>
<p>What's going on? Aren't energy levels of orbits supposed to be a fixed value given the kind of atom?</p> | 5,162 | [
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<p>In statistical mechanics, we usually think in terms of the Hamiltonian formalism. At a particular time $t$, the system is in a particular state, where "state" means the generalised coordinates and momenta for a potentially very large number of particles. (I'm interested primarily in classical systems for the sake of this question.) Since this state cannot be known precisely, we consider an ensemble of systems. By integrating each point in this ensemble forward in time (or, more often, by considering what would happen if we were able to perform such an integral), we deduce results about the ensemble's macroscopic behaviour. Using the Hamiltonian formalism is useful in particular because it gives us the concept of phase space volume, which is conserved under time evolution for an isolated system.</p>
<p>It seems to me that we could also consider ensembles within the Lagrangian formalism. In this case we would have a probability distribution over initial values of the coordinates (but not their velocities), and another distribution over the final values of the coordinates (but not their velocities). (Actually I guess these would need to be two jointly distributed random variables, since there could easily be correlations between the two.) This would then lead to a probability distribution over the paths the system takes to get from one to the other. I have never seen this Lagrangian approach mentioned in statistical mechanics. I'm curious about whether the idea has been pursued, and whether it leads to any useful results. In particular, I'm interested in whether the idea of phase space volume has any direct meaning in terms of such a Lagrangian ensemble.</p> | 5,163 | [
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<p>Assuming Higgs is found at 125 GeV.Is there any direct or indirect consequence on string theory ? Will it be a blow to string theory or models employing string theory ? </p>
<p>Thanks</p>
<p>Ps - I am just a curious pure maths student, so forgive me if my question makes no sense ! :)</p> | 5,164 | [
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<p>This might be hard to ask, but here goes nothing.</p>
<p>I recently poured a cup of water into a black coffee cup. There was a light source--not very bright--above the cup. Anyways, I was squeezing a lemon into my water mindlessly, and to make sure I got every last drop of lemon juice into the water, I watched the lemon juice hit the water. Then I looked closer as something pretty neat was happening.</p>
<p>When a single lemon drop hit the water, it dissipated into a shape that many would describe as a "smoke ring." Knowing some physics/fluids, I understood what was happening here was nothing out of the ordinary. But then I kept watching. As the ring dissipated, it eventually broke off into about 5 other smaller "smoke rings." I couldn't see further down, so who knows if it continued; but I would assume that the 5 smaller rings would turn into 5 + X amount more, etc.</p>
<p>At what point would they stop breaking up into smaller rings? Whats causing this to happen? Does this movement/shape have a scientific term? Does anything else do this that can be easily seen?</p>
<p>I'm more interested in what the movement/pattern that is happening here, not the chemistry aspect.</p> | 5,165 | [
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<p>For this circuit (a and b are connected by a battery), </p>
<p><img src="http://i.stack.imgur.com/CVtE3.png" alt="circuit diagram"></p>
<p>Will I be able to find the total resistance of the circuit by adding resistors that are in series and combining resistors that are in parallel without using the method of Kirchhoff's Voltage and Current Law?</p>
<p>I tried adding R3+R4 and this forms one resistor, which is in parallel with R2 and R1. And R2 is in parallel with R1 and R3, and R1 is in parallel with R2 and R3. This can be verified by Kirchhoff's Voltage laws. The problem is, I get stuck here. I know that if R1,R2 and R3+R4 can be combined into one resistor, then parallel-series would solve the problem neatly. But I can't see how these resistors be combined. </p> | 5,166 | [
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<p>I have some questions about the construction of $\mathcal{N}=2$ supermultiplets <strong>for chiral matter</strong>. I know that the supermultiplet should not include spin one states since they are always in the adjoint representation. So my first question is: <strong>why are spin-one modes always in adjoint representation?</strong></p>
<p>To avoid confusion, I will denote the four supersymmetry generators as $Q^A_\alpha$, where $\alpha$ is the spinor index and $A=1,2$.</p>
<p>Firstly, massless multiplet or short multiplet.
We know $Q^A_2$ generates states with zero norm. Then we can just focus on $Q^A_1$. The supermultiplet can be constructed in this way:
$$|\Omega_{-\frac{1}{2}}>\\
Q^{\dagger 1}_\dot{1} |\Omega_{-\frac{1}{2}}>, \, \, Q^{\dagger 2}_\dot{1} |\Omega_{-\frac{1}{2}}>\\
Q^{\dagger 2}_\dot{1} Q^{\dagger 1}_\dot{1} |\Omega_{-\frac{1}{2}}>$$
The two spin-zero states in the second line form an $SU(2)$ doublet.</p>
<p>Secondly, BPS states. In the presence of a central charge $Z$, one can write the four generators in terms of $A_\alpha$ and $B_\alpha$ after linear transformation with $\{ B_\alpha, B^\dagger_\beta\}=\delta_{\alpha\beta}(M-\sqrt{2}Z)$ and $\{A_\alpha, A^\dagger_\beta\}=\delta_{\alpha\beta}(M+\sqrt{2}Z)$. When $M=\sqrt{2}Z$, one can get $\{ B_\alpha, B^\dagger_\beta\}=0$. Thus, the multiplet will be:
$$|\Omega_0>\\
A^\dagger_\dot{1}|\Omega_0>, \, \, A^\dagger_\dot{2}|\Omega_0>\\
A^\dagger_\dot{1} A^\dagger_\dot{2}|\Omega_0>$$
Again there are only four states, the same as the short multiplet.</p>
<p>So do the two fermion states in BPS states form some kind of doublet as in the short multiplet? It's not so obvious since the relation between $A^\dagger_\dot{1}$ and $A^\dagger_\dot{2}$ are different from that of $Q^1_1$ and $Q^2_1$.</p>
<p>From the above discussion, <strong>is it legitimate to conclude that BPS states are different from the short multiplet in $\mathcal{N}=2$?</strong> For instance, in the short multiplet, the two scalars form a doublet while I don't see that in BPS states. But in Alvarez-Gaume's review paper hep-th/9701069 section 2.9, he mentioned that the BPS state belongs to a $\mathcal{N}=2$ short multiplet and the two scalar modes form a doublet while the two fermions are singlet. It's like Alvarez-Gaume was saying the two states are exactly the same. So what's missing in my thinking?
Thanks a lot. </p> | 5,167 | [
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<p>A ball is thrown straight up. The height of the ball above the ground in (m) is given by the function:</p>
<p>$$h(t)=-5t^2 + vt + c$$</p>
<p>What is the initial velocity if you want the ball to reach a max height of 100 m?</p> | 5,168 | [
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<p>Let's say you have a spherical charge distribution of radius R. This distribution has some charge density as a function of radius. I know that I can determine the electric field outside of the charge density by forming a spherical gaussian surface around the charge distribution and apply gauss' law. But what if I want to find the field directly on the surface. That requires that my surface intersects some of the charge of the distribution. In that case, when I apply Guass' Law. How do I determine the charge enclosed by the sphere. Do I discount the charge which exists on the surface or is that counted as enclosed by the sphere?</p> | 5,169 | [
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<p>It is written in my quantum physics book that the K shell contains only 2 electrons due to the Pauli principle.</p>
<p>I know that if $n = 1, l = 0, m = 0$, then the Hilbert space associated to the spin is of dimension $2$. I also know that Pauli principle says that if we have a vector $\lvert\psi\rangle$ which represents the state of N electrons, then $\lvert\psi\rangle$ must be antisymmetric by transposition. But how can we conclude please ?</p> | 5,170 | [
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<p><img src="http://i.stack.imgur.com/lVTfu.jpg" alt="Witten's Dog"></p>
<p>In an old episode (<a href="http://en.wikipedia.org/wiki/Mars_University" rel="nofollow">"Mars University"</a>) of Futurama which is a TV show, a character named Professor Farnsworth was trying to lecture "Superdupersymmetric String Theory" and <a href="http://www.google.com/search?q=Witten%27s+Dog" rel="nofollow">"Witten's Dog"</a> to some other characters.</p>
<p>He wrote on board a Feynman diagram, the equation under the diagram looks very much like the electron capture mechanism: $p + e^- \rightarrow n + \nu_e$, except for that wavy underline. However, the Feynman-like diagram above does not look like the electron capture. Also is there any meaning to that caption:"Neutron encrusted steaming hot dark matter"?</p>
<p>Are any of the concepts reference in this chalkboard real physical concepts? If so, what are they?</p> | 5,171 | [
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<p>Can anyone help me Convert a 200 mm linear stroke into 90 degrees motion with as much <a href="http://en.wikipedia.org/wiki/Mechanical_advantage" rel="nofollow">mechanical advantage</a> as possible or into two 90 degrees motions with as much mechanical advantage as possible? </p>
<p>Essentially I am trying to <em>convert the linear motion of a cylinder</em> into 90 degrees to bend a wire. Preferably, I'd like this setup to bend into two sides of the wire 90 degrees at the same time.</p>
<p>Here is a diagram of what I'm trying to do:
<img src="http://i.stack.imgur.com/w9B5j.png" alt="enter image description here"></p> | 5,172 | [
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<p>For a topological field theory to be a true “extension” of an Atiyah-Segal theory, the top two levels of its target (<em>ie</em> its $(n-1)^{\text{st}}$ loop space) must look like $\text{Vect}$. What other (physical) considerations constrain the choice of target category? The targets of invertible field theories are (by definition) $\infty$-groupoids and (therefore) can be represented as spectra. What constraints can we impose on target spectra of invertible theories; in particular, on their homotopy groups?</p> | 5,173 | [
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<p>I'm having some trouble with Exercise 5.1 in Shapiro's BH,WD&NS book, which goes as follows:</p>
<blockquote>
<p>Consider two particles of mass $m$ at distance $r$ and $r+h$, such that $h\ll r$, on the same vertical line from the center of earth. The particles fall freely from rest a the same time $t=0$ towards the earth surface. Show that an observer falling with one particle will see the separation between the particles gradually increase. Translate this into a quantitative statement about the observer's local inertial frame.</p>
</blockquote>
<p>This is the first exercise in this chapter, right after the first principle of GR. The author has not introduced curvature and other definitions yet. How can I understand this problem both quantitatively and intuitively? </p>
<p>Here is how I think about it:</p>
<p>I should solve this problem in two reference frames: the Earth, which is an inertial reference frame, with Minkowski metric, and the free falling particle frame. The two events are the other particle starting falling and after Earth time $dt$ the time and location of it. Compare in free falling frame if the particle is approaching to get the result. Am I conceptually correct?</p>
<p>Also I am confused because even if I treat the Earth as a stationary object it has gravitational field so how is distance $r$ and $r+h$ defined? I don't think the Earth frame is inertial.</p> | 5,174 | [
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<p>Why do physical bodies in universe are bound to follow a rule ( eg. laws of physics ) ? Is their any imaginary condition possible, where they stop following any rule. And everything becomes random ? ( For example, a flying butter fly moves randomly, diseases like cancer occur randomly, then why physical bodies too cannot be random ? Why universe is based on rules ?</p> | 179 | [
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<p>In classical integrable models, in the discrete case we have the <a href="http://www.google.com/search?as_epq=Sklyanin+algebra" rel="nofollow">Sklyanin algebra</a>, $$\lbrace T_{a}(u),T_{b}(v)\rbrace =[r_{ab}(u,v),T_{a}(u)T_{b}(v)].$$ How to prove that the conserved quantities are generated from $\ln(\tau(u))$ in the periodic case, which are the integrable models (long-range) or condition which is not this expression which provide us the local conserved quantities in the <a href="http://www.google.com/search?as_epq=arnold+liouville+theorem" rel="nofollow">Arnold-Liouville theorem</a>?</p>
<p>Reference: Hamiltonian Methods in the Theory of Solitons by Ludwig Faddeev, Leon Takhtajan and A.G. Reyman. In special Part 1 - Chapter 3 and Part 2 - Chapter 3 in this book follow more references.</p> | 5,175 | [
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<p>I know the equation for displacement as a function of time is $$\vec{s} = \vec{v_i} \Delta{t}+\frac{1}{2}\vec{a}\,(\Delta t)^2$$
I need to solve for $\Delta t$ I'm having problems rearranging to do this. As of now, I created a Python program to run through the displacement formula, adding a set amount of time in each iteration, until the displacement is equal to the displacement I am looking for. I feel that this is woefully inefficient, inaccurate, and horribly CPU intensive. I am confident that this is possible to rearrange, and that I have missed something.</p>
<p>If someone already know what this formula rearranged to solve for $\Delta t$ is, I'd much appreciate it if you would share it with me. In the meantime, I shall continue to look around for the rearranged formula and keep trying to solve for $\Delta t$.</p>
<p>Hopefully I have not missed something painfully obvious, and that I am not wasting anyone's time.</p> | 5,176 | [
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<p>In an electron double slit experiment, let's put two charged plates behind the slits in an attempt to move the pattern up and down on the the screen.</p>
<p>What will happen? Will it just shift the interference pattern on the screen or washes it out completely? </p>
<p>If it washes it out, what's the minimal field that doesn't affect the pattern? Since I don't believe the electron double slit experiment was performed in an environment where all fields were exactly zero, but they still managed to get the pattern.</p> | 5,177 | [
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<p>For a spinning top, the linearised equation in the angle $\theta$ when the top is spinning about its axis of symmetry, which is vertical, is of the form $$A\ddot\theta+\left(\frac{C^2n^2}{4A}-Mgh\right)\theta=0.$$</p>
<p>Why should we require that the bracket coefficient be positive if we want the top to be stable, i.e. if we want small disturbances to remain small?</p> | 5,178 | [
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<p>The title, I don't know whether it's correct or not, but I came across a video in youtube, <a href="http://www.youtube.com/watch?v=_PkgQQqpH2M" rel="nofollow">http://www.youtube.com/watch?v=_PkgQQqpH2M</a>.</p>
<p>The author of video used the title and hence I used the same.</p>
<p>The video doesn't seems to be fake because of the "noise" in water flow. But what kind of freq. and why it makes the water flow in this pattern?</p>
<hr>
<p>EDIT1: The title might create some confusion with the nature of the flow. The water drops <strong>are not</strong> stagnant. The water is moving/flowing. but the flow is sort of like standing wave pattern. There are nodes and anti-nodes in the flow. drop which seems stationary are nodes. But as such water seems to flow in and out of the nodal region.</p> | 5,179 | [
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<p>This question relates to <a href="http://physics.stackexchange.com/questions/74567/how-do-i-show-the-existence-of-a-conserved-ghost-number-with-brst-in-bosonic-str">this post</a></p>
<p>I tried to verify Eq. (4.2.7) in Polchinski's string theory book vol I p. 127 but I miserably miss a sign</p>
<blockquote>
<p>$$ \delta_B (b_A F^A) = i \epsilon (S_2 + S_3) \tag{4.2.7} $$</p>
</blockquote>
<hr>
<p>Necessary equations:
$$ S_2 =-iB_A F^A (\phi) \tag{4.2.4}$$
$$ S_3 = b_A c^{\alpha} \delta_{\alpha} F^A(\phi) \tag{4.2.5} $$
$$ \delta_B \phi_i = -i \epsilon c^{\alpha} \delta_{\alpha} \phi_i \tag{4.2.6a} $$
$$ \delta_B B_A=0 \tag{4.2.6b} $$
$$ \delta_B b_A = \epsilon B_A \tag{4.2.6c} $$
$$ \delta_B c^{\alpha} = \frac{i}{2} \epsilon f^{\alpha}_{\beta \gamma} c^{\beta} c^{\gamma} \tag{4.2.6d} $$</p>
<hr>
<p>My attempt to verify Eq (4.2.7),
$$ \delta_B b_A F^A = (\delta_B b_A) F^A + b_A (\delta_B F^A) $$
$$ = \epsilon B_A F^A + b_A (\partial^i F^A) \delta_B \Phi_i = \epsilon B_A F^A + b_A (\partial^i F^A) (-i \epsilon c^{\alpha} \delta_{\alpha} \phi_i) $$
$$ = \epsilon B_A F^A - i \epsilon b_A c^{\alpha} \delta_\alpha F^A = i \epsilon (S_2 - S_3) $$</p>
<p>Why did I get $-S_3$?</p> | 5,180 | [
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<p>in wikipedia, the weak interaction coupling constant is said to be 10^-13 times weaker than that of the strong interaction but hyperphysics (<a href="http://hyperphysics.phy-astr.gsu.edu/hbase/forces/couple.html" rel="nofollow">http://hyperphysics.phy-astr.gsu.edu/hbase/forces/couple.html</a>) says it's 10^-6. It seems wikipedia is right but why many other resources also provide a value similar to that of hyperphysics?</p> | 5,181 | [
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<p>You have three separate glass rods, and you know one is positive, one is negative and one is neutral. You also have an electroscope that is positively charged.</p>
<p>How can you determine which rod is positive, negative and neutral?</p>
<p>I think you will be able to determine which rod is negatively charged, as that rod will cause the electroscope leaf to rise. I'm not sure how to differentiate between the positive+neutral though, as both will cause the leaf to collapse. Could you place it next to the negative rod and see which repels?</p>
<p>Essentially it would be great if somebody could explain the actual physics behind the situation. Why will the negative rod cause the leaf to rise and vice versa?</p>
<p>Thanks!</p> | 5,182 | [
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0.0090528829023242,
0.015914486721158028,
0.014490875415503979
] |
<p>consider a system of three identical particles, A B ,and C.
Assume that each particle can be in one of three possible quantum states, 1,2 and 3.
For the following statistics listed below, enumerate the possible states of the system and calculate the probabilities that the state 1 has zero,one,two, and three particles.</p>
<p>a. Maxwell Boltzmann distribution
b. Bose-Einstein distribution
c. Fermi-dirac distribution</p> | 5,183 | [
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<p>How are photons being watched in the double slit experiment? What exactly does being observed mean, as it is obviously changes the state of the photon somehow - it must be depriving the photon of something or emitting something that interacts with the photon.</p> | 5,184 | [
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<p>I want to measure the avg. inner cross-section area of a flexible tube of outer diameter 5mm. Since the cross-section isn't a circle exactly, use of vernier caliper to measure inner diameter fails. The idea that I wish to try is, to take a tube of known length, fill it with water, then pour the water on to something like a volume flask and from the volume obtained compute the cross-section area.</p>
<p>My questions are,
1 - how valid does this method sound?
2 - Is there a better way to do this?
3 - How much error would the water stuck in the tube due to surface tension introduce?</p>
<p>Would appreciate any help. Thanks.</p> | 5,185 | [
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<p>These days I was studying the quantum theory.I found that some theories about that is similar to Fourier Transform theory.For instance, it says "A finite-time light's frequency can't be a certain value", which is similar to "A finite signal has infinite frequency spectrum" in Fourier analysis theory.I think that a continuous frequency spectrum can not be measured accurately, which is similar to Uncertainty principle by Hermann Weyl. How do you think about this?</p> | 5,186 | [
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<p>it is well known that the intrisic spin is closely related to the rotation in space. However, in 1d , it is impossible to define rotation, therefore it is meaningless to talk about spin in 1d.However, one can see many papers talking about electrons in 1d with spin Is this meaningful?</p> | 5,187 | [
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<p>So I'm trying to do this problem where I'm given the Lagrangian density for a piano string which can vibrate both transversely and longitudinally. $\eta(x,t)$ is the transverse displacement and $\xi(x,t)$ is the longitudinal one. (So a point at $[x,0]$ at some later time t would be at $[x+\xi(x,t),\eta(x,t)]$). The Lagrangian density is given by $$\mathcal{L} = \frac{\rho_0}{2}[\ddot{\xi}^2+\ddot{\eta}^2]-\frac{\lambda}{2}[\frac{\tau_0}{2}+\xi'+\frac{1}{2}(\eta')^2]^2$$ where the dot is a partial time derivative and a prime is a partial x derivative. Also $\lambda$, $\tau_0$ and $\rho_0$ are parameters such as the Young's modulus, density and tension of the string respectively. So applying the action principle in the case of a continuous system I get the following PDEs for $\xi$ and $\eta$: $$\ddot{\xi} = \frac{\lambda}{\rho_0}(\xi''+\eta'\eta'')$$ and $$\ddot{\eta} = \frac{\lambda}{\rho_0}[\eta''(\frac{\tau_0}{\lambda} + \xi'+\frac{3}{2}(\eta')^2)+\eta'\xi'']$$.</p>
<p>Linearization of the equations yields two simple wave equations, the $\eta$ wave traveling at a speed $c_T^2 = \frac{\tau_0}{\rho_0}$ and the $\xi$ wave at $c_L^2 = \frac{\lambda}{\rho_0}$.</p>
<p>Now the part that is stumping me is that the problem asks us to show that if a given pulse of the form $\eta(x,t) = \eta(x-c_Tt)$ propagates along the string then that induces a concurrent longitudinal pulse of the form $\xi(x-c_Tt)$. Obviously I can't use the linearized equations to prove this. I would have to use some sort of iterative process by using exact PDE's to get the next step. But I tried doing this and I just get a mess. Any ideas where to go from here? (for reference this is problem 1.10 in Stone and Goldbart's book).</p> | 5,188 | [
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<p>When we study electricity in high school we examine the resistance of conductors and its relation with temperature. Diagrams show the relationship at the beginning is pretty much a linear with temperature but at very low temperatures its starts to curve in such why it never reaches zero but it is close and the book explains it "At low temperatures it starts to curve because the conductor is not of pure material it has particles of other elements and their effect starts at low temperature and that is way it was linear beforehand"</p>
<p>My question is how do these other elements stop it from reaching 0 and why don't they have the same effect on our conductive the whole time why does it only show at 159.7K and even less?</p>
<p>The explanation is that as temperature is reduced, the movement of the atoms is slowed which attributes to a lower chance of the electrons' movement being slowed because of bumping into atoms increasing their speed which means it has less resistance.</p>
<p>Why can't the resistance reach zero?</p> | 5,189 | [
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