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<p>I read that thermodynamic entropy is a measure of the number of microenergy states. What is the derivation for $S=k\log N$, where $k$ is Boltzmann constant, $N$ number of microenergy states.</p>
<p>How is the logarithmic measure justified?</p>
<p>Does thermodynamic entropy have anything to do with information entropy (defined by Shannon) used in information theory?</p> | 3,722 | [
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<p>I am currently studying for an exam in Quantum Mechanics and came across a solution to a problem I have trouble with understanding.</p>
<p>The Problem:</p>
<p>A Particle sits in an infinite potential well described by</p>
<p>\begin{align}
V(x) &= 0, & 0 \leq x \leq L \\
V(x) &= \infty, & \text{otherwise}
\end{align}</p>
<p>We know that the energies are given by $E_n = \dfrac{n^2 \pi^2 \hbar^2}{2 m L^2}$ and $\Psi(x) = A_n \sin(n \pi x /L)$.</p>
<p>At time $t_0$ the potential well is suddenly doubled in size, such that the potential is now</p>
<p>\begin{align}
V(x) &= 0, & 0 \leq x \leq 2L \\
V(x) &= \infty, & \text{otherwise}
\end{align}</p>
<p>So the energies are now given by $\tilde{E}_n = \dfrac{n^2 \pi^2 \hbar^2}{2 \cdot 4 m L^2}$ and $\tilde{\Psi}(x) = \tilde{A}_n \sin(n \pi x /2L)$.</p>
<ol>
<li>If the particle is in the ground state of the potential well before the change, what is the probability to find the particle in the ground state of the new potential after the change?</li>
</ol>
<p><em>This is absolutely clear to me. We find a non vanishing probability as a result. But now it gets tricky:</em></p>
<ol>
<li>What is the expectation value of the energy of the particle directly after the change? How does the expectation value of the energy evolve in time?</li>
</ol>
<p><em>The solution suggests that the expectation value of the energy does not evolve in time, which is clear to me, since the Hamiltonian is time independent and thus energy is conserved. But it also suggests that the expectation value does not change after we double the width of the potential wall which I understand from the argument of energy conservation but not in terms of quantum mechanics. If the probability that the particle is in the state $\tilde{\Psi}$ does not vanish the particle could have the energy $\tilde{E}_n$ which is lower than $E_n$ and this would mean that the expectation value of energy could change (with a given probability).</em></p>
<p>What am I missing here, where is my mistake? Any help is appreciated!</p> | 3,723 | [
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<p>I was recently reading Atkins' Physical Chemistry, the topic of rotational energies of molecules. It states the degeneracies of spherical top, symmetric and linear molecules as being $(2J+1)^2$, $2(2J+1)$ and $(2J+1)$, respectively ($J$ being the rotational quantum number). </p>
<p>Atkins however does not discuss about asymmetric non-linear molecules. I would like to know the degeneracy for each energy level for asymmetric tops. Is it $1$, or $J$ or $(2J+1)$? Please let me know. If I assume $(2J+1)$ to be the degeneracy, I was working with ethylene molecule which is an asymmetric top with rotational constants $147$, $25$, $30$ GHz (A, B, C). So, rotational energy along "A-axis" for quantum number $J$ is $AJ(J+1)h$, if I take the degeneracy to be $(2J+1)$ along the "A-axis", the mean energy of rotation (from the dependence on partition function, Atkins 9th edition Chapter 16) comes out to be (at high temperature limit) kT. </p>
<p>Similarly, adding the mean energies for each axes A, B, C, the total rotational mean energy is $3kT$, while I was expecting it to be $1.5kT$ ($0.5kT$ for each axis - from quadratic contribition $\frac{1}{2}I\omega^2$). </p>
<p>May be I am mistaken somewhere can please help me?</p> | 3,724 | [
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<p>Suppose I have two charged capacitor plates that both are isolated and carry a charge density $D = \frac QA$. According to textbook physics the electric field between them is given by $E=\frac D {\epsilon\epsilon_0}$ and the voltage by $U = Ed = \frac {Dd}{\epsilon\epsilon_0}$ with $d$ the distance between the plates. According to the formula for the voltage from above I could set any voltage between the plates if I just separate them far enough from each other and also the electric field would be constant no matter how far the plates are apart which is also quite counter-intuitive. As far as I remember this is true as long as $d$ is small compared to the size of the charged plates.</p>
<p>But what if this condition no longer holds? What is happening then? Is there another formula for this case that is comparably simple? I would suppose that for very large $d$ the whole thing can be seen as two point charges which would give a $\frac1r$ dependency of the voltage. But what is happening in between?</p> | 3,725 | [
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<p>Suppose there are two objects in the universe. Earth, with a gravitational acceleration of g = 9.8m/s/s, and a typical electron. </p>
<p>The electron is dropped from a certain height, say 1000m above the Earth's surface.</p>
<p>The initial energy of the electron is only the potential energy, $mgh = m_eg\times1000$, where $m_e$ is the mass of the electron.</p>
<p>As the electron falls towards the earth, it will be accelerated and thus will radiate energy. Will this cause the electron to slow down, and thus will the electron take a longer time to hit the ground than that expected by the equation $s = 0.5at^2$, due to energy loss through radiation.</p>
<p>If so, what acceleration will the electron actually fall at? How long will it take to hit the ground?</p> | 3,726 | [
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<p>The experment would involve a small NIB magnet levitating between or on the diamagnetic material pyrolytic graphite, unlike other forms of levitation this doesn't require power to run such as electricity or that the levitating item be moving to maintain the effect. This does seem to require occasional adjustments, presumably as room changes temperature and two NIB magnets, one large and one small. The small magnet can levitate and rotate and tends to loose speed very gradually when set in motion, unlike a bearing that must make contact, or even some other magnetic levitations, the most significant force slowing the small magnet, at least at low speeds is air resistance.</p>
<p>The magnet could then be attached to the center of a necessarily very light object, for example a small disc, this is less straightforward than it sounds as cardboard tends to be too heavy and paper too flexible, I've found a thin, light, stiff plastic that works. Even smaller fins could then be attached to the edges of the disc at 90 degrees that are black on one side and white on the other, like a Crookes Radiometer.</p>
<p>At least the levitating part of this experement would be in a vacume and the rig would be made of non-magnetic materals to prevent attraction or eddy currents between the levitating magnet and the edge of the rig. A laser could then be directed at the edge of the disc to accelerate it to potentially vastly greater speeds than the more conventional blowing though a straw.</p>
<p>If this were constructed how quickly could the levitating material accelerate, pushed around by the laser.</p>
<p>For example how fast would be expected in RPM be given a 500mW laser, a typical vacuum for experiments and 10 hours?</p>
<p>edit: The tags won't let me choose magnetism or magnets, they keep changing to electromagnetism.</p>
<p><a href="http://www.youtube.com/watch?v=lHugJfFbJoc" rel="nofollow">http://www.youtube.com/watch?v=lHugJfFbJoc</a></p>
<p><a href="http://upload.wikimedia.org/wikipedia/commons/1/1d/Crookes_radiometer.jpg" rel="nofollow">http://upload.wikimedia.org/wikipedia/commons/1/1d/Crookes_radiometer.jpg</a></p>
<p><a href="http://1.bp.blogspot.com/_0pGYzSEhkxQ/TBUJZbewUUI/AAAAAAAAADg/e5mqQWk9_BQ/s1600/Floating_Magnet13.jpg" rel="nofollow">http://1.bp.blogspot.com/_0pGYzSEhkxQ/TBUJZbewUUI/AAAAAAAAADg/e5mqQWk9_BQ/s1600/Floating_Magnet13.jpg</a></p>
<p><a href="http://1.bp.blogspot.com/_0pGYzSEhkxQ/TBUJYZlEMfI/AAAAAAAAADQ/hVov2PRRtgo/s1600/Floating_Magnet11.jpg" rel="nofollow">http://1.bp.blogspot.com/_0pGYzSEhkxQ/TBUJYZlEMfI/AAAAAAAAADQ/hVov2PRRtgo/s1600/Floating_Magnet11.jpg</a></p>
<p><a href="http://4.bp.blogspot.com/_0pGYzSEhkxQ/TBUXZX-vXAI/AAAAAAAAAEU/xICBvsd24AA/s1600/floating_cuboid.jpg" rel="nofollow">http://4.bp.blogspot.com/_0pGYzSEhkxQ/TBUXZX-vXAI/AAAAAAAAAEU/xICBvsd24AA/s1600/floating_cuboid.jpg</a></p> | 3,727 | [
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<p>Whether it is necessary to search still for variants of an explanation of spontaneously breaking gauge symmetry, giving masses for a W, Z-bosons?</p>
<p>Goldstone bosons are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries, thus it is clear that the Higgs bogon - is a Goldstone boson. If LHC searches of a Higgs boson won't be a success, whether it will mean a theorem inconsistency for electroweak interactions?</p> | 3,728 | [
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<p>Magnets have a magnetic north and south pole. Solenoids too have north and south pole from which magnetic fields comes out and goes in respectively. But is it that every magnetic configuration have a north and south pole? Electrons have magnetic moment and they can be regarded as very tiny magnets. So, where is its north and south pole?</p> | 3,729 | [
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<p>As we know the eigenfunctions for a particle of mass m in an infinite square well defined by $V(x) = 0$ if $0 \leq x \leq a$ and $V(x) = \infty$ otherwise are:</p>
<p>$\psi_n (x) = \sqrt{(2/a)} sin(n \pi x/a)$</p>
<p>The question now is: How does the ground state function look like in momentum space? As far as i recall I have to integrate $\psi_n(x)$ over the whole of space with the extra factor $\frac{e^{(-i p x / \hbar)}} {\sqrt {2 \pi \hbar}}$ (everything for $n = 1$ )</p>
<p>In the solutions to this problem they integrated over $-a \leq x \leq a$ while I would've integrated from $0$ to $a$.
Am I somehow missing something or is this solution just plain wrong?</p>
<p>Further question: How would I check whether or not my resulting $\psi(p)$ is an eigenstate of the momentum operator? Just slap the momentum operator in front of my function and see if I get something of the form $c \psi(p)$, where $c$ is some constant? Or how does this work?</p>
<p>Thanks in advance!</p> | 3,730 | [
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<p>I am having trouble with a velocity-versus-time graph. I recently took a Physics test that asked this question: The graph shows the velocity versus time for a particle moving along the $x$ axis. The $x$ position of the particle at $t$=0 seconds was 8 meters. What was the $x$ position of the particle at $t$=2 seconds What was the $x$ position of the particle at $t$=4 seconds?</p>
<p><img src="http://i.stack.imgur.com/jsSFm.jpg" alt="VelocityvsTime"></p>
<p>So, in order to solve this problem, I used the position equation: $x(t)=x_o + v_ot + \frac{1}{2}at^2$. Starting with the first position, I plugged 2 seconds into $t$, the time; 8 meters into $x_o$, the initial position; 20 meters per second into $v_o$, the initial velocity; and -10 meters per second squared into $a$, the acceleration. I derived the acceleration from the graph using $\frac{\Delta v}{\Delta t}$.</p>
<p>$a = \frac{\Delta v}{\Delta t} = \frac{-20}{2} = -10$</p>
<p>The position equation, with everything in place, reads:</p>
<p>$x(2)=8 + 20(2) + \frac{1}{2}(-10)(2^2)$</p>
<p>This math works out to 28 meters, which is the answer I gave on the test. I did the same thing for the second part of the question--the position of the particle at 4 seconds, as well as the acceleration based on the change in velocity and the change in time.</p>
<p>$a = \frac{\Delta v}{\Delta t} = \frac{-35}{4} = -8.75$</p>
<p>The equation comes out to this:</p>
<p>$x(4)=8+20(4)+\frac{1}{2}(-8.75)(4^2)$</p>
<p>The math comes out to 18 meters. However, according to the curriculum I used, the answer to the first part is 38 meters and the answer to the second part is 16 meters. How can this be the case? Did I fail to apply the position equation correctly? Did I make a mistake when computing the acceleration of the particle? I can't figure out what I did wrong. Is it possible that I used the wrong equation?</p>
<p>I really have no idea. I would appreciate any insight you might have to offer. Thank you in advance for you consideration. </p> | 3,731 | [
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<p>Sometimes I need to look up a certain cross section, say the inclusive Z production cross section at $\sqrt{s}$ = 7 TeV. Is there a place where 'all the' cross sections are tabulated (experimental and theoretical), like the branching ratios are in the <a href="http://en.wikipedia.org/wiki/Particle_Data_Group" rel="nofollow">PDG</a>? I know it might be hard, because there is often not 'the' cross section and you have to take into account generator cuts and the likes, but still there should certainly be a review somewhere? </p> | 3,732 | [
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<p>A fairly common technique in experimental particle physics is event-by-event reweighting. The idea is that you have a sample of background-model events, either from a Monte Carlo simulation, or from a data-driven method. However, your background model doesn't describe the data well enough, so you go into a sideband region (where the signal you are looking for is neglegible), and look at a distribution $x$ which is incidental to the variables you are actually interested in $y$. Then you "fix" the model to fit the data in this distribution. For example, if the background is too low at $x=x_0$, all events with that value of $x$ get a weight slightly higher than 1.</p>
<p>The result is that the distribution of $x$ is now almost perfectly modeled, and variables correlated to $x$ are hopefully better modeled than before, and you have hopefully not overfitted your background to mask a possible signal.</p>
<p>Reweighting often seems like cheating to new users, but you get used to it quickly and stop worrying - and if used correctly it can be indeed a valuable tool. <strong>Now, I'm wondering if there is an accepted set of (thumb) rules when and how you are allowed to do reweighting.</strong></p>
<p>Points I'd like clarified:</p>
<ul>
<li>It seems $x$ and $y$ must not be too correlated. If $x = f(y)$ strictly, then it's cheating. If they are completely uncorrelated however, reweighting wrt. $x$ wouldn't change $y$.</li>
<li>Signal must be neglegible in the region where the reweighting factors are determined, but what is neglegible? A ratio of S/B = 1e-6 (before cuts) seems OK, but what if all signal events are in that region?</li>
<li>Sometimes the weights are determined bin-by-bin, $w,i = N_\mathrm{data,i}/N_\mathrm{BG,i}$. Sometimes you fit a function $w(x)$ to smooth out the weights. Any rules of thumb on how to determine the binning and/or the function?</li>
<li>How to calculate the systematic uncertainties from the procedure? Is it enough to vary the $w_i$ or the fit parameters by their uncertainties?</li>
</ul>
<p>Maybe there is also just a good reference you can point me to.</p> | 3,733 | [
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] |
<p>This is a past exam question from one of our lectures, and we have an issue with (i), I believe I need to use the equation $\rho=\frac{RA}{l}$, but I am not sure - could someone enlighten me on the issue?</p>
<blockquote>
<p>A mild steel ring of magnetic permeability 380, having a cross sectional area of $500mm^2$ and a mean circumference of $400mm$, has a coil of $200$ turns wound uniformly around it.</p>
<p>Given that the magnetic permeability of free space is $400 nH/m$ determine: <br /></p>
<ul>
<li><p>(i) The reluctance of the ring.</p></li>
<li><p>(ii) The current required to produce a flux of $800\mu$ Wb in the ring.</p></li>
</ul>
</blockquote> | 3,734 | [
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<p>What does the following statement mean and why is it true?</p>
<blockquote>
<p><em>The Weak Equivalence Principle (WEP) implies that in general curved space-time there is no privileged coordinate system.</em></p>
</blockquote>
<p>I have looked up the <a href="http://en.wikipedia.org/wiki/Equivalence_principle" rel="nofollow">WEP</a> -- as far as I can see, it is more or less the Universality of free fall (?) My (probably totally missing the point) interpretation of the statement is that in general curved space-time, you can't do away with the Gravitational Field everywhere simultaneously... But like I said, I am probably barking up the wrong tree here. Grateful if someone could explain!</p>
<p>Context: This was to justify the use of tensor calculus in GR.</p> | 3,735 | [
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<p>I am reading a paper, and I came across the Green-Kubo formulation, where the conductivity $\sigma$ of charged particles is related to the time correlation function of the $z$-component of the collective ionic current $J_z(t)$:</p>
<p>$$\sigma_{GK} = \frac{1}{V k_B T} \int_0^{\infty} dt \; C_{JJ}(t)$$</p>
<p>where $C_{JJ}(t) = \langle J_z(0) J_z(t) \rangle$ and the collective current along the $z$ direction is $J_z(t) = \sum_{i=1}^N q_i v_{z, i}(t)$. $V$ is the volume of the system and $q_i$ and $v_{z, i}$ are the charge and $z$-component of the velocity of the $i$th charged particle. $\langle ... \rangle$ is an equilibrium ensemble average.</p>
<p>My question is, what is the time correlation function? Is the time correlation function $C_{JJ}(t) = \langle J_z(0) J_z(t) \rangle$? Or is the time correlation function the integral: $\int_0^{\infty} dt \; C_{JJ}(t)$?</p> | 3,736 | [
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<p>This is not a joke. It is something I have wondered about for a long time. Nearly every picture of Werner Heisenberg shows him smiling. </p>
<p><img src="http://i.stack.imgur.com/1kvEV.jpg" alt="Heisenberg is happy">
<img src="http://i.stack.imgur.com/amoRb.jpg" alt="Heisenberg is still happy">
<img src="http://i.stack.imgur.com/tmsal.jpg" alt="Even in old age, Heisenberg is still smiling"></p>
<p>This is unusual. The difference between Heisenberg and other physicists is striking. Try an image search for Dirac or Schrödinger and you will see a lot of scowling, sour faces. It is possible that Niels Bohr smiled once but if so there does not seem to be any photographic evidence.</p>
<p>Was Heisenberg noted for being a particularly cheerful guy?</p> | 3,737 | [
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<p>Does the state of whether an object if moving or stationary affect the likelihood of it being struck by lightning?</p>
<hr>
<p>I suppose some things that could be considered would be:</p>
<p>Whether the movement means the object is not continually earthed, for example, a horse galloping across an open plain, during the gait there are moments when none of the horses hooves touch the ground.</p>
<p>Whether the movement affects the static charge of the object and whether this charge would be sufficient to affect the likelihood of attracting lightning strike.</p>
<p><em>Disclaimer</em></p>
<p><em>These are examples and limited, I am not interested in my own personal safety during a lightning storm, it is a scientific question. If we could move a large conductor are great speed (light and airplane), but on the ground, it would be interesting to see the results.</em></p> | 3,738 | [
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<p>I need some ideas on a problem. </p>
<p>The first part says: Whats the posible rise in the temperature of the water falling 49.4 m in the Niagara Falls? That one was easy, with answer 0.112 Kelvin. ($\Delta T = \frac{g*h}{c_{H_2 O}}$)</p>
<p>The second part asks what factors tend to prevent that rise in temperature? Im thinking kinetic energy, preassure, air conductivity but im not sure. </p>
<p>Thanks in advance </p> | 3,739 | [
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<p>I was going over my notes for an introductory course to electricity and magnetism and was intrigued by something I don't have an answer to. I remember my professor mentioning, to the best I can remember, that electric current is actually not the flow of electrons but the propagation of the electric field. My question is, how does the field "know" in which direction to travel, or even what to travel along? The electrons just move from atom to atom as they feel a force due to the potential difference (if I have it right). But what about the field? Thanks to all in advance.</p> | 3,740 | [
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<p>When beginning a study of the special theory of relativity, one discovers that the theory of special relativity has as an axiom that the laws of physics are invariant with respect to transformations between inertial frames. The theory then states that Maxwell's equations are laws of physics and thus invariant between transformations between inertial frames. Furthermore, from Maxwell's equations we find that the speed of light is a constant and therefore must be invariant between inertial frames. Thus there exists a finite, constant speed limit to any physical process. </p>
<p>This is the only physical argument on purely theoretical grounds that I've ever heard which argues that there must exist a finite speed for any physical process. </p>
<p>My question is this: Are there any other physical arguments for the existence of a finite speed limit on which relativity can use as an axiom without appealing to the existence of the constant of the speed of the wave in Maxwells equations? </p> | 3,741 | [
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<p>If you have some random object at rest and you apply a <a href="http://en.wikipedia.org/wiki/Couple_%28mechanics%29" rel="nofollow">couple</a> to it, the net force acting on it is zero. However because a moment acts on it, it starts to rotate.</p>
<p>So you had an object at rest, a net force of zero was applied to it, and it is now moving. Why doesn't this violate Newton's First Law?</p> | 3,742 | [
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<p>The Boltzmann equation in absence of external force reads:</p>
<p>$\frac{\partial f}{\partial t} + \vec{v} \cdot \frac{\partial f}{\partial \vec{r}} = \left( \frac{\partial f}{\partial t}\right)_{coll}$</p>
<p>Where the r.h.s. stands for the change in the distribution function of the velocities owing to collisions. I won't specify here a particular collision kernel as it is not necessary for my question...I think.</p>
<p>It is common to introduce </p>
<ul>
<li><p>The particle density $n(\vec{r},t) \equiv \int d^3v \:f(\vec{r}, \vec{v},t)$</p></li>
<li><p>The mean velocity field $\vec{u}(\vec{r},t) \equiv \frac{1}{n(\vec{r},t)}\int d^3 v \:\vec{v} f(\vec{r}, \vec{v},t)$</p></li>
</ul>
<p>It is then easy to integrate over the velocities the force-free Boltzmann equation above and find the famous conservation law:</p>
<p>$\frac{\partial n(\vec{r},t)}{\partial t} + \nabla_{\vec{r}}\cdot(n(\vec{r},t)\vec{u}(\vec{r},t)) = 0$</p>
<p>That is because, if the collisions conserve the number of particles, then there should not be any 'source' or 'sink' on the r.h.s.</p>
<p>My naive question is the following:</p>
<p>The above conservation law for matter can be recast as $\frac{\partial n(\vec{r},t)}{\partial t}+ \nabla_{\vec{r}}\cdot \vec{J}(\vec{r},t) = 0$ with the particle flux $\vec{J}(\vec{r},t) = n(\vec{r},t)\vec{u}(\vec{r},t)$.</p>
<p>This is fine but it does not seem very general. For instance, it does not seem to encompass a diffusive flux does it?</p>
<p>Hence, although this equation is very general, it does not seem to lead to Fick's laws of diffusion and yet I kind of remember that one can get the latter from some kind of expansion of at least a linearized Boltzmann equation...</p>
<p>Could somene unlighten me on this apparent paradox? </p> | 3,743 | [
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<p>In class my prof said that when showing a system is at equilibrium it suffices to show that the moment at one point is zero. Why? Why does showing the moment at a point is zero imply the moment of the whole system is zero?</p>
<p>If I misunderstood him and the statement above isn't true, how do you show the system is at moment equilibrium at all points of the system? You obviously can't show the moment is at equilibrium for infinite points individually. </p> | 3,744 | [
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<p>To the extent that I know:</p>
<p>There are symmetry groups like the rotation groups SO(3), the Groups of Poincare Transformations,... If the physics of a system has a symmetry group G, then it can be described by a representation of G and the vector space acted on.</p>
<p>Correct me if I'm wrong.</p>
<p>If I'm not that wrong, I want to know a simplest example how we can interpret the physics of this system by studying properties of G's representation. (because I have been learning in the reverse way: first is the Hilbert Space of states, then the group of symmetry operators)</p>
<p>EDIT:
I think the process of making a physics theory would be as following:</p>
<p>Corresponding to a specific "physics", there is particularly a Lie group (called G) of symmetry. Then we can build a framework by representing this Lie group as a group of linear transformations acting on a vector space V.</p>
<ul>
<li>Each element of V would be a state of physics.</li>
<li>Each element of the Lie algebra (corresponding to G) would be an observable (this is what I want to know if it's true or false for sure)</li>
</ul>
<p>Then we can apply quantum concepts as eigenstate, eigenvalue, distribution,...</p>
<p>Am I wrong? If I am wrong, how can I be fixed?</p>
<p>(I just read accidentally about representation theory last week and I'm kind of excited about the idea of promoting a theory from a somewhat simple (fundamental) object as a group of symmetry)</p>
<p>I have found a paper describing the way to construct quantum physics from symmetry group and representation theory:</p>
<p><a href="http://www.math.columbia.edu/~woit/QM/qmbook.pdf" rel="nofollow">http://www.math.columbia.edu/~woit/QM/qmbook.pdf</a></p> | 3,745 | [
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<p>Say you have a sphere, and you have several torque vectors acting on it, all at different points. Say you have the vector (6i + 3j + 5k) originating from point A, and the vector (3i + 1j + 9k) originating at point B, and (7i + 2j + 9k) acting on point C.</p>
<p>Summing the vectors gives you (16i + 6j + 23k) which is the resultant moment/torque vector. But at what point does the moment act on - A,B, or C?</p>
<p>The point it acts on has to matter right? I mean if you think of the moment vector as an axis the sphere revolves around, placing it in the center of the sphere and rotating the sphere around that is clearly different from placing it at the far left of the sphere and rotating it around that.</p> | 3,746 | [
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<p>Where I can find photos of nuclear explosions just after detonation (before 5-10 ms, the shorter the better)?</p> | 3,747 | [
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<p>I am having trouble using the <a href="http://en.wikipedia.org/wiki/Right-hand_rule" rel="nofollow">right hand rule</a> properly and often find myself putting my hand in awkward orientations. I know you point your hand in the direction of $r$ and then point your fingers in the direction of $F$ but that doesn’t really help me. Can anyone explain how to use it correctly and without awkward hand positions and apply it to the image below for F1, F2, and F3?
<strong>I don't get the orientation of the thumb and how that dictates direction. In the first example for for F1 my thumb is pointed towards me to the right, F2 my thumb is pointed away from me, and F3 it is pointed towards me to the left. I dont get how this Shows me how F1 is clockwise, F2 counter, and F3 counter</strong></p>
<p><img src="http://i.stack.imgur.com/WGpTw.png" alt="Link to example 1"></p>
<p>Also in <a href="http://i.stack.imgur.com/ze8OP.png" rel="nofollow">this example</a> I would have thought all of the forces are counterclockwise except D but they aren't, Help! :? </p> | 3,748 | [
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<p>Bohr-Sommerfeld quantization provides an approximate recipe for recovering the spectrum of a quantum integrable system. Is there a mathematically rigorous explanation why this recipe works? In particular, I suppose it gives an exact description of the large quantum number asymptotics, which should be a theorem.</p>
<p>Also, is there a way to make the recipe more precise by adding corrections of some sort?</p> | 3,749 | [
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<p>In a type Ia Supernova, the carbon accumulated in the earlier stages of a stars death fissues to create even heavier elements. Could this be used by humans aswell? Is it theoretically possible to build a Carbon Bomb?</p>
<p><a href="http://physics.stackexchange.com/questions/30755/pressures-necessary-for-carbon-detonation">This thread</a> explains a bit about it; I guess these pressure values cannot even be achieved with a Hydrogen Bomb as a fuse?
In turn, could it be used for commercial reasons (in the far future, e.g. as spacecraft engine?)</p>
<p>Does anyone know further reference?</p>
<p>How much damage would such a bomb deal? (approximate scale of course, would it destroy our entire planet or just scratch my nose)</p> | 3,750 | [
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<p>This may be a very basic question but I am not seeing how it works.
Consider the standard example of an ice skate rotating about his/her center of mass and pulling in his/her arms. The torque is zero so we have conservation of angular momentum. This implies that $\omega$ increases to keep $I\omega$ constant, but then $K_{rot}=\frac{1}{2}I\omega^2$ doesn't stay constant, it increases. This implies that there is work done, but what force is doing this work?</p> | 594 | [
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<p>Suppose we are given a mechanical frame consisting of two points. How can we prove that assuming any initial conditions there is an inertial frame of reference in which these points will be in a static plane?</p> | 3,751 | [
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<p>I have heard numerous times when getting x-rays, MRIs, CAT Scans, etc. that each one is equivalent to a cross country airplane trip. Disregarding the different types of radiation as asked in <a href="http://physics.stackexchange.com/questions/20665/can-x-ray-radiation-be-compared-to-background-radiation">this question</a>, I was wondering if they are mixing total dosage over the entire body with the same dosage in a concentrated area.</p>
<p>To illustrate by example, if the total radiation received on a plane trip is 100 'rad' units and the body is 100 area units each body part would only receive 1 rad unit of exposure. Whereas if an x-ray is the same 100 rad units but your arm (for example) is only 10 area units then your arm is actually receiving 10 rad units of exposure - or 10 times the amount received on a plane trip.</p> | 3,752 | [
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<p>The quantum mechanics of Coloumb-force bound states of atomic nuclei and electrons lead to the extremely rich theory of molecules. In particular, I think the richness of the theory is related to the large mass ratio between the nucleon and the electron. This mass ratio leads to the Born-Oppenheimer approximation which gives rise to a complicated effective potential for the nuclei which posses many local minima: the molecules.</p>
<p>I wonder whether there are analogical phenomena in which the bound states are gravitational. It seems that if we take a collection of electrically neutral molecules and neutrinos, it should be possible to form a large number of bound states, in particular because of the large (not so long ago deemed infinite) mass ratio between molecules and neutrions. Of course neutrinos are highly relativistic and I can't tell how it affects things.</p>
<p>Now, even if we leave neutrinos alone, the typical size of such a bound state is</p>
<p>$$\frac{\hbar^2}{G m^3}$$</p>
<p>where $m$ is proton mass. Google calculator reveals this to be 3.8 million light years. Holy moly! However, this is still much smaller than the observable universe. Can there be places in the universe sufficiently empty to contain such bound states? What would be the effect of general relativistic phenomena (expansion of space)? EDIT: I guess no place is sufficiently empty at least because background radiation is everywhere. Maybe these creatures will become relevant in a very distant future when the background radiation cools off considerably?</p>
<p>Summing up:</p>
<blockquote>
<p>What is known about the quantum mechanics of gravitational bound states of electrically neutral molecules and neutrinos?</p>
</blockquote>
<p><em>I'm tagging the question as "quantum gravity" since it involves gravity and quantum mechanics. Of course it is not quantum gravity in the usual sense of studying Planck-scale phenomena. I think the tag is still appropriate</em></p>
<p>EDIT: Gravitional bound states of molecules will often be unstable with respect to collapse to a van der Waals bound state (thx Vladimir for bring up the issue of vad der Waals interaction in the comments). However the lifetime of these states is very long.</p> | 3,753 | [
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<p>I realised, reading another Phys.SE question about <a href="http://physics.stackexchange.com/q/86774/">balloons moving forwards in an accelerating car</a> that I don't really understand how <a href="http://en.wikipedia.org/wiki/Buoyancy" rel="nofollow">buoyancy</a> works. Particularly concerning, for a SCUBA diver.</p>
<p>The top answers to that question seem to claim that balloons get their "sense of down" from a pressure differential. They continue: when a car accelerates, the air at the back of the car becomes more dense, and at the front less dense, changing the plane of the pressure differential and so also, the balloon's sense of up. I find that extremely hard to credit. However, I realised that I don't really know why less dense things float in more dense things.</p>
<p>I'm fairly sure it's something to do with displacement of heavier things by lighter things, and I think pressure acting on the lighter thing's surface has something to do with it, but that's about it.</p> | 3,754 | [
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<p>Starting from a homework problem:</p>
<blockquote>
<p>An aluminum cup of $100 cm^3$ capacity is completely filled with glycerin at $22°C$. How much glycerin, if any, will spill out of the cup if the temperature of both the cup and the glycerin is increased to $28°C$? (The coefficient of volume expansion of glycerin is $5.1x10^4/C°$.)</p>
</blockquote>
<p>I find that I have the for efficient of linear expansion for aluminum, but I need to know how the volume of the cup changes. Worse, I don't know the dimensions of the cup.</p>
<p>I think I use the linear expansion equation for metal rod $\Delta L = L \alpha \Delta T$ to find how much taller the cup is after the temperature changed and the volume expansion equation for a solid of liquid $\Delta V = V \beta \Delta T$ but not knowing any of the dimension of the cup I do not see how to determine this?</p> | 3,755 | [
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<p>I had a question on one of the details of the derivation of the second law of thermodynamics starting from the phase space volume. I'll type out what I understand so far:</p>
<p>Letting the Hamiltonian depend on some external parameter $a$ so that $H=H(a)$. The phase space volume can be written as </p>
<p>$$\bar{\Omega}(E,a)=\int d\Gamma \Theta(E-H(a)),$$</p>
<p>where $\Theta$ is the heaviside function. The total differential is </p>
<p>$$d\bar{\Omega}(E,a)=\int d\Gamma \delta(E-H(a))(dE-\frac{\partial H}{\partial a}da)=\Omega(E,a)(dE-\langle\frac{\partial H}{\partial a}\rangle da).$$</p>
<p>In the equation above $$\Omega=\int d\Gamma \delta(E-H(a)).$$</p>
<p>Using the logarithmic derivative </p>
<p>$$d \log\bar{\Omega}=\frac{\Omega}{\bar{\Omega}}(dE-\langle\frac{\partial H}{\partial a}\rangle da).$$</p>
<p>Using the definition of entropy $S=k\log\bar{\Omega}$</p>
<p>$$k\ dS=\frac{\Omega}{\bar{\Omega}}(dE-\langle\frac{\partial H}{\partial a}\rangle da).$$</p>
<p>At this point the book jumps to </p>
<p>$$ dS=\frac{1}{T}(dE-\langle\frac{\partial H}{\partial a}\rangle da)$$ </p>
<p>without much explanation. My question is how the $1/T$ follows from the previous line?</p> | 3,756 | [
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<p>Consider a particle in a box system.Assume its state to be a superposition of the ground and the first excited energy states.Consider two observers A and B (rest of the world).A made the measurement of the energy of the system and got energy corresponding to one of the states.
Consider two scenarios from now.</p>
1.A made the measurement and B is not aware of it at all.For B would the state still be the superimposed state?</p>
2.A made the measurement and B know(s) but is unaware of the result.Would it be ok to say that for B the state of the system is still the same as it was initially?
</p>Are these two scenarios equivalent?</p> | 3,757 | [
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<p>As you know the active noise cancellation technology used in many application such as protection of aircraft cabins and car interiors to reduce engine noise also some headphones use this feature to reduce unwanted ambient sounds.</p>
<p>Actually I don't know the details of how this feature works, but I do know that it has input (microphone) that receives noise waves, control unit that invert the incoming noise waves and output (speakers) to emit inverted noise waves and therefore they will cancel each other.</p>
<p>What I'm not understanding is that If we considered the sound waves are traveling in air, then the <strong>inverted</strong> noise waves emitted from the speaker will reach the listener before they combine with the original noise waves so they cancel each other while the <strong>inverted</strong> noise waves are traveling in a solid-media (components of the microphone, speaker, etc.) faster than the original noise waves traveling in air.</p>
<p><img src="http://i.stack.imgur.com/t5EhA.jpg" alt="enter image description here"></p>
<p>I'm I missing something or there is another feature they use to adjust the speed of sound in the active noise cancellation?</p> | 107 | [
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<p>Hi Guys generally when you evaluate the 3 open string tachyon tree level amplitude in CFT, you do a conformal transformation mapping the worldsheet to the upper half of the complex plane and the incoming and outgoing strings become points on the real axis. However in evaluating the string field theory tree level amplitude the problem involves a different procedure where the 3 string worldsheet are conformally transformed in to 3 different worldsheets which are then glued together to form a circle, with the incoming and outgoing strings being the points on the circumference of the circle. I don't understand why the same amplitude is seen in two different ways. </p> | 3,758 | [
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<p>Suppose I was designing an apparatus which needed to lift 250kg 5cm high, hold it there for a few seconds, and then lower the object back to the original height. Such a process would need to be repeated every certain time interval.</p>
<p>What kind of mechanism (pneumatic, hydraulic, electric, etc) is likely to have the least overall energy losses? Please also describe the type of energy reservoir for the solution you propose (e.g. for an electric motor, would you use a certain type of battery or capacitor etc.)</p>
<p>Update: it's impossible in my system to add any sort of counterweight due to the size and weight restrictions on the apparatus.</p> | 3,759 | [
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<p>Does Coulomb's law apply to Plasma?</p> | 3,760 | [
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<p>I have been exploring for some time both the Special and General Relativity, hoping to glean at least a conceptual grasp of their basic tenets.
In reading the book "Gravitation" by Misner, Thorne and Wheeler, the authors stress that Riemann came very close to make a decisive connection between gravitation and curvature of space, but he failed to do so, they say, precisely because he thought of SPACE and curvature of SPACE instead of curvature of SPACETIME and this makes the whole difference! </p>
<p>Can anybody explain in conceptual terms, as far as possible, why spacetime, unlike SPACE alone, can be seen and understood as curved? </p>
<p>I firmly believe that mathematics is only a language, albeit a complex one, which facilitates our understanding of reality, but that the same reality is not hopelessly beyond reach without maths. We should remember after all that Einstein's mathematical formulations of the Special and General Relativity are rooted in thought experiments and in a basic conceptual grasp, which preceded its mathematical formulation.</p> | 3,761 | [
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<p>I am trying to do a homework problem where we re-write the mass, momentum and energy conservation formulas for downward flow in a vertical pipe and it says "where all hydrodynamic entrance effects have disappeared". What exactly does this mean? Also I am using an incompressible substance if that helps.</p>
<p>Someone told me this meant that $$\frac{\partial \vec V}{\partial z}=0$$ and $$\frac{\partial \vec V}{\partial t}=0$$</p>
<p>I don't see how the z-component of velocity doesn't change with time or with position in the z-direction if gravity is acting on the fluid particles. Gravity should be accelerating the fluid particles which means the z-component of velocity should change with time, and if its accelerating then shouldnt the profile also change?</p> | 3,762 | [
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<p>My question essentially revolves around multi-electron atoms and spectroscopic terms. I understand the idea that the total wavefunction for Fermions should be antisymmetric. Consider as an example, the $2p^2$ electrons in a partially filled p shell; that is, the outer shell of Carbon. The two electrons both have $l=1$, and hence total orbital angular momentum takes the values:</p>
<p>$L = L1+L2, L1+(L2-1),...,|L1-L2| = 0,1,2$
and
$S = 0,1$</p>
<p>I can sort of intuitively see that $L=2$ must refer to a symmetric spatial wavefunction and hence an antisymmetric spin wavefunction. I can handwave and say that for $L=2$, we must have $m_{l1}=m_{l2}=\pm1$ and hence they must have opposing spin to satisfy PEP which gives S=0 - but I'm not sure how to express that in terms of an actual wavefunction and it seems to be a bit of a circular argument. However, I don't see why $L=1$ must have $S=1$ (a triplet) and $L=0$, $S=0$ (another singlet). </p>
<p>Can anyone shed some light on this?</p>
<p>Thanks!</p> | 3,763 | [
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<p>In order to find the magnetic field generate by an infinitely long straight wire of radius R, I have to use local Ampère's Law :</p>
<p>$\boxed{ \oint_L B\cdot dl = \mu_0 \sum_j I_j}$</p>
<p>If $r > R$ then $ \oint_L B\cdot dl = B(r)\cdot 2\pi r = \mu_0 I \Rightarrow B(r) = \dfrac{\mu_0 I}{2\pi r} $</p>
<p>(I'm okay with this expression)</p>
<p>But if $r<R $ I'm supposed to find $ \oint_L B\cdot dl = B(r)\cdot 2\pi r = \mu_0 I \cdot \dfrac{\pi r^2}{\pi R^2} \Rightarrow B(r) = \dfrac{\mu_0 I r}{2\pi R^2} $</p>
<p>I don't understand where come from the $\dfrac{\pi r^2}{\pi R^2}$ factor.</p>
<p>Thanks for your help (and sorry for my bad english)</p> | 3,764 | [
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<p>The <a href="http://i.stack.imgur.com/Qzc6I.png">following question</a> was on a quiz in physics class:</p>
<blockquote>
<p>If the net work done on a particle is zero, which of the following statements must be true?</p>
<p>a) The velocity is zero<br>
b) The velocity is decreased<br>
c) The velocity is unchanged<br>
d) The speed is unchanged<br>
<strong>e) There is no displacement for the object</strong> </p>
</blockquote>
<p>The correct answer was e. In what scenario would the speed change?</p>
<p>There was an explanation beside the question:</p>
<blockquote>
<p>Since the work done is zero, it indicates that the applied force is zero. Since Force = Mass X Acceleration, and the mass is not zero, this implies that the acceleration is zero.</p>
</blockquote>
<p>When asked about this question, the teacher responded:</p>
<blockquote>
<p>If there is no displacement, then only work done is zero. If the speed is unchanged, then there is no acceleration. This will lead to an absence of a force. Hope it explains the situation. </p>
</blockquote> | 3,765 | [
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<p>I was wondering how <a href="http://www.wickedlasers.com/phosforce" rel="nofollow">this lens</a> works. It converts blue laser light into white light and effectively turns a portable laser into a flashlight. The info mentions phosphor coating. I used my Google Fu and found this: <a href="http://www.photonics.com/Article.aspx?AID=33972" rel="nofollow">Phosphor-Coated LED Converts Blue Light to White</a>.</p>
<p>Could someone please give me a more conceptualized description? </p> | 3,766 | [
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<p><img src="http://i.stack.imgur.com/KaLim.jpg" alt="enter image description here"></p>
<p>Ans:
Applying Gauss’s law the net flux can be calculated. And for option (B), I guess the flux will be 0. But not sure. Can anyone explain all the 3 options?</p>
<p>For left and rignt face, EA = 300*(0.05)^2 = 0.75 Nm^2/c , but this does not match with the answer.
And for top, bottom, front and back i guess it should be 0.</p> | 3,767 | [
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<p>As we all know that anticommutator of one set of supercharges in massive extended supersymmetry is something like $$\{b_\alpha, b_\beta^\dagger \} = \delta_{\alpha \beta} (M-\sqrt{2} Z).$$
My question is that everyone says that it is obvious that <a href="http://en.wikipedia.org/wiki/BPS_state" rel="nofollow">BPS states</a> annihilate half of the supersymmetric charges. Why is this so? It may be trivial but I don't know how. May be it is just because the annihilation operator acts on the lowest energy state and annihilate BPS States? But why only half? Can anyone clear that up.</p> | 3,768 | [
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<p>Currently in my last year of high school, and I have always been told that <a href="http://en.wikipedia.org/wiki/Centrifugal_force">centrifugal force</a> does not exist by my physics teachers. Today my girlfriend in the year below asked me what centrifugal force was, I told her it didn't exist, and then she told me her textbook said it did, and defined it as "The apparent force experienced towards the outside of a circle is the centrifugal force and is due to the mass of the object resisting the inward centripetal acceleration that the object is experiencing". I was pretty shocked to hear this after a few years of being told that it does not exist.</p>
<p>I did some reading and found out all sorts of things about pseudo forces and reference frames. I was wondering if someone could please explain to me what the hell is going on (in high school student terms), is it wrong to say that centrifugal force does not exist? </p>
<p>This has always nagged me a bit as I often wonder that if every force has a reaction force then a centripetal force must have a reaction centrifugal force, but when I asked my teachers about this they told me that centrifugal force does not exist. Now I'm just confused. Help.</p> | 3,769 | [
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<p>I am curious due to the course of the idle research that I am doing (my hobby), and I am curious for various reasons as to the answer of this question. Please forgive me in advance for possible convoluted language, as I occasionally have trouble explaining my questions. How could one go about calculating a <a href="http://en.wikipedia.org/wiki/Roche_limit" rel="nofollow">Roche limit</a> for a black hole? I have asked this question in other places, and please leave off pedantic responses about black holes not existing. Basically my problem is thus: current Roche limit formulae are solved with respect to primary and secondary density, and, as we currently understand them, black holes have infinite density, due to them being zero-dimensional. I have seen some suggestions that mass can be replaced for density in the calculation, but this is often presented with no justification, or modifications in the formula. Currently, for the purposes of this, I am using Sag A* as the primary, and 3 different test particles (The moon, earth, and sun, because in my own little math world, I can destroy all 3) as the secondaries, and possibly other masses as well. I am aware of the limitations of the limit, that they only really hold if a body is held together only with gravity. So, if someone can give me a pointer or two, I'd be eternally grateful.</p> | 3,770 | [
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] |
<p>On page 1 of this recent <a href="http://arxiv.org/abs/1105.3796" rel="nofollow">paper</a> by Bousso and Susskind we read.</p>
<blockquote>
<p>This question is not about philosophy.
Without a precise form of decoherence,
one cannot claim that anything really
"happened", including the specific
outcomes of experiments. And without
the ability to causally access an
infinite number of precisely decohered
outcomes, one cannot reliably verify
the probabilistic predictions of a
quantum-mechanical theory.</p>
</blockquote>
<p>According to the <a href="http://en.wikipedia.org/wiki/Law_of_large_numbers" rel="nofollow">Law of Large Numbers</a> there is no requirement that I need to perform an infinite number of experiments in order to verify that results will asymptotically approach an expected value. Why would Quantum Mechanics be any different?</p>
<p>UPDATE: I wanted to find an relatively simple proof of the strong law of large numbers, best I can come up with is the combination of the following two wikipedia entries:</p>
<p><a href="http://en.wikipedia.org/wiki/Kolmogorov%27s_three-series_theorem" rel="nofollow">Kolmogorov's Three series Theorem</a></p>
<p><a href="http://en.wikipedia.org/wiki/Kronecker%27s_lemma" rel="nofollow">Kronecker's lemma</a></p>
<p>Additional links are available in articles.</p> | 3,771 | [
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] |
<p>Is there software available that can analyse a 5MHz RF pulse to give a plot of frequency spectrum.
The signal data is visible on a LCD screen or a print out could be obtained.</p> | 3,772 | [
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<p>Spin echo experiments have been able to reverse the motions of all the molecules in a gas in statistical mechanics in the manner of Loschmidt. The Fermi-Ulam-Pasta model has solutions with a single mode dispersing, only to recohere after quite some time has elapsed. Can the same thing happen for decoherence? What are the conditions fyor decoherence to be irreversible?</p> | 3,773 | [
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<p><strong>Question</strong>: are they mathematically possible at all? physically?</p>
<p>with finite mass systems, usually the binding energy contributes to the rest-mass of the system. It would seem that even if you could bind two massless fields/particles, the coupled system would still have a finite rest mass because of the binding energy</p>
<p>by any chance, is this how higgs boson gives 'mass' to particles?</p> | 3,774 | [
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<p>Using the renormalization group approach, coupling constants are "running". If we apply this to the fine structure (coupling) constant, we do know that, e.g., at energies around the Z mass, $$\alpha \approx 1/128$$ instead of 1/137. We know that $$\alpha =e^2/ \hbar c$$. Therefore, if alpha is running with energy, at least one of the 3 "constants" there (electric charge, the Planck's constant or the speed of light is varying with energy). I find hard to see (due to gauge invariance) why e should vary, but the remaining options are not much better. Making c vary with energy drives us to varying speed of light theories, and I believe that energy variations of the speed of light are well bound from different experiments. A varying Planck constant? I can not see a physical meaning of it! Therefore, my question is: </p>
<p><strong>HOW PEOPLE can not find "disturbing" the issue of a "running coupling constant" like alpha?</strong> And related to this: Is there some experiment to search for energy variations of the Planck constant beyond those with respect to the speed of light? An about a varying electric charge with energy? I find it difficult due to gauge invariance! So, how can people live with " a varying fine structure constant withoug being "puzzled" too much?</p> | 3,775 | [
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<p>I know that $\frac {dv}{dt}=a$ is acceleration,
but:</p>
<ol>
<li><p>what is convective acceleration of a flow velocity?</p></li>
<li><p>what is difference between $(v\cdot \nabla) v$ and $v\cdot (\nabla v)$, ?</p></li>
</ol> | 3,776 | [
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<p>I'm quite sure that similar questions like this have been asked for more than thousands of times on here but <strong><em>since each person's background and interests are unique</em></strong> I believe questions like this should not be considered duplicates.</p>
<p>I'm a first year pure math undergradudate student. I have a decent knowledge of multi-variable calculus, mathematical analysis, ODE's, linear algebra and probability theory. Recently I've been teaching myself vector analysis because apparently physicists tend to use vector analysis way more often than mathematicians do (As Daniel Fisher on math.SE said: theoretically mathematicians don't need to know vector analysis because they study differential forms on manifolds in details, that's why vector analysis is never taught in math curriculum). I feel comfortable with index notations, Einstein's summation convention and tensor algebra, however, I have no intuition about how physicists think about tensors.</p>
<p>My knowledge of physics is limited to the Physics I and Physics II courses that I've taken. Both of these courses were taught from the book "Fundamentals of Physics, 8th edition" written by "Halliday, Resnick and Walker". Before I go to university, I used to self-study Resnick's introduction to Special Relativity on my own. I feel quite comfortable with the mathematical ideas behind SR to the level explained in that book. I guess I have to add that our physics I course covered all chapters of Halliday's book except the last chapters on thermodynamics and our physics II course covered up to the RLC circuits (It was a summer course and we had shortage of time).</p>
<p>So, having said all this, my main question is this: I think if I ever decide to get into a graduate program in my current university, I prefer to focus on mathematical physics. I like to study physics from a mathematical point of view. Fortunately, my university allows students to take courses from the Physics department and jointly work with Physics professors during a Master degree. I want to strengthen my physics background during my undergraduate studies. I know that physics undergrad students study a lot of physics and I don't have time to cover all those topics, so I prefer to jump directly to the topics I like to study during my master degree. I'm fascinated by General Relativity and Quantum Mechanics and I hope one day I'd be able to understand string theory.</p>
<p>What is the fastest "mathematically oriented" approach to study GR and QM without learning Lagrangian and Hamiltonian mechanics or Advanced Electromagnetism or such complicated stuff?</p>
<p>Thanks for spending your time reading my question. I hope that my question is not closed as off-topic or duplicate because I really need guidance for my own specific situation.</p> | 108 | [
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<p>I have a question... Do you round with significant digits during each subcalculation of a problem or only when the entire problem is complete?</p>
<p>Example:</p>
<p>multiply the following number:</p>
<p>$$1.8 \times 2.01 \times 1.542$$</p>
<p>saving rounding until the end:</p>
<p>$$(1.8 \times 2.10) \times (1.542) = (3.78)\times(1.542) = (5.82876) \to 5.8$$</p>
<p>rounding at each sub-calculation:</p>
<p>$$(1.8 \times 2.10) \times (1.542) = (3.8)\times(1.542) = (5.8596) \to 5.9$$</p>
<p>I also have the strong feeling that if you round at each sub-calculation then multiplication is no longer commutative (although after experiencing matrices that no longer seems to be too much of a problem)</p> | 3,777 | [
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<p>In Classical Mechanics, both Goldstein and Taylor (authors of different books with the same title) talk about the centrifugal force term when solving the Euler-Lagrange equation for the two body problem, and I'm a little confused about what it exactly means - is it a real centrifugal force or a mathematical consequence of using polar coordinates for solving the Euler-Lagrange equation.</p>
<p>Their derivations of the Langrangian $L=\frac{1}{2}\mu(\dot{r}^{2}+r^{2}\dot{\theta}^{2})-U(r)$ would lead to one motion of equation (theta) showing that angular momentum is constant and one radial equation of motion shown as
$\mu\ddot{r}=-\frac{dU}{dr}+\mu r\dot{\phi}^{2}=-\frac{dU}{dr}+F_{cf}$. They call $\mu r\dot{\phi}^{2}$ the fictitious force or the centrifugal force. I'm quite hazy on my memory of non-inertial frames, but I was under the assumption that fictitious forces only appear in non-inertial frames. The frame of reference in the two body problem was chosen such that the Center of Mass of the two bodies would be the origin so that would be an inertial frame, and I'm assuming that there are no non-inertial frames involved since neither author had talked about it in the previous chapters. </p>
<p>Would calling $\mu r\dot{\phi}^{2}$ an actual centrifugal force be incorrect then? Isn't it a term that describes the velocity perpendicular to the radius? From this two-body problem, it appears as though if I were to use polar coordinates when solving the Euler-Lagrange equations for any other problem, the centrifugal force term will always appear, so it would be a mathematical consequence of the choice of coordinate system rather than it being an actual fictitious force. Is that term being called a centrifugal force because it actually is a centrifugal force or is it because it has a mathematical form similar to it?</p> | 3,778 | [
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<p>I want to know if my solution to a textbook problem has any major problems with it. Here is the problem:</p>
<blockquote>
<p>Ethanol has a given density of 0.789 g/mL at 20 degrees Celsius and isopropanol has a given density of 0.785 g/mL at 20 degrees Celsius. A chemist analyzes a substance using a pipet that is accurate to $\pm 0.02$ mL and a balance accurate to $\pm 0.003$ g. Is this equipment precise enough to distinguish between ethanol and isopropanol?</p>
</blockquote>
<p>And here is my solution:</p>
<blockquote>
<p>We can calculate with tolerances in the same way we calculate measurements. The mass tolerance of $\pm 0.003$ g has three significant figures. The volume tolerance of $\pm 0.02$ mL has two significant figures. The density tolerance will therefore have two significant figures. $\pm 0.003 \text{ g} / \pm 0.02 \text{ mL} = \pm 0.15 \text{ g/mL}$.In order to distinguish between ethanol and isopropanol, whose densities differ by 0.789 - 0.785 g/ mL, or 0.004 g/ mL, we need a precision smaller than half the difference, or 0.002 g/mL. But we can only measure density to within 0.15 g/mL of the actual value. Therefore, this equipment is not precise enough to distinguish between ethanol and isopropanol.</p>
</blockquote>
<p>But what I don't like or feel is right about dividing the tolerances like that is that having a smaller tolerance (more precise) for volume in the denominator blows up (bad) your density tolerance. Shouldn't higher precision (smaller tolerance) of either pipet OR balance result in higher precision (smaller tolerance) of density measurement?</p>
<p>Edit: I tried to give just as much information as is relevant to my problem, but I guess one detail from part (a) of the textbook problem (what I described was part (b)), which said the nominal sample volume was 15.00 mL, must carry over to part (b). So I think I assume the nominal volume of my sample is 15.00 mL plus or minus 0.02 mL.</p> | 3,779 | [
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<p>A particle moves with force</p>
<p>$$F(x) = -kx +\frac{kx^3}{A^2}$$</p>
<p>Where k and A are positive constants.</p>
<p>if $KE_o$ at x = 0 is $T_0$ what is the total energy of the system?</p>
<p>$$ \Delta\ KE(x) + \Delta\ U(x) = 0$$</p>
<p>$$F(x) = -\frac{dU}{dx} = m\frac{dv}{dt} = m v\frac{dv}{dx}$$</p>
<p>Integrating to get U(x) and 1/2mv^2 I get </p>
<p>$$\Delta\ U(x) = \frac{kx^2}{2} - \frac{kx^4}{4A^2}$$</p>
<p>$$\Delta\ KE(x) = -\frac{kx^2}{2} + \frac{kx^4}{4A^2}$$</p>
<p>Which Makes sense. But how do I find the function KE(x) where KE(0) = $T_0$? Do I Even need to? The total energy in the system is $T_0$ Correct?</p>
<p>Also a kind of side note. What is really confusing me, is when should I add limits of integration and under what circumstances should I just use an indefinite Integral?</p> | 3,780 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/11398/double-light-speed">Double light speed</a><br>
<a href="http://physics.stackexchange.com/questions/30505/someting-almost-faster-than-light-traveling-on-something-else-almost-faster-than">Someting almost faster than light traveling on something else almost faster than light</a> </p>
</blockquote>
<p>Well I've been wondering quite a long time about this problem. If you had let's say 10 rockets, each of them having a length enough to provide space for an acceleration of another rocket inside. Each rocket would accelerate from 0, to 1/10 of speed of light. From the observers point of view, wouldn't the last rocket achieve the speed of light (since the speeds add up).</p> | 109 | [
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<p>$$\langle \hat A \rangle \langle \hat B \rangle=\langle \hat A\hat B \rangle,$$</p>
<p>$$\langle \hat A \rangle + \langle \hat B \rangle=\langle \hat A + \hat B \rangle,$$</p>
<p>$$\langle \hat A^2 \rangle \langle \hat B^2 \rangle=\langle \hat A^2 \hat B^2 \rangle,$$</p>
<p>$$\langle \hat A^2 \rangle + \langle \hat B^2 \rangle=\langle \hat A^2 + \hat B^2 \rangle,$$
Which one of them is not true!?</p> | 3,781 | [
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<p>Even if many interesting similarities between the classical and the quantum mechanical framework have been worked out, e.g. in the subject of deformation quantization, in general, there are some mathematical problems. And in the conventional formulation, you don't want to make things like $\hbar\rightarrow 0$ for the expression $P=-\text i\hbar\tfrac{\partial}{\partial x}$. </p>
<p>In special relativity there are many formulas where one optains the non-relativistic formula by taking the naive limit $c\rightarrow \infty$, e.g.</p>
<p>$$\vec p=\frac{m\vec v}{\sqrt{1-|v|/c}}\ \rightarrow\ \frac{m\vec v}{\sqrt{1-0}}=m\vec v.$$</p>
<p>I wonder if it is know that you can always do that. Is there a formulation of special relativity (maybe it's the standard one already), where the starting assumptions/axioms/representations of objects of discourse involve the constant $c$, and as you take them with you to do all the standard derivations, you always end up with results which reduce to the Newtonian mechanics if you take that limit?</p> | 3,782 | [
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<p>Is black hole entropy, computed by means of quantum field theory on curved spacetime, the entropy of matter degrees of freedom i.e. non-gravitational dofs? What is one actually counting?</p> | 3,783 | [
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] |
<p>What is the physical meaning/significance of the <a href="http://en.wikipedia.org/wiki/Classical_electron_radius" rel="nofollow">classical radius of the electron</a> if we know from experiments that the electron is point like?</p>
<p>Is there similarly a classical radius of the photon? The W and Z bosons?</p> | 3,784 | [
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] |
<p>I'm working through <a href="http://natureofcode.com/" rel="nofollow">The Nature of Code</a>, which is an awesome book, lots of fun.</p>
<p>I've come across Exercise 3.13 and I'm not sure how to solve it. I'm assuming that the force of gravity and the force of friction are provided. I'm not sure in which directions to draw my right triangles to solve for the normal force. Can someone provide an explanation?</p>
<blockquote>
<p>Using trigonometry, what is the magnitude of the normal force in the
illustration on the right (the force perpendicular to the incline on
which the sled rests)? Note that, as indicated, the “normal” force is
a component of the force of gravity.</p>
</blockquote>
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<p>I wanted to make sure I understand induction well enough.</p>
<p>Assume we have two wires running parallel to each other. Wire A has a signal of $f(t)$, wire B has a signal of $\hat{f}(t)$.</p>
<p>Let's connect a signal generator to wire A, therefore putting
$$f(t) = A \cdot sin(2\pi f_{c} t)$$
where $A$ is the amplitude of the wave, and $f_{c}$ is its frequency.</p>
<p>This will induce changing current $\hat{f}(t)$ in wire B. My question is: How will $\hat{f}(t)$ look like?</p>
<p>My guess is that it will have the form
$$\hat{f}(t) = \hat{A} \cdot sin(2\pi f_{c} t + \phi)$$
where $\hat{A} \leq A$ and is proportional to $A$ and $\phi$ is an additional phase factor due to the fact that radio waves travel at finite speed.</p>
<p>Is that a correct guess? Does it only happen if the wires are infinitely long? I haven't derived this properly and I don't really need a detailed derivation (although it wouldn't hurt). I just want to know if there is a good way to describe the received $\hat{f}(t)$.</p> | 3,786 | [
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<p>I am reading on magnetic monopoles from a variety of sources, eg. <a href="http://arxiv.org/abs/hep-th/9603086/" rel="nofollow">the Jeff Harvey lectures.</a>. It talks about something called the winding $N$, which is used to calculate the magnetic flux. I searched the internet but am not being able to understand the calculation done in this particular case. </p>
<p>$g=-\frac{1}{8}\int_{S^2_\infty} Tr([d\hat{\Phi},d\hat{\Phi}],\hat{\Phi})$</p>
<p>Then the author says that</p>
<blockquote>
<p>Now $\Phi$ restricts to a map $\Phi : S_\infty^2 → S^2$ , where the target is the unit sphere in $su(2)$. This map has some degree $N$ , and it is easy to verify that the right-hand side of the above equation is $−2\pi$ times this. Therefore $g = −2\pi N$ .</p>
</blockquote>
<p>What is $N$, the winding number also called as the degree on the map? By what i have learnt, it is the number of times you wind an object unto the another, then shouldn't the integral be $N*4\pi$, as $4\pi$ is the surface area of $S^2$.</p> | 3,787 | [
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] |
<p>Is there any scientific evidence that demonstrates why time passes?
Or is it just an opened question?</p> | 3,788 | [
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<p>From what I gather the interstellar medium has about about 1 atom per cubic centimeter. But on the other hand, as they say, "Space is big, really really big" So if it is known (or at least theorized about) what percent of the universes matter is in the following: black holes, stars, and interstellar medium?</p> | 3,789 | [
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<p>I'm doing a paper for school about space navigation. It's an interesting subject, but upon searching google, all I could find were non-detailed explanations of astrophysics that I think are meant for children, and a few somewhat-more-detailed articles about other subjects.</p>
<p>I remember learning about the subject in high school physics class, and I remember there were people that were credited for there work in space exploration. Who are those people?</p>
<p>My paper is specifically about 'navigating', ie determining when to leave the Earth, where the spacecraft is headed, and where certain astronomical objects will be when the spacecraft gets there. I'm looking for articles (or any information) on the work that goes into that, and the people who should be credited for that work, and for pioneering it.</p> | 3,790 | [
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<p>Since we see the <a href="http://en.wikipedia.org/wiki/New_moon" rel="nofollow">new moon</a> at least once in a month when the Moon gets in between of the Sun and the Moon at the night and as far as I know if this happens during the day, you'll get to see a <a href="http://en.wikipedia.org/wiki/Solar_eclipse" rel="nofollow">solar eclipse</a>. Why don't we get to see this often or in the day? </p>
<p>Does it mean that in some part of world there's a solar eclipse when we are seeing a new moon? I'm looking for a diagram or interactive way to understand this if possible as I'm not a native English speaker, but I'll try my best to do so. </p> | 110 | [
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<p>Suppose a string element oscillating transversely in <em>SHM</em> having both kinetic energy and elastic potential energy travels with velocity $u$. Let $y(m)$ be the amplitude. When it reaches $y(0)$, it has max. $K.E.$ and elastic $P.E.$ together at the same time. But in <em>SHM</em>, when the kinetic energy is max. the potential energy is min. but in the string it is contradicted. Does transverse wave violate <em>SHM</em> and if not why at the same time it has both K.E. and P.E. maximum unlike <em>SHM</em>?</p> | 3,791 | [
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<p>An often quoted figure is that the LHC magnets take a month to completely cool and a month to warm. There is never an explanation as to why that is. I can conjure any number of reasons (slow changes to prevent stress, very low temperature deltas, gremlins can hold their breath for 29 days, etc) but I can't find any facts.</p>
<p>I don't suspect the answer will be complex, but I'd just like to know why it takes a month and not, say, 14.6 hours.</p> | 3,792 | [
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<p>Diamond is one of the best thermal conductors you can get. If the diamond is crushed into dust and spread out over a flat surface, but still held fairly compact (for instance in a small petri dish), would it still conduct as well as before?</p> | 3,793 | [
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<p>Let's say we have a meter stick with a single rope attached to it. One end of the rope is attached to one end of the meter stick, the other end of the rope is attached to the opposite end of the meter stick. By hanging the rope from a beam in the center of the rope, the meter stick (which is attached to the rope) balances perfectly flat in equilibrium as it hangs from the rope. Now, say the rope is not hung perfectly in its center, and as a result the meter stick accelerates from its perfectly horizontal position to an almost vertical position. (Does this make sense so far?) At this point where the meter stick is now at rest at an angle which is at a diagonal (the actual angle is irrelevant, we could say perhaps 5 degrees off the vertical, just to clarify to the reader), is the tension in the rope still constant among the two halves of the rope? How can it be, when it appears that so much more of the meter stick's weight is being held up by one end of the rope?</p> | 3,794 | [
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<p>When I boil water in the kettle on my electric stove, sometimes it rocks back and forth making an annoying sound at a frequency of about 6Hz. When that happens, I move the kettle slightly to make it stop rocking. It occurred to me that although my motivation is to stop the annoying sound, it might also make the water boil slightly faster because the energy that had been wasted in maintaining the oscillation of the kettle is now being directed elsewhere. I am supposing that this energy is transferred to the kettle because the bottom of it is now in more constant contact with the electric heating element.<br>
To test this theory, I am thinking that I could experiment by boiling the water each morning (as I already do) and note the time it takes to boil, the ambient temperature and the temperature of the water, and make sure that the exact same amount of water is in the kettle each morning. By noting whether the kettle vibrates or not, but without adjusting it, I could see if I can statistically disprove the null hypothesis that there is no correlation between the kettle boiling and the time it takes to boil.<br>
Is there anything missing from the proposed design of the experiment, and/or is the answer already well known?</p> | 3,795 | [
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<p>Say I have a pot of water that boils in 20 minutes, at whatever temperature.</p>
<p>If I leave the fire on, take the pot off, pour the hot water into a container, refill the pot with tap water and put it back on the fire.</p>
<p>How long will the 2nd pot take to boil and why?</p> | 3,796 | [
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] |
<p>Single photons: Is there a 90° offset of the electric to the magnetic component in the direction of propagation?</p>
<p><img src="http://i.stack.imgur.com/ksHYn.png" alt="Electromagnetic wave"></p> | 3,797 | [
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<p>I've taken a class on elementary number theory (for fun), but now I wonder: was it at all useful to learn number theory for my future career in physics?</p>
<p>More to the point, are there any applications of elementary number theory (the kind that would be taught in a first or second undergrad-level course) in the natural sciences (especially physics, but also chem, bio, or <em>maybe</em> geology)?</p> | 111 | [
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<p>The universe is expanding at an ever increasing rate. Virtual particles are being created and then destroyed everywhere at every moment. As space expands even faster, will a time come when these virtual particles will be ripped apart before they can destroy each other, much like what happens at gravitational event horizons. Does this mean that the universe will at this point find itself suddenly full of Hawking radiation, almost as if a new "Big Bang" had occurred?</p> | 3,798 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/1801/why-space-expansion-affects-matter">Why space expansion affects matter?</a> </p>
</blockquote>
<p>Imagine two tiny spacecrafts that are moving with the Hubble flow and so are moving away from each other. Let's assume that they've been that way since the very early universe, never firing their engines, just drifting along their Hubble flow geodesics in a homogenous isotropic universe.</p>
<p>They then momentarily fire their engines so that they "cancel" the Hubble flow and have a fixed proper separation. Will they now start drifting apart again (presumably due to expansion)? Or will they stay at fixed proper separation, and maybe very slowly move towards each other due to their mutual gravitational attraction?</p>
<p>I guess this is another way of asking whether expansion is kinematic and hence can be forgotten (so that "Brooklyn isn't expanding" because gravitational collapse and structure formation have erased memory of the expansion). Or maybe someone will help me refine this question and make me realize I just haven't thought things through completely?</p> | 56 | [
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<p>In the paper, <a href="http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.52.997" rel="nofollow">"Density-functional theory for time-dependent systems" Physical Review Letters 52 (12): 997</a> the authors mentioned that the action
$$ A= \int_{t_0}^{t_1} dt \langle \Phi(t) | i \hbar\partial / \partial t - \hat{H}(t) | \Phi(t) \rangle \tag{1} $$</p>
<p>provides the solution of time-dependent Schrodinger equation at its stationary point. <a href="http://en.wikipedia.org/wiki/Time-dependent_density_functional_theory" rel="nofollow">Wikipedia </a> called (1) as the Dirac action without further reference. </p>
<p>If I do a variation, indeed the stationary point of action (1) gives
$$ i \hbar\partial / \partial t | \Phi(t) \rangle = \hat{H}(t) | \Phi(t) \rangle $$</p>
<p>However, from path-integral point of view, the least action principle is only a limiting case when $\hbar \rightarrow 0$. In general, there is no least action principle in quantum mechanics. </p>
<p>My question is, how to reconcile these two aspects? What does vary of action (1) mean?</p> | 3,799 | [
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<p>A positron is odd under charge conjugation and parity reversal but nevertheless even with respect to time reversal. Is a theoretical positron which would be odd under all three symmetries (C, P, T) physical?</p> | 3,800 | [
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<p>I thought light like singularity is where the geodesics end on a lightlike hypersurface and can't be extended anymore. I guess its different than light cone singularity. Lot's of places have mention of it, but I wanted to know its definition for sure.</p> | 3,801 | [
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<p>Suppose a simple circuit with a <strong>DC voltage</strong> source and a <strong>resistor</strong>. The voltage of the source will be situated over the resistor.
So the <strong>electric field</strong> (which is the gradient of the potential) will be constant in the resistor (if you assume a linear potential function in the resistor), and will be equal to <strong>zero in the conducting connections</strong>.</p>
<p>Since <strong>electrons</strong> are <strong>drifted</strong> because of an <strong>electric field</strong> (with Newton's second law and Lorentz' law for the force), what keeps them drifting in the ideal conductors? Or do they just keep their velocity they got in the resistor, and don't decelerate because there's no resistance there? What would mean that the electrons <strong>obtain their velocity within the resistor</strong>, which sounds a bit paradoxal...</p>
<p>Where's the loop in my argumentation, or am I just right?</p> | 3,802 | [
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<p>In particle physics, when you read $J^P$, does it mean Spin parity or total angular momentum parity?</p>
<p>I know that the letter $J$ is used for TOTAL angular momentum but I think I read somewhere that sometimes it is meant spin when used in this $J^P$ notation.</p> | 3,803 | [
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<p>Sorry if this is rather simple, but I've only just started learning about using logarithms in experimental physics. </p>
<p>I did an experiment to test the amount of time it would take for an amount of water to leave a burette. I used the starting volume of water in the burette as a control variable, $50cm^3$. I recorded the time it took for the a given volume of water to be left in the burette. For example, $10cm^3$ left took a time of roughly $71\mbox{s}$; $45\mbox{cm}^3$ left took roughly $6\mbox{s}$, and then many values in between.</p>
<p>I would expect this to represent exponential decay, seen as different concentrations and masses of water in the burette would have different effects on the speed of the water leaving the burette. (Correct me if I'm wrong.)</p>
<p>So I plotted a graph of volume against time and it showed exponential decay, but it was only very slightly curved, but curved nonetheless. </p>
<p>So I decided then to plot a graph of $\ln\left(V/\operatorname{cm}^3\right)$ against time/s. However, this did not produce a straight line. If I were to follow the plotted points with a curve, the gradient of the line would have been negative and increased in negative 'magnitude'. </p>
<p>I'm meant to analyse the extent of whether or not my experiment shows exponential decay. I'm quite stuck, because my original graph shows very slight decay, whereas my log graph isn't a straight line. Does the fact that the log graph doesn't produce a straight line show that there isn't exponential decay? Does it not matter? Would it have been straight had there been very few experimental errors/uncertainties (there would have been a lot)?</p>
<p>So I guess, fundamentally, my question is:</p>
<p>What does the curved line on my natural log graph suggest? </p> | 3,804 | [
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<p>I've recently heard about this new material called Quantum Stealth which is suppost to be a replicate of Harry Potter's 'Invisibilty cloak' . Apparently the material, bends light waves around a target which allows complete invisibility. Is all this true?</p> | 3,805 | [
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<p>I'm dealing with a problem here. Today my professor asked us a question :</p>
<p>How should a metallic wire move in Earth's magnetic field such that she gets maximum potential difference in it's ends:</p>
<p>a) Horizontally
b) Vertically
c) Normal in the magnetic field lines </p>
<p>Can anyone help me with this question?</p>
<p>Thank you !</p> | 3,806 | [
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<p>Fermi surfaces are surfaces of constant energy in reciprocal space. They provide information about the properties of a material in solid state physics.</p>
<p>Constant mean curvature surfaces are a superset of minimal surfaces, which minimize area and have zero mean curvature. Soap bubbles are surfaces of constant mean curvature.</p>
<p>The two certainly coincide in the case of a sphere, but must they always coincide? If so, why? Is there a sharper constraint that Fermi surfaces must obey as well?</p>
<p>I found a few papers that mention the two as connected, but they never establish the exact connection between the two.</p>
<p>Preliminary literature review:</p>
<p>Mackay, Alan L. Periodic minimal surfaces, 1985.</p>
<blockquote>
<p>Minimal surfaces are found, as mentioned
above, in soap films . . . The surfaces of
constant energy in reciprocal space, used in solid
state physics for finding the Fermi surface, are
very similar.</p>
</blockquote>
<p>Mackay, Alan L. Periodic minimal surfaces from finite element methods, 1994.</p>
<blockquote>
<p>Fermi surfaces, which are surfaces in reciprocal space, are closely related to nodal surfaces.</p>
</blockquote> | 3,807 | [
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<p>In $\phi^3$ theory, the generating functional for interacting field theory is given by:
$$ Z_1(J) = \sum_{V=0}^{\infty} \frac{1}{V!} \Big[ \frac{iZ_g g}{6} \int \Big( \frac{1}{i}\frac{\delta}{\delta J}\Big)^3 d^4 x \Big]^V \times
\sum_{P=0}^{\infty} \frac{1}{P!} \Big[ \frac{i}{2} \int J(y) \Delta(y-z) J(z) \, d^4 y \, d^4z \Big]^P $$</p>
<p>[Reference: Srednicki: eqn. (9.11)]</p>
<p>Let, for specific values of $V$ and $P$ we get some terms from it. One of them is a disconnected diagram consisted of two connected diagrams $C_1$ and $C_2$. The disconnected diagrams symmetry factor is, say, $S$; that is the term for disconnected diagram has a numerical coefficient: $\frac{1}{S}$. Now we write the term for disconnected diagram according to the eqn (9.12):
$$ D = \frac{1}{S_D} \prod_I (C_I)^{n_I}$$</p>
<p>where $n_I$ is an integer that counts the number of $C_I$ ’s in $D$, and $S_D$ is the additional symmetry factor for $D$. Here, $S_D = \prod_I n_I !$</p>
<p>In this case is this true: $S=\frac{1}{n_1!} \times \frac{1}{n_2!} \times C_1$'s symmetry factor $\times C_2$'s symmetry factor?</p> | 3,808 | [
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<p><a href="http://en.wikipedia.org/wiki/Energy" rel="nofollow">Energy</a> seems to me to be a very abstract thing, and while it clearly works out very nicely, I don't understand how anyone would have thought to come up with it. Where does the concept of energy find it's roots, and how was it settled down on as a 'useful' quantity as opposed to something else?</p> | 3,809 | [
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<p>I've noticed something curious about the rotation of a rectangular prism. If I take a box with height $\neq$ width $\neq$ depth and flip it into the air around different axes of rotation, some motions seem more stable than others. The 3 axes which best illustrate what I mean are:</p>
<p>(1) Through the centre of mass, parallel to the longest box edge.</p>
<p>(2) Through the centre of mass, parallel to the shortest box edge.</p>
<p>(3) Through the centre of mass, parallel to the remaining box edge.</p>
<p>It's "easy" to get the box to rotate cleanly around (1) and (2), but flipping the box around (3) usually results in extra twisting besides the rotation around (3) that I'm trying to achieve (obviously a "perfect" flip on my part would avoid this twisting, which is why I call it an instability). If you're not quite sure what I'm talking about, grab a box or book with 3 different side lengths and try it out (but careful not to break anything!).</p>
<p>What's special about axis (3)?</p>
<p><img src="http://i.stack.imgur.com/yzD8c.png" alt="enter image description here">
Image taken from <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.111.1838">Marsden and Ratiu</a>.</p> | 3,810 | [
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<p>I've read sci-fi stories in which a spaceship crew, moving at some significant fraction of 'c', observes stars ahead as bluer, stars behind as redder, stars to port and starboard as...well, you get the idea.</p>
<p>This seems wrong, though I'm not sure if it is. Seems to me the red/blue shifts would only be apparent with spectroscopic analysis, not to the naked eye. Yes? No?</p> | 3,811 | [
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<p>I may be wrong, but it seems that only logarithmic divergences need to be retained when using the Callan-Symanzik equation, finding running couplings, etc. Why is this the case? Is there some simple intuitive understanding for why the logarithmic divergences are most important for these applications?</p> | 3,812 | [
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<p>I was thinking about the Google XPrize for Space Travel the other day.</p>
<p>In order to claim the prize of building a robot that goes to the moon, travels 500m, and relays data, I had the idea of building a tiny vessel the size of a marble and shooting it at the moon with a railgun.</p>
<p>The theory is that since the mass of the marble-sized craft is only a fraction of the size of a regular shuttle, it should take exponentially less energy to get it to the moon.</p>
<p>So, if you could aim it right, and shoot it with enough power to escape Earth's gravity (Let's say it weight 1oz) couldn't you shoot your own vessel to the moon with essentially a potato launcher? </p>
<p>Why does this not work?</p> | 3,813 | [
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<ul>
<li><p>In this paper, <a href="http://arxiv.org/abs/1301.4504" rel="nofollow">http://arxiv.org/abs/1301.4504</a> in equation 4.1 in what sense are the two states a "9-qubit state"? I did not understand this counting. </p></li>
<li><p>Can someone explain what are the different $X_i$ and $Z_i$ in 4.2? How is say $X_1$ Pauli matrix different from $X_8$ and so on? </p></li>
<li><p>And what is the non-trivial thing that happened in equation 4.2? (aren't the two kets on the LHS and the RHS the same?) </p></li>
</ul>
<hr>
<p>It would be great to hear of any other general insights/explanation people might have about this section 4. </p> | 3,814 | [
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