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<p>Hello again question board.
I'm in need of some help with my b) question for homework I have. Following question is verbatim:</p>
<p>"How thick should a wall of shielding iron be to absorb a 10 GeV/c pion beam. Use the PDG table and the lecture notes to find the necessary parameters. a) Compute the thickness if the ionization loss would be the only source of energy loss. (Hint: make a simple model or use plots from the lecture.) b) The pion may lose energy also via nuclear interaction. The nuclear interaction length is $\text{132.1 g/cm}^2$. Compute the thickness for pure nuclear interaction effects. (Use an intensity reduction of $10^9$.)"</p>
<p>I have been struggling with this assignment for the past few days, and for the life of me I can't find any information (aside from it being a real thing, and that the nuclear interaction stopping power being far lower than the ionization stopping power) on how exactly to answer the question, meaning, I can't find any sort of formula to use. Any hints or tips would be much appreciated!</p> | 2,249 | [
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] |
<p>I have several doubts about solving circuits.</p>
<ol>
<li>Can any circuit be solved using <a href="http://en.wikipedia.org/wiki/Nodal_analysis" rel="nofollow">Nodal Analysis</a>?</li>
<li>If some circuit can be solved using Nodal Analysis, can it be solved using Mesh Analysis too?</li>
<li>Why do we need these techniques to solve circuits?</li>
</ol> | 2,250 | [
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<p>I need a nudge in the right direction, i guess (this is not a homework question).</p>
<p>I want to calculate the total length of a ray from an emitter to a target which passes through a slab with known properties.</p>
<p>Given:</p>
<ul>
<li>Position of Emitter and Target $P_\text{Source}, P_\text{Dest}$</li>
<li>Refractive indices of the slab and surrounding medium $n_1, n_2$</li>
<li>position and thickness of the slab $d_1$, $d_2$, $d_3$</li>
</ul>
<p><img src="http://i.stack.imgur.com/6RHfd.png" alt="Double refraction on glass slab"></p>
<p>Question:</p>
<ol>
<li>What angle of incidence $\theta_i$ is required for a ray cast from $P_{source}$ to intersect $P_\text{dest}$?</li>
<li>How long is the path the ray actually takes?</li>
</ol>
<p>I know the incident ray is displaced parallely depending on angle of incidence and refractive indices. I have also found some equation for the determiantion of the offset, but i am still not sure how to apply it to my problem:
$$
\begin{equation}\tag{1}
\Delta y = d_2 \tan \theta_i \left( 1- \frac{\cos \theta_i}{\sqrt{n^2 - \sin^2 \theta_r}}\right)
\end{equation}
$$</p>
<p>where $\Delta_y$ is the offset of the ray emerging from the slab.</p>
<p>Could you give me a few directions?</p> | 2,251 | [
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<p>A satellite at 8000 km from Earth where gravity is 6.4 m/s^2. What is the velocity of satellite? I don't the formula to find speed of satellite.</p> | 2,252 | [
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<p>We want to study the magnetic field at point $P$. So, from the figure we take that:</p>
<p><img src="http://i.stack.imgur.com/xWx3I.png" alt="enter image description here"></p>
<ol>
<li>$\oint_{L_1} B\cdot dl=\mu_0 I_1$</li>
<li>$\oint_{L_2} B\cdot dl=\mu_0 I_2$</li>
<li>$\oint_{L_3} B\cdot dl=\mu_0 I_2$</li>
</ol>
<p>The magnetic field contribution of the current $i_1$ at the point $P$ is: $B_1=\frac{\mu_0I_1}{2\pi r}$. And the magnetic field contribution of the current $i_2$ at the point $P$ is: $B_2=\frac{\mu_0I_2}{2\pi r}$. Then, the resulting magnetic field at point $P$ is: $$B = \frac{\mu_0(I_1+I_2)}{2\pi r}$$</p>
<p>But, my question is what happen if I take into account the line $L_3$? Because it also has a contribution $B_3=\frac{\mu_0I_2}{2\pi r}$ at point $P$. Would be erroneous to say that $B = B_1+B_2+B_3$. </p>
<p>Does the problem is that $L_3$ is in the same current that the $L_1$ line? If it is so, what happen if $i_1=i_2$ and both wires are in the same circuit? would it be wrong to separately calculate contributions from $L_1$ and $L_2$?</p>
<p>I'm a little confused about it.</p> | 2,253 | [
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<p>It is difficult to imagine the infiniteness of space and how it itself is expanding rather than the universe expanding into something else. A helpful analogy is that of drawing little dots (representing galaxies or some other sub-universal structure) onto a deflated balloon and then blowing it up. The surface expands in all directions, with each dot moving away from every other dot. Although the analogous surface (the outside of the balloon) is effectively 2 dimensional, it's possible to imagine its translation into 3 dimensions.</p>
<p>As for time, though, I have a hard time picturing its "before / during / after" states, and I realize those words aren't even accurate. Time supposedly began at the Big Bang and may end at the Big Crunch. But I'm wondering if anyone knows of an analogy for time, similar to the balloon analogy that applies to space. Is there a way to imagine time in some comprehensible way?</p> | 2,254 | [
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<p>Animals that have an eye on each side of their head have 360 degree vision so just like with the inverted vision glasses you can buy I'm wondering how you make 360 degre e glasses?
I'm thinking of a half periscope setup with a lense on the end on each eye.</p> | 2,255 | [
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<p>I have situation where I am simulating discrete time quantum walk (DTQW) for various graphs. I have two quantum walkers on the graph and they can interact with each other by the fact that where the two walkers are co-located on the same node of the graph a relative phase can be introduced.</p>
<p>I then measure the entanglement of the two quantum walkers using the Von Neumann entropy. What you find is that two non entangled walkers become entangled over time however the degree of entanglement varies over time in a complex manor. </p>
<p>I am investigating if there is any 'quantum chaos' exhibited by the system. So I have been trying to read '<em>Quantum Signatures of Chaos</em>'by Haake. As I understand it there will not be any exponential sensitively to initial conditions as this is Quantum Mechanics. There could however be such a sensitivity to the phase of the interaction since this is changing the Hamiltonian.</p>
<p>So my question is, what would be an appropriate measure or tool to determine if this system exhibits any quantum chaos in relation to the interaction phase as the parameter of the Hamiltonian which is varied?</p>
<p>Here is one of the many papers which explains the background framework of DTQW: <a href="http://arxiv.org/pdf/quant-ph/0504042.pdf" rel="nofollow">http://arxiv.org/pdf/quant-ph/0504042.pdf</a></p> | 2,256 | [
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<p>An ergodic dynamical system $(\Omega,\phi^t,\mu)$ is such that the time average $\bar{f}$ of every function $f\in L_1(\Omega,\mu)$ equal the <em>space</em> average $\langle f \rangle_\mu$, i.e. the system densely cover all the phase space ($\mu$-almost everywhere). Another equal condition of ergodicity is that the only invariant sets ($\phi^t(B)=B$) are the trivial ones (no way of partitioning the phase space), or looking at the $L_1$ space, the only invariant functions ($f\circ\phi^t=f$) are the constant functions. Moreover, we have a stronger property than ergodicity, namely mixing that implies the former.
A system is said to be mixing if $$\mu(\phi^{-t}(A)\cap B)\rightarrow\mu(A)\mu(B),\qquad\text{as }t\rightarrow\infty.$$
At the end we have the recurrence, i.e. the system pass through all the points of the phase space infinitely many times.
My question arise because in my mind for a system that approaches the equilibrium, there exist a time $T>0$ such that for all $t>T$ the system will spend its future time in a smaller region of the phase space, i.e. there exist a partition of $\Omega$ and then the system cannot be ergodic. </p> | 2,257 | [
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<p>The graph below (see attached) shows the measured magnetization against temperature at room pressure for the material Gd and for another material for use in a magnetic refrigerator at room temperature.</p>
<p>1)Calculate the expected change of temperature of a thermally isolated piece of Gd when the field in which it is located slowly reduced from an induction of $H_1$ = 1.2T to $H_2$ = 0.8T starting at a temperature 300K. You may assume that the $M$ versus $T$ curve does not change appreciably with $H$ for $H$ in the range 0.8-1.2T. Additionally, assume that the molar specific heat capacity of Gd is $C_p = 3R J/K/mol$ of atoms, independent of temperature and field. The formula mass of Gd is 157.25g/mol.</p>
<p>In a previous question, I showed that $$dT = - \frac{T}{C_{P}} \left( \frac{\partial \mathcal{M}}{\partial T}\right)_{H,P} dH,$$
where $\mathcal{M}$ is the magnetic moment. So, given $C_p$ I need to find the final temperature of the material and also the quantity $\partial \mathcal{M}/\partial T$. The assumptions in the questions indicate that the gradient of the M versus T curve is approximately constant, but I am not sure how to find this value. We are also to assume that $\mathcal{M} = MV$ The graph shows $M/\rho$ vs $T$ which is equivalent to $\mathcal{M}/m$ vs $T$. </p>
<p>Many thanks<img src="http://i.stack.imgur.com/Qrd3K.jpg" alt="enter image description here"></p> | 2,258 | [
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<p>Gravitational attraction and electrostatic attraction/repulsion are intrinsic properties of matter, any particle (electron, proton) for some unknown reason can produce KE at a distance.</p>
<p>But magnetic attraction/force is not an intrinsic property of matter, a charged particle generates a magnetic field/flux and a magnetic force only when it is moving: higher velocity = much higher force. The definition of KE says that it is <em>'work done to accelerate an object'</em>, energy spent only to make it move, and does not mention the generation of other forces/energy or doing 'extramural' work.</p>
<p><em>[Electrons moving in a current produce magnetic induction that makes an electric appliance do work, who/what is spending the necessary energy to produce such work?
Voltage (the difference of potential) coming from the mains provides the energy to accelerate the electrons, not the energy to blend your fruit. Same voltage would produce equivalent v/Ke if the electron is accelerated in a vacuum (such as in a synchrotron, where there is no fruit). But this is difficult to prove.]</em></p>
<p><strong>Edit</strong>: I will not reply to any comment. They are attributing to me statements I never made. Now, to avoid technical complications (<em>voltage</em>, wattage etc,) that could trick me, forget the previous example, <strong>let's consider another, simpler case:</strong></p>
<p>An electron is travelling at high speed (say,0.9 C). If is moving near another electron, (proton, positron or a live wire) <em>it can make anything move, acquire KE</em>, it <strong>can do work</strong>. When v approaches c, the attractive magnetic force gets so great that it <em>equals the huge electrostatic repulsion.</em></p>
<p>All the issues you have raised (intrinsic spin, magnetic field, electromagnetism etc., some comments have been deleted) are irrelevant here. If those properties exist they exist even when the electron is at rest. When the electron is at rest, it still has spin, nevertheless it is <strong>not able any more</strong> to do work. </p>
<p>If this is an irrefutable fact, then, how come it can do work when it is moving, where is the necessary energy coming from?</p> | 2,259 | [
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<p>When there is a wave, something is undulating. In the example of a rope, the rope is what undulates. In the case of a ripple on a pond, the water is undulating, and when a sound wave propagates, the air is undulating. The question is: in the case of a particle, what undulates?</p> | 2,260 | [
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<p>Say you want to calculate degree of ionization for different gases in atmosphere of a star with abundances similar to those in Sun (let's assume you only have hydrogen, helium and sodium) over the temperature range (from 2000 K to 45000 K for example) using Saha equation: </p>
<p>$$\frac{n_{i+1}}{n_i}=\frac{g_{i+1}}{g_i} \frac{2}{n_e} \frac{{(2\pi m_e)}^{3/2}}{h^3} {(k_B T)^{3/2}} e^{-\chi /k_B T}$$,</p>
<p>which you write down for all three elements and of course next to abundances and temperature you also know ionization potentials $\chi$ for each element.</p>
<p>How can one calculate electron density in that case and how does it change? I understand, that at lower temperatures number of electrons is equal to number of ionized sodium atoms since it is the easiest to ionize (and in general $n_e=n_H^1+n_{He}^1+n_{Mg}^1$, where 1 means first level of ionization) but that doesn't help much. And additional question: should higher levels of ionization be included, given the temperature? </p> | 2,261 | [
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<p>When you spray gas from a compressed spray, the gas gets very cold, even though, the compressed spray is in the room temperature.</p>
<p>I think, when it goes from high pressure to lower one, it gets cold, right? but what is the reason behind that literally?</p> | 2,262 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/3081/why-cant-a-piece-of-paper-of-non-zero-thickness-be-folded-more-than-n-times">Why can't a piece of paper (of non-zero thickness) be folded more than "n" times?</a> </p>
</blockquote>
<p>On skeptics.stackexchange, there is a question on the <a href="http://skeptics.stackexchange.com/questions/5912/is-it-impossible-to-fold-a-sheet-of-paper-in-half-more-than-seven-times">maximum limit on the folds of a paper</a>. The referenced answer said that <a href="http://en.wikipedia.org/wiki/Britney_Gallivan" rel="nofollow">Britney Gallivan</a> derived an equation to estimate the number of maximum folds possible for a given sheet of paper. The answer justified the 'only'-folding scenario but when the same scenario is changed to folding the paper preceded by tearing the paper off into two equal halves, it fails.</p>
<p>I tried to tear an A4 sheet off in two equal halves and kept on tearing it in two equal halves in alternate directions by first folding the paper and then tearing it. After the 7th tear-off, I found that I could still fold the paper for the 8th time. Applying the same reasoning here, the A4 should fail to fold for the 8th time but it doesn't. Why?</p> | 41 | [
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<p>It is known that all observers will agree on the position of the black hole event horizon. But what about the cosmic horizon of the de Sitter space? Can one say that the horizon of scientist1 is different from the horizon of scientist2?</p>
<p>If so, it turns out that the scientists are researching different universes: the information available for scientist1 is forever lost for scientist2. This may lead to the information loss paradoxes if the observers can communicate.</p>
<p>This question was sparked by the answers to this question: <a href="http://physics.stackexchange.com/questions/93428/can-matter-leave-the-cosmic-horizon">Can matter leave the cosmic horizon?</a></p> | 2,263 | [
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<p>I'm new here. I have a (maybe dumb) question: What are <a href="http://www.google.com/search?as_q=nonlinear+capacitor" rel="nofollow">non-linear capacitors</a>? I'm given a circuit including a capacitor and the question says </p>
<blockquote>
<p><em>The given capacitor is non-linear with the characteristic equation $g=v^3$.</em> </p>
</blockquote>
<p>But I don't know what this equation means.</p> | 2,264 | [
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<p>Wolfram as well as Aldrovandi and Freitas <a href="http://arxiv.org/abs/physics/9712026" rel="nofollow">1</a> maintain that iterated functions $f^t(x)$ are a valid alternative to PDEs for modelling physics. Instead of just citing <a href="http://arxiv.org/abs/physics/9712026" rel="nofollow">1</a>, I want to be able to cite the initial paper and author who justified using iterated functions in physics. I do not mean simply modeling a problem in physics, I mean modeling physics itself. I study the mathematics and structure of $f^t(x)$. It is my contention that if $f^t(x)$ has universal properties, then those properties must occur in physics. </p>
<p>Schroeder wrote Über iterirte Functionen, the first paper on dynamical systems in 1871, but this appears to be a paper of pure mathematics. Poincare is the first person I think used the dynamical systems of iterated functions to study physics. </p>
<p>R. Aldrovandi and L. P. Freitas,
<a href="http://arxiv.org/abs/physics/9712026" rel="nofollow">Continuous iteration of dynamical maps</a>,
J. Math. Phys. 39, 5324 (1998)</p> | 2,265 | [
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<p>I have unitary matrix and I would find the quantum circuit associated. There are 3 qubits input so it's a 8x8 matrix but it's not a simple operation. The number of gates is not specified. </p>
<p>Is there a universal method ? Matrix decomposition with a software ?</p>
<p>Thank you!</p> | 2,266 | [
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<p>Different frequencies of light travel at different speeds through solids, which along with Snell's law allows for rainbows. Has this phenomenon of variable speeds been predicted through derivations? What does it tell us about the interactions that occur when light travels through a solid?</p> | 2,267 | [
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<p>Erm. This question got shot-down in electronics.stackexchange.com and somebody recommended I raise it on here, so ... </p>
<p>Do metals with the property of strong diamagnetism also exhibit inductance?
Would a fluctuating magnetic field induce a weaker current in a strongly diamagnetic metal as compared against a ferromagnetic/paramagnetic metal?</p>
<p>I remember reading metal-detectors are based upon the property of inductance. So if a metal detector were to hover over a silver ingot/plate, would it succeed or fail?</p> | 2,268 | [
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<p>Consider the following scenario that I actually encounter frequently: I just finished washing a pot in the sink, and now I want to boil some water for cooking noodles. If I fill the pot with hot water from the pipes, my stove will require less energy to bring the water to boiling, but my tankless water heater will kick in and heat an equivalent volume of water to displace the water taken from the pipes. Or, I can fill the pot with cold water, which won't trip the water heater, but will instead require more energy from the stove to heat.</p>
<p>It seems to me that if my goal is to minimize energy spent on heating my water, I need to know which is more efficient at heating water - the tankless water heater, or the stove. I have seen <a href="http://physics.stackexchange.com/questions/21370/which-is-more-efficient-heating-water-in-microwave-or-electric-stove">this question</a>, but it is about heating water on the stove vs. in the microwave, which is a different situation.</p>
<p>Which is the more efficient solution?</p> | 2,269 | [
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] |
<ol>
<li><p>What is the relationship between bound states and scattering length? </p></li>
<li><p>What is the relationship between scattering states and scattering length? </p></li>
<li><p>When we say, potential is 'like' repulsive for positive scattering length and viceversa, are we talking with respect to scattering states or bound states?
(though the answer should be scattering states, but the literature everywhere assumes you to know it on your own.) </p></li>
</ol>
<p>All these concepts are found in the theory of BCS-BEC crossover.</p> | 2,270 | [
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<ol>
<li><p>I have some confusion over Vectors, Its components and dimensions. Does the number of vector components mean that a vector is in that many dimensions? For e.g. $A$ vector with 4 components has 4 dimensions?</p></li>
<li><p>Also, how can a Vector have a fourth dimension? How can we graphically represent vectors with more than 3 dimensions? Its hard for me to visualize such a vector, Can anyone point me to some resource where they explain this graphically and in detail?</p></li>
</ol> | 2,271 | [
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<p>How things radiate electromagnetic radiation? I don't ask why they radiate (higher temperature than 0K) but how they radiate this electromagnetic waves?</p> | 2,272 | [
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<p>I read that the physical properties of a sound wave correspond to its audible qualities: pitch, volume, and timbre. However, an oscilloscope uses only two-dimensions to accurately depict the physical properties of a wave. Intuitively, pitch and volume seem more basic than timbre does, so I surmise that timbre must consist in those two properties. </p>
<ol>
<li>Does timbre consist in those two properties?</li>
<li>How do two-dimensional waveform diagrams depict three properties of sound waves?</li>
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<p>What would the double-slit experiment or something analogous to it look like implemented as a quantum circuit/program? Also, what about the delayed choice quantum eraser, how would that look in a quantum circuit?</p>
<p>For example, if implemented in this quantum computing simulator: <a href="http://www.davyw.com/quantum/?example=Toffoli" rel="nofollow">http://www.davyw.com/quantum/?example=Toffoli</a> .</p>
<p>I couldn't find any direct answer via Google, so even good search terms would be useful.</p> | 2,274 | [
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<p>When introducing the fine-tuning problem, a sharp cut-off as a regulator in the calculation of the Higgs mass corrections is used. Since this regulator breaks translational and gauge invariance, up to which point can we trust this cut-off as the true validity limit of our theory?</p>
<p>If we do not trust this regulator, there is no way of seeing that the corrections to the Higgs mass are quadratically divergent, right? So how can we know that, in a renormalizable theory, an elementary scalar should have his mass around the highest possible scale in the theory? Moreover, does it even make sense of talking about the cut-off of a theory when it is renormalizable?</p>
<p>Since the notion of a cut-off, the quadratic corrections to the Higgs mass and naturalness are the driving principle to almost every BSM particle physics model I'm pretty puzzled with this.</p>
<p>Any thoughts, hints or literature that could clarify these ideas?</p> | 2,275 | [
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<p>From what (little) I know about physics, I understand that the universe is expanding due to dark energy, and I understand that no one quite understands it yet. I also understand that the cosmic background radiation hints that there were quantum fluctuations in the universe's infancy.</p>
<p>Could the background radiation fluctuations be explained by assuming dark energy consists of force carrying particles that obey quantum mechanics? That is, they generally appear to follow the path of least action, but due to their quantum-mechanical probablistic nature, may fluctuate slightly?</p> | 2,276 | [
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<p>I apologize if this seems like a quack question, but I need some insights by those who know much more than me in Physics.</p>
<p>Anyway, the gravitational "force" (not really a force) is a manifestation of the curvature of space-time, caused due to presence of mass. Suppose I conjecture that ANY force (electroweak, strong...) is also a manifestation of some property of space-time effected by "something" (as in mass in the case of gravity, and different for different types of forces).</p>
<p>So is there any blatant fault with this line of thought? (I'm asking for blatant faults, since it's a wild supposition anyway)</p>
<p>Thanks.</p> | 537 | [
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<p>If you boil water inside a pot the outer rim bubbles first, I imagine because its hotter. Does that same concept apply for the inside of a refrigerator for example. Is the outer rim always more extreme, or at least at first? </p> | 2,277 | [
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] |
<p>I was wondering what is the difference between the Clausius-Clapeyron equation and the Van't Hoff equation. They appear to have the exact same physical meaning and are often used interchangeably.</p> | 2,278 | [
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<p>I like to play inside the lift (elevator). For instance, there are bars attached at the side of the lift, and I like to hold my body up using my two hands on the bar. I realised that I actually feel lighter when the lift is decelerating and heavier when it is accelerating. Takes the upward motion as positive. </p>
<p>Also, I tried weighing myself inside the lift using a balance ;D As most people know, my weight increases as the lift is accelerating, and vice versa. Does this mean that when I am holding an object inside the lift, the object will actually feel heavier when the lift is accelerating? </p>
<p>Lastly, and this question is quite stupid. My friend once told me that if you want to survive an elevator crash, just jump the moment it hit the floor. I told him that's a stupid idea, but I can't really explain why it wouldn't work! Please help me. </p> | 2,279 | [
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<p>Wanted to understand the physics behind usage of passive antennae and matched load combination, to absorb, control and reduce the Electromagnetic-Field (s.a. due to microwave radiation from cellular phone towers), within a confined area (s.a. a room). Also, does the shape / size / material used for the antenna have a role to play ? If this kind of absorption does work, what might be the range / shape of the area that such a device can reduce / remove the radiation from ?</p>
<p>I have come across EMF shielding/reduction solutions which use such combination, and also some sort of Faraday-cage effect to keep out EMF radiation, which require "grounding". Are these two based on similar / related principles ?</p>
<p>Finally, I read in an article that not grounding the Faraday-cage (or the metallic protective shield), results in the metal starting to becoming radioactive, or maybe even emit X-rays.</p>
<p>Please do excuse the rather layman approach to the questions, a dumbed-down but factual (i.e. can be backed by theory and empirical data, if required) explanation would be highly appreciated.</p> | 2,280 | [
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<blockquote>
<p>Does spontanous symmetry breaking affect the existence of a conserved charge? </p>
</blockquote>
<p>And how does depend on whether we look at a classical or a quantum field theory (e.g. the weak interacting theory)?</p>
<p>(In the quantum case, if we don't want to speak of the Noether theorem, the question can be worded as how does the field breaking the symmetry affect the identities resulting from the Lagrangian symmetry.)</p>
<hr>
<p>The question came up as I wondered if you can make a gauge theory out of every "kernel" of the process in which you compute obervables. In the sense that if</p>
<p>$$\langle A \rangle_\psi=\int (\psi^*A\psi)\text d V,$$</p>
<p>the transformation $T:\phi\rightarrow \text e^{i\alpha}\phi$ is in the kernel of $\langle - \rangle_\psi$.</p> | 2,281 | [
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<p>Let imagine a tunnel that connect two distant places at the globe (eastern-western or north-south)</p>
<p>There are a lot of posible "distances" or metrics, defined by maps, routes, "as the crow flies", etc.. but none of those distance can be shorter than the distance of the tunel.</p>
<p>So if two trains travels at same speed, one inside the tunnel and other above in the surface, the one on the tunnel will reach first.</p>
<p><img src="http://i.stack.imgur.com/Rydj2.jpg" alt="enter image description here"></p>
<p>If this is possible, then perhaps it's possible to have differents coexisting metrics with differents dispositions or topologies, within the same system. </p>
<p>Of course that if we describe a space-time metric surrounding a sphere, then "holes" in it would change the metric (just because it's not a sphere anymore). But it's strange for me that making a hole we could change in some way the space-time shape.</p>
<p>In an extreme case. Could be an euclidian space <em>of same dimension</em> be build within a non-euclidian space?</p>
<p>I would like to have a view from people familiar with general theory of relativity, thanks</p> | 2,282 | [
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<p>This may be an elementary question, but if gravity causes a curvature in spacetime, then why isn't everything distorted when looking down on earth, or up at the moon? Shouldn't there be a <a href="http://en.wikipedia.org/wiki/Distortion_%28optics%29#Radial_distortion" rel="nofollow">pincushion effect</a> when viewing an object that is bending spacetime? </p>
<p>I understand that being here on the surface of the earth, everything relative appears normal, but far away, shouldn't it all look warped? </p> | 2,283 | [
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<p>I like the lectures by <a href="http://web.mit.edu/physics/people/faculty/lewin_walter.html" rel="nofollow">Walter Lewin</a> 8.0x. However the quality of the videos is pretty bad. Is there any way (DVD, web,...) to get the lecture videos in a good quality, best in HD? </p> | 2,284 | [
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] |
<p>I see many learned contribution about the role of a <a href="http://en.wikipedia.org/wiki/Theory_of_everything" rel="nofollow">TOE</a>, what it
might do or not do, what kind of answer it might provide, and what
not.</p>
<p>But I do not know what a TOE is, how I would recognize it if I met it
in the street. Is it an axiomatic theory, a collection of equations, a
sacred book, an incoherent dream, a good topic for fun discussions on
a forum, or just a unification of QM and GR, which might not even tell
us everything about either.</p>
<p>It would be nice for naive users like me to know what is being talked
about, and whether it is the same for all who talk.</p>
<p>Personally, I naively took it to mean a complete description of the
physical universe, which does remain very vague, and may not be
meaningful.</p> | 2,285 | [
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<p>Ok so this might be a very stupid and trivial question but I have spent a couple of hours on this little problem.</p>
<p>I am trying to derive a simple formula in a paper. We have a real commuting spinorial variable $y_\alpha$ ($\alpha = 1,2$) ($y^\alpha = \epsilon^{\alpha\beta}y_\beta$, $\epsilon^{\alpha\beta}=-\epsilon^{\beta\alpha}$, $\epsilon^{12}=\epsilon_{12}=1$).</p>
<p>The linear space of functions of the variables $y_\alpha$ is endowed with the structure of the algebra with the following $*$ product law:</p>
<p>$$(f*g)(y) = \int_{\mathbb{R}^4} \! d^2u~d^2v~
\exp \left[i u_{\alpha} \epsilon^{\alpha\beta} v_{\beta}\right]
f(y+u)g(y+v).$$</p>
<p>I am trying to compute the $*$ commutator </p>
<p>$$[y_\alpha,y_\beta]_{*} := y_\alpha * y_\beta - y_\beta * y_\alpha = 2i\epsilon_{\alpha\beta},$$</p>
<p>but when I try to use the formula for lets say computing $y_1*y_2$, I get a divergent integral</p>
<p>$$y_1*y_2 = \int d^2u\,d^2v \,\,\, (y_1+u_1)(y_2+v_2)\exp[i (u_1v_2-u_2v_1)].$$</p>
<p>I feel like I am applying the formula wrong but I can't understand where. Can someone help me?</p> | 2,286 | [
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<p>I met an approval that massless particle moves with fundamental speed c and this is the consequence of special relativity. Some authors (such as L. Okun) like to prove this approval by the next reasoning.</p>
<blockquote>
<p>Let's have $$ \mathbf p = m\gamma \mathbf v ,\quad E = mc^{2}\gamma \quad \Rightarrow \quad \mathbf p = \frac{E}{c^{2}}\mathbf v \qquad (.1) $$ and
$$ E^{2} = p^{2}c^{2} + m^{2}c^{4}. \qquad (.2) $$ For the massless
case $(.2)$ gives $p = \frac{E}{c}$. By using $(.1)$ one can see, that
$|\mathbf v | = c$.</p>
</blockquote>
<p>But as for me, this is non-physical reasoning. Relation $(.1)$ is derived from the expressions of impulse and energy for the massive particle, so it's scope is limited to massive cases.</p>
<p>We can show, that massless particle moves with speed of light by introducing hamiltonian formalism: for the free particle</p>
<p>$$
H = E = \sqrt{p^{2}c^{2} + m^{2}c^{4}},
$$
for massless particle
$$
H = pc,
$$
and by using Hamilton equation it's easy to show, that
$$
\dot {|r|} = \frac{\partial H}{\partial p} = c.
$$
But if I don't like to introduce hamiltonian formalism, what can I do for proving an approval about speed of massless particle? Maybe, expression $\mathbf p = \frac{E}{c^{2}}\mathbf v$ can be derived without usage the expressions for massive case? But I don't imagine, how to do it by using only SRT. </p> | 2,287 | [
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<p>I am really stuck on a problem in my textbook:</p>
<p>Water is heated in an open pan where the air pressure is one atmosphere. The water remains a liquid, which expands by a small amount as it is heated. Determine the ratio of the work done by the water to the heat absorbed by the water.</p>
<p>MY ATTEMPT:</p>
<p>We are given that:</p>
<p>$P = 1.013 \cdot 10^5 Pa$</p>
<p>We then have:</p>
<p>$$\frac{W}{Q} = \frac{P \Delta V}{cm \Delta T} = \frac{P \beta V_0 \Delta T}{cm \Delta T} = \frac{P \beta m \Delta T}{ cm \rho \Delta T} = \frac{P \beta}{c \rho} = \frac{1.013 \cdot 10^5 \cdot 207 \cdot 10^{-6}}{4186 \cdot 1} = 5 \cdot 10^{-3}$$</p>
<p>But according to the textbook, the solution should be $4.99 \cdot 10^{-6}$. If anyone can help me by pointing out what I'm doing wrong here, I would be extremely grateful!</p> | 2,288 | [
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<p>I witnessed a phenomenon that I couldn't conclude its cause. Please bear with me for the length of the recall, for I merely want to include any details that might help us to investigate. I had a <strong>cooking glass lid</strong> sat on a wooden shelf that is <strong>away from</strong> the stove and oven and other heating objects. The shelf is nailed on the wall and is situated just above my eye level, and a counter top is also on the same side of the wall where the shelf is installed. </p>
<p>Now here comes the surprise. In a winter afternoon 2011, my room had almost the same temperature as an autumn morning, and while I was cutting my lettuce on that counter top which I pointed out in above passage, <strong>a pounding sound</strong>, as if a heavy car door slam or a tree trump falling on top of the roof, knocked its introduction from the shelf that was just above my eye level. First, I thought I may had knocked something around me off(which I didn't believe that for there wasn't anything around me to knock off); then I thought it may be my neighbor next door dropping a heavy box; last, I suspected somewhere my roof top collapsed. </p>
<p>But it was my third suspicion directed me to meet that glass lid I mentioned above, and <strong>I found it had ruptured completely like glacier creaked BUT still having all broken pieces bounded without any pieces scattering toward random direction! Only the nob of the lid popped out partially.</strong> Before this happened, I hadn't used that lid for cooking for years, and I didn't removed it from any heating object nor there was something on top of the lid that day, and I believe what the lid had maybe just an invisible layer of dust.</p>
<p>I was glad my face hadn't been stung by any glass residues, but ponder what really happen to that glass lid and why it ruptured without collapsed. Below, I attached 2 pictures of the scene from that day. If you have any similar experience or know the theory behind it, may you please drop me an explanation to this incidence? Thank you in advance.</p>
<p>I'm new and need reputation to post pictures. I definitely post that 2 pictures once I earn enough reputation point.</p> | 2,289 | [
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<p>For my daughter's science experiment, she placed six beverages (cola, diet cola, milk, chocolate milk, apple juice, and water) in the exact same amount in the exact same type and size of plastic cups, and placed all of the cups in a refrigerator at the same time and allowed them to remain in the refrigerator for the same period of time (4 hours). When the beverages were taken out and set on the same counter, in a room at approx. 70 degrees, a cooking thermometer was then used to measure the temperature of each of the beverages at different time intervals (the intervals were the same for all beverages)---i.e., the temperature of each was measured at 15 minutes, 45 minutes, 1 hour, and 90 minutes. The results were that all but two of the beverages held the same temperatures throughout the measuring period. The two beverages with different (colder) temperatures were diet cola and (white) milk. Of course, my daughter is supposed to cite to references that support (and explain) the basis of the test results/conclusions, and we are finding nothing that addresses this. "Viscosity" is the closest principal that we have found, but we cannot find anything that provides the viscosity levels for the separate beverages (hence, no conclusion can be drawn). Any ideas on where to find research references related to this? thanks,</p> | 2,290 | [
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<p>Spin-1/2
The eigenspinor , $X=aX_++bX_-$
$$X_+=\left( \begin{array}{cc}
1\\
0\end{array} \right) $$$$X_-=\left( \begin{array}{cc}
0\\
1\end{array} \right)$$
They are define like this because they work well in the following?
$S_zX_+={\hbar}/2X_+$ and $S^2X_+={\frac{3}{4}}{\hbar}X_+$.</p>
<p>But for $|s m \rangle$, I don't understand why do we need to put $|1 0\rangle = \frac{1}{\sqrt2} | \uparrow \downarrow + \downarrow \uparrow\rangle$
Because without $\frac{1}{\sqrt2}$ we can prove the eigenvalue of $S^2$ is 2har.
Why $\frac{1}{\sqrt2}$?</p> | 2,291 | [
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<p>I was hoping someone could give an overview as to how the Lie groups $SO(3)$ and $SU(2)$ and their representations can be applied to describe particle physics?</p>
<p>The application of Lie groups and their representations is an enormous field, with vast implications for physics with respect to such things as unification, but I what specifically made these groups of physical importance and why there study is useful.</p>
<p>I have just started studying these two groups in particular, but from a mathematical perspective, I'd very much appreciate understanding some sort of physical motivation.</p> | 2,292 | [
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<p>I want to determine how many minutes a satellite is in a circular orbit around the Earth at about $1000 km$ altitude. I assumed that the Sun-Earth vector lies exactly in the orbital plane of the satellite. Also, in this case, the Sun can be seen as a point light source and the distance to Earth is infinite. Is it possible to make an approximation of the duration that the satellite is on the 'dark' side of the Earth? Or do I need more information, like the speed of the satellite? The radius of the Earth is $6378 km$. And one orbital period is ofcourse $24$ $hours$.</p> | 2,293 | [
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<p>Alright so I really want to put some art into science for my project.project. I'm thinking of creating a beautiful image,preferably in air.I read about the visible lime-green lasers but i still want to be able to use other colors,and I need to make it visible.How can i make "fog" for it so that a pretty image is seen yet at the same time the inspectors won't be suffocating.Any ideas on how should I make it and what other things I can use to create pretty images?Please explain in details as I might not know some terms in optics.Thanks in advance!</p> | 2,294 | [
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<p>The <strong><a href="http://en.wikipedia.org/wiki/amplituhedron" rel="nofollow">Amplituhedron</a></strong> has recently been popular; it supposedly encodes <strong>perturbative scattering amplitudes</strong> in a simple, <strong>geometric</strong> fashion. </p>
<p>What happens to it in a <strong>non-perturbative</strong> context? Is there still some sort of amplituhedron, somehjow? </p>
<p>If so, a side question: </p>
<p>Can the amplitudihedron be able to solve the problem of 5-Brane scattering amplitudes in M-Theory? </p> | 2,295 | [
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<p>Consider a hydraulic jack with massless pistons as follows.</p>
<p><img src="http://i.stack.imgur.com/MVJaU.png" alt="enter image description here"></p>
<p>The famous equation for this system is </p>
<p>$$
\frac{F_1}{A_1}=\frac{F_2}{A_2}
$$</p>
<p>My question is why isn't the equation as follows?</p>
<p>$$
\frac{F_1}{A_1}=\frac{F_2}{A_2} + \rho g h
$$</p>
<p>It is based on my understanding that any points in the same horizontal line have the same pressure.</p>
<p>Could you spot my misconception?</p> | 2,296 | [
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<p>I'm trying to develop some basic intuition here, so this comes mostly as a jumble of commentary/questions. Hope its acceptable.</p>
<p>Helmholtz Free Energy: $A = -{\beta ^{-1}}lnZ$. I find this statement to be incredible profound. Granted, I found it yesterday.</p>
<p>Suppose my system has one energy state with no degeneracy. $Z = e^{-\beta E_1}$, then $A = E_1$, which I suppose says if the system consists of one particle, all its internal energy is available for work. That's nice.</p>
<p>Now, if we introduce some degeneracy $\gamma$, we get $Z = \gamma e^{-\beta E_1}$, and so $A = E_1 - \beta ^{-1}ln \gamma$, and we have clearly lost some of our free energy to the degeneracy (ie. to the fact that there are multiple microstates for our given macrostate, and so we have limited information about the actual configuration of the system, which is free to explore its micro states, limiting the energy we can get from it). So that's nice too.</p>
<p>We can go further by introducing more energies, so $Z = \Sigma \gamma_i e^{-\beta E_i}$, but nice analysis is confounded by my inability to deal coherently with sums in a logarithm. Though I managed to show that $A$ for such a multi-state system is strictly less than $\Sigma [E_i-\beta ^{-1}ln\gamma _i]$, ie. less than the sum of the free energies for independent systems of a given energy $E_i$ and degeneracy $\gamma_i$ . This result, however, requires $E_i > 0$, which I take for granted, but makes plenty sense.</p>
<p>Now, what does it mean for A to be negative? Perhaps more importantly, how does one simply go about obtaining work from a system with some A (a practical question)? Or, perhaps even more importantly, is it this requirement that there be a second final state, seemingly of lower free energy, that makes $A$ itself not so significant, but rather $\Delta A$? And if so, what happens to the intuition about a system with only one state having exactly its energy as free-energy?</p>
<p>Your insights on these and related matters pertaining to legendary $Z$ and its relation to $A$, as well as pointers on where my thinking may be flawed or enlightened, are much appreciated.</p> | 2,297 | [
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<p>Given a metric of the form $$ds^2=dr^2+a^2\tanh^2(r/b)d\theta^2$$
why does it follow that $a=b$?
I can't quite spot a constraint condition...</p> | 2,298 | [
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<p>I'm trying to answer the following question:</p>
<blockquote>
<p>Air consists of molecules Oxygen (Molecular mass = 32$amu$) and Nitrogen (Molecular mass = 28$amu$). Calculate the two mean translational kinetic energies of Oxygen and Nitrogen at 20($^\circ C$)</p>
</blockquote>
<p>To solve it I have done:</p>
<p>Use $E = \frac{3}{2}kT$</p>
<p>Energy = $\frac{3}{2} \times (1.38 \times 10^{-23}) \times (20+273) = 6.07 \times 10^{-21}$</p>
<p>Use $KE = \frac{1}{2}mv^2$:</p>
<p>For oxygen: $\sqrt{\frac{6.07 \times 10^{-21}}{2 \times (32 \div 6.02\times 10^{23})}} = 15.11$</p>
<p>However, 15.11 isn't the answer in the textbook (the answer is 480m/s)</p>
<p>For nitrogen: $\sqrt{\frac{6.07 \times 10^{-21}}{2 \times (28 \div6.02\times 10^{23}) \div 32}} = 16.12$</p>
<p>16.12 isn't the answer either (it's 510m/s)</p>
<p>I know that my answers are wrong (gas molecules don't move as slow as I calculated at room temperature) but I can't see why my method doesn't work. Any help?</p> | 2,299 | [
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<p>Can someone explain me the meaning of the A/W unit of the photosensivity when reading a spectral response function of the wavelength characteristic of a photodiode?</p> | 2,300 | [
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<blockquote>
<p><em>A trolley of mass 300kg carrying a sand bag of 25kg is moving uniformly with speed of 27km/h on a frictionless track. After a while, sand starts leaking out of a hole on the floor of the trolley at the rate of 0.05 kg/s.</em></p>
<p><em>What is the speed of the trolley after the entire sand bag is empty?</em></p>
</blockquote>
<p>I was so surprised when I read this question. It doesn't make sense to me. I can't comprehend how the loss of sand creates an <strong>external</strong> unbalanced force on the trolley such that it affects its velocity.</p>
<p>Maybe I haven't analyzed the question enough but I find this a bit conceptually challenging for me. Maybe I have to consider how the sand particles affect the back wheels of the trolley or maybe consider the sand to be a propellant?</p> | 2,301 | [
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] |
<p><img src="http://i.stack.imgur.com/1YRyD.png" alt="enter image description here"></p>
<p>The overtones of a vibrating string. </p>
<p>These are eigenfunctions of an associated Sturm–Liouville problem.
The eigenvalues 1,1/2,1/3,… form the (musical) <a href="http://en.wikipedia.org/wiki/Vibrating_string" rel="nofollow">harmonic series</a>.</p>
<p>How can Hilbert spaces be used to study the harmonics of vibrating strings?</p> | 2,302 | [
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<p>I'm reading electrodynamics notes and come across that:</p>
<p>$$\int_\text{all space} d\vec r \; \nabla \cdot(\vec A \times \vec B)=0$$ in case of magnetostatics and:
$$\int_\text{all space} d\vec r \; \nabla \cdot(\phi \vec E)=0$$ in case of electrostatics.</p>
<p>($A$ is the magnetostatic potential, $B$ the magnetic field and $\phi$ the electrostatic potential)</p>
<p>My question: is why are they equal to $0$?</p>
<hr>
<p>The above mentioned formulas are used to show that:
$$W=\frac{\epsilon_0}{2 \ } \int_\text{all space} d \vec r\vec E ^2$$
$$W=\frac{1}{2 \mu_0} \int_\text{all space} d \vec r\vec B ^2$$</p>
<p>starting from </p>
<p>$$W=\frac{1}{2 \ } \int_\text{all space} \phi(\vec r) \rho(\vec r) $$
$$W=\frac{1}{2 } \int_\text{all space} d \vec r \vec j \cdot \vec A$$</p>
<p>with $\rho$ the charge density, and $\vec j$ the current density</p>
<hr>
<p>I've tried using Gauss's Theorem:
$$\int_{\partial V} d\vec r \; (\vec A \times \vec B)\cdot d\vec S=0$$</p>
<p>and:
$$\int_{\partial V} d\vec S\; \cdot (\phi \vec E)=0$$
but this doesn't bring me any further to solving my problem.</p> | 2,303 | [
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] |
<p>Let's have Fock state for fermions:
$$
| \mathbf p_{1} , \mathbf p_{2}\rangle = \frac{1}{\sqrt{2}}\hat {a}^{+}(\mathbf p_{1})\hat {a}^{+}(\mathbf p_{2})| \rangle , \quad | \mathbf p_{2} , \mathbf p_{1}\rangle = -| \mathbf p_{1} , \mathbf p_{2}\rangle .
$$
How to get the expression for Slater determinant by starting from definition above?</p> | 2,304 | [
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<p>Why do we get information about position and momentum when we go to different representations. Why is momentum, which was related to time derivative of position in classical physics, now in QM just a different representation brought about by some unitary transformation. Is Ehrenfest's theorem the only link?</p>
<p>I just started studying QM. So please suggest some references explaining the structural aspects and different connections.I don't want to start with noncommutative geometry. I would like something of an introductory nature and motivating.</p> | 2,305 | [
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<p>A mass of point performs an effectively 1-dimensional motion in the radial coordinate. If we use the conservation of angular momentum, the centrifugal potential should be added to the original one. </p>
<p>The equation of motion can be obtained also from the Lagrangian. if we substitute, however, the conserved angular momentum herein then the centrifugal potential arises with the opposite sign. So if we naively apply the Euler-Lagrange equation then the centrifugal force appears with the wrong sign in the equations of motion. </p>
<p>I don't know how to resolve this "paradox".</p> | 2,306 | [
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<p>I am a reading a book for beginners of the quantum mechanics. In one section, the author shows the <a href="http://en.wikipedia.org/wiki/Inner_product_space" rel="nofollow">inner product</a> of two wave functions $\langle\alpha\vert\beta\rangle$. I am wondering what's the significance of that product? I Googled that and someone call that probability amplitude, but that product could be complex, so does it tell any phyical significance?</p>
<p>In terms of quantum interference, shall we add the wavefunctions or the probability amplitudes before taking the square of modulus? Sorry I am just starting on learning the quantum mechanics and many concepts are pretty confusing to me.</p> | 2,307 | [
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<p>QCD is the best-known example of theories with negtive beta function, i.e., coupling constant decreases when increasing energy scale. I have two questions about it:</p>
<p>(1) Are there other theories with this property? (non-Abelian gauge theory, principal chiral field, non-linear sigma model, Kondo effect, and ???)</p>
<p>(2) Are there any simple (maybe deep) reason why these theories are different from others? It seems that the non-linear constraint of the non-linear sigma model (and principal chiral model) is important, but I have no idea how to generalize this argument to other theories.</p> | 2,308 | [
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<p>I was testing in my lab with water and found that it starts to solidify when it is stationery at 0 degree centigrade.but when I move the liquid with some velocity it dosent change its phase.my interpretation of this happening because when I move liquid it has kinetic energy which dosent allow it to crystallize.but at certain velocity and further lowered temperature it should start crystallising again.is there any graph which shows at what velocity and temperature it crystallize with atmospheric pressure?as in phase transition of water under temperature and pressure?</p> | 2,309 | [
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<p>I spent a while working with MgF2-windowed xenon flash / discharge lamps. Primarily, I characterized their spectra with two normal-incidence spectrometers against a calibrated Deuterium lamp. In this particular case, it included a Czerny-Turner-type design. </p>
<p>As side investigation, I rotated one of the xenon flash lamps by 360° in about 45° steps and took time-resolved spectra in the range of 200 to 500 nm for each rotation step. Time-resolved means, that I took measurements / spectral scans at an interval of 0.1 µs with an exposure time of 0.168 µs. Because the exposure time is longer that the interval, I took only one spectral scan per one flash, for each flash delayed by another 0.1 µs. I assume this is ok because the intensity of the light emitted by the discharge over its entire spectrum is stable with $$ \sigma = 0.0294 $$ So for each rotation step and each interval, I integrate over an entire calibrated spectrum. If I plot the result, it looks somewhat like this: <img src="http://i.stack.imgur.com/zyb88.png" alt="some plot"></p>
<p>(The actual flash is "visible" for about 1.5 µs, while there is an "afterglow" mostly in longer wavelengths with less than 5% of the maximum intensity for about another 40 µs.)</p>
<p>My understanding is, that normal-incidence spectrometers behave like polarization filters. So I would expect perfect ellipses - point symmetric - if the the light is (partially) polarized. What I see looks different, unsymmetrical. </p>
<p>If have also done another type of plot. The first one shows how it theoretically should look like: <img src="http://i.stack.imgur.com/C8roc.png" alt="some plot"></p>
<p>And this is what I get: <img src="http://i.stack.imgur.com/icVSv.png" alt="another figure"></p>
<p>I assume, that I see ellipses. The semi-major-axis / maximal radius is indicated red (1), the semi-minor-axis / minimum radius is indicated blue (-1). </p>
<p>My questions ... is there any physical context, that could explain the unsymmetrical behaviour (other than uncertainties in measurements?). If there is not, would it be appropriate to fit ellipses into the data? </p>
<p><strong>EDIT (1):</strong> <img src="http://i.stack.imgur.com/X2sSK.png" alt="another one">
Based on the reply by @akhmeteli, I looked into different wavelengths. Top left: 230 nm; top right: 260 nm; bottom left: 362 nm; bottom right: 461 nm. The first three represent spectral lines, the fourth one is a random choice without any specific feature. My data has a resolution of 0.2nm. Here, I integrated from lambda-0.2 to lamda+0.2 nm, kind of as narrow as possible.
<strong>My impression is, that the observed effect does not (very much) depend on the wavelength.</strong></p> | 2,310 | [
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<p>This is probably trivially related to the question:
<a href="http://physics.stackexchange.com/questions/45922/action-for-a-point-particle-in-a-curved-spacetime">Action for a point particle in a curved spacetime</a> , but am a bit unsure how to write it as a Lagrangian density.</p>
<p>In curved spacetime the action is related to the Lagrangian density by:
$$ S = \int \sqrt{-g} \, \mathrm{d}^4 x \ \mathcal{L}(x^\mu)$$</p>
<p>The simpilest way I can think of describing a point mass taking a path through space time would be something like:</p>
<p>$$ \mathcal{L}(x^\mu) = -m\int d\alpha \ \delta^4(x^\mu-s^\mu(\alpha))$$</p>
<p>Where $\alpha$ is parameterizing a path $s^\mu(\alpha)$ through spacetime. Is that the correct Lagrangian density?</p>
<p>Can someone show me how to manipulate these mathematical structures to check that this simplifies somehow to the known action for a free particle:</p>
<p>$$ S = - m \int d\tau $$</p>
<p>or how it relates to what was written in the other question?</p>
<p>$$\mathcal S =- m \int\sqrt{g_{\mu\nu}(x)\frac{dx^\mu(\xi)}{d\xi} \frac{dx^\nu(\xi)}{d\xi}} \ \ d\xi$$</p> | 2,311 | [
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<p>From what I have learned in my chemistry course, Electrons with similar quantum numbers but with opposite spin are attracted to each other. What does this mean when there is a covalent bond being formed between lets say hydrogen and fluorine?</p>
<p>I can think of four different results:</p>
<p>A) A bond is not formed until an unpaired fluorine electron meets an unpaired hydrogen electron with the same energy shift (in this case, 0).</p>
<p>B) The fluorine or hydrogen electron forces the other electron to have the same energy shift.</p>
<p>C) The electrons don't change, but pair up regardless of quantum numbers.</p>
<p>D) The electrons from both atoms enter into a different state that I have not learned yet.</p>
<p>Thanks in advance for any help received!</p> | 2,312 | [
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<p>Assuming the solid is less dense than the liquid, will a solid object float on a frictionless liquid?</p>
<p>I can imagine that due to the pressure gradient the object will float, but I can also imagine that without friction there would be no upwards force on the object and it would sink.</p>
<p>What would happen?</p> | 2,313 | [
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<p>Are observations of Hawking radiation at the acoustic event horizon in Bose-Einstein condensates consistent with Gravastars?</p>
<p>To reconcile the second law of thermodynamics with the existence of a black hole event horizon, black-holes are necessarily said to contain high entropy while Gravastars not at all. An Event-Horizon forming out of a collapsing star's intense gravity sufficient enough to force the matter to phase change transforming into Bose-Einstein condensate would be such that nearby matter would be re-emitted as another form of energy, and all matter coming into contact with the Event-Horizon itself would become incorporated.</p>
<p>So, it seems reasonable to wonder if Black-Holes are distinguishable from Gravastars since Gravastars appear to be better emitters, and Black-holes better entropy sinks. What do observations of Hawking radiation from acoustic black holes from Bose-Einstein condensate seem to suggest?</p> | 2,314 | [
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<p>I know that the main difference between the two is that a battery can provide a constant voltage whereas a capacitor's voltage decreases as the charge stored decreases.
But what about the internal structure ?
A battery also has chemical reactions going inside of it while a <strong>polarized</strong> capacitor also has an electrolytic structure.
So is there or is there not any difference between the internal workings of a <strong>polarized</strong> capacitor and a battery?</p> | 2,315 | [
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<p>Here is a quote from <a href="http://en.m.wikipedia.org/wiki/Hilbert_space#Hilbert_dimension" rel="nofollow">http://en.m.wikipedia.org/wiki/Hilbert_space#Hilbert_dimension</a> (accessed: Nov. 22, 2013) : </p>
<blockquote>
<p><em>As a consequence of Zorn's lemma, every Hilbert space admits an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same cardinality, called the Hilbert dimension of the space.</em></p>
</blockquote>
<p>But it seems to me that for a lot of quantum systems this is not so. For example, the one dimensional harmonic oscillator has a countable basis of energy eigenstates but an uncountable basis of position eigenstates (whose wave functions are delta functions). </p>
<p>So where am I wrong? I assume the answer has to do with the fact that delta functions are not actually part of the Hilbert space. Is that it? </p> | 42 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/22120/is-the-universe-a-quantum-computer-is-light-speed-barrier-a-computational-cons">Is the universe a quantum computer - is light speed barrier a computational constraint</a> </p>
</blockquote>
<p>Cross-posting this question, since physics.stackexchange has not provided any answers. </p>
<p>There is currently a debate ongoing on leading maths blog Gödel’s Lost Letter, between Gil Kalai and Aram Harrow, with the former arguing that building a quantum computer may not be possible due to noise propagation, and the latter arguing to the contrary.</p>
<p>I am wondering if there is any argument to show that building a quantum computer is possible, by virtue of showing that quantum computation is evident in the physical world.</p>
<p>So the question is:</p>
<p>(A) Are there any known examples of physical interactions where macro level state transitions could be determined to only be in correspondence with an underlying quantum computation? I.e. similarly to Shor's algorithm being exponentially faster than any known classical factoring algorithm, are there any examples of known physical processes, for example perturbation stabilization in a very large particle cluster, that could be shown, assuming P<>NP, to only be efficiently solved by a quantum computation.</p>
<p>Some, I admit highly speculative, additional questions would then be:</p>
<p>(B) Is the speed of light barrier possibly a natural computational limit of our particular universe, so that for the computational complexity class of quantum mechanics, working on an underlying relational network-like spacetime structure, this is the maximum speed that the computational rules can move a particle/wave representation through a network region of the lowest energy/complexity (i.e. a vacuum)?</p>
<p>(C) Is quantum mechanics an actual necessity for the universe to follow classical physical laws at the macro level? The informal argument being that in many-to-many particle quantum level interactions, only the capability of each particle to compute in parallel an infinite or quantum-quasi-infinite number of paths is what allows the universe to resolve a real-time solution at the macro level.</p>
<p>Requesting references to research along these lines, or any arguments to support or contradict these speculations. </p> | 43 | [
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<p>The answer to <a href="http://physics.stackexchange.com/q/6457/2359">a previous question</a> suggests that a moving, permanently magnetized material has an effective electric polarization $\vec{v}\times\vec{M}$. This is easy to check in the case of straight-line motion, using a Lorentz boost.</p>
<p>I suspect this formula is still correct for motion that is not in a straight line, but I'm not interested in reinventing the wheel. Does anyone know of a textbook or journal article that derives this $\vec{v}\times\vec{M}$ term? Even better, does anyone know of experimental observation of this effect?</p>
<p>EDIT:<br>
Followup question:
<a href="http://physics.stackexchange.com/questions/6581/what-is-the-electric-field-generated-by-a-spinning-magnet">What is the electric field generated by a spinning magnet?</a></p> | 2,316 | [
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<p>In general relativity, we use the term "time-like" to state that two events can influence one another. In fact, in order for an event to physically interact with another one, they have to be inifnitely close both in time and space.</p>
<p>As far as I know (correct me if I'm wrong) this principal of "near action/causality" is conserved in all branches of modern physics and that is one of the reasons people are looking for "force carriers".</p>
<p>If this is the case, then would it be accurate to say that causality is simply a measure of continuity in all dimensions - and not only the time dimension?</p>
<p>(I don't know anything about continuum mechanics other than its name, but it may have something to do with what I'm asking)</p> | 2,317 | [
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<p>If you compress a large mass, on the order of a star or the Earth, into a very small space, you get a black hole. Even for very large masses, it is possible in principle for it to occupy a very small size, like that of a golf ball.</p>
<p>I started to think, how would matter react around this golf ball sized Earth? If I let go of a coffee mug next to it, it would go tumbling down toward the "golf ball". Isn't that exactly how magnets work, with paperclips for example? </p>
<p>Magnets are cool because they seem to defy the laws of gravity, on a scale that we can casually see. Clearly, the force carrier particles that produce electromagnetic attraction are stronger than gravity on this scale (or are at least on par: gravity plays some role in the paperclips path, but so does electromagnetism). </p>
<p>My question is, why do we try to consider gravity as anything different than magnetism? Perhaps "great mass" equates to a positively (or negatively) charged object. Pull so much matter in close and somewhere you've crossed the line between what we call electromagnetic force and gravity force. They are one in the same, no?</p> | 2,318 | [
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<p><img src="http://i.stack.imgur.com/5bscW.jpg" alt="enter image description here"></p>
<p>Like the order behavior shown in the image, is it due to the universality of some fundamental mathematic theory? Is there some general physics explanation for it? </p>
<p>-
edit: This question comes after I read Terence Tao's talk on universality (<a href="http://terrytao.files.wordpress.com/2011/01/universality.pdf" rel="nofollow">pdf</a>).</p> | 2,319 | [
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<p>As far as I know, there is a smooth transition between quantum and classical regimes, so that even classical particle like a massive object has a wavefunction associated with it. However, the double slit experiment can either show quantum character, where the particle supposedly passes through both slits and interferes with itself, or classical character, where the particle passes through one slit with no interference. What would a semi-classical particle do when faced with this situation?</p>
<p>In addition, I have a problem imagining a semi-classical particle in general. What is a good example of one?</p> | 2,320 | [
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] |
<p>A neutron outside the nucleus lives for about 15 minutes and decays mainly through weak decays (beta decay). Many other weakly decaying particles decay with lifetimes between $10^{-10}$ and $10^{-12}$ seconds, which is consistent with $\alpha_W \simeq 10^{-6}$.</p>
<p>Why does the neutron lives so much longer than the others? </p> | 2,321 | [
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<p><img src="http://i.stack.imgur.com/z8fR1.png" alt="enter image description here"></p>
<p>I'm making a toy for my kids and this problem came up. I have a channel on a slight angle (angle is between ground and length of channel) and I'm pouring water into it. I want to know how quickly I need to pour water in to make it flow continuously at a given height.</p>
<p>So the water is (meant to be) .5m wide, .2m high, and 1m long, angled at 5 degrees. How fast is that water going to fall out, in other words, how fast do I have to pour it in?</p>
<p>I'm particularly looking for the technique to do this, not just an answer.</p>
<p>The part I find very confusing is that the water at the top of the box will accelerate a little, and have a low velocity, while the water at the bottom of the box will have accelerated for a while, and have a higher velocity. But intuition tells me the water will stay cohesive, for lack of a better word. Can someone explain what's going on here and how this can be calculated?</p> | 2,322 | [
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<p>At quantum scale, gravity is the weakest force. <a href="http://physics.stackexchange.com/a/31484/2170">Its even negligible in front of weak force, electromagnetic force, strong force.</a></p>
<p>At macroscopic scale, we see gravity everywhere. Its actually ruling the universe. Electromagnetic force is also everywhere, but its at rank 2 when it comes to controlling motion of macroscopic bodies. And, there's no luck finding strong force and weak force.</p>
<p>How can that be? Is that because gravity only adds up but others cancelled out too? I am unable to understand how resultant of weakest force can be so big. Can you please show it with calculation?</p> | 2,323 | [
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<p>Of course it is expected. But how to prove it analytically?</p>
<p>Slater determinant is mentioned in almost every quantum mechanics textbook. But it is necessary to warn the undergraduate students that not every fermionic wave function is a Slater determinant. </p>
<p>So is there any model whose ground state can be easily and rigorously shown to be not a Slater determinant?</p>
<p>I had this problem since my paper: <a href="http://arxiv.org/abs/1309.1848" rel="nofollow">http://arxiv.org/abs/1309.1848</a>, in which we considered the best Slater approximation of a fermionic wave function. </p> | 2,324 | [
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<p>I'll apologize in advance if this is not an appropriate place for my question. My background is not in physics, and my understanding of quantum mechanics is extremely rudimentary at best, so I hope you'll be forgiving of my newbish question.</p>
<p>Given a system of entangled particles (eg, 2 or more electrons), possibly in a superposition state: if the particles interact with each-other, what effect does this have on their quantum state? Is their state now determined (but perhaps unknown until observed)?</p> | 2,325 | [
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<p>I recall faintly from my quantum theory lecture that there was a really neat way to derive Brillouin-Wigner perturbation theory for the special case of two coupled subspaces that involved a geometric series in reverse. </p>
<p>I know that the beginning was to have the Hilbert space split into subspaces 0 and 1 so that the Hamiltonian reads </p>
<p>$$H = H_{00} + H_{01} + H_{01}^\dagger + H_{11}$$
where the two indices indicate what subspace goes "in" and what subspaces comes "out" of each of the components. </p>
<p>If $P$ is a projector onto subspace $0$ and $Q$ a projector onto subspace $1$, this means, for example, that
$$H_{00} = PH_{00}P, H_{01} = PH_{01}Q$$
and so on.
In matrix notation
$$\begin{pmatrix}
H_{00} & H_{01} \\
H_{01}^\dagger & H_{11}\end{pmatrix}
\begin{pmatrix} \psi_{0} \\ \psi_1\end{pmatrix}
= E \begin{pmatrix} \psi_0 \\ \psi_1\end{pmatrix}$$</p>
<p>Now we can formally solve the Schrödinger equation for $\psi_1$ only:
$$|\psi_1\rangle = \frac{1}{E - H_{11}} H_{01}^\dagger |\psi_0\rangle$$
and insert that back into the SG for $\psi_0$ to obtain</p>
<p>$$H_{00} |\psi_0\rangle + H_{01} \frac{1}{E-H_{11}} H_{01}^\dagger |\psi_0\rangle
= E|\psi_0\rangle$$</p>
<p>Now I know that some cool trick with geometric series and an inverted matrix was going on to immediately write down the expansion of $|\psi\rangle$. Usually in the literature one first shows the iterative formula and THEN notes that this is a geometric series, but here it was done in the reverse, somehow an operator of the type
$1/(A-B)$ with $A$ and $B$ matrices, was found and then expanded in the geometric series, but no matter how I massage the equations, I cannot seem to make it work, because there are so many different points where one could substitute one form into the other etc. </p>
<p>Any ideas on how this works? I'm also relatively sure that the work was done using operators, not for "individual" matrix elements. Personally, I find this more elegant because it makes it clearer what's going on on a more abstract level instead of dealing with the low-level matrix elements. </p>
<p>It could also be that at this point already the approximation $E \approx E_0$ was made where $E_0$ is a typical energy of subspace $0$, assuming that this subspace is near-degenerate and well-separated from subspace 1.</p> | 2,326 | [
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<p>Suppose you have as many electrically charged particles as needed (even countably many) and consider the open unit ball centered at some point in space. For every continuous real valued function on the unit ball, is there a configuration of the particles (outside of the ball) that would generate an electric potential corresponding to said function on the ball? </p>
<p>Does an answer somehow trivially follow from Gauss' law?</p> | 2,327 | [
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<p>I'm not talking about an ideal wire in a circuit (a wire with infinite conductance).</p>
<p>I'm talking about an ideal wire in the case of the magnetic field of an infinite current carrying wire. What dimensions must an ideal wire have to better approximate an infinite current carrying wire in terms of its magnetic field?</p>
<p>For example, for a parallel plate capacitor, the field better approximates two infinite plates as $$A >> d$$ where $A$ is the area and $d$ is the length.</p>
<p>Or for a solenoid, the field better approximates an infinite solenoid as</p>
<p>$$L>> d$$ where $L$ is the length and $d$ is the diameter.</p>
<p>For a current carrying wire, what dimensions must it have to better approximate an infinite wire?</p> | 2,328 | [
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] |
<blockquote>
<p>A troop $5$ meters long starts marching. A soldier at the end of the file steps out and starts marching forward at a higher speed. On reaching the head of the column, he immediately turns around and marches back at the same speed. As soon as he reaches the end of the file, the troop stops marching, and it is found that the troop has moved by exactly $5$ meters. What distance has the soldier travelled? </p>
</blockquote>
<p>I thought that some info is lacking until my friend showed it in a book. How to get the answer?</p> | 2,329 | [
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<p>In $1g$ the average adult human walks 4-5 km in an hour. How fast would such a human walk in a low gravity environment such as on the Moon $(0.17g)$ or Titan $(0.14g)$? </p>
<p>Let's ignore the effects of uneven terrain (regolith or ice/snow/sooth); suppose our human walks on hardened pavement.</p> | 2,330 | [
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<p>Given the situation that $B=-\nabla A$ where B is magnetic field and A is some Scalar Field. How can I calculate the scalar field, A.</p>
<p>We are dealing with current free region, here.</p>
<p>I know we can calculate B for a infinitely long thin wire given $I$. But I am not sure how we can calculate $A$.</p> | 2,331 | [
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<p>I actually posted this to <a href="http://math.stackexchange.com/questions/657752/question-on-using-leibniz-formula-to-derive-thin-film-equation-from-navier-stoke">math.stackexchange.com</a> a few months ago but never got any answers.</p>
<p>I am trying to work through the derivation in <a href="https://online.unileoben.ac.at/mu_online/voe_main2.getVollText?pDocumentNr=63375&pCurrPk=29022" rel="nofollow">this paper</a> by Petr Vita, which derives a thin-film simplification of the Navier-Stokes equation, similar to the <a href="http://en.wikipedia.org/wiki/Reynolds_equation" rel="nofollow">Reynolds or Lubrication Equation</a>, but including inertial terms as well. To walk through the major steps to the point where I have questions:</p>
<ol>
<li>Start with basic N-S equation (eq 1 in the paper):
$$\rho \left( \frac{\partial \mathbf{u}}{\partial t} + \nabla \cdot(\mathbf{u u}) \right) = -\nabla p + \rho \mathbf{g} + \nabla \cdot \mathbf{\underline{T}} $$
Here $ \mathbf{\underline{T}}$ is the deviatoric stress tensor, and I have left off the final body force term since it isn't used in the rest of the paper.</li>
<li>Use the thin-film assumption that $u_z=0$ and define (eq 4 in the paper) $$\bar{\mathbf{u}}=\frac{1}{h}\int_0^h \mathbf{u}\, dz$$</li>
<li><p>Integrate the N-S equation with respect to $z$ from $0$ to $h$ (equation 8 in the paper). $$ \rho \frac{\partial}{\partial t} (h \bar{\mathbf{u}}) + \rho \int_0^h \nabla \cdot (\mathbf{u u})dz = -h \nabla p -\left.\mu \frac{\partial \mathbf{u}}{\partial z}\right|_{z=0} $$
Obviously there are a few steps being skipped over here. For the time-derivative term, I use the Leibniz formula to derive the following:
$$ \int_0^h \rho \frac{\partial \mathbf u}{\partial t} dz = \rho \frac{\partial}{\partial t} \int_0^h \mathbf u\,dz - \rho \frac{\partial h}{\partial t} \mathbf u (x,y,h,t) + \rho \frac{\partial 0}{\partial t} \mathbf u (x,y,0,t)$$
Obviously the last term is $0$ and can be dropped. Also $ \rho \frac{\partial}{\partial t} \int_0^h \mathbf u\,dz = \rho \frac{\partial}{\partial t} (h \bar{\mathbf{u}}) $, giving the form seen in the equation. However, I don't see how $ \color{blue}{\rho \frac{\partial h}{\partial t} \mathbf u (x,y,h,t)} $ can be taken to be $0$. The height of the film is certainly changing with time, and the top surface has a von Neumann boundary condition, not a Dirichlet no-slip boundary. Any insight here?</p>
<p>Also, the deviatoric stress has to be integrated as well. I think the divergence theorem can be used here:
$$ \int_V \nabla \cdot \mathbf{\underline{T}}\, dV = \int_S \mathbf{n} \cdot \mathbf{\underline{T}} \, dS$$
In this case that should come out to be
$$ \int_0^h \nabla \cdot \mathbf{\underline{T}}\, dz =\left.\mu \frac{\partial \mathbf{u}}{\partial z}\right|_{z=h} - \left.\mu \frac{\partial \mathbf{u}}{\partial z}\right|_{z=0} $$
The top surface stress is $0$, leaving the bottom stress term as is found in the derived equation, right?</p></li>
<li><p>Now we get to my main question, the integration of the $\nabla \cdot(\mathbf{u u})$ term. The author is able to evaluate this term by using a combination of <a href="http://web.mit.edu/fluids-modules/www/highspeed_flows/ver2/bl_Chap2/node12.html" rel="nofollow">Pohlhausen's method</a> of assuming a cubic profile for the liquid flow, and the <a href="http://en.wikipedia.org/wiki/Reynolds-averaged_Navier%E2%80%93Stokes_equations" rel="nofollow">Reynold's Averaged Navier-Stokes</a> method of splitting the velocity into an average velocity and deviation from that average.</p>
<p>For the cubic profile he defines:
$$ \mathbf{u}(x,y,z) = u(x,y,\xi) \text{, where} $$
$$ u(x,y,\xi) = a_0 + a_1\xi + a_2\xi^2 + a_3\xi^3,\quad \xi \in \langle 0,1
\rangle,\; z=h\xi $$
Then he applies the boundary conditions and integral relation to obtain
$$ u(x,y,\xi) = \mathbf{u}_{disk} + (\bar{\mathbf{u}}-\mathbf{u}_{disk})\left( \frac{12}{5}\xi - \frac{4}{5}\xi^3 \right) $$</p>
<p>This step is fine, I had no problems figuring it out. Then the author defines the velocity fluctuation (with respect to the vertical direction) $\mathbf{\tilde u}$ as
$$ \mathbf{u} = \mathbf{\bar u} + \mathbf{\tilde u} \text{. This makes:} $$
$$ \int_0^h \mathbf{\bar u}\, dz = h\,\mathbf{\bar u}\text{, and } \int_0^h \mathbf{\tilde u}\, dz = 0 \text{.}$$</p>
<p>Anyway, so the author does this to integrate the advection term:
$$ \int_0^h \nabla \cdot ( \mathbf{uu} )\,dz = \nabla \cdot \left( \int_0^h \left[ \mathbf{\bar u} \mathbf{\bar u}+\mathbf{\bar u} \mathbf{\tilde u} + \mathbf{\tilde u} \mathbf{\bar u} + \mathbf{\tilde u} \mathbf{\tilde u} \right]\,dz \right)$$</p>
<p>So on the RHS the 1st term is just a constant, the 2nd and 3rd terms become $0$, and then he uses the derived polynomial form of $u$ to evaluate the last term. However, how was he able to pull the divergence operator out of the integral? He didn't use the divergence theorem, and I don't know if you can use the Leibniz formula on a divergence operator. If you could do that though, wouldn't you have a term that's something like $\nabla \cdot h \left.(\mathbf{u}\mathbf{u})\right|_{z=h}$ leftover as well? </p></li>
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<blockquote>
<p>Why is effective length of a bar magnet shorter than its geometric
length?
Blockquote</p>
</blockquote>
<p>in a textbook, author wrote that, effective length is equal to 0.85 times geometric length. But didn't mention the mathematics behind it. I don't understand which keyfactor can determine the differences between geometric length and effective elgth? </p> | 2,333 | [
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<p>When working with path integrals of both bosonic and fermionic field variables, I'm a bit unsure of how to do the usual complete the square trick when an interaction between the two is concerned. Say you have a generic partition function like
\begin{equation}Z=\int D\phi D\bar\psi D\psi \,e^{iS[\phi,\bar\psi,\psi]}\end{equation}
where the action has your standard quadratic part, but also an interaction term like $\mathcal{L}_{int}\sim \bar\psi \phi\psi$. If you wanted to integrate out the Bose fields first, you would complete the square but that would involve making a substitution like
$\phi\rightarrow \phi'= \phi-\bar\psi\psi $ and integrating over $\phi'$ (I may be wrong here). I'm confused on what it means to:</p>
<ol>
<li>Subtract a product of Grassmann numbers from an ordinary number (they aren't the same kind of number)</li>
<li>Integrate over the new Bose field, which is technically a function of the Grassmann variables (are there any subtleties with that process?)</li>
</ol>
<p>Any help would be greatly appreciated. </p> | 2,334 | [
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<p>I was talking my professor about entanglement swapping between light and matter and it is briefly described here:</p>
<p>You start out with a crystal capable of doing parametric down conversion of incoming photons. When they go in, they undergo a physical process which produces two entangled photons that come out. At the same time, you get two atoms of the same type, say two Hydrogen atoms that are trapped in a harmonic oscillator potential in the, let's say x direction. Then, you send each entangled photon pair to each of the atoms along the x direction and have them interact. You keep on sending entangled photon pairs until the two atoms come to a steady state where they oscillate in a synchronous fashion. You essentially transfer the momentum of the photon to the atoms and do the opposite of laser cooling.
The atoms are entangled in position and momenta. When one is measured at x1 = 1, the other is at x2 = -1. Their momenta are equal. p1 = p2.</p>
<p>So, if we are to imagine two people. These two people would be the Hydrogen atoms. The photons that bounce off each of the two people are like the entangled photons. With their vision, absorption of photons, the two people can come into sync. When two people dance mirror images of each other, this can be viewed as if they are in "opposite" positions but since they are moving at the same velocity, they share equal momenta. You can't take the analogy too far, but would this be good?</p> | 2,335 | [
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<p>My understanding of relativity isn't very sophisticated, but it seems to me that relative to a photon moving at the speed of light, we are moving at the speed of light. Is this the case?</p> | 2,336 | [
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<p>If a photon hits a proton, would it have a color? What color would it be?</p> | 2,337 | [
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<p>Even at the core of the sun, the temperature of $\sim 10^7$ K only results in $kT\sim1$ keV, which is about a thousand times less than the electrical potential energy of $\sim1$ MeV needed in order to bring two hydrogen nuclei to within the ~1 fm range of the strong nuclear force. Therefore nuclear fusion reactions can only occur inside the sun, or in any other normal star, through the process of quantum-mechanical tunneling. The low probability of this tunneling, along with the need for a weak interaction in order to fuse two protons into a deuterium nucleus, are the two factors that make stars have lifetimes billions of years long.</p>
<p>How is the tunneling probability calculated?</p> | 2,338 | [
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<p>Can one understand Newton's law of gravitation using the holographic principle (or does such reasoning just amount to dimensional analysis)?</p>
<p>Following an argument similar to one given by <a href="http://arxiv.org/abs/1001.0785">Erik Verlinde</a>, consider a mass $M$ inside a spherical volume of space of radius $R$.</p>
<p>The holographic principle says that the mass $M$ must be completely described on the surface of the sphere in terms of the energy in elementary entities or strings.</p>
<p>By the equipartition theorem, the total energy, $E=Mc^2$, on the surface is given by the number of strings, $N$, times the number of degrees of freedom per string, times $1/2\ k_BT$ per degree of freedom:</p>
<p>$$E = N \times d_f \times \frac{1}{2} k_B T.\ \ \ \ \ \ \ \ \ \ (1)$$</p>
<p>The number of strings on the surface is given by the <a href="http://en.wikipedia.org/wiki/Black_hole_thermodynamics#Black_hole_entropy">Bekenstein-Hawking formula</a>:</p>
<p>$$N = \frac{A}{4},$$</p>
<p>where $A$ is the sphere surface area in units of the Planck area $G\hbar/c^3$. In terms of the radius $R$ the number $N$ is given by:</p>
<p>$$N = \frac{\pi c^3 R^2}{G \hbar}.$$</p>
<p>By substituting into equation (1) we obtain an expression for temperature $T$:</p>
<p>$$M c^2 = \frac{\pi c^3 R^2}{G \hbar} \times d_f \times \frac{1}{2}k_BT$$</p>
<p>$$T = \frac{4}{d_f}\frac{\hbar}{2\pi ck_B}\frac{GM}{R^2}.$$</p>
<p>If there are $d_f=4$ degrees of freedom per string then the above formula gives the Unruh temperture for an object falling through the surface with acceleration $g$:</p>
<p>$$g = \frac{GM}{R^2}.$$</p>
<p>This of course is the acceleration given by Newton's law of gravity due to the presence of a mass $M$.</p>
<p>P.S. Could $d_f=4$ come from the 4 dimensions of spacetime?</p> | 2,339 | [
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<p>Question 1:Most science textbooks have appendixes that have a value for some physical property of some object. This includes diameter of electrons, viscosity of fluids, boiling points, etc. My question is, are the values presented in such appendixes (or other data bases) averages?</p>
<p>Question 2: Also, suppose I try to calculate the boiling point of water using equations. I then compare my result with data obtained from the CRC Handbook. How would I determine how accurate my result is? Would I use percentages or is there another way to calculate the error? Note: by percentages I mean $$\vert\frac{result-reference\,\,value}{reference\,\,value}\vert\cdot100$$</p> | 2,340 | [
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<p>I would like to know what are these formulas used for. There is no intro about it in my book at all, and I am reading Heat Transfer book.
<img src="http://i.stack.imgur.com/nHffh.jpg" alt="enter image description here"></p>
<p>If needed Q. can be edited.</p> | 2,341 | [
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<p>I'm studying statistical mechanics, in particular classical regime for Fermi Dirac and Bose Einstein gases. Time average value for occupation numbers in FDBE statistics:
$$ \langle n_\epsilon\rangle_{FB} = \frac{1}{e^{(\epsilon-\mu)\beta}\pm1} $$
For Boltzmann Statistics:
$$ \langle n_\epsilon \rangle_B = e^{(\mu-\epsilon)\beta} $$
How can one work out a nice condition of classical regime in which $ \langle n_\epsilon\rangle_{FB} \rightarrow \langle n_\epsilon\rangle_B $ ?</p>
<p>An obvious option is $e^{\frac{(\epsilon-\mu)}{kT}}\gg1$. However, I don't really like it, since it implies convergence at low temperature. Moreover I'm expecting an $\epsilon$-free asymptotic expression in terms of temperature and density.</p>
<hr>
<p>@Adam : i've read your comment again and things are much more clear now :)!
Here's what i've got:</p>
<p>I'll assume $ \beta|\mu|>>1 $ and $\mu<0 $ or $z \rightarrow 0 $.</p>
<p>In terms of z:</p>
<p>$$ \langle n_\epsilon\rangle_{FB} = \frac{1}{\frac{e^{\epsilon\beta}}{z}\pm1} \, \,\underrightarrow{z\rightarrow0} \, \,\langle n_\epsilon\rangle_{B}$$</p>
<p>Being $z=\lambda^3_t \rho$, i can say FDBE gases behaviour like classical one when the particle's thermal wavelenght is small if compared to the typical particle distance. Almost the "low density, high temperature" condition i was looking for.</p>
<p>At low temperature Boltzmann statistics lose physical mean (for example it's easy to recover the classical Sackur–Tetrode entropy from his thermodynamic) . Approximating in this scenary, although it may look mathematically legitimate, is conceptually wrong. Quantum statistcs have to be handled carefully on their own.</p>
<p>Am i doing it right :)?</p>
<p>Sorry for the poor english.
Thanks you so much</p> | 2,342 | [
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<p>So at the end of one of my prof's lectures he gives us something to think about:</p>
<blockquote>
<p>Both electric and magnetic dipoles tend to line up with their
respective fields.</p>
<p>Materials made out of electric dipoles cause the electric field that
turns the dipoles to be <strong>reduced</strong>.</p>
<p>Materials made out of magnetic dipoles cause the magnetic field that
turns the dipoles to be <strong>increased</strong>. </p>
<p>Why are the two types of dipoles different in this regard?</p>
</blockquote>
<p>Now, this isn't a homework assignment or anything. It is just something to ponder on. I'm curious why this happens.</p> | 2,343 | [
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<p>If $\{|\psi_{i}\rangle\}$ is an orthonormal basis for a bipartite system, will $E(|\psi_i\rangle) = E(|\psi_j\rangle)$ for all $i, j$, where $E$ is some entanglement measure?</p> | 2,344 | [
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