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<p>An <a href="http://en.wikipedia.org/wiki/Alcubierre_drive" rel="nofollow">Alcubierre Drive</a> is a hypothetical device that can move a place someplace else faster than the speed of light without violating known laws of physics.
<a href="http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20110015936_2011016932.pdf" rel="nofollow">This paper</a> provides some equations as to the energy density in such a field, as well as proposals about testing the theory experimentally and finding modes of applications.</p>
<p>One application is described as follows:</p>
<blockquote>
<p>the spacecraft departs earth and establishes an initial sub-luminal velocity $v_i$, then initiates the field. When active, the field’s boost acts on
the initial velocity as a scalar multiplier resulting in a much higher apparent speed, $v_{eff}= γ v_i$ as measured by either an Earth bound observer or an observer in the bubble. Within the shell thickness of the warp bubble region, the spacecraft never locally breaks the speed of light and the net effect as seen by Earth/ship observers is analogous to watching a film in fast forward. Consider the following to help illustrate the point – assume the spacecraft heads out towards $\alpha$ Centauri and has a conventional propulsion system capable of reaching $0.1c$. The spacecraft initiates a boost field with a value of 100 which acts on the initial velocity resulting in an apparent speed of $10c$. The spacecraft will make it to $\alpha$ Centauri in 0.43 years as measured by an Earth observer and an observer in the flat space-time volume encapsulated by the warp bubble.</p>
</blockquote>
<p>A formula to arrive at $\gamma$ is given in the paper. Now ...</p>
<ul>
<li><p>I assume that the lower limit to create such a "warp bubble" is independent of the method or device or whatever used, there's a hard lower physical limit - is this even correct?</p></li>
<li><p>Now, as an example, how much energy would be needed to create a boost factor $\gamma$ of 100 for a golf ball (or any other arbitrary sized volume)?</p></li>
<li><p>Where does this energy go?</p></li>
</ul>
<p>While White mentions an experimental setup to test the theory with only a ring of capacitors, others mention exotic matter to be neccessary. I don't want to go into details of implemementation in this question. Since my question is about the physical limits, it's here on Physics SE and not over at <a href="http://space.stackexchange.com/">Space exploration</a>.</p> | 5,285 | [
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<p>In classical computer simulations such as molecular dynamics (MD) simulations, one integrates Newton's equations of motion to determine particle trajectories. If we think of Newton's Second Law as simply $$\textbf{f}_i = \frac{d \textbf{p}_i}{dt}$$ where $\textbf{f}_i$ is the net force acting on a particle $i$ and $\textbf{p}_i$ is the momentum of the particle, then it is clear that to ultimately integrate this and propagate the trajectory, we will need to calculate the force on every particle in the system.</p>
<p>However, in practice, I think that the <em>potential energy</em> $u$, rather than the force, is initially calculated (a force field specifies the form of the potential energy $V$). Now, I know that there is a relationship between potential energy $u$ and force $f$: $$\textbf{f} = -\boldsymbol{\nabla} u$$</p>
<p>In other words, force is the negative gradient of the potiental energy.</p>
<p>I am reading <a href="http://rads.stackoverflow.com/amzn/click/0122673514" rel="nofollow">Understanding Molecular Simulation by Frenkel and Smit (Second Edition)</a>, and on page 69 (Google Books has some pages <a href="http://books.google.com/books?id=5qTzldS9ROIC&pg=PA553&source=gbs_selected_pages&cad=3#v=onepage&q=69&f=false" rel="nofollow">here</a>), I see this paragraph:</p>
<blockquote>
<p>Suppose that we wish to compute the $x$-component of the force:
$$f_x(r) = -\frac{\partial u(r)}{\partial x} = -\left(\frac{x}{r}\right)\left(\frac{\partial u(r)}{\partial r}\right)$$</p>
</blockquote>
<p>I am a little afraid this is a silly question, but how do the authors go from $-\frac{\partial u(r)}{\partial x}$ to $-\left(\frac{x}{r}\right)\left(\frac{\partial u(r)}{\partial r}\right)$? Is this the chain rule? Could you please help me see the step that the authors are making? </p> | 5,286 | [
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<p>Weinberg claims that it is obvious that the $\sigma = 0$ component of $u^\mu$ at zero spatial momentum points in the 3-direction. This is supposed to follow from (5.3.6). Unfortunately I am not seeing it. I thought that the $J=1, m = 0$ component is conventionally aligned in the 2-direction?</p>
<p>Any help would be appreciated.</p>
<p>(5.3.20)</p>
<p>$u^\mu(0,0) = (2m)^{-1/2} (0,0,1,0)$</p>
<p>(5.3.6)</p>
<p>$\sum_{\bar{\sigma}} u^\mu(0,\bar{\sigma}) \mathbf{J}^{(j)}_{\bar{\sigma}\sigma} = {\mathbf{\mathcal{J}}^\mu}_\nu u^\nu (0,\sigma)$</p>
<p>(5.3.8)</p>
<p>$(\mathcal{J}_k)^0_0 = (\mathcal{J}_k)^0_i = (\mathcal{J}_k)^i_0 = 0$</p>
<p>(5.3.9)</p>
<p>$(\mathcal{J}_k)^i_j = - i \epsilon_{ijk}$</p> | 5,287 | [
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<p>I want to know how the flow of electrons will change when I change the tempertature from 100 F to 250 F in a silicon semi-conductor (ex: computer mouse)</p>
<p>How can I find this out?</p> | 5,288 | [
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<p>I have again an old question from a comprehensive exam I took a couple of months ago. Lucky for me one could pick 5 out of 8 questions, because on some of the problems I didn't even know how to start. Now that classes are over I've now the time to revisit those problems I was dumbfounded by, such as this one: (Abriged version)</p>
<blockquote>
<p>Life on earth would be impossible if we were constantly exposed to charged solar particles. Luckily, earth's magnetic field protects us from them. The solar particles have a typical energy spectrum of $d\Phi / dE \propto E^{-3}$ particles/$m^2/s/J$. What is the <em>minimal</em> field strength of the earth's magnetic field based on the anthropic principle, i.e., it couldn't be weaker or else we wouldn't live to observe it.</p>
</blockquote>
<p>Well, this quantity $d\Phi/dE$ looks like a flux, so I guess the general setting is that of a scattering problem. But first, $d\Phi / dE$ isn't given completely, only a rough form of its energy dependence. And second, I'm not sure what a reasonably simple model for this entire process would be. Easiest in terms of calculation would be to assume some sort of homogeneous magnetic field aligned with the earth's magnetic axis, because I guess it's a pain to calculate the path of a particle in a dipole field...</p>
<p>Maybe the idea of this is to calculate the total cross section of earth's magnetic field and then demand that it should "cover" the earth? Or they want me to solve an equation of motion for incoming solar particles and show that all of them are deflected? My problem right now is that I don't even know how to interpret the $d\Phi/dE$ quantity whose energy dependence I'm given. I guess it makes more sense to someone with a background in elementary particle physics?</p>
<p>Right now I'm trying to write a vector potential $\vec{A} = \mu_o/(4\pi r^2) \vec{m} \cdot \vec{e}_r$ where $\vec{e}_r$ is the unit vector in $r$-direction in spherical coordinates, and then try to get equations of motion from the Hamiltonian $$H = \frac{(\vec{p} + q\vec{A})^2}{2m}$$ but I am not sure if I'll be able to solve whatever comes out of that, or if I'm completely on the wrong track with this.</p>
<hr>
<p>EDIT</p>
<hr>
<p><img src="http://i.stack.imgur.com/jbZDA.png" alt="Schematic of Deflection of Solar Particles">
Image taken from <a href="http://www.astronomynotes.com/solarsys/radiationbelts.png" rel="nofollow">here</a></p>
<p>Maybe it helps trying to understand this schematic, but I cannot easily see how the Lorentz force would create such a trajectory.</p>
<hr>
<p>ANOTHER EDIT</p>
<hr>
<p>From further searching, I know suspect that this has something to do with how a plasma current (the charged particles) interact with a magnetic field. That would mean that I have to calculate the radius of the ensuing magnetosphere and then demand, via the anthropic principle, that it should be at least of the same size (or larger) as the radius of earth. So the Lorentz force would probably not directly have anything to do with it. But I also have no training in plasma physics. (Some of the problems in the exam were specifically geared towards Astronomy students, so I guess they'd find it a breeze).</p> | 5,289 | [
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<p>In my physics book "University Physics", there is a chapter on relating linear and angular kinematics.</p>
<p>I understand the parts where it shows $v = r\omega$ and $a_{\text{tan}} = r\alpha$.</p>
<p>However in the part where they show $a_{\text{rad}} = r\omega^2$, they use the formula $a_{\text{rad}} = \frac{v^2}{r}$ which had previously been shown for when $v$ is constant. It mentions that it is also true when $v$ is not constant, but does not provide proof.</p>
<p>Is this hard to prove, or is it something I should easily be able to see? Can anyone help make it clear?</p>
<p>Thanks!</p> | 5,290 | [
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<p>I'm working on an inertial measurement unit with some MEM'S component. I want to retrieve data on a micro-controller. I have 3-axis accelerometer sensor. however I want retrieve data in $mg$ ( $g=9.81$ $m/s^2$ so $mg = 0.001g = 0.00981m/s^2$) .</p>
<p>I just would like to calculate velocity in $meters/s$ .</p>
<p>how can i do that?</p> | 5,291 | [
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<p>I am reading a book on fundamental physics for mechanics. There is a truss shown in the figure. A train of mass $M=56$ ton is resting at the middle point of AC, ignore all the mass of truss and rails. All triangles are equilateral. I am trying to find the force exerted on strut AB. </p>
<p><img src="http://i.stack.imgur.com/mRaBG.png" alt="enter image description here"></p>
<p>I know this issue is all about static equilibrium on force. I am looking at point A, I have the force $F_{AB}\sin(60) = 0.5*Mg$. I use "half" of the mass because there are two strut AB and BC sharing the weight of the train. But this doesn't match the answer ($F_{AB}$ should be $7.9\times 10^4 $N). What's wrong with my calculation. </p> | 5,292 | [
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<p>In <em>Philosophy of Language</em> by William G. Lycan, there are the lines:</p>
<blockquote>
<p><em>Even apparent truths of logic, such as truths of the form "Either P or not P", might be abandoned in light of suitably weird phenomena in quantum mechanics.</em></p>
</blockquote>
<p>I really don't know much about quantum mechanics, is this statement valid?</p> | 5,293 | [
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<p>If you recorded data which represents some physical value in space (e.g. electron density) and you need to explore this dynamically in 3d (say you have a isosurface and you change the value it represents) what toolkit/program would you use?</p>
<p>Personally I used MayaVi and it has all the all the features I could need. Unfortunately it is sometimes unstable and often slow. So in principle I am looking for equally powerfull alternatives.</p>
<p>Edit: I know this should probably not be in the physics SE but I think the answer to my question could be relevant for many physicists.</p> | 5,294 | [
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<p>Is there a way to extend or reduce the half-life of a radioactive object? Perhaps by subjecting it to more radiation or some other method.</p> | 5,295 | [
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<p>Can someone please explain why Indirect band gap semiconductor can not be used for LED creation. Can you also please give me some reference link for details.</p> | 5,296 | [
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<p>Are there any other facilities that would be capable of independantly verifying the opera result? In other words, a completely different source/detector for $\nu_{\mu}$ beams?</p>
<p>Alternatively, there might be another source that could send a beam to the same detector at Gran Sasso, with a different baseline, looking to verify the value of $\frac {v-c}{c}$ with a different value of $\delta t$?</p> | 5,297 | [
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<p>In some basic physics homework I have, it asks what the resulting vector would be for a plane traveling at x m/s in y direction that is affected by a tailwind going in z direction at w m/s. How would I calculate this? I know basic vector addition, but this has stumped me.</p> | 5,298 | [
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<p>In the 2011 movie <a href="http://www.imdb.com/title/tt1527186/" rel="nofollow">Melancholia</a>, a planet, also called Melancholia, enters the solar system and hits the Earth. I want to leave aside the (also unreasonable) aspect that planet "hides behind the Sun" and is undetected until it's a few weeks from Earth. I want to focus on the final stages. In the movie, the planet passes close to the Earth, starts receding, doubles back and hits the Earth, all in a few days. My question: Is that physically possible? I guess not, but I was never good at orbital mechanics.</p>
<p>Assume the givens are as in the movie. One of the main characters finds on the internet a path for the planet that seems to what it eventually does. Mass of the planet is four times that of the earth?
<a href="http://en.wikipedia.org/wiki/File%3aEarthAndMelancholiaDanceOfDeath.png" rel="nofollow"><img src="http://i.stack.imgur.com/qoskR.png" alt="enter image description here"></a></p> | 5,299 | [
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] |
<p>I am referring to <a href="http://pages.swcp.com/pcaskey/feynman.html" rel="nofollow">this</a>:</p>
<blockquote>
<p>Within a week I was in the cafeteria and some guy, fooling around,
throws a plate in the air. As the plate went up in the air I saw it
wobble, and I noticed the red medallion of Cornell on the plate going
around. It was pretty obvious to me that the medallion went around
faster than the wobbling.</p>
<p>I had nothing to do, so I start to figure out the motion of the
rotating plate. I discover that when the angle is very slight, the
medallion rotates twice as fast as the wobble rate - two to one. It
came out of a complicated equation! Then I thought, ``Is there some
way I can see in a more fundamental way, by looking at the forces or
the dynamics, why it's two to one?''</p>
<p>I don't remember how I did it, but I ultimately worked out what the
motion of the mass particles is, and how all the accelerations balance
to make it come out <strong><em>two to one</em></strong>.</p>
</blockquote>
<p>Anyone knows how to derive the two-to-one relationship?</p> | 5,300 | [
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<p>Hey so I've just learned about angular velocity and momentum and how torque changes it.</p>
<p>Looking at a <a href="http://www.youtube.com/watch?v=8H98BgRzpOM" rel="nofollow">wheel spinning around an axis, with one end being held up</a> by a rope, what causes the wheel to rotate downwards over time, and eventually fall?</p> | 5,301 | [
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<p>Lasers are used in various industrial processes that need intense, localised, heat (3d printers and laser cutters come to mind).</p>
<p>My question is: why use lasers? There are many other (cheaper, brighter) light sources. There are even other monochromatic and coherent light sources (LEDs and mercury vapour lamps respectively), and <a href="http://www.makerbot.com/blog/2011/06/24/solar-sinter-by-markus-kayser/">this video shows someone sintering desert sand using a Fresnel lens and sunlight</a>, which is of course neither monochromatic nor coherent.</p>
<p>So, what is it about lasers that make them so better than a conventional light source combined with appropriate focusing elements?</p> | 5,302 | [
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<p>What type of magnetic fields does a Hall effect semi-conductor pick up on? AC or DC fields? How would one go about building a device that measures AC Magnetic fields?</p> | 5,303 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/6450/is-there-an-accepted-analogy-conceptual-aid-for-the-higgs-field">Is there an accepted analogy/conceptual aid for the Higgs field?</a> </p>
</blockquote>
<p>A <a href="https://www.youtube.com/watch?v=ASRpIym_jFM" rel="nofollow">video</a> I watched explains the <a href="http://en.wikipedia.org/wiki/Higgs_mechanism" rel="nofollow">Higgs mechanism</a> as follows.</p>
<p>Take massless particles. These can only ever travel at the speed of light, but they can bounce off things. Now take a particle with mass. Imagine that it can also only travel at the speed of light, but it interacts with the (very dense) Higgs field all the time, by constantly bouncing back and forth off the field. Thus it appears to travel slower than the speed of light.</p>
<p>Would you say this is a reasonable explanation of the mechanism? Does the described process happen in any real sense, or is this one of those interpretations where we can't really pick one over the other because they are mathematically equivalent?</p>
<p>For example, if the particles were really travelling at the speed of light but constantly bouncing off the field, wouldn't GR predict that they would never decay as observed in any frame of reference?</p> | 183 | [
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<p>I am given the information that a parcel of air expands adiabatically (no exchange of heat between parcel and its surroundings) to five times its original volume, and its initial temperature is 20° C. Using this information, how can I determine the amount of work done by the parcel on its surroundings?</p>
<p>I know that $dq = 0$, and that $du + dW = dq = 0$, but I don't know what to do with this information. $dW = pdV$, which seems like it should be helpful, but I don't know what to do for the pressure.</p> | 5,304 | [
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<p>When EM radiation with fixed intensity and frequency strikes the metal plate, are the outgoing electrons at higher energy if the plate were charged to some potential than if the plate were simply neutral?</p> | 5,305 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/32120/why-do-we-still-not-have-an-exact-definition-for-a-kilogram">Why do we still not have an exact definition for a kilogram?</a> </p>
</blockquote>
<p>I was thinking about SI units. I found the following definition for the base units:</p>
<ul>
<li><strong>Meter</strong>: distance covered by light in 1/299792458 seconds.</li>
<li><strong>Second</strong>: duration of 9192631770 periods of the radiation corresponding to the transition between two levels of fundamental state of 133-Cesium.</li>
<li><strong>Kelvin</strong>: 1/273.16 of the thermodynamic temperature of water triple point</li>
<li><strong>Mole</strong>: Number of atoms in 0.012 kg of carbon 12</li>
<li><strong>Candela</strong>: [...]</li>
<li><strong>Ampere</strong>: [...]</li>
</ul>
<p>I searched the definition of the kilogram and I found only this one:</p>
<blockquote>
<p>Mass of the international prototype of kilogram.</p>
</blockquote>
<p>Why such definition? It is impossible to well define the kilogram? Why?</p>
<p>By my point of view the mass of the prototype will change a little with time. Is this effect considered? And what about definition of mole: it is based on kilograms, so also mole definition is "impossible"?</p> | 184 | [
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<p>I live on the first floor of a seven story apartment building. We are considering our hot water options and one of them is a solar-powered water heater in which water is heated in a specialised device similar to an air conditioning condenser under the sun (very common in Israel).</p>
<p>Ignoring the fact that the device will need electric heating in winter, the heat loss through the pipe on the way down, and the wasted water while we wait for the hot water to make it's way to the faucet (all real concerns), I would like to disprove the imaginary concern of pressure loss after the round trip. My argument is that the pressure lost on the way up will be exactly compensated by the pressure gain on the way down, excluding losses due to friction in the pipe.</p>
<p>That leaves the question of how much friction is in the pipe. Googling the question I have found all means of formulae, diagrams, equations, etc, most consisting of variables that I don't have (such as the Hazen-Williams coefficient of all the pipe and fittings, the initial pressure and the flow rate). <strong>The truth is, I really don't need such an accurate assessment but rather just to know if the pressure drop would be "not noticeable", "noticeable but not a problem", or possibly "you can no longer do laundry".</strong> So for seven floors up and back down, through a flexible plastic pipe with an outside diameter of 16 mm (I don't know what the inside diameter is) and a few fittings, how can I estimate the percentage of pressure drop in layman's terms, not engineering terms?</p>
<p>In other words, I'm not looking for someone to do the calculus for me and give me a number, but rather I would like some help in deciding how to go about figuring out a rough estimate.</p> | 5,306 | [
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<p>I wonder if someone could tell me where my logic is going wrong here?</p>
<ol>
<li>If two particles both have definite energy, then they have indefinite position.</li>
<li>As their positions could literally be anywhere in the universe, we cannot tell them apart.</li>
<li>If two particles are indistinguishable, then we have to consider the probability of interference due to the particles exchanging places (as in <a href="http://www.feynmanlectures.info/docroot/III_04.html" rel="nofollow">Feynmann Vol 3 Ch 4</a>)</li>
<li>Does this mean no two electrons in the entire universe can share the same energy due to the Pauli exclusion principle?</li>
<li>Does this break locality?</li>
</ol> | 5,307 | [
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<p>According to Faraday's law of induction, volts = -Number of coils in a solenoid * change in strength of magnet / change in time. This doesn't take into account distance or speed, only time. If amps = volts / ohms, and ohms is 0, it seems like amps should be infinity. If there are infinite amps, then wouldn't the alternator just generate a constant voltage without the magnet needing to move at all?</p> | 5,308 | [
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<p>why streams of energy-matter are emited along the axis of rotation of a compact object?</p>
<p>(or) what is the reason that they are emited along the axis of rotation of a compact object?</p>
<p><img src="http://upload.wikimedia.org/wikipedia/commons/3/38/Protoplanetary_disk_HH-30.jpg" alt="polar jet herbig haro object"></p> | 5,309 | [
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<p>I am studying Statistical Mechanics and Thermodynamics from a book that i am not sure who has written it, because of its cover is not present.</p>
<p>There is a section that i can not understand:</p>
<p>${Fj|j=1,..,N}$</p>
<p>$S= \sum_{j=1}^{N} F_{j}$</p>
<p>$<S>=< \sum_{j=1}^{N} F_{j}> = \sum_{j=1}^{N} <F_{j}>$</p>
<p>$\sigma^{2}_{S} =<S^{2}>-<S>^{2}$</p>
<p>line a:</p>
<p>$=\sum_{j=1}^{N}\sum_{k=1}^{N} <F_{j}F_{k}> - \sum_{j=1}^{N} <F_{j}>\sum_{k=1}^{N}<F_{k}>$</p>
<p>I can not understand why these terms are different</p>
<p>First term is:</p>
<p>$\sum_{i}\sum_{j}A_iA_j=(A_1A_1+A_1A_2+A_1A_3+\dots+A_1A_n)+\dots+(A_nA_1+A_nA_2+\dots+A_nA_{n-1}+\dots+A_nA_n)$</p>
<p>Second term is</p>
<p>$A_1(A_1+A_2+A_3+\dots+A_n)+\dots+A_n(A_1+A_2+\dots+A_n)$</p>
<p>I can see they are same.</p>
<p>But as you can see above it is different, how? Where am i wrong?</p>
<p>The question <a href="http://physics.stackexchange.com/questions/59832/math-for-thermodynamics-basics">Math for Thermodynamics Basics</a> 's answer have some clue but i could not understand exactly. Thanks.</p>
<p>Thanks</p> | 5,310 | [
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<p>Which are the best introductory books for topology, algebraic geometry, manifolds etc, needed for string theory?</p> | 185 | [
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<p>I'm trying to understand <a href="http://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment" rel="nofollow">Stern-Gerlach experiment</a> on a computational level. Suppose we have a neutral particle with magnetic moment (e.g. a neutron), and apply an inhomogeneous magnetic field to it (let it change linearly with coordinate). As I understand, its Hamiltonian would look like:</p>
<p>$$\hat H=-\frac{\hbar^2}{2m}\nabla^2+\left(\frac e{mc}\right)\hat{\vec s}\vec B$$</p>
<p>Now the spin operator is
$$\hat s_i=\frac{\hbar}2\sigma_i,$$</p>
<p>where $\sigma_i$ is $i$th <a href="http://en.wikipedia.org/wiki/Pauli_matrices" rel="nofollow">Pauli matrix</a>.</p>
<p>So, for magnetic field $\vec B=\vec e_x B_0 x$ we'd have Schrödinger 1D (Y and Z directions can be separated due to translation symmetry) equation:</p>
<p>$$-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}+\left(\frac {\hbar e}{2mc}\right)\sigma_x B_0 x\psi=i\hbar \frac{\partial\psi}{\partial t}.$$</p>
<p>I now try to solve this equation numerically, taking initial wave function in the following form:</p>
<p>$$\psi(x,t=0)=\begin{pmatrix}\psi_0(x)\\ \psi_0(x)\end{pmatrix},$$</p>
<p>where $\psi_0(x)$ is a gaussian wave packet with zero average momentum.</p>
<p>The problems start when I select $\sigma_x$ as is usually given:</p>
<p>$$\sigma_x=\begin{pmatrix}0&1\\1&0\end{pmatrix}.$$</p>
<p>The solution appears to look like showed below. I.e. both wave function components accelerate left!</p>
<p><img src="http://i.stack.imgur.com/Rrbsk.gif" alt="enter image description here"></p>
<p>I thought, what if I choose another axis as $x$, so I tried doing the same with $\sigma_y$:</p>
<p>$$\sigma_y=\begin{pmatrix}0&-i\\i&0\end{pmatrix}.$$</p>
<p>The result in the animation below. Now it's a bit better: the wavefunction at least splits into two parts, one going left, another right. But still, both parts are composed of a mix of spin-up and spin-down states, so not really what one would expect from Stern-Gerlach experiment.</p>
<p><img src="http://i.stack.imgur.com/8hyNA.gif" alt="enter image description here"></p>
<p>Finally, I tried the last option — using $\sigma_z$:</p>
<p>$$\sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$</p>
<p>The result is again showed below. Finally, I get the splitting into "independent" spin parts, i.e. one spin part goes left, another one goes right.</p>
<p><img src="http://i.stack.imgur.com/KPe1s.gif" alt="enter image description here"></p>
<p><strong>Now, the question</strong>: how to interpret these results? Why does choice of active axis result in such drastic differences in results? How should I have done instead to get meaningful results? Shouldn't permutation of Pauli matrices not affect results?</p> | 5,311 | [
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<p>Assume that you fixed a speaker to an inclined pipe as well the torch. You can hear sound from the other end of the pipe, but can't see the light from other end of the pipe, why? </p>
<p><img src="http://i.stack.imgur.com/Cc7tp.png" alt="enter image description here"></p> | 186 | [
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<p>I know that there's a difference between relativistic rest mass. Relativistic mass is "acquired" when an object is moving at speeds comparable to the speed of light.Rest mass is the inherent mass that something has regardless of the speed its moving at.</p>
<p>When fusion happens, a certain percentage of the rest mass of two hydrogen atoms is converted into energy. I understand that $E = m_{rest}c^2$. </p>
<p><strong>How does the rest mass of an object turn into energy?</strong> Does this involve particle physics? Doesn't this suggest that rest mass can be viewed as some sort of potential energy? It seems to me that it's a sort of locked energy that has to undergo a certain process to expel its energy to surrounding particles.</p>
<p>Also, <strong>Why does only a small percentage of rest mass turn into energy? Why not all of it. What dictates how much rest mass gets converted into energy?</strong></p> | 46 | [
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<p>In a thermodynamic turbine using air as an ideal gas, given that you have a known inlet temperature value $T_i$, a known exit pressure value $P_e$, a known inlet and exit velocity $V_i$ and $V_e$, a known value for the actual work of the turbine $w_{actual}$, and an isentropic efficiency $\eta$, how can you find the other state parameters?</p>
<p>I know the efficiency of the turbine is</p>
<p>$$\eta = \frac{w_{actual}}{w_{isentropic}} = \frac{h_i - h_e}{h_i - h_{e)s}}$$</p>
<p>Through getting $w_{isentropic} = \frac{w_{actual}}{\eta}$, we'll know that value. My approach was then to find $h_i$.</p>
<p>$$h_i = w_{actual} + h_e = w_{isentropic} + h_{e)s}$$
$$w_{actual} - w_{isentropic} = h_e - h_{e)s}$$</p>
<p>And then I'm stuck there. I tried a different approach.</p>
<p>$$s_e - s_i = C_{p0}ln\frac{T_e}{T_i} - Rln\frac{P_e}{P_i}$$</p>
<p>$C_{p0}$ and $R$ are known here because they are constants. If I assume an isentropic process, then we'll have $s_e - s_i = 0$. But our unknowns are still $T_e$ and $P_i$. I can use the ideal gas equation:</p>
<p>$$\frac{T_e}{T_i} = (\frac{P_e}{P_i})^\frac{k-1}{k}$$</p>
<p>And $k$ is known here because it's air and I assumed it's isentropic, and it's a constant. So with two equations and two unknowns, I'll find $P_i$ and $T_e$ assuming the process is isentropic. So then, from here, how do you find $h_{e)s}$? I think I should be able to find it, but with what relationship?</p>
<p>I see that once I find $h_{e)s}$ I'll be able to solve for $h_i$, and then $h_e$. Once I have $h_e$, what relationship can I use with my known $T_i$ to derive the other state parameters?</p>
<p>The two key things here are the $h_i$ and $h_e$, I think, because I already have $P_e$ and $T_i$ given to me. And if I have those four, I'll be able to find all the other state parameters ill both the inlet and exit states. But what relationship can I use to find them?</p>
<p>My thermodynamic tables don't have any values for air, just steam and other elements and refrigerants. I do have a table for air, but only for a 1-bar pressure, and I suspect I can't assume that pressure in this question. I suspect that since it's an ideal gas, I won't need the tables to solve for it.</p> | 5,312 | [
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<p>I read that magnetic fields perpendicular to a current shoot out and expand all the way to infinity.<br>
Additionally a gravitational wave, no matter how small will also expand to infinity at the velocity of light.<br>
Are these correct? If yes, why don't waves "lose" energy while "traveling"? </p>
<p><strong>Update</strong></p>
<p>After reading the answers: </p>
<p>I did not read this statement in a physics book. I read it in a "philosophical" book where the author tries to make a point taking this assertion for granted.<br>
To be specific his exact example is: </p>
<blockquote>
<p>We shall take a short piece of wire, connect both its ends to a
battery through a switch and close the switch. Three things will
happen: </p>
<ol>
<li><p>That electrical current will run from one side of the battery to the other</p></li>
<li><p>A magnetic field perpendicular to the current will shoot out and expand all the way to infinity at the velocity of light </p></li>
<li><p>The wire will heat up slightly</p></li>
</ol>
</blockquote>
<p>So is this description correct only in a theoretical manner? In reality, due to other objects, will point 2 be invalid?</p> | 5,313 | [
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<p>We know of geomagnetic flip in Earth's history by studying geologic data. Given other planets in the system also possess a magnetic field leads to the assumption that such polarity reversal may not be unique to Earth. </p>
<p>Is it possible to determine geo-astrophysically whether such geomagnetic polarity reversal occurs on other planets? </p> | 5,314 | [
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<p>I was watching a set of lectures on effective field theory and the lecturer said that you can always integrate the covariant derivative by parts due to gauge symmetry. For example, if I understand correctly, we can write:
\begin{equation}
\int {\cal D} \phi \exp \left\{ i \int d ^4 x D _\mu \phi D ^\mu \phi + \, ...\right\} = \int {\cal D} \phi \exp \left\{ i \int d ^4 x - \phi D ^2 \phi + \,...\right\}
\end{equation}
where $D_\mu = \partial_\mu + i g T ^a G_{a, \mu} $. This would be obvious if we didn't have the gauge boson contribution but why does integration by parts hold for the covariant derivative?</p> | 5,315 | [
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<p>I do understand why we are using the double cover, but why exactly do we make the transition to complex Lorentz transformations? Where and why are they needed? </p>
<p>To be precise:<br>
The double cover of the Lorentz group is given by $SU(2)$. Where and why do we make the transition from $SU(2)$ to $SL(2,C)$? ($SL(2,C)$ is the complexification of $SU(2)$ as far as i understand) </p>
<p>Complexification means that we allow the vector space spanned by the generators of the group to be complex, i.e. we allow complex linear combinations of the generators. $SU(2)$ are all unitary 2x2 matrices with unit determinant, whereas $SL(2,C)$ are all complex matrices with unit determinant. As far as i understand this has something to do with allowing complex Lorentz transformations and therefore extending the Lorentz group to the complex Lorentz group. The double cover of the <strong>complex</strong> Lorentz group is then $SL(2,C)$, whereas the double cover of the (real) Lorentz group is given by $SU(2)$?! </p>
<p>If someone could clarify this i would be very thankful.</p> | 5,316 | [
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<p>I'm studying for my upcoming physics course and ran across this concept - I'd love an explanation.</p> | 152 | [
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<p>I keep reading in the Physics World focus issue on vacuum technology about scientists creating high temperatures in the vacuums etc.</p>
<p>If heat is caused by thermal energy being radiated from particles due to their energy, then how can there be heat in a vacuum, as there are no particles present?</p> | 5,317 | [
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<p>I have been given the Lorentz generator
$$J_{ab}=\lambda_a \frac{\partial}{\partial \lambda^b}+a\leftrightarrow b$$</p>
<p>I know $\frac{\partial}{\partial \lambda^b}\lambda_a=\epsilon_{ba}$ and $A_a=A^b\epsilon_{ab}$ (? not sure about these two).
I want to show that this acting on a spinor bracket gives zero as this is supposed to be Lorentz invariant. Say I have a bracket $\langle A B \rangle = A_a B_b \epsilon^{ab}$ and now I act with $J_{ab}$ on it w.r.t $A$, i.e.</p>
<p>$$\begin{split}J^A_{ab}\langle A B \rangle &=(A_a \frac{\partial}{\partial A_b}+a\leftrightarrow b) A_cB_d\epsilon^{cd}\\&=A_a\epsilon_{bc}B_d\epsilon^{cd} + A_b\epsilon_{ac}B_d\epsilon^{cd}\\&=A_a\epsilon_{bc}B_{c}+A_b\epsilon_{ac}B_c\\&=?\end{split}$$</p>
<p>Where do I go wrong? Could someone please help me out? Why doesnt it sum to zero?</p> | 5,318 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/21051/why-are-continuum-fluid-mechanics-accurate-when-constituents-are-discrete-object">Why are continuum fluid mechanics accurate when constituents are discrete objects of finite size?</a> </p>
</blockquote>
<p>When we apply differentiation on charge being conducted with respect to time,i.e dq/dt, we consider the charge flown to be infinitely small, but q cannot be less than 1.602*10^-19. So how can we assume this to be infinitely small?</p> | 187 | [
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<p>I read about the near-earth asteroid <a href="http://en.wikipedia.org/wiki/99942_Apophis" rel="nofollow">99942 Apophis</a>. It is in the first place of Potentially Hazardous Asteroids (visiting earth at an altitude of about 36,000km on 2028). But, scientists have calculated that the asteroid would impact earth on 2036. What is the fact behind this? I mean, Why is this difference in period of 8 years for a close-encounter and then impact? Is Apophis, a part of the asteroid belt between Mars and Jupiter or is it a part of the solar system? Would we actually get an "Armageddon"? Even if it doesn't impact, it could cause some damages either to our atmosphere or even our environment, Nah?</p> | 5,319 | [
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<p>Just wondering, what would happen in this experiment.</p>
<p>In the experiment you would first have two entangled particles.</p>
<p>Then you fire one of the particles, lets say "Particle A", at a double slit towards a detector.</p>
<p>While in transit to the detector, what if the other entangled particle, lets call it "Particle B" was observed / had it's wave function collapsed?</p>
<p>Would "Particle A" still generate a wave-like interference pattern or would the wave function for both be collapsed?</p>
<p>In theory you cannot send classical data by entanglement, so this experiment must somehow fail, but I can't quite figure out why. If this experiment were to succeed, then you could read and send data about wave function states over entangled particles.</p> | 5,320 | [
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<p>In quantum mechanics when we talk about the wave nature of particles are we referring in fact to the wave function? Does the wave function describes the probability of finding a particle (ex: photons) at some location? So do the "waves" describe probabilities just the way in classical physics the electromagnetic waves describe the perturbations of the electric and magnetic fields?</p> | 5,321 | [
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<p>where could i find an explanation (appart from Wikipedia) of how the Hawking's effect is obtained from quantum field theory GR and thermondynamic :D </p> | 5,322 | [
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<p>I recently found a problem that looked like this:</p>
<blockquote>
<p>A box sits on a horizontal wooden ramp. The coefficient of static friction between the box and the ramp is <code>.30</code>. You grab one end of the ramp and lift it up, keeping the other end of the ramp on the ground. What is the Angle between the ramp and the horizontal direction when the box begins to slide down the ramp?</p>
</blockquote>
<p>The only thing this question gives me is the Coefficient of static friction between the box and the ramp. I don't think it's enough to solve the problem, is this true?</p>
<p>Equations Related:</p>
<p>$F_f=μ_sN$</p>
<p>$N=mg$</p> | 5,323 | [
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<p>I know it's more than 70% water. But what has it got to do with earth's colour ?</p> | 5,324 | [
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<p>I've been trying for days, but I just can't understand why escape velocities exist. I've searched the web and even this site, and although I've read many explanations, I haven't been able to <em>truly</em> understand them. Most of the explanations I've seen involve calculus; I only know very little calculus. Could somebody provide a more intuitive explanation?</p>
<p>Here's what I know & understand:</p>
<ul>
<li>Escape velocity is the speed at which the kinetic energy equals the gravitational potential energy of an object</li>
<li>Escape velocity isn't the same as orbital velocity, a satellite never reaches escape velocity, escape velocity is only an initial velocity. And all the other common misconceptions.</li>
</ul>
<p>What I do not understand, is why an object that has reached escape velocity will never return back to the planet it was launched from. To help you understand where I'm stuck, here's a short "proof":</p>
<ol>
<li>Our whole universe consists of only two bodies. A teapot of mass m, and a planet of mass M. M is many million times larger than m, so the gravitational force acting on the planet is trivial.</li>
<li>We launch the teapot from the planet, giving it an initial velocity U (relative to the planet). <strong>U > escape velocity for that particular planet.</strong></li>
<li>Now, the only force acting on the body (teapot) is the gravitational force from the planet, resulting in a negative acceleration (relative to the planet). The force, and therefore the resulting deceleration, will be decreasing quadratically as distance increases. The negative acceleration will get very close to, but never quite reach, zero.</li>
<li>Therefore, the speed of the teapot will never stop decreasing. <strong>The teapot will keep decelerating forever.</strong></li>
<li>We conclude that at some point, <strong>the speed of the teapot will reach zero</strong>. The teapot will then fall back to the planet, even though U was greater than the escape velocity.</li>
</ol>
<p>I suspect that my implication <em>(4) => (5)</em> is false. Somebody please explain why, using as little Calculus as possible. Could this be similar to the <em>Achilles and the Tortoise</em> paradox?</p>
<p>Thanks in advance! </p> | 5,325 | [
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<p>This question is a continuation from <a href="http://physics.stackexchange.com/a/34677/10531">this one</a>.</p>
<p>A material disk have two sides, one that is reflective and another absorptive of electromagnetic radiation in the range where the background cosmic radiation spectrum is significative.</p>
<p>It is pretty clear that the disk will not accelerate any further after thermalizing with the background radiation.</p>
<blockquote>
<p><strong>Question:</strong> what is the condition for the net exchange of momentum between the background radiation and the disk to become zero?</p>
</blockquote>
<p>will the disk accelerate until the momentum of the blueshifted radiation absorbed at the front equates the momentum of redshifted photons being reflected at the back side?</p>
<p>Will the reflective side stop reflecting efficiently after the disk reaches the same temperature as the background radiation?</p> | 5,326 | [
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<p>How far would I have to go to see a fully rounded Earth?</p>
<p>Recently, I saw a video on Youtube in which a sky diver called Felix Baumgartner ascends to $120,000$ feet (= $39$ miles) in a stratospheric balloon and make a freefall jump, rushing toward earth at supersonic speeds.<br>
I could see the full face of the Earth in this video from a height of approximately 39 miles. This prompts me to ask this question.</p> | 5,327 | [
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<p>I'm studying fluid mechanics in more depth during my Ph. D. and there is something related with the diffusive term that has been bothering me for a long time. Looking at the convection diffusion equation:
$$
\frac{\partial u}{\partial t} + a\cdot\nabla v - \nabla(\nu \nabla v )=f,
$$</p>
<p>and thinking in Fick's law, it's not hard for me thinking in diffusion as the process by which, for example, the particles of a solution with a concentration gradient get apart one from each other to "reduce" the energy of the system; being, each one of these particles, more "comfortable" inside the solvent, which is the same as saying "with the bigger free mean path possible".</p>
<p>But now when I look to the Navier Stokes equation (incompresible and viscous):</p>
<p>$$
\rho \left(\frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}\cdot\nabla\mathbf{v}\right)=-\nabla p+\mu\nabla ^2\mathbf{v}+\mathbf{f}
$$</p>
<p>I can easily see the viscous term as a diffusive one, but there's no way I can relate it with Fick's law. So, somebody can explain me how can I see viscosity as a diffusive process?</p>
<p>If you are interested in why I'm asking this, it's because in FEM there is a stabilization method called artificial viscosity that add some viscosity to increase diffusion and make the model more stable. So I understand a) why artificial viscosity increases the diffusion, b) why diffusion stabilizes the equation; but I don't understand why viscosity is diffusive (besides the fact that $\mu$ is multiplying $\nabla^2v$)</p> | 5,328 | [
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<p>Say you want to store hot coffee in a container surrounded by a vacuum. To remove all sources of conductive energy loss the container is suspended in the vacuum by a magnetic field and does not have a physical connection to the sides of the vacuum chamber,</p>
<p>My question is would the magnetic field be a path for energy to be conducted out of the suspended container?</p>
<p>Another way to look at this question would be two magnets are suspended in a vacuum with their poles aligned. A heat source is attached to one of the magnets. Would the second magnet show a corresponding increase in temperature, excluding radiated heat transfer?</p> | 5,329 | [
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<p>I would like to ask if someone knows the physical meaning of Boltzmann's definition of entropy. of course the formula is pretty straightforward </p>
<p>$$S=K_b\ln(Ω)$$</p>
<p>but what in the heck is the natural logarithm of micro states of the systems? I mean what does it express?</p>
<p>Maybe it is a dumb question but I am very curious to know the answer!</p>
<p>Thanks</p> | 5,330 | [
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<p>The background: I'm doing some simulation work involving the diffusion equation in 1D. Specifically I have some temperature profile, constant thermal conductivity and fixed temperature at each end of the system.</p>
<p>I know that we can write:</p>
<p>$$
\tau = \frac{L^2}{\kappa}
$$</p>
<p>where $\tau$ is the characteristic time scale, $L$ is the characteristic length scale and $\kappa$ is the thermal conductivity. In this case, $\kappa = 1$ so the time scale is equal to the square of the length scale.</p>
<p>I know that in a gas, the time scale corresponds to something like the amount of time it takes a particle to diffuse over the length scale of interest, but I'm not sure what it means in the context of temperature.</p>
<p>Could anyone enlighten me? Thanks!</p> | 5,331 | [
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] |
<p>I have a working knowledge of wave-particle duality, I think. I know the de Broglie wavelength is a sort of probability of finding a particle in a specific position, and is calculated by $\lambda=\frac{h}{\vec{p}}$. I have a couple questions I'm hoping to have cleared up, though.</p>
<p>First, since $\vec{p}$ is momentum, a vector, what happens to the direction in the above formula? Is it meant to be the magnitude of momentum instead? $\lambda$ having a direction doesn't seem to make sense.</p>
<p>Second, clearly if $\vec{v}=\vec{0}$, then $\vec{p}=m \vec{v}=\vec{0}$, and $\lambda$ is undefined. Does this mean particles that are stationary have no wave character? This also doesn't make sense, so I think this is not the case.</p> | 5,332 | [
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<p>I'd like to ask some questions about flipping two coins related to statistical mechanics, e.g. microcanonical distribution, phase space distribution function etc... after I rephrase the coin flipping problem into the language of statistical mechanics.</p>
<p>In probability theory, given the following problem</p>
<ul>
<li>Random experiment: Toss two coins</li>
<li>Example of an Outcome: $10 = (Heads, Tails)$</li>
<li>Sample space: $S = {11,10,01,00}$, $|S| = 4$</li>
<li>Examples of Events: 2 Heads $= 2H = \{11\}$, $|2H| = 1$, $1H = \{10,01\}$, $|1H| = 2$, $0H = \{00\}$, $|0H| = 1$ </li>
</ul>
<p>I'd like to translate this to statistical mechanics as much as possible, in which:</p>
<ul>
<li>A microstate is an element of the sample space (right?), e.g. $10$ or $01$.</li>
<li>A macrostate is an event (a subset of the sample space, right?), e.g. $1H = \{10,01\}$.</li>
<li>The statistical weight (statistical probability) of a macrostate is
the cardinality of the event, e.g. $|1H| = 2$.</li>
<li>The equilibrium distribution is the most likely macrostate which is
the macrostate with the highest statistical weight which is the event
with the highest cardinality, e.g. $1H = \{10,01\}$ since $|1H| = 2$.</li>
</ul>
<p>we can find the Maxwell-Boltzmann distribution function $n_i$ for this system by extremizing </p>
<p>$$w(n) = "number \ of \ heads \ in \ n" = \tfrac{2!}{n!(2-n)!}= \tfrac{2!}{n_1!n_2!}$$</p>
<p>with respect to $n_i$ given the constraint equation $n_1 + n_2 = 2$, showing $n_1 = e^0 = 1$ maximizes $w$, $w(1) = |1H| = |\{10,01\}| = 2$ is the maximum, thus the entropy $S = \ln(w) = \ln(2!)$ is at it's largest and the system is most disordered. </p>
<ol>
<li>What is the microcanonical distribution of a random experiment in
which you toss 2 coins?</li>
<li>What is the canonical distribution for this experiment?</li>
<li>Can I choose both, or is there an example of when I should use one
or the other, related to this example?</li>
<li>How do I find it's phase space distribution function $\rho = \rho(p,q)$?</li>
<li>Are these stupid questions? If they are, why? Is there a way to make
it so that I can derive the phase space distribution function? As far as I can see I think I'm supposed to add another Lagrange multiplier as a way to incorporate the energy, then after expressing $n$ in terms of the $\varepsilon _i$'s I can take a derivative of $w$ with respect to $E$ (so long as I replace $\varepsilon _1 = (E - n_2 \varepsilon _2)/n_1$ and $\varepsilon _2 = (E - n_1 \varepsilon _1)/n_2$ in the Maxwell-Boltzmann distribution that I'm plugging back in to $w$, but that seems crazy and seems to make no sense since if $E$ varies then shouldn't $\varepsilon _1$ and $\varepsilon _2$ also vary? This idea assumes they stay fixed...) to get $\frac{dw}{dE} = \int \delta [H - E]dpdq$, but it seems logically flawed as I've described and even if it worked how would you use it to determine $\rho$ inside the integral? I'm not even sure if this set up allows for the micro or just canonical distribution anyway, hence the question...</li>
</ol>
<p>Thanks!</p> | 5,333 | [
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<p>One of the most common gauges in QED computations are the $R_{\xi}$ gauges obtained by adding a term
\begin{equation}
-\frac{(\partial_\mu A^{\mu})^2}{2\xi}
\end{equation}
to the Lagrangian. Different choices of $\xi$ correspond to different gauges ($\xi=0$ is Landau, $\xi=1$ is Feynman etc.) The propagator for the gauge field is different depending on the choice of gauge. The choice of Landau gauge forces $\partial_\mu A^\mu=0$, but I have never seen a similiar statement for the other gauges. I would like to know what constraint on the gauge field is produced by the other covariant gauges. For instance, what is the constraint on $A_\mu$ when $\xi=1,2,3,...$ etc. Is it still $\partial_\mu A^\mu=0$ or something different (it seems like it should be different)?</p> | 5,334 | [
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<p>In Minkowski spacetime, two observers, A and B, are moving at uniform speeds u and v, respectively, along different trajectories, each parallel to the y-axis of some inertial frame S. Observer A emits a photon with frequency $\nu_{A}$ that travels in the x-direction in S and is received by observer B with frequency $\nu_B$. Show that the Doppler shift $\nu_B/\nu_A$ in the photon frequency is independent of whether A and B are travelling in the same direction or opposite directions. </p>
<p>Relevant equations: $$\lambda/\lambda' = \nu_B/\nu_A = \gamma(1-\beta\cos\theta)$$</p>
<p>Aberration formula: $\cos\theta' = (\cos\theta - \beta)/(1-\beta\cos\theta) = -\beta$<br>
(for transverse case)
The answer is apparently that the Doppler shift is independent of the relative direction of motion. I have tried to transform to the frame S' where B is stationary, finding the velocity of A using the addition of velocities formula - to then get gamma. I have used the abberation formula to insert $\cos\theta'$ into the Doppler shift formula above to get $\nu_B/\nu_A = \gamma(1+\beta^2)$. Plugging in the velocity of the emitter A in frame S' doesn't seem to get the required result. </p> | 5,335 | [
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<p>I am having some trouble understanding three-phase alternating current. I realize that most houses are not three-phase but single phase. Would that not mean that at some point when the flow of electricity switches direction it will stop and the electrical motor would stop under single phase? </p>
<p>Also, I also realize that most transmission lines are three-phase but again houses are single-phase. Does the three-phase transmission towers split off into three separate single-phase transmissions towers? </p>
<p>If anyone can provide me with any of this information it would be great. </p>
<p>--</p>
<p>Also is the wye or delta connection inside the alternator? Like the magnetic poles are the alternator symbols?</p>
<p><img src="http://upload.wikimedia.org/wikipedia/commons/thumb/4/48/3-phase_flow.gif/352px-3-phase_flow.gif" alt="picture"></p> | 5,336 | [
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<p>Hi all I'm a little confused </p>
<p>So I have that in special relativity time is included as a coordinate so that in 1 spatial dimension we have 2 space time coordinates. The most basic metric is the Minkowski metric given by $\left[ {\begin{array}{*{20}{c}}
{ - 1}&0 \\
0&1
\end{array}} \right]$ so now if I have a vector $\left[ {\begin{array}{*{20}{c}}
1 \\
1
\end{array}} \right]$ and I use Einstein Summation Convention with the metric I get $-1 + 1 =0$ ... so what does this tell me about the length squared of the vector.. why is the $0$ significant ? </p>
<p>EDIT: I found out that if the length squared = $0$ the interval is then light-like and light-like particles have world-lines confined to the light cone and the square
of the separation of any two points on a light-like world line is zero. Is this correct ?</p>
<p>Help is very much appreciated</p>
<p>Thank you =)</p> | 5,337 | [
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<p>I am trying to calculate the expectation value of an infinite quantum well in one dimension (L).</p>
<p>Given:<br>
$$\phi_n = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi}{L}x\right)$$ and $$E_n=\left[\frac{\hbar^2\pi^2}{2mL^2}\right]n^2$$</p>
<p>I am trying co calculate $\langle\phi_n|\hat{T}|\phi_n\rangle$, where $\hat{T}$ is the kinetic energy operator.</p>
<p>So far I have:<br>
$$\int_{-\infty}^\infty\sqrt{\frac{2}{L}}\sin\left(\frac{n\pi}{L}x\right)\left[\frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\sqrt{\frac{2}{L}}\sin\left(\frac{n\pi}{L}x\right)\right]dx$$</p>
<p>which I reduce to:<br>
$$\left(\frac{2}{L}\right)\left(\frac{\hbar^2}{2m}\right)\left(\frac{n\pi}{L}\right)^2\int_{-\infty}^\infty\sin^2\left(\frac{n\pi}{L}x\right)dx$$</p>
<p>Then I get stuck at this point. I'm not sure what to do with the integral as I'm pretty sure it diverges. What am I misunderstanding?</p> | 5,338 | [
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<p>This question probably verges on pseudo-science and probably sounds like gibberish, so please pardon me. But I'll ask it anyway.</p>
<p>In an ideal lab experiment there is generally a separation between the system to be studied and the apparatus (environment). The two are considered 'unentangled' before the decoherence occurs, right? </p>
<p>However, the environment in itself can also be considered an entangled system (like the previous system) to be observed by humans.</p>
<p>We could then naively conclude that our flesh and form constitutes the true observer.</p>
<p>But all our bodily substance is also coupled to the environment. So who then is the one truly causing the decoherence?</p>
<p>Separate all the entangled material from the human body, what do you get?</p>
<p>What physical structure within us constitutes the eternally unentangled and irreducible observer?</p>
<p>Perhaps my line of thinking is wrong, maybe the entity does not exist outside the Hilbert Space. Then is the the observer entangled with the observed? So would that not mean that the observer can cause his own decoherence?</p>
<p>It might seem like gibberish, but I really don't know what the present status on this question is int the physics community. If there are any papers on the matter please do provide the link. This question nags me all the time.</p> | 5,339 | [
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<p>In antenna technology we distinguish between nearfield and widefield. In the nearfield the electric and the magnetic fields are shifted by 90°. If you look closer you can see that there are two possibilities of this shift, 90° and -90°. </p>
<p>To explain it you have to remember what this 90° means. Let (in vacuum) the coordinate systems X axis be parallel to the E field, let the Y axis be parallel to the B field and the Z axis is parallel to c * t . In Z equal zero let the E field be maximum and directed in the X direction. The B field is zero. </p>
<p>90° later (in terms of E = E(max) * cos α and B = B(max) * sin α) and this is a quarter of the wavelength the B field can be directed to the left or to the right. And this is natural because B = B(max) * - cos α is the second possible state of the nearfield radio waves.</p>
<p>Regardless of the approach to see radio waves as one electromagnetic wave (statistical method) it is obvious that all radio waves are made from photons which are emitted during the electrons acceleration in the antenna rod.</p>
<p>My question is, do these photons all have the same sequence of the E and B fields? The same question appears for quantum dots which produce single photons.</p>
<p>Edit: There has to be a right or left hand rule because if half the photons have B-field with 90° to E and half have -90° then there wouldn't be magnetic field at all.</p>
<p>Update: I get it. It's the right hand grip rule (conventional direction of current) because there is no principal difference to a straight wire.</p> | 5,340 | [
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] |
<p>I'm not familiar with any complicated physics equation, however I do understand some basics. Suppose there is two objects, both of them are moving away from each other in a 3-dimensional space, which they both have the speed of half the speed of light ($c/2$). Relative to the reference frame of object A, object B would be moving away from itself equal to the speed of light.</p>
<pre><code> A B
<----- ----->
-c/2 c/2 to a 3rd observer
0 c to A
-c 0 to B
</code></pre>
<p>This seems to break the law of "nothing can move faster than the speed of light". I believe that there is some other explanation for this case. I have done some research, and there are quite a few questions on this topic. Take <a href="http://physics.stackexchange.com/q/7446/">this one</a> and <a href="http://physics.stackexchange.com/q/128161/">this one</a> for example. However, they are about adding velocities together, which isn't quite the same as in the case I was describing.</p>
<p><a href="http://physics.stackexchange.com/q/107820/">This question</a> is also similar to my case, however the two objects described are both moving toward the same direction, and I'm not shooting any beam or laser toward the other object.</p>
<p>Most probably in reality there are some extremely complex laws and equations which makes this question more complicated. However, the two objects are not interacting with each other and I'm not trying to "detect" the speed of the other object in the reference frame of A or B. Also none of the objects are moving at a significant fraction of the speed of light relative to the third observer, so special relativity doesn't apply(?) I just could not think of any reason why wouldn't B be moving away from A in the speed faster than light.</p>
<p>A simple explanation would be great. </p> | 5,341 | [
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<p>$$\vec{F}_\text{mag} = \frac{\mu_0 I_1 I_2}{4\pi } \oint\oint\frac{\mathrm{d}\vec{l}_1\times(\mathrm{d}\vec{l}_2\times\hat{r})}{r^2}$$</p>
<p>Is there any derivation for <a href="http://en.wikipedia.org/wiki/Amp%C3%A8re%27s_force_law" rel="nofollow">Ampère's force law</a> or is it just derived empirically?</p>
<p>If there is any derivation please include it in your answer.</p>
<p>But if it is derived empirically then Ampère must have made this formula from the following facts:</p>
<ol>
<li><p>$F_\text{mag} \propto I_1$</p></li>
<li><p>$F_\text{mag} \propto I_2$</p></li>
<li><p>$F_\text{mag} \propto \frac{1}{r^2}$</p></li>
<li><p>$F_\text{mag} \propto \oint\oint[\mathrm{d}\vec{l}_1\times(\mathrm{d}\vec{l}_2\times\hat{r})]$</p></li>
</ol>
<p>The first three can easily be seen experimentally:</p>
<ol>
<li><p>The current in the first circuit is directly proportional to magnetic force.</p></li>
<li><p>The current in the second circuit is directly proportional to magnetic force.</p></li>
<li><p>The square of the distance is inversely proportional to magnetic force.</p></li>
</ol>
<p>But what about the last one?</p>
<p>How did Ampere came to the conclusion that this complicated math stuff $\oint\oint[\mathrm{d}\vec{l}_1\times(\mathrm{d}\vec{l}_2\times\hat{r})]$ is directly proportional to magnetic force?</p>
<p>What if the magnetic force is larger for a smaller value of $\oint\oint[\mathrm{d}\vec{l}_1\times(\mathrm{d}\vec{l}_2\times\hat{r})]$ and the magnetic force is smaller for a larger value of $\oint\oint[\mathrm{d}\vec{l}_1\times(\mathrm{d}\vec{l}_2\times\hat{r})]$?</p> | 5,342 | [
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<p>Through various raids and acts of sabotage during WWII, the Allies succeeded in preventing Germany from coming into possession of large quantities of heavy water produced by the Norsk Hydro plant in Vemork Norway. These actions were carried out at considerable expense in terms of Allied troops and civilian casualties, presumably because the heavy water was perceived as a providing Germany with a substantial "leg up" in producing an atomic bomb.</p>
<p>But from my (admittedly limited) knowledge of the construction of Fat Man and Little Boy, heavy water was not a factor (at least not in the final weapons). Why was it believed that heavy water would make a German atomic bomb much more likely?</p>
<p>(The reason I ask is that I recently saw an old (1965) movie ("The Heroes of Telemark") about the sabotage operations. While a bit too "Hollywood" (with Kirk Douglas and Richard Harris), it was apparently reasonably accurate. And several years back I saw something on PBS about the sinking of the ferry carrying the heavy water.)</p> | 5,343 | [
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<p>I am a chemist with a passion for astrophysics and particle physics, and one of the most marvellous things I have learned in my life is the process of stellar nucleosynthesis. It saddens me how my colleagues point to the periodic table and draw compound structures so often, yet never seem to be curious about where the elements on which they rely came from, and completely miss the beauty underneath.</p>
<p>As far as I understand, there are several types of nucleosynthesis conditions (big bang, stellar, supernova, black hole accretion disk, artificial, pycnonuclear, and perhaps others), the first three of which are most important when discussing the composition of the Universe, but I wish to focus on an aspect of big bang nucleosynthesis.</p>
<p>The Universe became extremely hot after inflation (or so it is thought), and while cooling down there was a very brief period in which the temperatures and densities were adequate for protons and neutrons to exist and fuse into heavier elements. However, this period of nucleosynthesis was so brief that most of the nucleons formed in baryogenesis didn't even have time to fuse, ending up as hydrogen. Of the amount that did manage to fuse, almost all of it stopped at helium, helped by <a href="http://en.wikipedia.org/wiki/File%3aBinding_energy_curve_-_common_isotopes.svg">the exceptional stability of the $^4He$ nucleus</a> relative to its neighbours. Only a tiny amount of lithium is said to have been made, and I've rarely heard anyone discussing elements heavier than that. It is often said that "metals" (in the astronomical sense) only really appeared after stellar nucleosynthesis began, but there was already <em>a lot</em> of matter around after baryogenesis ended (about the same amount as the $10^{80}$ baryons present now?), so even the tiniest atomic fraction of an element could correspond to galactic masses' worth of it.</p>
<p>It would be really neat if it turned out that carbon, the very stuff of life as we know it, didn't exist <em>at all</em> until <em>over a hundred million years after the big bang</em>, when the first generation of stars began to light up. However, I don't know if this is the case, as confirmation or refutation relies on precise quantitative modelling of the big bang nucleosynthesis. Is it possible to figure out when the first carbon atoms came into existence? I tried using a search engine, but my specific query is completely drowned in links covering the more general aspects of nucleosynthesis.</p>
<p>Note: I suppose that if one were to consider the matter outside the observable Universe, then there probably were carbon atoms even the tiniest timestep after nucleosynthesis began, because the sheer ridiculous amount of space expected to be out there could well overwhelm the nigh-impossible odds of carbon formation. Thus, the calculation is more interesting if limited to our observable volume.</p> | 5,344 | [
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<p>When dealing with fluid mechanics of viscous fluids, both theoretically and numerically, I've always been told that the boundary condition applied at solid walls has to be a no-slip one. My teachers or textbooks never really explained why, except sometimes "well, the fluid's viscous, so it sticks to solid" which is far from an explanation to me. Therefore I'm reading a little bit to understand the real origin of this condition, and so far there is on thing that I don't understand in what I've found: in Volume II of <em>Modern Developments in Fluid Dynamics</em> by S. Goldstein, it is written that:</p>
<blockquote>
<p>"[...]; finally he [Navier] decided on the first [hypothesis on the behaviour of a fluid near a solid body], on the grounds that <strong>the existence of slip would imply that the friction between solid and fluid was of a different nature from, and infinitely less than, the friction between two layers of fluid</strong>, and also that the agreement with observation of results obtained on the assumption of no slip was highly satisfactory."</p>
</blockquote>
<p>I do not understand what is in bold: what does the "different nature" of friction means, and why would it be "infinitely less"?</p>
<p>NB: After this statement, Goldstein refers to a bibliographic entry, but I don't know if it's possible to find such archive on the Internet. Here it is anyway: <em>Trans. Camb. Phil. Soc. <strong>8</strong> (1845), 299, 300; Math and Phys. Papers, <strong>3</strong>, 14, 15.</em></p> | 5,345 | [
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<p>Today I have by accident thrown a AAA battery into a bucket of water. I fished it out of the water immediately (within 20 seconds or so) and nothing notable had happened and the battery is still full according to a battery test device. As the water should have short circuited the battery I would have expected that something should have happened, at least that the battery should have been emptied rather quickly.</p>
<p>Is my expectation wrong, is there something that I didn't think of or was the time the battery spent in water was just too short to substantially drain it?</p> | 5,346 | [
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<p>The first working point-contact transistor made in 1947 by Bell Labs. I'm looking for specific dimensions, all I've been able to find is "Fits in the palm of your hand".</p> | 5,347 | [
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<p>To check the correlation between Hidden Variable Theory and Quantum Mechanics, Bell calculated the expectation value </p>
<p>$<\sigma_{e}(\vec a,\vec V) \sigma_{p}(\vec b,\vec V)> = \int d^n V \rho(\vec V) \sigma_{e}(\vec a,\vec V) \sigma_{p}(\vec b,\vec V)$</p>
<p>Here I am assuming that "Alice" is measuring the spin of an electron e along $\vec a$ and "Bob" is measuring the spin of the positron p along $\vec b$. Then $\sigma_{e}(\vec a,\vec V)$ and $\sigma_{p}(\vec b,\vec V)$ are the resulting spin values ($\pm \frac{1}{2}$) of the electron and positron, respectively. The vector $\vec V$ is an n-dimensional vector containing the hidden variables and $\rho(\vec V)$ is a probability distribution for the hidden variables.</p>
<p>But does this not assume QM is probabilistic? I thought Einstein disagreed with the probabilistc nature of Quantum Mechanics (God does not throw dice). </p> | 5,348 | [
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<p>It seems that water generally dampens sound waves. Is there any way one could attach a speaker to a body of water in such a way that the water would actually amplify some frequencies (for nearby listeners in the air, not under water)?</p>
<p>Imagine a speaker on a boat on a small creek. Listeners are on the bank.</p> | 5,349 | [
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<p>Say you have a charged particle in a region that contains a fluid that will produce a drag force that goes as $F=-kv$ where $v$ is the speed and $k$ is some constant. The region also contains a uniform magnetic field.
Suppose you give the particle some initial velocity $v_0$ in the plane perpendicular to the magnetic field. What will be the particle's subsequent motion? Please provide semi-quantitative answers. Note that this is not a homework question.</p> | 5,350 | [
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<p>I asked on <a href="http://math.stackexchange.com/q/306253/11127">Math.SE</a> and was advised to try here instead.</p>
<p>I need to draw from a Maxwell-Boltzmann velocity distribution to initialise a molecular dynamics simulation. I have the <a href="http://en.wikipedia.org/wiki/Probability_density_function" rel="nofollow">PDF</a> but I'm having difficulty finding the correct <a href="http://en.wikipedia.org/wiki/Cumulative_distribution_function" rel="nofollow">CDF</a> so that I can make random draws from it.</p>
<p>The PDF I am using using is:</p>
<p>$$f(v)=\sqrt \frac{m}{2\pi kT} \times exp \left( \frac{-mv^2}{2kT}\right) $$</p>
<p>I am told that to find the CDF from the PDF we perform:</p>
<p>$$CDF(x)= \int_{-\infty}^x PDF(x) dx $$</p>
<p>After integrating $ f(v) $ I get:</p>
<p>$$ CDF(v)= \sqrt \frac{m}{2\pi kT} \times \left( \frac{\sqrt\pi\times erf \left( \frac{mv}{2\pi kT} \right) }{2\times \left( \frac{m}{2kT} \right)} \right) $$</p>
<p>$$CDF(v)= _{-\infty} ^{x} \left[ {\sqrt \frac{m}{2\pi kT} \times \left( \frac{\sqrt\pi\times erf \left( \frac{mv}{2\pi kT} \right) }{2\times \left( \frac{m}{2kT} \right)} \right)} \right] $$</p>
<ol>
<li><p>After I reach this point I am unable to proceed as I do not know how to evaluate something between $x$ and ${-\infty}$.</p></li>
<li><p>I am also concerned that I have not done the integration correctly.</p></li>
<li><p>I want to implement the CDF in C++ in the end so I can draw from it. Does anyone know if there will be a problem with doing this because of the erf, or will I be alright with <a href="http://www.gnu.org/software/gsl/manual/html_node/Error-Function.html" rel="nofollow" title="this">this GSL implimentation</a> ?</p></li>
</ol>
<p>Thanks for your time.</p>
<p>@bryansis2010 on Math.SE says that I can evaluate in the range $x$ to $0$ instead of $-\infty$ as we do not drop below 0 Kelvin.</p>
<p>Would this then make the CDF:</p>
<p>$$ CDF(v)= \sqrt \frac{m}{2\pi kT} \times \left( \frac{\sqrt\pi\times erf \left( \frac{mv}{2\pi kT} \right) }{2\times \left( \frac{m}{2kT} \right)} \right) $$</p>
<p>as $erf(0)=0$</p>
<p>Is this correct?</p> | 5,351 | [
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<p>I see there are different accounts on the number of physical constants. Is there any theory predicting what the number of physical constants is? I agree there should be at least one but I have no justification why there should be more than one constant. </p> | 5,352 | [
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<p>I have always been confused by the relationship between the <a href="https://en.wikipedia.org/wiki/Theoretical_and_experimental_justification_for_the_Schr%C3%B6dinger_equation" rel="nofollow">Schrödinger equation</a> and the <a href="https://en.wikipedia.org/wiki/Electromagnetic_wave_equation" rel="nofollow">wave equation</a>.</p>
<p>$$ i\hbar \frac{\partial \psi}{\partial t} = - \frac{\hbar^2}{2m} \nabla^2+ U \psi \hspace{0.25in}\text{-vs-}\hspace{0.25in}\nabla^2 E = \frac{1}{c^2}\frac{\partial^2 E}{\partial^2 t} $$</p>
<p>Because of the first derivative, the Schrödinger equation looks more like the <a href="https://en.wikipedia.org/wiki/Heat_equation" rel="nofollow">heat equation</a>. </p>
<p>Some derivations of the Schrodinger equation start from <a href="https://en.wikipedia.org/wiki/Wave%E2%80%93particle_duality" rel="nofollow">wave-particle duality</a> for light and argue that matter should also exhibit this phenomenon. </p>
<p>In some notes by <a href="http://rads.stackoverflow.com/amzn/click/0226243818" rel="nofollow">Fermi</a>, it was derived by comparing the <a href="https://en.wikipedia.org/wiki/Fermat%27s_principle" rel="nofollow">Fermat least time principle</a> $\delta \int n \;ds = 0 $ and Maupertuis <a href="https://en.wikipedia.org/wiki/Principle_of_least_action" rel="nofollow">least action principle</a> $\delta \int 2T(t) \; dt = 0 $.</p>
<p>Was this ever clarified? How can we see the idea of a matter-wave more quantitatively?</p>
<hr>
<p>To summarize, I am trying to understand why the Electromagnetic wave equation is hyperbolic while the Schrodinger equation is parabolic.</p> | 188 | [
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<p>I am having trouble understanding how to solve some theoretical physics problems I have come across. Specifically how to convert the Hamilton-Jacobi equation:
$$(\partial_\mu S+e A_\mu)^2=m^2$$
From Minkowski-space notation to conventional form:
$$(\frac{\partial S}{\partial t}+\frac{\alpha}{r})^2-(\nabla S)^2=m^2$$
where $A_0=-\frac{\alpha}{er})$ and $A_x=A_y=A_z=0$
Any help would be appreciated.</p> | 5,353 | [
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<p>In an experiment where the type of metal,intensity of light and potential difference across a battery is kept constant at 2V the results show that an increase in wavelength, obviously in turn decreases the frequency, causes the current of the circuit to decrease (eventually to 0A).
what would be the direct cause for the current to become zero? i understand that with the increased wavelength there will be less energy supplied to the metal surface causing the emitted electrons to decrease but does this lack of energy cause the current to become zero?</p> | 5,354 | [
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<p>If you want to fly a spaceship with human passengers as close to the Sun as possible, then what effects would the spaceship have to be designed to counteract in order to keep the passengers alive and how close to the Sun could you get before there would be no way to counteract the effects ?</p> | 5,355 | [
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<p>The footnote (bottom) for the following paragraph from “Creative Tension: Essays On Science & Religion” by Michael Heller, has “<em>such a projection, being always “onto,” switches off all possible “short distance correlations</em>.” I’ve looked at the references and cannot see how this “switching off” process occurs. Is it possible to provide a less-mathematical explanation for this putative “switching off” process?</p>
<hr>
<p>"All these nonlocal phenomena find their natural explanation within the noncommutative approach. Because the fundamental level is totally nonlocal, it is no wonder that some quantum phenomena (such as the EPR type of experiment) that are rooted in that level exhibit some nonlocal effects; they are but the tip of the iceberg of the “fundamental noncommutativity” that somehow survived the “phase transition” to the usual physics.22”</p>
<p>“22. To see why certain nonlocal effects survive the Planck threshold and can be observed, one should look at some mathematical properties of the noncommutative regime. Generally speaking, the transition from the noncommutative level to the ordinary physics in our model has the character of a projection. <em>Such a projection, being always “onto,” switches off all possible “short distance correlations,</em>” but can leave “long-distance correlations.” See, for instance, M. Heller and W. Sasin, “Einstein-Podolski-Rosen Experiment from Noncommutative Quantum Gravity,” <a href="http://arxiv.org/abs/gr-qc/9806011" rel="nofollow">http://arxiv.org/abs/gr-qc/9806011</a> or M. Heller and W. Sasin, “Noncommutative Unification of General Relativity and Quantum Mechanics,” <a href="http://arxiv.org/abs/gr-qc/0504014" rel="nofollow">http://arxiv.org/abs/gr-qc/0504014</a></p> | 5,356 | [
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<p>Suppose I have two rigid bodies A and B and they are connected by a spring which is attached off-center (thus possibly causing torques). Due to the spring a force $f$ acts on A and a force $-f$ acts on B (at the respective attachment points) in direction of the spring as in Fig. 1. How can I show the conservation of momentum? $\frac{\rm d}{\rm dt} p_A + p_B = 0$ (where $p_A$ and $p_B$ are the linear momenta of A and B respectively) is missing the angular part and $\frac{\rm d}{\rm dt} p_A + p_B + L_A + L_B = 0$ (where $L_A$ and $L_B$ are the angular momenta of A and B around their center of masses respectively) seems to be wrong. Is $\frac{\rm d}{\rm dt} p_A + p_B + L_A^0 + L_B^0 = 0$ (where $L_A^0$ and $L_B^0$ are the angular momenta of A and B around the origin respectively) the correct ansatz?</p>
<p>What if the forces are opposite but not in the direction of the spring as in Fig. 2?</p>
<hr>
<p><img src="http://i.stack.imgur.com/V602Y.png" alt="Fig. 1: Opposite forces along the line between the points where the forces act."></p>
<p>Fig. 1: Opposite forces along the line between the points where the forces act.</p>
<hr>
<p><img src="http://i.stack.imgur.com/zhg5j.png" alt="Fig. 2: Opposite forces but *not* along the line between the points where the forces act."></p>
<p>Fig. 2: Opposite forces but <em>not</em> along the line between the points where the forces act.</p> | 5,357 | [
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<p>I met some problem about the Virasoro operator in "old covariant quantization" in Polchinski's string theory vol I p 123.</p>
<p>It is given
$$L_0^{\rm m}=\alpha' p^2 + \alpha_{-1} \cdot \alpha_1 + \cdots \tag{4.1.11a} $$
But on p 59,
$$ L_0 = \frac{ \alpha' p^2 }{4} + \sum_{n=1}^{\infty} ( \alpha^{\mu}_{-n} \alpha_{\mu n} ) + a^X \tag{2.7.7} $$</p>
<p>Why there is a factor of $1/4$ difference in the first terms in Eqs. (2.7.7) and (4.1.11a)? Is that just a convention?</p> | 5,358 | [
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<p>I am confused with adiabatic expansions. I have a homework problem wherein 2.75 moles of an ideal gas at 375 K expands adiabatically with an initial pressure of 4.75 bar and a final pressure of 1.00 bar with a C(p,m)=5R/2. I need to calculate the work done by the expansion. </p>
<p>Where do I start (I don't want just the answer, as I could have just copied that from the back of the book)?</p> | 5,359 | [
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<p>Basically, I would really like it if somebody just explained to me what is going on here. Please use any physics lingo you feel is necessary, but explain what you mean. I am just having trouble finding a nice explanation without having to know tons of physics. ( I have a fifth grade physics level)</p>
<p>So the reference I have states that the symmetric stress-energy tensor is given by $T=a_1(dx)^2+2a_2dxdy+a_3dy^2$, which I understand as being a section of $S^2(T^{\ast}\mathbb{R}^2)$, the 2nd-symmetric power of the cotangent space to $\mathbb{R}^2$. Why does this have anything to do with energy or momentum?</p>
<p>If we change to $\mathbb{C}$ rather than $\mathbb{R}^2$, then $T=(a_1-a_3-2ia_2)\,dz^2+2(a_1+a_3) dz d\bar{z}$. Conformal invariance is implied by the condition $\mbox{tr}\,T=a_1+a_3=0$. (What does this sentence mean?)</p>
<p>Lastly, we have some kind of expansion (in a conformal field theory with central charge $c=0$, whatever that means): $$T(z)=\sum_{n\in\mathbb{Z}}L_n z^{-n}\, \big(\frac{dz}{z}\big)^2,$$ where I take it that the $L_n$ are the generators for the centerless Virasoro algebra. What type of object is $T(z)$? And how is obtained from $T$?.</p> | 5,360 | [
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<p>I was doing some exercises the other day, when I came across this question in my book:</p>
<blockquote>
<p><em>A proton weighs about 1.66 x 10<sup>-24</sup> g and has a diameter of
about 10<sup>-15</sup> m. What is its density in g/cm<sup>3</sup>?</em></p>
</blockquote>
<p>As you can see, a really simple and standard question, but... does it even make sense to say that a proton has a <em>mass</em>? And calculate its density?! If so, do I consider it a sphere?</p>
<p>It seems to me, subatomic particles are so small it doesn't really make sense to talk about <em>mass</em>, <em>volume</em> and <em>density</em> as if we were talking about... tennis balls!</p> | 5,361 | [
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<p>In biology, it is stated that more surface area and less volume helps for keeping bodies cool. How can one explain this phenomena in terms of physics?</p> | 5,362 | [
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<p>I hope this question is simple and can be quickly cleared up.</p>
<p>In a 1D conservative dynamical system, I've always been taught that the potential function is the function $V(x)$ such that:</p>
<p>$$F=-\frac{dV}{dx}$$</p>
<p>That makes sense to me, simply derived from the definitions of work and conservation of energy. </p>
<p>However, just reading through the book 'Nonlinear dynamics and chaos' by Steven Strogatz, in the first chapter he defines the potential for the most basic 1D system:</p>
<p>$$\dot{x} = f(x)$$</p>
<p>To be</p>
<p>$$f(x)=-\frac{dV}{dx}$$</p>
<p>What is happening here??? Usually, the value $-\frac{dV}{dx}$ is proportional to $\ddot{x}$ (because, with constant mass, $F$ is proportional to $\ddot{x}$) but now this value is equal to $\dot{x}$. is this an alternative definition of potential? I don't think the 2 definitions are equivalent.</p>
<p>Thanks!</p> | 5,363 | [
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<p>Recently I have discovered the method of constructing of GR from massless field with helicity 2 theory. It is considered <a href="http://arxiv.org/abs/gr-qc/0411023" rel="nofollow">here</a>, in an article "Self-Interaction and Gauge Invariance" written by Deser S. </p>
<p>By the few words, the idea of the method is following. When starting from massless equations for field with helicity 2 we note that it doesn't provide stress-energy momentum tensor conservation:</p>
<p>$$
\tag 1 G_{\mu \nu}(\partial h , \partial^{2}h) = T_{\mu \nu} \Rightarrow \partial^{\mu}T_{\mu \nu} \neq 0
$$
(here $h_{\mu \nu}$ is symmetric tensor field).</p>
<p>But we may change this situation by adding some tensor to the left side of equation which provides stress-tensor conservation:
$$
G \to \tilde {G}: \partial \tilde{G} = 0.
$$</p>
<p>Deser says that it might be done by modifying the action which gives $(2)$ by the following way:
$$
\eta_{\mu \nu} \to \psi_{\mu \nu}, \quad \partial \Gamma \to D \Gamma .
$$
Here $\eta $ is just Minkowski spacetime metric, $\Gamma$ is the Christoffel symbol with respect to $h$, $\psi^{\mu \nu}$ is some fictive field without geometrical interpretation, and $\partial \to D$ means replacing usual derivative to a covariant one with respect to $\psi $ (this means appearance of Christoffel symbols $C^{\alpha}_{\beta \gamma}$ in terms of $\psi$). By varying an action on $\psi$ we can get the expression for correction of $(1)$ which leads to stress-energy conservation.</p>
<p>Here is the question: I don't understand the idea of this method. Why do we admit that we must to introduce some fictive field $\psi_{\mu \nu}$ for providing conservation law? Why do we replace partial derivatives by covariant ones using $\psi $? How to "guess" this substitution? I don't understand the explanation given in an article.</p> | 5,364 | [
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<p>Now this might be a silly question but it's actually bugging me, this one might be easier to understand if you have kids that watch (or used to watch) <a href="http://en.wikipedia.org/wiki/Peppa_Pig">Peppa Pig</a>. In one of the episodes, about shadows, the kids try to run away from their shadows, they try to move faster and faster and of course failing. Then comes in Mr Elephant saying something along the lines</p>
<blockquote>
<p>It doesn't matter how fast you go, you can't run away from your shadow</p>
</blockquote>
<p>At this point I would just like to say that I'm aware that this is a kids cartoon and all but in theory: is it possible?</p>
<p>For example if you would be traveling at a higher than speed of light (without anything in your pathway and with a constant source of light) would the position of the shadow be offset or moving away from ones current position? Or do elephants actually know a thing or two about physics?</p> | 993 | [
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<p>Say I use the metric signature $(-+++)$. Then
$\partial_a=(\partial_0,\partial_i)=(-\partial^0,\partial^i)$,
but
$\partial^a=(\partial^0,\partial^i)=(-\partial_0,\partial_i)$.</p>
<p>The same goes for $p^a$ and $p_a$, I suppose. I know that we need to contract, say, $x^a$ with $p_a$ to give $-x^0p_0+x^ip_i$. So my question is: Does this convention of whether to put ``$-$'' on quantities with superscripts or subscripts apply to all quantities? $x^a$ and $x_a$? $A^a$ and $A_a$?</p> | 5,365 | [
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<p>First of all, what is the mathematical relationship between measured linewidth (usually in units of magnetic field) and spin relaxation time? I see papers talk about spin relaxation times in terms of linewidths but I have no idea how to correlate the size of the linewidth to an actual time. </p>
<p>Second, is the measured linewidth in NMR or ESR experiments related to $T_2^{\ast}$ only. If you want just $T_2$, you would have to do spin-echo, right? If that is the case, how is $T_1$ determined?</p> | 5,366 | [
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<p>It is stated that "the formula for the energy stored in the magnetic field is: $$E = \left(\frac{1}{2}\right)(LI)^2$$ and the energy stored in the magnetic field is equal to the <em>work</em> done to produce the current.</p>
<p>What is the formula for <strong><em>work</em></strong> , that shows they are equal?
I understand they are equal due to conservation of energy, but I'd like to see the mathematical operation to support my understanding. </p> | 5,367 | [
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<p>My question is the following:</p>
<p><em>What is the shape of the rope which holds a kite flying?</em> (Steady state.)</p>
<p>I am not a physicist (I am a mathematician), so I can not work the physics part of the question. </p>
<p>My guess is that it should be the same as catenary (hyperbolic cosine) </p> | 5,368 | [
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<p>I am currently a first year Ph.D. student in Applied Math. As an undergrad, I did not have the opportunity to take a calculus based physics course and the one I took was a general physics course where emphasis was on the concepts (most of which I already forgot).</p>
<p>Given my mathematical background, I honestly think that I would've understood and appreciated physics more had it been taught to me from a calculus perspective. Now I am still unsure about my research interests but I definitely want to work on applications of pde, statistics, and probability (all 3) in engineering applications. Right now, I am hoping to work in stochastic differential equations or maybe stochastic inverse problems, totally unsure but I am exploring my options!</p>
<p>That said, I am taking graduate level courses in the math fields relevant to my interests. But because a lot of courses is required to learn pdes/probability, I do not have time to take courses in engineering or at least a physics course! So during my winter break, I am hoping to read up on a calculus based physics book to enhance my understanding of physics as well as to be able to have a grasp of the engineering applications I will be working on in the future. At the moment, the book I have is University Physics by Sears, Zemansky, Young, Freedman, and Ford.</p>
<p>Now I am primarily looking at civil engineering applications, specifically in earthquake engineering and materials science involved in structures. I am also open to fluid mechanics. For those unfamiliar, this book is divided into major fields in physics: Mechanics, Waves/Acoustics, Thermodynamics, Electromagnetism, Optics, and Modern Physics. Which fields should I focus on and which ones do you recommend I should skip? Honestly, I feel that electromagnetism is not relevant but I am unsure about the others. </p>
<p>Any suggestions? I'd greatly appreciate it! Thanks for the help.</p> | 5,369 | [
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<p>Is there a way to find a solution for the positions (as a function of time) of multiple free bodies that push and pull on each other?</p>
<p>Say for instance I have a collection of cells which can individually contract or expand, and are joined to one another. For simplicity, I'm assuming two dimensions; no gravity, friction, or other external forces; and the cells are circular, and therefore connect at a point. When a cell contracts or expands, it simply reduces or increases its radius, respectively. When it contracts, it pulls on any cells that are attached to it. When it expands, it pushes on any attached cells, and in both cases it does so with a particular force. The mass of each cell is known.</p>
<p>I suspect this may be related to an N-body problem (perhaps even more difficult) and therefore cannot be feasibly solved exactly. If that's the case, can anyone suggest any approach for reasonably approximating this system?</p> | 5,370 | [
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<p>We are looking at White Dwarfs in Quantum and specifically how they are stabilized between gravitation pressure inwards balanced by the Fermi pressure due to electrons. </p>
<blockquote>
<p>To start the problem off we are asked to calculate the inward pressure due to gravity in terms of the total gravitational energy. We are examining a white dwarf with the mass of our Sun (ignoring where this is probable/possible). </p>
</blockquote>
<p>If we take the total gravitation energy to be $U_g$ then I figured $U_g=PV$ and that the inward pressure due to gravity would be $P=\frac{U_g}{V}$ where $V={4 \over 3}\pi r_{sun}^3$.</p>
<p>However when I use this later on to calculate the radius of a white dwarf with mass equal to our sun I get an obscenely small number. While I understand the densities of white dwarfs are enormous I think I made a mistake in the inward pressure due to gravity part. Should it be that $V={4\over 3}\pi K f^3$ and then...? Not sure where to go or what I did wrong. Ideas?</p> | 5,371 | [
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<p>There are plenty of formulas that use gravity acceleration of Earth. This is represented with the symbol $g$. In my school work (I am a high school student) we usually take it as $g= 9,8 \,\text m/\text s^2$. </p>
<p>This thing is obviously a number that is only usable on Earth. What I want to know is that, what if I want to make my calculations according to another planet? How the number is going to change? </p> | 5,372 | [
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<p>Q3 from a mechanics exam past paper:<img src="http://i.stack.imgur.com/95nD6.png" alt="enter image description here"></p>
<p>I can do parts i) and ii) but for iii) in finding the angular acceleration, i used $C=I\alpha$, where $C$ is the applied couple or torque, $I$ is the moment of inertia for the lamina about A and $\alpha$ is the angular acceleration.</p>
<p>At the instant the object is released the only force acting on it is its own weight. Hence, $6g*0.8=9\alpha$ which yields $\alpha=5.227$. </p>
<p>However the mark-scheme says the answer is $\alpha=1.65$, as they have taken into account the frictional couple. Here is the mark-scheme:<img src="http://i.stack.imgur.com/l1s2i.png" alt="enter image description here"></p>
<p><strong>It was my understanding that at the moment of release no friction <em>(and hence no frictional couple)</em> can act as there is no movement (yet).</strong> </p>
<p>So could someone please kindly explain what the mark-scheme is talking about?</p> | 5,373 | [
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<p>I am a programmer and I want to pursue the Game Design field. From talking to my teachers about it, they said that physics plays a major role. My Question is: How does physics transfer to a virtual computer based environment. Is it fundamentally the same or can you create a world that defies our rules.</p> | 5,374 | [
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<p>I am reading section 3.2.1 of Griffiths 3ed which explains how to calculate potential using first uniqueness theorem.</p>
<p>Griffiths/3.2.1
<img src="https://i.imgur.com/hTmCHFX.png" alt=""></p>
<p>Griffiths/First Uniqeness theorem (Its corollary actually)</p>
<h2><img src="https://i.imgur.com/JlGMUEl.png" alt=""></h2>
<ol>
<li>As I understand we are taking volume $V = R^3$ , $\Phi (\vec{r})=0\ $on the boundary of $V$. But what does it mean by "boundary of $R^3$" ?</li>
<li>Since $\ \rho_1(\vec{r}) \ne \rho_2(\vec{r})$ which does not satisfy the prequisites of the theorem, we cannot use the theorem. Moreover no other distribution can be used which makes this theorem useless which it is not. So what am I missing here ?</li>
<li>Does not single charge distribution also satisfies the same boundary condititons ? So should not $\Phi$ also be $\frac{1}{4\pi \epsilon_0}\frac{q}{R}$ ?</li>
</ol> | 5,375 | [
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