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<p>I know that the maximum shear stress </p>
<p>$$\tau = \frac{T}{J}\rho$$</p>
<p>where $\rho$ is the radial distance from the center of the cross section. I have also determined the torsional constant $J$, which is equal to $\frac{\pi}{2}(R_o^4-R_i^4)$ for this particular cross section. The following boundary condition applies: </p>
<p>$$\nabla^2\phi = F$$ </p>
<p>I know that $\phi$ would be zero for a solid cross section, but I do not understand how I should calculate it with torsion equivalence. Any help is appreciated.</p> | 1,590 | [
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<p>Frequently when trying to solve cosmology questions physicists turn to computer simulations of the universe (albeit massively simplified) in order to verify or disprove their hypotheses. This got me thinking.</p>
<p>My question is about the theoretical maximum possible complexity of these systems.</p>
<p>Let me give an example, if we imagine a tennis ball bouncing on a flat surface if we want to accurately simulate and measure the results of every single facet of the collision right down to the atomic and quantum effects you could actually find a tennis ball and drop it over your surface. In this case the universe is "simulating" the collision for you.</p>
<p>Would it be possible to simulate this same event just as accurately using a computer? Is there a theoretical reason why the computer would need to have more mass than the two colliding objects? (in this case a tennis ball and the planet!)</p>
<p>Now I have always assumed that the answer to this question is "yes you need a more massive computer to simulate any object with total physical accuracy" because if that were not the case there would be no reason why a computer less massive than the universe could not simulate the entire universe with total accuracy, which seems to me to be counterintuitive.</p>
<p>I hope I was clear with my question! I would like to find a hard and fast answer to this question with a reason preferably, been mulling it over for a little while now. Thanks for any help.</p> | 1,591 | [
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<p>I'm reading through Novotny and Hecht's book on Nano-Optics (Principles of Nano-Optics), and I've come across a subtlety in the boundary conditions for evanescent wave generation (via total internal reflection) that I don't understand.</p>
<p>In the 2nd edition of the book the equation I'm struggling with is labelled 2.104, but exactly the same content is available online <a href="http://www.photonics.ethz.ch/fileadmin/user_upload/optics/Courses/NanoOptics/foundations.pdf" rel="nofollow">here</a> where it's labelled 2.91.</p>
<p>My question is this: where does the negative sign come from in the x component of the transmitted field $\textbf{E}_2$? I understand that the parallel components of the electric field must be conserved across the boundary, but where does the factor of $-1$ arise?</p>
<p>Many thanks!</p> | 1,592 | [
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<p>As far as I understand it the photon propagator, $P(A\rightarrow B)$, described in Feynman's QED book, gives the amplitude that a photon moves from spacetime point A to spacetime point B.</p>
<p>I was wondering if in quantum field theory terms $P(A\rightarrow B)$ is made up of the product of the following two amplitudes:</p>
<ol>
<li>the amplitude that a photon is created at $B$ given that there is a photon at $A$.</li>
<li>the amplitude that a photon is annihilated at $A$ given that there is a photon at $B$.</li>
</ol>
<p>Is this the correct approach to describing a photon moving from $A$ to $B$ using creation and annihilation operators?</p> | 1,593 | [
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] |
<p>Although I have informed myself in string theory through reading books and watching videos, I do not know or understand what superstrings are made of. History has definitely shown us that when we think something is fundemental, there is always something smaller.</p>
<p>What are superstrings made of?
Can we go smaller than strings?</p> | 24 | [
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<p>2+1D lattice gauge theory can emerge in a spin system through fractionalization. Usually if the gauge structure is broken down to $\mathbb{Z}_N$, it is believed that the fractionalized spinons are deconfined. However in general, $\mathbb{Z}_N$ gauge theory also have a confined phase. The question is how to determine if the discrete emergent gauge theory is really deconfined or not?</p>
<p>For example, I am considering a $\mathbb{Z}_3$ gauge-Higgs model defined on the Kagome lattice with the Hamiltonian $H=J\sum_{\langle i j\rangle}\cos(\theta_i-\theta_j-A_{ij})$, where $\theta_i=0,\pm2\pi/3$ is the matter field and $A_{ij}=0,\pm2\pi/3$ is the gauge field. If the matter field is in a ferromagnetic phase, then I can understand that the gauge field will be Higgs out. But the matter field here is a Kagome antiferromagnet, which is strongly frustrated and may not order at low temperature. So in this case, I would suspect that the effective $\mathbb{Z}_3$ gauge theory will be driven into a confined phase. Is my conjecture right? How to prove or disprove that?</p>
<p>Thanks in advance.</p> | 1,594 | [
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<p>I read that the field that have simultaneously minimum divergence and minimal transversal extension is the fundamental Gaussian mode. What are the good references in this subject, to prove the assertion above?</p> | 1,595 | [
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<p>I know how to derive below equations found on <a href="http://en.wikipedia.org/wiki/Creation_and_annihilation_operators#cite_note-4" rel="nofollow">wikipedia</a> and have done it myselt too: </p>
<p>\begin{align}
\hat{H} &= \hbar \omega \left(\hat{a}^\dagger\hat{a} + \frac{1}{2}\right)\\
\hat{H} &= \hbar \omega \left(\hat{a}\hat{a}^\dagger - \frac{1}{2}\right)\\
\end{align} </p>
<p>where $\hat{a}=\tfrac{1}{\sqrt{2}} \left(\hat{P} - i \hat{X}\right)$ is a annihilation operator and $\hat{a}^\dagger=\tfrac{1}{\sqrt{2}} \left(\hat{P} + i \hat{X}\right)$ a creation operator. Let me write also that:</p>
<p>\begin{align}
\hat{P}&= \frac{1}{p_0}\hat{p} = -\frac{i\hbar}{\sqrt{\hbar m \omega}} \frac{d}{dx}\\
\hat{X}&=\frac{1}{x_0} \hat{x}=\sqrt{\frac{m\omega}{\hbar}}x
\end{align}</p>
<p>In order to continue i need a proof that operators $\hat{a}$ and $\hat{a}^\dagger$ give a following commutator with hamiltonian $\hat{H}$: </p>
<p>\begin{align}
\left[\hat{H},\hat{a} \right] &= -\hbar\omega \, \hat{a}\\
\left[\hat{H},\hat{a}^\dagger \right] &= +\hbar\omega \, \hat{a}^\dagger
\end{align}</p>
<p>These statements can be found on wikipedia as well as <a href="http://quantummechanics.ucsd.edu/ph130a/130_notes/node170.html#section%3aHOraise" rel="nofollow">here</a>, but nowhere it is proven that the above relations for commutator really hold. I tried to derive $\left[\hat{H},\hat{a} \right]$ and my result was: </p>
<p>$$
\left[\hat{H},\hat{a} \right] \psi = -i \sqrt{\frac{\omega \hbar^3}{4m}}\psi
$$</p>
<p>You should know that this this is 3rd commutator that i have ever calculated so it probably is wrong, but here is a <a href="http://www.2shared.com/photo/9tGw_Zv_/Image1646.html" rel="nofollow">photo</a> of my attempt on paper. I would appreciate if anyone has any link to a proof of the commutator relations (one will do) or could post a proof here.</p> | 1,596 | [
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<p><a href="http://www.google.com/search?as_q=invariable" rel="nofollow">Invariable</a> means which is not variable i.e. can't be changed.</p>
<p>Recently I have seen a sentence when reading a chapter based on measurement:</p>
<blockquote>
<p><em>The accepted standards must be accessible to those who need to calibrate their secondary standards, and they must be <strong>invariable</strong> to change with the passage of time or with changes in the environment (temperature, humidity, etc.).</em></p>
</blockquote>
<p>What does 'invariable to change' mean here?</p> | 1,597 | [
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<p>Studying quantum mechanics, I've found an exercise I don't know how to solve it. Given the radial Schrödinger equation, </p>
<p>$$\left [ \frac{d^2}{dr^2}+k^2-\frac{2m}{\hbar^2}\lambda U\left ( r \right ) \right ]\psi_\lambda {\left ( r \right )}= 0$$</p>
<p>and doing whatever integrals are required, I have to show that:</p>
<p>$$\int_{0}^{R}\psi_\lambda {\left ( r \right )}U\psi_\lambda {\left ( r \right )}dr=\left [ \psi_\lambda \frac{\partial^2 \psi_\lambda}{\partial r\partial\lambda}-\frac{\partial \psi_\lambda}{\partial r}\frac{\partial \psi_\lambda}{\partial \lambda} \right ]_{0}^{R}.$$</p>
<p>I tried to see if integration by parts, or <a href="http://en.wikipedia.org/wiki/Hellmann%E2%80%93Feynman_theorem" rel="nofollow">Hellmann-Feynman theorem</a> may help, but no luck. Any ideas?</p>
<p>Updated:
Following the indications given by @joshphysics (thanks!), I get, for the first step:</p>
<p>$$ \frac{\partial }{\partial \lambda} \frac{d^2\psi_\lambda}{dr^2}+k^2\frac{\partial \psi_\lambda}{\partial \lambda}-\frac{2m}{\hbar^2}\lambda U\left ( r \right ) \frac{\partial \psi_\lambda}{\partial \lambda}-\frac{2m}{\hbar^2}U\left ( r \right ) \psi_\lambda =0$$
And using the original equation, to eliminate the lambda-dependent term, I get:
$$\frac{\partial }{\partial \lambda} \frac{d^2\psi_\lambda}{dr^2}+k^2\frac{\partial \psi_\lambda}{\partial \lambda}-\frac{d^2\psi_\lambda {\left ( r \right )}}{dr^2}-k^2\psi_\lambda {\left ( r \right )}-\frac{2m}{\hbar^2}U\left ( r \right ) \psi_\lambda= 0$$. How do I eliminate the k dependent terms?</p> | 1,598 | [
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<p>Let's say you have two dipole type antennas. Antenna A has a gain of 2.15 dBi, a horizontal beam width of 360 deg and a vertical beam width of 45 deg. Antenna B is similar to antenna A, but has a horizontal beam width of 360 deg and a vertical beam width of 42 deg. Can you use the ratio of the vertical beam widths to predict the gain of antenna B?</p>
<p>Note: In the application I'm asking about I'm not sure what method they used to calculate the beam widths. Maybe someone else knows which methods are most commonly used for dipole antennas.</p> | 1,599 | [
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<p>How much memory would we need to represent a human? How would each atom be stored as? Bytes? Something more complex?</p> | 1,600 | [
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<p>The Feynman lectures are universally admired, it seems, but also a half-century old.
Taking them as a source for self-study, what compensation for their age, if any, should today's reader undertake? I'm interested both in pointers to particular topics where the physics itself is out-of-date, or topics where the pedagogical approach now admits attestable improvements.</p> | 997 | [
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<p>Ok, although this question arises out of the global warming debate, this is a question purely for physicists and not intended to branch into that particular debate.</p>
<p>We are told that LWIR from the Earth to the atmosphere is absorbed by greenhouse gasses and then re-radiated (in the form of backradiation) partially to the surface of the planet, "...thus warming the surface...". </p>
<p>My specific question is this:</p>
<p>How exactly can the re-radiated LWIR from a cooler atmosphere warm the original source of its absorbed energy? </p>
<p>Edit: I am editing this question as several people have stated this is a duplicate question. I disagree. The other question quoted three alternatives and some answers stated the 2nd Law of Thermodynamics was not compromised. Please re-read my question: can the backradiated energy possibly warm a surface which was the original source of that energy? In other words, how can the surface heat itself further when there is no other energy source? I hope this clarifies, and thanks to all that have answered.</p> | 25 | [
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<p>If I look through the microwave window I can see through, which means visible radiation can get out. We know also that there is a mesh on the microwave window which prevents microwave from coming out.</p>
<p>My question is how does this work? how come making stripes or mesh of metals can attenuate microwave radiation yet allow visible radiation?</p>
<p>Looks like an electrodynamics problem to me with periodic boundary conditions (because of the partitions on the microwave oven window). Is it discussed in any textbook?</p> | 1,601 | [
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<p>My apologies if this question has already been answered. Using the notation of Sakurai, we have an energy eigenket N and an eigenvalue n.</p>
<p>$$N | n \rangle = n | n \rangle$$</p>
<p>For the number operator $N = a^\dagger a $. My question is precisely why does n have to be an integer. I do understand that it must be positive definite due to the norm condition on $ a | n \rangle $:</p>
<p>$$\langle n | a^\dagger a | n \rangle \geq 0$$</p>
<p>I can see that we can show:</p>
<p>$$ a | n \rangle = \sqrt{n-1} | n-1 \rangle $$ and similarly for $a^\dagger | n \rangle$</p>
<p>Sakurai argues that the sequential operation of $a$ operators leads to the form which he does not write, but I will write here for clarity:</p>
<p>$$ a^k | n \rangle = \left(\sqrt{n}\sqrt{n-1}...\sqrt{n-k+1}\right) | n - k + 1 \rangle $$ </p>
<p>Since</p>
<p>$$ a|0 \rangle = 0 $$</p>
<p>the sequence must terminate, but can only terminate if n is an integer. How can we make that jump? Is there a formal mathematical proof that the sequence $\left(\sqrt{n}\sqrt{n-1}...\sqrt{n-k+1}\right)$ will terminate if and only if n is an integer?</p> | 1,602 | [
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<p>Can someone show me, without glossing over anything, why $F = E - TS$ is minimized when $p_i = e^{-U_i/k_bT}/\sum_ie^{-U_i/k_bT}$? I understand it conceptually, but am having difficulty showing it formally.</p> | 1,603 | [
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<p>On my textbook is written that gravitational force is the force that attracts bodies with mass. But I've seen on a book that It actually attracts bodies with energy. I'm having a class tomorrow and I would like to know some argumments to use with - against my professor.</p> | 1,604 | [
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<p><a href="http://physics.stackexchange.com/questions/8626/if-all-conserved-quantities-of-a-system-are-known-can-they-be-explained-by-symme">This question</a> resulted, rather as by-product, the discussion on how to count degrees of freedom (DOF). I extend that question here:</p>
<ul>
<li>Are necessary<sup>1</sup> derivatives such as velocities counted as individual DOFs or together with the respective coordinate<sup>2</sup>?</li>
<li>Should complex valued DOFs be counted twice as in "two real-valued DOFs" or once as in "one complex-valued DOF"? (I mean, when one does not want to specify this explicitly)</li>
</ul>
<p>Please answer with some reference, unless it turns out this is actually a matter of taste rather than a strictly defined thing.</p>
<hr>
<p><sup>1)</sup> I mean those a value of which is <em>required</em> as an initial condition<br>
<sup>2)</sup> I count fields in QFT as coordinates as well, while space-time-coordinates are parameters to me, if that matters. I know a field actually has $\infty$ (or rather, $2^\aleph$) DOFs itself, but let's say e.g. "one $\mathbb R^3$ continous DOF" in that case</p> | 1,605 | [
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<p>Is there a formal definition of <a href="http://en.wikipedia.org/wiki/Drag_%28physics%29" rel="nofollow">drag</a>, say, as some surface integral of normal and shear forces? There seem to be a lot of formulas for specific cases, but is there a general one?</p>
<p>I need to accurately calculate the drag of three cylinders placed between two plates. </p> | 1,606 | [
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<p>The world is full of nuclear warheads being stockpiled. Controlled fusion power seems a long way away. Could we put these warheads to better use by exploding them in a controlled way and capturing the energy they produce?</p>
<p>By useful work I mean the power is then available for the national grid to boil kettles or have a shower!</p>
<p>Extra points for looking at the practicalities of building a facility to do this (although I guess that would get the question closed :-(... )</p> | 1,607 | [
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<p>At present the only way we can produce anti-matter is through high powered collisions. New matter is created from the energy produced in these collisions and some of them are anti-matter particles such as positrons, anti-protons etc.</p>
<p>My question is, normal matter and anti-matter are so similar. They both appear to react to the fundamental forces in the same way. Is it conceivable that we could find a way of "switching" matter to become "anti-matter" through some low energy technique? E.g. find a way to flick a few "quantum switches" and voila your proton is now an anti-proton, ready for annihilation and limitless energy for humanity!</p>
<p>Would we be breaking a great deal of the fundamental laws of physics if this was found to be possible (i.e. should I give up now)?</p> | 1,608 | [
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<p>It is easy to observe that on a windy day, the wind does not blow for several hours at constant speed, then gradually subside. Instead, on a time scale of seconds or tens of seconds, there are stronger gusts of wind followed by lulls.</p>
<p>Presumably this is an effect of turbulence. If so, is this turbulence due to the complicated geometry around me - buildings, trees, hills, etc.? If we removed these features and had wind blowing over a flat ocean surface or a flat plain for hundreds of miles, would we still observe the wind to blow in the same sorts of gusts, or should I expect a more steady flow?</p> | 1,609 | [
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] |
<blockquote>
<blockquote>
<blockquote>
<blockquote>
<p>Here the problem states to find the steady state charges on the condensers.<<<<
<img src="http://i.stack.imgur.com/5MwH2.png" alt="enter image description here"></p>
</blockquote>
</blockquote>
</blockquote>
</blockquote>
<p>According to me the charges on second at steady must be zero else there would be some current in the resistance in it's parallel(which can't be true because then $C$ too will have current in it and it will not be steady state) . Is this argument correct or there would be something else in this case?</p>
<h2>EDIT:additional question</h2>
<p>How does the charge initially that started accumulating on the $2C$ capacitor goes away at steady state? </p>
<p>Initially the capacitor does not pose resistance so, maximum charges pass through the capacitor instead of the resistance.But finally the charge on it is zero,please explain what happens in mean time.</p> | 1,610 | [
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<p>I’m trying to convince my boss that the mixers we are using are too much. I’m trying to prove that we are over-mixing our product. Our product is ink…just your basic ink found in your printer at home. We mix in a 23 inch diameter 50 gallon vessel using a 2.75 inch diameter axial flow impeller going at 1050 rpm with a 3 inch rotor stator 5 inches below it. The inks have a viscosity of around 3. What Reynolds number should be sufficient for this process or can you give me any further input? Any help would be GREATLY appreciated. Thanks.</p> | 1,611 | [
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<p>I would like to write a Gaussian state with density matrix $\rho$ (single mode) as a squeezed, displaced thermal state:
\begin{gather}
\rho = \hat{S}(\zeta) \hat{D}(\alpha) \rho_{\bar{n}} \hat{D}^\dagger(\alpha) \hat{S}^\dagger(\zeta) .
\end{gather}
Here,
\begin{gather}
\rho_{\bar{n}} = \int_{\mathbb{C}} P_{\bar{n}}(\alpha) |\alpha\rangle\langle\alpha|d\alpha \text{ with } P_{\bar{n}}(\alpha) = \frac{1}{\pi \bar{n}} e^{- |\alpha|^2 / \bar{n}}
\end{gather}
is a thermal state with occupation $\bar{n}$,
\begin{gather}
\hat{S}(\zeta) = e^{(\zeta^* \hat{a}^2 + \zeta \hat{a}^{\dagger 2}) / 2}
\end{gather}
is the squeezing operator, and
\begin{gather}
\hat{D}(\alpha) = e^{\alpha^* \hat{a} - \alpha \hat{a}^\dagger} .
\end{gather}
is the displacement operator. I prefer to use the convention $\hat{x} = (\hat{a} + \hat{a}^\dagger) / \sqrt{2}$ and $\hat{p} = (\hat{a} - \hat{a}^\dagger) / \sqrt{2} i$.</p>
<p>I assume that the way to accomplish this is to derive the mean and variance of our Gaussian state $\rho$ and thereby determine $\zeta$ and $\alpha$. However, I have been unsuccessful in doing so. That is, given the mean and variance of our Gaussian state $\rho$, what are $\zeta$ and $\alpha$?</p>
<p>On a side note, I was also wondering if there is a standard result for the commutator of $\hat{S}(\zeta)$ and $\hat{D}(\alpha)$?</p> | 1,612 | [
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<p>If we describe spacetime with a Lorentzian manifold, it is always possible to choose a coordinate system such that at any particular point $x^\alpha$, the components of the metric are:
$$ g_{\mu\nu}(x^\alpha) =
\left( \begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 &-1 & 0 & 0 \\
0 & 0 &-1 & 0 \\
0 & 0 & 0 &-1 \end{array} \right) $$</p>
<p>But our freedom is much greater than this. In curved spacetime, the equivalence principle suggests we can choose coordinates such that the metric is of that form for every point along a chosen time-like geodesic. And in flat spacetime, we can see an explicit example that it doesn't even need to be a geodesic, for the Rindler metric has that form on every point of a particular worldline with constant proper acceleration. I have a feeling this is possible for any time-like worldine.</p>
<p>So my question is:</p>
<p>Given a coordinate system and metric for a Lorentzian manifold and a time-like worldline on this spacetime, is it always possible to find a coordinate transformation such that <em>for every point on the world line</em> the components of the metric in this coordinate system are just (1,-1,-1,-1) on the diagonal?</p>
<p>I realize that even for simplified cases (say a geodesic on the Schwarzschild background), such a coordinate system could be incredibly complicated. So if someone creates an incredibly complicated explicit construction, please also show that the solution to the equations you setup exist with some kind of existence proof.</p>
<p>This originally started from trying to find such a local coordinate chart for a free falling observer towards a blackhole, but realized I didn't know a good mathematical way to represent such coordinate freedom to get me started. Eventually I ended up pondering this current question. So even if you can't give a full answer, but can suggest some mathematical tools or where to read up on them, any help would be appreciated.</p> | 1,613 | [
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<p><em>[Again I am unsure as to whether this is appropriate for this site since this is again from standard graduate text-books and not research level. Please do not answer the question if you think that this will eventually get closed. Since otherwise the software doesn't allow me to delete questions once it has answers!]</em> </p>
<ul>
<li><p>A priori the technique of quantization using the background gauge field seems to be a "mere" redefinition of the good old Lagrangian whereby one "artificially" seems to double the number of variables by writing all of the fileds like say the fermions $\psi$, the gauge field $A_\alpha^\mu$, the ghosts $\omega_\alpha$ and $\omega_\alpha ^*$ as a sum of two fields one with a prime (the quantum fluctuation) and one unprimed (the classical background) - like replace every $A_\alpha ^\mu$ by $A_\alpha ^\mu + A_\alpha^{\mu'}$. Till this step nothing seems to distinguish the primed from the unprimed fields. </p></li>
<li><p>Then one defines the a new set of covariant derivatives where the the connection is defined only by the classical background gauge field (the unprimed fields) only. </p></li>
</ul>
<p>Then one rewrites the Lgrangian in terms of these new covariant derivatives and these doubled the number of fields. But I would think that the Lagrangian is still exactly the same and nothing conceptually has changed. </p>
<p>Now two statements come in which seem to be somewhat "magical" and thats where it seems that the crux lies, </p>
<ol>
<li><p>Firstly one says that it is possible to set the background fields such that all the unprimed gauge fields are constant and every other unprimed field is 0. </p></li>
<li><p>Then one says that the 1-loop contribution to the running of the coupling constant (and also the effective action?) is completely determined by looking at only that part of the orginal action which is quadratic in the primed fields ("quantum fluctuations"?)</p></li>
</ol>
<p>I would like to understand why the above two steps are consistent and correct and how to understand them. Also if there is some larger philosophy from which this comes out and if there are generalizations of this. </p>
<p>{This technique and idea seems quite powerful since after one says the above it is almost routine calculation to get the beta-function of a $SU(N_c)$ Yang-Mill's theory with say $N_f$ flavours - and hence asymptotic freedom of QCD! - things which I would think are corner stones of physics! } </p> | 1,614 | [
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<p>First off, let me just say that I am unsure if this question is appropriate for this site, and if the community deems it necessary, the question should be closed.</p>
<p>So right now I am a fourth year mathematics and physics major whose research interests lie in quantum field theory / quantum gravity. In particular, I would like to go into theoretical physics. However, I have found that most physicists do not achieve a level of rigor I would strive for in my work, and so I am reluctant to apply for graduate school in physics. On the other hand, it seems that in most mathematics departments in the U.S., there are at most a couple of mathematicians working in what you might all theoretical physics, and I would not feel comfortable applying to a graduate school banking on the fact I could work with a specific one or two persons.</p>
<p>What possible routes are there that I could take if my interests lie in theoretical physics, but I wish to practice theoretical physics as a mathematician would? Furthermore, are there any (not necessarily American) universities with a joint mathematics-physics department (like the DAMTP at Cambridge), or, the next best thing, a mathematics and physics department that collaborate frequently? What about theoretical physicists working in quantum field theory/quantum gravity who don't shy away from rigor?</p>
<p>Any and all input is greatly appreciated!</p> | 1,615 | [
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<p>In probability theory, <a href="http://en.wikipedia.org/wiki/Schramm%E2%80%93Loewner_evolution" rel="nofollow">the Schramm–Loewner evolution</a>, also known as stochastic Loewner evolution or SLE, is a conformally invariant stochastic process. It is a family of random planar curves that are generated by solving Loewner's differential equation with Brownian motion as input. The motivation for SLE was as a candidate for the scaling limit of "loop-erased random walk" (LERW) and, later, as a scaling limit of various other planar processes. </p>
<p>My question is about connections of SLE with theoretical physics, applications of SLE to theoretical physics and also applications of (other) theoretical physics to SLE. I will be happy to learn about various examples of such connections/applications preferably described as much as possible in a non-technical way. </p> | 1,616 | [
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<p>Are we fooled into thinking that expansion of the universe is accelerating, when in fact, time itself is slowing?</p>
<p>Or if dark energy does exist?</p> | 26 | [
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<p>I am a little bit perplexed as to how to compute the three-point correlation function for a massive scalar field, I know that it should be equal to zero.</p>
<p>I need to show that: $\lim_{T\rightarrow \infty (1-i\epsilon)} \int D\phi \phi(x_1) \phi(x_2) \phi(x_3) = 0$</p>
<p>How to show this?</p>
<p>Edit: I forgot to mention without using the generating founction, $Z[J]$.</p> | 1,617 | [
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] |
<p>Why are the strong and weak nuclear forces short range?</p>
<p>Are quarks confined or welded together?</p>
<p>Why are elementary particles confined at short range?</p>
<p>Or is color confinement color welding?</p>
<hr>
<p>the quarks are glued together by strong nuclear force, (but not at high energy) it seems to me that its a kind of elementary welding process.</p> | 1,618 | [
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<p>I think of an NMR experiment, but with a single spin half nucleus initially set to the excited state.</p>
<p>When the nucleus finally returns to its ground state, it will emit a photon. An observer in the lab frame will see a photon in the Larmor frequency. An observer in a rotating frame, very close to the Larmor frequency, will see a photon with a very very low frequency. </p>
<p>My question is: Because of the frequency difference, will it take the detector of the rotating frame observer longer time to tick (i.e., detect the photon) compared to the lab frame observer?</p>
<p>Also, and i know it sounds silly, if there is a different, does this somehow relates to time dilution in general relativity? </p> | 1,619 | [
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<p>These facts are taken for granted in a QM text I read. The purportedly guaranteed non-normalizability of eigenfunctions which correspond to a continuous eigenvalue spectrum is only partly justified by the author, who merely states that the non-normalizability is linked to the fact that such eigenfunctions do not tend to zero at infinity.</p>
<p>Not a very satisfying answer. What I'm really after is an explanation based in functional analysis. I believe there is a generalized result about inner products being finite for discrete spectra but infinite for continuous spectra.</p>
<p>Can anyone shed some light on this?</p> | 1,021 | [
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<p>A 2+1D topological field theory (topologically ordered states), implies that the topological ground state degeneracy (GSD) on $T^2$ torus (2D manifold <strong>without boundary</strong>). For example a level k U(1) Chern-Simons theory implies a GSD$=k$.</p>
<p>If we put the topological field theory on a 2D manifold <strong>with 1D boundary</strong>, we expect 1+1D gapless edge states; and there are central charges $c$, which roughly measures the degree of freedom of the gapless edge states. </p>
<p>My question is: are there some explicit formula relates: <strong>topological ground state degeneracy GSD</strong> and <strong>central charges c</strong>?</p>
<p>Say,</p>
<p>$$\text{GSD}=\text{GSD}(c,\dots)$$</p>
<p>and </p>
<p>$$c=c(\text{GSD},\dots)$$</p>
<p>here $\dots$ are other possible data. RHS are the desired functions of my questions. It will be better to take some examples of <strong>non-Abelian</strong> topological field theory (topologically ordered states) to test its formula's validity.</p> | 1,620 | [
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<p>Related: <a href="http://physics.stackexchange.com/q/79074/2451">What is meant by boiling off electrons in a heater coil?</a></p>
<p>In the Thomson tube we used in our class to produce an electron beam, the lab manual stated that the tube was filled with a low pressure argon gas. When the electrons collided with the argon gas, electrons in the gas molecules were excited and when they de-excite, green light is emitted. However, when one looks at the atomic spectra of argon, there are two green lines, as shown below.</p>
<p><img src="http://i.stack.imgur.com/kihYy.jpg" alt="enter image description here"></p>
<p>Is one able to distinguished between these two green lines in this experiment or am I actually seeing both of these lines? I assume that because the Thomson tube sends in a range of electron velocities, that maybe its both green lines that I am seeing. However, I don't know for sure.</p> | 1,621 | [
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<p>Imagine an eruption of energy/mass $E$ from a singularity $O$, as in a Big Bang. After the energy/mass $E$ is all at more than a distance $d$ from $O$, is it for some value of $d$ possible that there could be a new eruption of energy/mass from $O$, i.e. a new Big Bang? If yes, is there an upper limit to the number of succeeding Big Bangs? </p> | 1,622 | [
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<p>I'm writing my first MD simulation (ever) for liquid Argon. The code is up and running. I am supposed to do the calculations in the <a href="http://en.wikipedia.org/wiki/Microcanonical_ensemble" rel="nofollow">NVE ensemble</a>. Having implemented a 4th order <a href="http://en.wikipedia.org/wiki/Symplectic_integrator" rel="nofollow">symplectic integrator</a> (forest-ruth) the total energy of my system (approx 1000 atoms) oscillates (as expected).</p>
<p>Now I am "measuring" the heat capacity of my system, which I do by calculating the mean total energy and it's fluctuations for several distinct simulations (using the same parameters).</p>
<p>This is exactly the point that I don't get (conceptually). By definition, the total energy of a system is fixed in the microcanonical ensemble. Am I in the microcanonical or in the canonical ensemble when I do an MD simulation? The literature (I have) is somewhat sloppy when it comes to this.</p> | 1,623 | [
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<p>I have the following problem to solve:</p>
<blockquote>
<p>A particle of mass $m$ and charge $e$ moves in the laboratory in crossed, static, uniform, electric and magnetic fields. $\mathbf{E}$ is parallel to the $x$-axis; $\mathbf{B}$ is parallel to the $y$-axis. Find the EOM for $|\mathbf{E}|<|\mathbf{B}|$ and $|\mathbf{B}|<|\mathbf{E}|$.</p>
</blockquote>
<p>I was planning on using the following:
$$\vec{E}'=\gamma(\vec{E}+\vec{\beta}\times\vec{B})-\frac{\gamma^2}{\gamma +1}\vec{\beta}(\vec{\beta}\cdot \vec{E})$$
$$\vec{B}'=\gamma(\vec{B}-\vec{\beta}\times\vec{E})-\frac{\gamma^2}{\gamma +1}\vec{\beta}(\vec{\beta}\cdot \vec{B})$$
along with
$$t'=\gamma (t-\vec{\beta}\cdot\vec{x})$$
$$ \vec{x}'=\vec{x}+\frac{(\gamma -1)}{\beta^2}(\vec{\beta}\cdot \vec{x})\vec{\beta}-\gamma \vec{\beta}t$$
with the addition constraints
$$\frac{d\vec{p}}{dt}=e\left[ \vec{E}+\frac{\vec{u}}{c}\times \vec{B}\right]$$
and
$$\frac{dU}{dt}=e\vec{u}\cdot \vec{E}$$
To solve this I am going to switch to a frame with
$$\vec{\beta}=\frac{E}{B}\hat{z}$$for the first case. With this case $dU/dt=0$ and i can solve the equations of motion to find $\vec{x}(t)$ directly, and then boost back to get the trajectories in the original frame. However for the the second case I was wondering if my procedure is correct. I am going to switch to a frame with $\vec{\beta}=(B/E) \hat{z}$ to remove the magnetic field. Now here it seems that $\vec{u}_0$ is perdepdicular to $\vec{E}$ to start in the new frame, but that it will be accelerated in the $x$ direction and hence $dU/dt\neq 0$ and I can't just straightforwardly solve the EOMs. How would I proceed from here?</p>
<p>Thanks,</p> | 1,624 | [
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] |
<p>Let the free electromagnetic current $J_\mu(x)$ be = $:\bar{\psi}(x)\gamma_\mu Q \psi(x):$ where $::$ is the normal ordering. </p>
<ul>
<li>In this expression why is $Q$ thought of as a "charge operator" instead of just a number?...its quite pesky to keep track of this operator while doing the current-current OPEs though I don't see anything changing conceptually if I just thought of it as a number...</li>
</ul>
<p>After a lot of (I found it very subtle!) calculations one can show that in the light-cone limit $x^2 \rightarrow 0$ the commutator, $[J_\mu(x),J_\nu (0)]$ has one of its terms (say X), </p>
<p>$$X = \frac{iTr[Q^2]}{\pi^3}\{\frac{2}{3}g_{\mu \nu}\delta''(x^2)\epsilon(x_0) + \frac{1}{6}\partial_\mu \partial_\nu [\delta'(x^2)\epsilon(x_0)]\}$$</p>
<p>Now one wants to compare the contribution of this term to two different situations, </p>
<ul>
<li>The total hadronic cross-section, </li>
</ul>
<p>$\sigma(e^+e^- \rightarrow hadrons) = \frac{8\pi^2\alpha^2}{3(q^2)^2}\int d^4x e^{iq.x}<0|[J_\mu(x),J^\mu(0)]0>$</p>
<ul>
<li>The inclusive hadronic tensor in deep inelastic lepton-nucleon scattering, </li>
</ul>
<p>$W_{\mu \nu}(p,q) = \frac{1}{M} \sum _{\sigma} \int \frac{d^4x}{2\pi} e^{iq.x}<p,\sigma|[J_\mu(x), J_\nu(0)]|p,\sigma>$</p>
<p>In the derivation/argument for the expression for $W_{\mu \nu}$ it is kind of clear that the initial and final states have to be $|p,\sigma>$ - the initial state of the proton. </p>
<ul>
<li><p>But I am unable to pin down as to exactly why the initial and final states in the first case had to be vacuum $|0>$. It would be great if someone can explain this conceptual point about the difference in the initial and final states. </p></li>
<li><p>Hence if someone can explain as to why the term $X$ contributes (and is infact the leading contributor!) to $\sigma$ but does not contribute to $W_{\mu \nu}$!?</p></li>
</ul>
<p>(..my vague understanding is that this difference stems from the difference in the initial and final states..but can't make this precise..) </p>
<ul>
<li>Though initially $W_{\mu \nu}$ is defined in terms of the correlator $[J_\mu(x), J_\nu(0)]$, often I see that during calculations one is in practice evaluating $[J_\mu(\frac{x}{2}), J_\nu(-\frac{x}{2})]$. Why this change? </li>
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<p>I have this question that I dont know how to solve correctly :</p>
<p><img src="http://i.stack.imgur.com/HsYdc.jpg" alt="enter image description here"></p>
<p>My question is, how do I find $V_B$ ? I will find the angular velocities myself, but I want to know the method to get $V_B$ ?</p>
<p>I know I can start by using $V_C=V_B+V_{C/B}$, but then I'm not sure what to do next...is the direction of $V_B$ and $V_{C/B}$ the same ? I'm using vector algebra (cross product with i and j etc).</p>
<p>More specifically : </p>
<p>$$\vec{V}_C=\vec{V}_B+\vec{V}_{C/B}$$</p>
<p>$$\vec{V}_C = -1 \hat{j}$$</p>
<p>$$\vec{V}_B= ?$$</p>
<p>$$\vec{V}_{C/B}= \omega_{CB} \hat{k} \times \vec{r}_{CB}$$</p>
<p>Shouldn't $\vec{V}_B = \omega_{CB} \hat{k} \times \vec{r}_CB$ ??? If not, then what should it be ?</p>
<p>If you need more info please let me know.</p> | 1,626 | [
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<p>I'm confused about the terminology in the two contexts since I can't figure out if they have a similar motivation. Afaik, the definitions state that quantum processes should be very <em>slow</em> to be called <a href="http://en.wikipedia.org/wiki/Adiabatic_theorem">adiabatic</a> while <a href="http://en.wikipedia.org/wiki/Adiabatic_process">adiabatic thermodynamic processes</a> are supposed to be those that don't lose heat. Based on my current intuition, this would mean that the thermodynamic process is typically <em>fast</em> (not leaving enough time for heat transfer). What gives, why the apparent mismatch?</p> | 1,627 | [
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<p>So I was writing a lab report for Physics I, where I was describing projectile motion. I had written </p>
<blockquote>
<p>Since gravity on Earth does not have a horizontal force, the object
will not lose horizontal velocity (velocity in the x direction).</p>
</blockquote>
<p>Then this made me think, is there any gravitational force found in nature that has both vertical <em>and horizontal</em> components? </p> | 1,628 | [
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<p>As the title suggests, Can the effects of a person's mass upon the local gravitational field be detected and measured remotely?</p>
<p>I am aware any mass produces and effects gravity but couldn't find anything in my searching if it is possible or theoretically possible to detect this effect remotely.</p> | 1,629 | [
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<p>For my research work i am trying to calculate band gap of zinc oxide theoratically and found this <a href="http://dx.doi.org/10.1002/adma.200304904" rel="nofollow">paper (Determination of the Particle Size Distribution of Quantum Nanocrystals from Absorbance Spectr)</a>. What i am doing is I am making a small matlab programm which will reporduce the results using the same values from the above paper. I used the values given in the paper to make the program</p>
<p><img src="http://i.stack.imgur.com/v8Fbp.jpg" alt="enter image description here"></p>
<p><strong>Absorbtion onset</strong> </p>
<p>$\lambda ^{onset}=\frac{c*h}{E^{*}}$</p>
<pre><code> Values used for calculating Band gap
8.85418782 × 10^-12 permittivity of free space
1.60217657 × 10^-19 charge of an electron
9.11 × 10^-31 free electron mass
1.05457173 × 10^-34 hcut
0.59 effective mass of holes
0.26 effective mass of electrons
8.5 relative permittivity
3.2 zno bandgap
c velocity of light
</code></pre>
<p>But after running the programm i got these results</p>
<pre><code>Size Absorbtion Onset
1 6.21E-26
1.5 6.21E-26
2 6.21E-26
2.5 6.21E-26
3 6.21E-26
3.5 6.21E-26
4 6.21E-26
4.5 6.21E-26
5 6.21E-26
5.5 6.21E-26
6 6.21E-26
6.5 6.21E-26
7 6.21E-26
7.5 6.21E-26
8 6.21E-26
8.5 6.21E-26
9 6.21E-26
9.5 6.21E-26
10 6.21E-26
</code></pre>
<p>The results i got doesnt match with tha paper and I belive that my coding in matlab is correct but i am doubtful about the values i have taken for calculating band gap. </p>
<p>Can you help me with this</p> | 1,630 | [
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<p>If I had a semi infinite, 1-D object and a finite 1-D object, both heated at the same constant rate at one end each for the same time period and both begin at the same initial temperature, is it physically meaningful for me to integrate along the length of the object and consider this integral as a function of time and a measure of the 'amount' of heat on the object?</p> | 1,631 | [
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<p>The discussion on <a href="http://www.physicsforums.com/showthread.php?t=178383" rel="nofollow">this</a> webpage mentions that shining a laser beam at a hair produces an effect like that of the double-slit experiment. Does classical physics predict the effect you observe when you do this (since light is a wave)?</p> | 1,632 | [
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<p>I am trying to create a machine that moves on two points (Wheels or legs). Because of the extremely difficult nature of perfectly balancing the parts, I am wondering is there any way to create a mechanical mechanism to balance it. I know that it can be done with many different electric circuits, but I am wondering is it even theoretically possible to create such a system? And if it's possible, does anyone know of any systems that do that?</p>
<p>P.S. Could someone retag this appropriately? I'm new to this SE, and I'm not sure how to tag it.</p> | 1,633 | [
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<p>If you take a giant whale out of the water and put it on land for long enough, <a href="http://books.google.com/books?id=8bFcJgX0lIUC&pg=PA185&lpg=PA185&dq=orca%20crush%20itself%20under%20its%20own%20weight&source=bl&ots=M0cCc-fMyD&sig=gRvN795BpgGYwLIMd9GCEmbvzog&hl=en&sa=X&ei=M36sT8SGDsqSiALp0pmwBw&ved=0CJAFEOgBMAM#v=onepage&q=orca%20crush%20itself%20under%20its%20own%20weight&f=false" rel="nofollow">it will crush itself under its own weight</a>. Why doesn't the animal get crushed under its own weight when it's in water?</p> | 1,634 | [
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<p>After studying the definition (& derivation) of the potential to an electric field and the Poisson equation I'm currently wondering whether the following is possible:</p>
<ul>
<li>Can one give an example of a physical setup where the Poission equation fails to provide the electric field?</li>
</ul>
<p>I have tried to come up with an example but failed. What I thought of was, that considering a valid solution of the Poisson equation, the electric field derived from the solution would have to be differentiable (since the second derivative of the potential appears in the differential equation). If one were to find a setup with a discontinuous electric field, would the Poisson equation fail?</p> | 1,635 | [
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<p>I heard at least three claims about the development of the heat of the sun.
In an old book, I read, that nothing dramatically will happen in the next
few billion years. Wikipedia states, that the average temperature will hit
30° in a billion years, 100° a billion years later, and in an internet forum,
someone claimed that in 500 million years, the oceans would cook.</p>
<p>Which of this claims is true? How reliable are the claims about events in such far future?</p> | 1,636 | [
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<p>Lorentz length contractions states that the length of any moving object gets divided by the <a href="https://en.wikipedia.org/wiki/Lorentz_factor" rel="nofollow">Lorentz factor</a> equal to the Lorentz factor for that object (always $\geq 1$), equal to
$$
\gamma=\frac{1}{\sqrt { 1-\frac { { v }^{ 2 } }{ { c }^{ 2 } } } }
$$
However, in massless particles $v=c$, so the Lorentz factor becomes $\infty$, meaning that an object traveling at $c$ will have $0$ length. However, photons and obviously all forms of electromagnetic waves move at c when traveling through a vacuum, such as from a space shuttle to a space station or back to Earth. Does this mean that photons have no length? How does this affect wavelength?</p> | 1,637 | [
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<p>The most simplified version of the energy equation (which is also the most known) is $E=mc^2$<br>
However, I understand that this <em>only</em> applies to objects with non-zero mass and zero velocity. I also read that "relativistic mass" is equal to the "rest mass" multiplied by the Lorentz factor:
$$
m=\gamma m_0 =\frac { { m }_{ 0 } }{ \sqrt { 1-\frac { { v }^{ 2 } }{ { c }^{ 2 } } } }
$$
What exactly <em>IS</em> the "complete" equation for objects with non-zero mass that can be applied to objects with any velocity?</p>
<p>My knowledge would suggest that the equation is:
$$
E=\sqrt { { (\gamma m{ c }^{ 2 }) }^{ 2 }+{ (\gamma pc) }^{ 2 } }
$$
where E is energy, $\gamma$ is the Lorentz factor, $m$ is the mass, $c$ is obviously the speed of light, and $p$ is the momentum of the object (defined as $p=mv$ but multiplied by the Lorentz factor because the mass increases with velocity)</p> | 1,638 | [
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<p>I'm quite lost what $B$ and $H$ is. It seams to me that most of the texts I read do quite poor job in explaining them properly. They are explained only in cases when magnetic susceptibility is constant in the space. </p>
<p>First <em>law</em> says that $M = \chi H$, $M$ is magnetization and to my understanding $H$ is something like external magnetic field (for example Earth magnetic field when I do experiment in some lab).</p>
<p>Next <em>law</em> says that magnetic field is $B = \mu_0(M+H) = \mu H$. This has to be wrong when $\chi$ is not constant because than we are not guarantied that divergence of $B$ is zero.
$$
\nabla \cdot B = \nabla \cdot (\mu H ) \neq \mu \nabla \cdot H = 0
$$</p>
<p>Can you please direct me to some book when this is explained properly?</p>
<p>My aim: I want to calculate magnetic field around paramagnetic material.
I cannot use formula $B = \mu H$ because outside paramagnet is $\mu = \mu_0$ so magnetic field would be unchanged.</p> | 1,639 | [
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<ol>
<li><p>If any configuration of matter can fall into a black hole and hit the singularity, and ditto for the big crunch, and there is time reversal CPT invariance, does it mean anything can pop out of the big bang singularity, and there is unpredictability at the big bang? </p></li>
<li><p>How do we explain the very low entropy of the big bang then?</p></li>
</ol> | 1,640 | [
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<p>I am reading David J. Griffiths and have a problem understanding the concept of discontinuity for E-field.</p>
<p>The E-field has apparently to components. (How does he decompose the vector field into the following?) The tangential and the normal component. The normal component is discontinuous? (Question 1: Why?) But the tangential components are <em>not</em> discontinuous? (Question 2: Why?).</p>
<p>My suggestion for question 1: I tried drawing it, and the way I see it, the size of the E-field is continuous, but the direction is not a continuous!</p>
<p>My suggestion for question 2: The tangential components has a direction that changes continuously. </p>
<p>The understanding of the word Continuous, that I am applying is this: If I can take the vector field, imagine it in a $\mathrm{R}^2$ graph, and if it is continuous (not abruptly stopped), then.. That's it!</p>
<p>Can anyone please give me the explanation I am missing or another source? </p>
<p>Also why does this have anything to do with ANYTHING:
$$
{E}_{above} - E_{below} = \frac{\sigma}{\epsilon_0}\hat{n}
$$</p> | 1,641 | [
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<p>dewitt claimed in his paper </p>
<blockquote>
<p>Bryce S. DeWitt. Quantum theory without electromagnetic potentials, <em>Phys. Rev.</em> <strong>125</strong> no. 6 (1962), pp. 2189-2191, <a href="http://dx.doi.org/10.1103/PhysRev.125.2189" rel="nofollow">DOI: 10.1103/PhysRev.125.2189</a>,</p>
</blockquote>
<p>that the discovery of the Aharonov and Bohm that electromagnetic potentials play primary role in quantum mechanical theory is false. Who won? What are the errors in the argument of the losing side in this battle?</p> | 1,642 | [
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<p>Understanding <a href="http://en.wikipedia.org/wiki/Atmospheric_refraction" rel="nofollow">atmospheric refraction</a>, particularly of ultraviolet, and into the blue part of the visible spectrum is of great interest to me. Although, I have a strong background in trigonometry and geometry, I am very much interested in how these are applied in the context mentioned.</p>
<p>Are there any references (books, papers and the like), that are mathematically rigourous, but also have examples of how the mathematics is applied to atmospheric refraction? Are there any other mathematical knowledge that would be required?</p> | 1,643 | [
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<p>This is a problem from school. I will show my attempt.</p>
<p>The question:</p>
<p>"The gas constant for dry air R is 287 $\frac{m^2}{s^2*K}$. Assuming the temperature is 330 K and the pressure is 1050 hPa, what is the atmospheric density."</p>
<p>The professor said DO NOT produce an answer by finding a formula, but to use the magic of unit conversion to try to solve things.</p>
<p>I know density is measured in kg/m^3 or thereabouts so I tried the following:</p>
<p>1050 hPA = 105, 000 Pa</p>
<p>1 Pa = 1 kg/m*s^2</p>
<p>105,000 $\frac{kg}{m*s^2}$ * 330 K * 287 $\frac{m^2}{s^2*K}$.</p>
<p>This cancels some units... but not enough...in fact it cancels just K, so far as I understand, far from what I need for my density unit.</p>
<p>Any ideas on what Im doing foolishly here?</p> | 1,644 | [
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<p><a href="http://en.wikipedia.org/wiki/Poynting%27s_theorem" rel="nofollow">Poynting's theorem</a> is given by</p>
<p>$$\frac{\partial}{\partial t}\int_{v}Udv
+ \oint_{A}\vec S\cdot \vec {dA}
+\int_{v}\vec E\cdot\vec J dv
=0
$$</p>
<p>Where,</p>
<ul>
<li>the total electromagnetic energy inside the volume v is $U = \frac 1 2 (\vec E\cdot\vec D+\vec B\cdot\vec H)$</li>
<li>the Poynting vector $\vec S=\vec E\times \vec H$</li>
</ul>
<p>This equation is interpreted as the conservation of electromagnetic and mechanical energy for a volume of space with each term representing respectively the rate at which</p>
<ol>
<li>electromagnetic energy changes inside the volume</li>
<li>electromagnetic energy crosses the boundary of the enclosing surface</li>
<li>mechanical work is done on charges inside the volume </li>
</ol>
<p>Now take the case of a charge accelerating from rest by a static electric field, and initially at the centre of a spherical volume fixed in space with radius cT where c is the speed of light and T the time taken for electromagnetic fields to propagate from the center to the spherical boundary. For 0 < t < T both the magnetic and mechanical energy inside the volume increases without electromagnetic energy crossing the boundary.</p>
<p>So during this time, where does the negative term come from to maintain the RHS = 0 in Poynting's theorem?</p> | 1,645 | [
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<p>Can particles emit a virtual Higgs boson in a similar manner to the way a virtual photon is emitted?</p> | 1,646 | [
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<p>Why did they used to make the mill chimneys so tall?</p>
<p>This question was asked in an Engineering Interview at Cambridge University.</p> | 1,647 | [
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<p>The context is that I'm building a simulation of a starfield, as seen from a relativistic spaceship. (EDIT: the simulation can now be found <a href="http://tinyurl.com/pt8h6j4" rel="nofollow">here</a>.)</p>
<p>One reference that I'm using is this paper by John M. McKinley (1980) "Relativistic transformation of solid angle":</p>
<p><a href="http://adsabs.harvard.edu/abs/1980AmJPh..48..612M" rel="nofollow">http://adsabs.harvard.edu/abs/1980AmJPh..48..612M</a></p>
<p><a href="http://cartan.e-moka.net/content/download/248/1479/file/Astronave%20relativistica.pdf" rel="nofollow">http://cartan.e-moka.net/content/download/248/1479/file/Astronave%20relativistica.pdf</a></p>
<p>There are two formulas, one with D^2, and one with D^4 (D being the Doppler factor). The logic stated by the above paper seems to be the following:</p>
<p>Use D^2 for:</p>
<ul>
<li>comparing two detectors in relative motion</li>
<li>plane waves</li>
<li>stars</li>
</ul>
<p>Use D^4 for:</p>
<ul>
<li>comparing a moving source of radiation, relative to a fixed source</li>
<li>extended sources: galaxies, cosmic microwave background, astrophysical jets, and so on</li>
</ul>
<p>Question: is the above breakdown of when to use D^2 versus D^4 correct?</p>
<p>Here is the last paragraph of the paper in its entirety:</p>
<blockquote>
<p>It is hoped that this presentation can clarify the role of
the solid-angle transformation. Because it is a necessary
consequence of aberration, there is always a change in the
solid angle associated with light when one changes reference
frame (except for the case of idealized plane waves). Its
effect effect upon the transformation of intensity must be included
in the study of cosmic microwave background radiation, where the
angle is associated with convergence into the detector of
radiation from different parts of the primordial fireball. It
must also be included when a moving source is compared
to an equivalent fixed source, both at the same apparent
distance measured in the reference frame of the detector.
In this case the angle represents divergence from the compact
source to the detector. The change in solid angle does
not contribute to the transformation of intensity measured
at the same event point by detectors in relative motion. Note
particularly that the amount of divergence of the light, or
equivalently the curvature of the wavefronts, does not
determine which intensity transformation is correct. Thus, in
considering the appearance of stars from a moving spaceship,
a typical source distance is 1-1000 pc but the intensity
transformation is equivalent to that for plane waves. On the
other hand, typical receding galaxies are 10^6 times further
away (so their light can is actually more nearly plane), but
their light cannot be treated as plane waves in finding its
intensity transformation.</p>
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<p>Car batteries are usually 12 V. What is the difference between buying a car battery and hooking up a bunch of cheap household batteries in series? Both would register at 12 V. I assume that cars need much more current to start an engine then regular household things like lamps and toys. Does that mean that a car battery holds more charge within it?</p>
<p>If we think of a simple cell battery, where there are two electrodes and an electrolyte. One electrode eventually, through chemical reaction, becomes positive and the other becomes negative. Thus in a car battery, does that mean the electrodes (if we can reduce a car battery to a primitive cell), have more charge separated on each electrode? But if that was the case, wouldn't a greater charge separation mean that the voltage would be greater between the terminals as well?</p>
<p>In general, what is the relation between charge and voltage? I know the equation V = U/q, just like E = F/q (similar form, in the limit that the test charge is small as to not effect the PE or electric field. V = U/q is sorta useless too, since differences matter, but nevertheless that is how we defined it). If charges are separated further, does that mean greater voltage? If more charge is separated, does that mean greater voltage?</p>
<p>Lastly, what is the difference b/w the charge in a 12 V car battery, and the charge in a 12 V makeshift, series strung battery from home? </p> | 1,649 | [
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<p>So, we're fairly sure that the universe is infinitely large. We're also fairly sure that it is expanding. My question is how do these two facts relate to the multiverse? My favorite interpretation was given by Michio Kaku, who stated that in the multiverse each individual universe is like a bubble, and all of these bubbles are floating around in some space (and sometimes interact?). </p>
<p>How can there be an infinitely large object in a space with other presumably infinitely large objects? Is this in like math where there are different "sizes" of infinities, where some are larger than others? What is the space that the universe is expanding into? Does that have a boundary, or is it also some infinity? </p> | 1,650 | [
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<p>I'm mostly wondering about radio frequencies. I understand that voltage is the movement of electrons, and that the antenna acts as a light bulb, emitting at radio frequencies, following the reverse square law, some materials are opaque, some are transparent. Yet, at the receiver end, it's almost the same as having the two antennas connected, except with a voltage drop. Are photons and electrons the same thing? (It is called the electromagnetic spectrum). It's obviously not quite the same as electrons moving through the air, as this creates lightning, or something similar.</p>
<p>Is anything I've said incorrect? What happens when an electron goes through an antenna? </p> | 1,651 | [
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<p>I am learning about nuclear fissions and learned about the fission fragment distribution.
It was interesting to see that the fission fragments have unequal masses.</p>
<p>I was wondering as to what governs which fission products are generated in each fission?</p>
<p>What parameters decide which fission products will have higher yield?</p> | 1,652 | [
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<p>The name of the question is rather contradictory and counter-intuitive since sound is <em>produced</em> by vibration. However, very low frequencies around 32Hz and receding are bass. </p>
<p>From what I have read online, you can feel frequencies below 20Hz which is what we normally hear, yet you can hear as low as 12Hz while amplifying very high frequencies (above 20k Hz? I'm not an expert). So again, which frequency(ies) should I amplify in my equalizer to achieve this effect?</p> | 1,653 | [
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<p>So in this Youtube video <a href="http://youtu.be/ZjaaSUHG7Xo" rel="nofollow">http://youtu.be/ZjaaSUHG7Xo</a>
The guy says, around 6:15, that the total energy of the universe is zero because of gravitational potential (or something like that i dont understand completely which is why I'm asking this question) so since the totally energy of the universe is zero, the law of conservation would not have been violated by the universe coming into existence at the big bang. I heard about a similar argument made in one of Stephen Hawking's books. Could someone explain the idea to me in its fullness? Thanks. And if you disagree with the idea that the universe could have created itself, hopefully this goes without saying in a forum like this, please no responses based on religious dogma.</p> | 27 | [
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<p>Before I am inundated by myriad and vociferous claims that conservation of energy is the single most well-attested and experimentally verified principle in all of science, let me say that I am well aware of the ever-growing body of evidence which seemingly bears this principle out. However there seems to be, in my view at least, an abiding problem inherent to a principle of conservation with respect to how we could ever truly verify such a thing empirically. That no humanly constructed device can be perfectly accurate or precise is an incontrovertible and obvious truth. As a result, I am unsure as to how we could ever be sure that energy (or any other conserved quantity) is not being lost or gained (in defiant violation of physical principles) at length and/or time scales orders of magnitude below the resolution of our best instruments. I mean, how could we ever know? Ontologically speaking, it's almost inconceivable that energy could not be conserved. But epistemologically speaking, I am at a loss as to how we might ever verify—via experimentation or measurement—that energy is in fact being conserved to the precision guaranteed by a conservation law.</p> | 1,654 | [
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<p>This <a href="http://arxiv.org/pdf/hep-th/9303048v2.pdf" rel="nofollow">paper</a> describes a way to find the entanglement entropy of $N$ entangled harmonic oscillators, after tracing out the first $n$. A few statement made within have royally confused me, and I haven't been able to find a solution.</p>
<p>It starts at equation (10): I understand that $\Omega$ has to be a symmetric matrix with positive eigenvalues. I suppose it's useful to split it into parts, based on which oscillators are traced over and which aren't. I think this is the reason for the split into $A$ and $C$. Is this all that goes into the construction of $\Omega$, or am I missing something?</p>
<p>Then equation (11): comparison with equation (3) shows a lot of similarities, with $x$ and $x'$ now vectors and $\beta$ and $\gamma$ now matrices. However, I do not understand by what method the integrals are 'carried out explicitly', or how the matrices $\beta$ and $\gamma$ are constructed from the entries of $\Omega$.</p>
<p>At the top of page four, the author writes:</p>
<blockquote>
<p>To find [the eigenvalues], we note that the appropriate generalization of Eq. (4) implies that (det $G$) $\rho_{out}(Gx, Gx')$ has the same eigenvalues as $\rho_{out}(x, x')$, where $G$ is any nonsingular matrix.</p>
</blockquote>
<p>Equation (4) is the eigenvalue equation for the case of two oscillators:</p>
<p>$$\int_{-\infty}^{+\infty} dx' \rho_{out}(x, x') f_n(x') = p_n f_n(x)$$</p>
<p>where each possible state is indexed by $n$. I have no clue what this appropriate generalisation is, nor how the quoted statement follows. I first thought it was simply subsituting $x$ and $x'$ for their $N-n$ vectors, but I can't even begin to visualise what $\rho_{out}$ looks like if that's the case.</p>
<p>I have interpreted all the matrix transforms between here and equation (13) as just a way to eventually write the new reduced density matrix (of $N-n$ oscillators) as a product of reduced density matrices of 2 oscillators. If that is the case, then I understand this part.</p>
<p>My last question relates to the block of text directly beneath equation (13), specifically the entropy. The statement is that since the new density matrix can be written as a product, the entropy is simply the sum of the entropies of its parts. I've heard this statement before, but I run into trouble when I try to derive it. The statement is that</p>
<p>$$S = \sum_i S(\xi_i)$$</p>
<p>So lets say we look at the simple case where the total reduced density matrix $\rho$ is just the product of two reduced density matrices $\rho_1$ and $\rho_2$.</p>
<p>$$S = Tr[\rho \; log(\rho)] = Tr[\rho_1 \rho_2 \; log(\rho_1 \rho_2)] = Tr[\rho_1 \rho_2 \; log(\rho_1) + \rho_1 \rho_2 \; log(\rho_2)] \neq Tr[\rho_1 \; log(\rho_1)] + Tr[\rho_2 \; log(\rho_2)] = S_1 + S_2$$</p>
<p>Is this incorrect?</p>
<p>I realise this is a lot of work to answer, since you'd probably need to read the paper as well. Hence, I would be grateful for an answer to any of my questions.</p> | 1,655 | [
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] |
<h2>Background</h2>
<p>Let me start this question by a long introduction, because I assume that only few readers will be familiar with <a href="http://en.wikipedia.org/wiki/Coherence_theory" rel="nofollow">the theory of partial coherent light</a> and concepts like a mutual coherence function or a mutual intensity. The <a href="http://en.wikipedia.org/wiki/Polarization_(waves)#Coherency_matrix" rel="nofollow">coherency matrix</a> and <a href="http://en.wikipedia.org/wiki/Stokes_parameters" rel="nofollow">Stokes parameters</a> descriptions of partially polarized light are related concepts which are more widely known.</p>
<p>Correct treatment of partial coherent light is important for an appropriate modeling of optical pattern transfer in computer simulations of proximity and projection lithography as currently used by the semiconductor manufacturing industry. When I came to this industry, my previous optics "training" was insufficient in this area. I found chapter X "Interference and diffraction with partially coherent light" in <a href="https://archive.org/details/PrinciplesOfOptics" rel="nofollow"><em>Principles of Optics</em></a> by Max Born and Emil Wolf most helpful for filling my gaps in this area. Later, I also "browsed" through "Statistical Optics" by Joseph W. Goodman, which has a nice paragraph in the introduction explaining why insufficient familiarity with statistical optics is so common:</p>
<blockquote>
<p>Surely the preferred way to solve a problem must be the deterministic way, with statistics entering only as a sign of our own weakness or limitations. Partially as a consequence of this viewpoint, the subject of statistical optics is usually left for the more advanced students, particularly those with a mathematical flair.</p>
</blockquote>
<p>The interesting thing is that Hermitian matrices and eigenvalue decompositions like the Karhunen-Loève expansion are used quite routinely in this field, and they somehow feel quite similar to modeling of coherence and decoherence in quantum-mechanics. I know that there are important obvious (physical) difference between the two fields, but my actual question is what they have in common.</p>
<h2>Question</h2>
<p>Some elementary experiments like the double slit experiment are often used to illustrate the particle wave duality of light. However, the theory of partially coherent light is completely sufficient to describe and predict the outcome of these experiments. There are no particles at all in the theory of partially coherent light, only waves, statistics and uncertainty. The global phase is an unobservable parameter in both theories, but the amplitude of a wave function is only important for the theory of partial coherent light and is commonly normalized away in quantum-mechanics. This leads to a crucial difference with respect to the possible transformations treated by the respective theories. But is this really a fundamental difference, or just a difference in the common practices of the respective theories? How much of the strange phenomena of quantum-mechanics can be explained by the theory of partial coherent light alone, without any reference to particles or measurement processes?</p>
<hr>
<h2>More information on what I would actually like to learn</h2>
<p>One reason for this question is to find out how much familiarity with partial coherence can be assumed when asking questions here. Therefore it explains why this familiarity cannot be taken for granted, and is written in a style to allow quite general answers. However, it also contains specific questions, indicated by question marks:</p>
<blockquote>
<ul>
<li>How is the theory of partial coherent light related to quantum-mechanics?</li>
<li>... the amplitude of a wave function ... But is this really a fundamental difference, or just a difference in the common practices of the respective theories?</li>
<li>How much of the strange phenomena of quantum-mechanics can be explained by the theory of partial coherent light alone, without any reference to particles or measurement processes?</li>
</ul>
</blockquote>
<p>Don't be distracted by my remark about the double slit experiment. Using it to illustrate the particle wave duality of light seemed kind of cheating to me long before I had to cope with partial coherence. I could effortlessly predict the outcome of all these supposedly counter-intuitive experiments without even being familiar with the formalism of quantum-mechanics. Still, the outcome of these experiments is predicted correctly by quantum-mechanics, and independently by the theory of partial coherent light. So these two theories do share some common parts.</p>
<p>An interesting aspect of <a href="http://en.wikipedia.org/wiki/Coherence_theory" rel="nofollow">the theory of partial coherent light</a> is that things like the mutual intensity or the Stokes parameters can in principle be observed. A simple analogy to the density matrix in quantum-mechanics is the coherency matrix description of is partial polarization. It can be computed in terms of the Stokes parameters
$$J=\begin{bmatrix}
E(u_{x}u_{x}^{\ast})&E(u_{x}u_{y}^{\ast})\\
E(u_{y}u_{x}^{\ast})&E(u_{y}u_{y}^{\ast})
\end{bmatrix}=\frac12\begin{bmatrix}
S_0+S_1&S_2+iS_3\\
S_2-iS_3&S_0-S_1
\end{bmatrix}
$$
and hence can in principle be observed. But can the density matrix in quantum-mechanics in principle be observed? Well, the measurement process of the Stokes parameters can be described by the following Hermitian matrices $\hat{S}_0=\begin{bmatrix}1&0\\0&1\end{bmatrix}$, $\hat{S}_1=\begin{bmatrix}1&0\\0&-1\end{bmatrix}$, $\hat{S}_2=\begin{bmatrix}0&1\\1&0\end{bmatrix}$ and
$\hat{S}_3=\begin{bmatrix}0&i\\-i&0\end{bmatrix}$. Only $\hat{S}_0$ commutes with all other Hermitian matrices, which somehow means that each individual part of the density matrix can be observed in isolation, but the entire density matrix itself is not observable. But we don't measure all Stokes parameters simultaneous either, or at least that's not what we mean when we say that the Stokes parameters can be measured in principle. Also note the relation of the fact that $\hat{S}_0$ commutes with all other Hermitian matrices and the fact that the amplitude of a wave function is commonly normalized away in quantum-mechanics. But the related question is really a serious question for me, because the <a href="http://en.wikipedia.org/wiki/Mueller_calculus" rel="nofollow">Mueller calculus</a> for Stokes parameters allows (slightly unintuitive) transformations which seem to be ruled out for quantum-mechanics.</p> | 1,656 | [
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<p>I've been trying to find papers that list various properties of idealized cigarette smoke( ex. the molar mass, gas compressibility factor, density of smoke exiting a cigarette) but I have had very little luck.</p>
<p>Does anyone know what these properties are or where I should be able to find them?</p>
<p>I'm building a volumetric measurement tool for use in psychological studies in case anyone wonders why I'm looking for this information.</p>
<p>I posted this question in chemistry.stackexchange first but I think physics is a better forum for it.</p> | 1,657 | [
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<p>I would like some help to find</p>
<p>good and detailed books on the history of physics.</p>
<p>Which are the classics in this domain? Which are your favorite?</p> | 1,658 | [
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<p>I've just begun learning capacitance, and my lecture notes have a section on calculating capacitance for capacitors in vacuum of various shapes, e.g. two parallel plates and concentric spherical shells.</p>
<p><img src="http://i.stack.imgur.com/0s7wd.png" alt="the cylindrical capacitor"></p>
<p>For a cylindrical capacitor, comprising of a <em>long cylindrical conductor</em> with radius $r_a$ and linear charge density $+\lambda$, and a <em>coaxial cylindrical conducting shell</em> with radius $r_b$ and linear charge density $-\lambda$, to calculate its <em>capacitance per unit length</em>, the notes lay out the steps as follows:</p>
<ul>
<li>A point outside a long line of charge a distance r from axis has potential $V = \frac{\lambda}{2\pi\epsilon_0}ln\frac{r_0}{r}(1)$. Holds here also because charge on outer cylinder doesn't contribute to field between cylinders.</li>
<li>Then $V_{ab} = V_a - V_b = \frac{\lambda}{2\pi\epsilon_0}ln\frac{r_b}{r_a}$</li>
<li>Total charge $Q = \lambda L$ in a length $L$, so $C = \frac{Q}{V_{ab}} = \frac{2\pi\epsilon_0L}{ln(r_b/r_a)}$</li>
<li>Therefore $\frac{C}{L} = \frac{2\pi\epsilon_0}{ln(r_b/r_a)}$</li>
</ul>
<p>There are a couple of things in here which I don't understand:</p>
<ul>
<li>The first bullet mentions "charge on outer cylinder doesn't contribute to field between cylinders". Is this true? Shouldn't the electric field strength double since there is both a positive charge $+\lambda$ and a negative charge $-\lambda$?</li>
<li>The second bullet assigns $r_b$ as $r_0$ and $r_a$ as $r$ in equation $(1)$. Why is this so? As far as I understand, $r_0$ denotes the arbitrary distance where $V_b = 0$, which suggests that the potential on the coaxial shell is 0. Why is this not, say, the other way around? Why don't we take the potential at the edge of the inner cylinder as 0? Would that cause $r_b$ and $r_a$ to switch places in the fraction? Why is it possible to assign the potential on the coaxial shell as 0 in the first place?</li>
</ul> | 1,659 | [
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<p>I think of <a href="http://en.wikipedia.org/wiki/Coulomb%27s_constant" rel="nofollow">Coulomb's constant</a> as a conversion factor (not sure if this is correct).
Kind of like how you would do calculations in kg and then times it by the conversion constant to convert your answer to pounds.
The conversion factor would be $2.2\: \mathrm{lbs/kg}$.</p>
<p>Since the units for Coulomb's constant is $\mathrm{N \cdot m^2/C^2}$, would it make sense to define the Newton as:</p>
<p>$1\:\text{Newton} = \frac{1}{1/1\: \mathrm{meter^2} \cdot 1\: \mathrm{Coulomb^2}}$</p>
<p>Would the above definition be valid?</p>
<p><strong>EDIT:</strong> So if $k$ is not a conversion factor since the above definition for a Newton is invalid and $k$ is not just a scaling factor, since it has units, then what is it? If its just a proportionality constant to adjust the magnitude then why does it have units? Shouldn't it be a unit less constant?</p>
<p><strong>EDIT:</strong> So $k$ is not just a scaling factor (since it has units) and its not a conversion factor since a Newton can't be expressed as the other units. So if its unit just exists so that things cancel out "nicely" doesn't this make dimensional analysis useless since you can add in random constants and units to cancel out whatever you want? </p>
<p><strong>My question is not about the meaning of $k$. Its about its units.</strong> </p> | 1,660 | [
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<p>Suppose you have a quantum system with a Hamiltonian having some number (greater than 2, possibly infinite) of eigenfunctions, and that the system is prepared in the ground state.</p>
<p>When can you approximate it as by two-level system (using just the ground state and first excited state)? Is there some property that will make it better approximated by a two-level system (e.g., something like a bigger energy gap between the second and third energy levels)?</p> | 1,661 | [
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<p>I'm trying to calculate the 'instantaneous' semi-major axis of a binary system with two equal (known) mass stars for an $N$-body simulation. I know their velocities and positions at a given time, but am unsure how best to calculate the semi-major axis. I tried using the <a href="http://en.wikipedia.org/wiki/Vis-viva_equation" rel="nofollow">vis-viva equation</a> </p>
<p>$$v^2 ~=~ G(M_1 + M_2)\cdot(2/r - 1/a) $$</p>
<p>However I'm not sure if $r$ should be the distance between the two masses or the distance to the centre of mass and whether the $M$'s should be the masses or the sum and the reduced mass? Whatever I try the semi-major axis seems to oscillate over the orbit at around the correct value, but surely it should be constant? Any advice would be brilliant, thanks.</p>
<p>edit: Not allowed to answer my own question yet, but I'm an idiot and wasn't doing relative velocities, thanks guys.</p> | 1,662 | [
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<p>Are there any ways to convert an <a href="http://en.wikipedia.org/wiki/Autocorrelation" rel="nofollow">autocorrelation function</a> to a <a href="http://en.wikipedia.org/wiki/Radial_distribution_function" rel="nofollow">pair correlation function</a>, and vice versa?</p> | 1,663 | [
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<p>If a gas, such as hydrogen, is pressurized into an air tight container, a force in terms of pascals (or whatever unit you want to use) is exerted, correct? That is what pushes against every surface within the container. But what I don't understand is how the gas can constantly push against the walls without being supplied more energy. Does the force of pressure not need energy, or am I missing something? What about when the force is used to move something, such as in a hydraulics system? </p> | 1,664 | [
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<p>I worked in my masters thesis with $^{87}Rb$ and $^{40}K$, really small beta emitters. But there are so many other things around in the lab, that I want to keep track on all the things I might get in contact with.</p>
<p>Is there any computer program to calculate the dose of the whole decay chain to get a picture of the artificial radiation and supports logging. I don't want to look up all the individual numbers and calculate it manually.</p>
<p>Also in my apparatus various clusters (Cr, Ni, Co, Cu, Ag, Pd, Ca....) are produced. E.g. I know chromium(VI) is carcinogen, but the pure metal is not. But in the nanoworld things may change. So is there a database around where I can lookup the toxity of various substances, with an emphasize on nanoparticles? I use nitrile rubber gloves and try to do not inhale something if I clean the apparatus. But this might be not enough precaution. The laser dyes are not healthy too.</p> | 1,665 | [
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<p>So I know re-normalization has bean "beaten to death". I want to understand something a bit specific which might seem trivial. Independence of the bare parameters on $\mu$ and relevance to the beta function derivation. This seems to be an important assumption or rather fact that is built upon in making further renormalization and renormalization group arguments. In fact, even explicit statements about locality trace to this assumption. Again there is a chance this is an extremely lousy question, but can someone give me a precise mathematical argument describing this independence.</p>
<p>I am also looking to put together some reading lists comprising of the original papers treating the subject at a pedagogic level($\phi ^{\text{something}}$ theories). Not those directly addressing the QED problems yet still assuming ignorance of the reasons behind the divergences. Papers making explicit and hopefully proven statements accounting for some of the assumptions taken.</p>
<p>I want to know the deal with independence of the bare parameters. </p> | 1,666 | [
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<p>Just a simple question regarding $\epsilon$, the absolute permittivity. I read that it measures the resistance to of certain medium to 'permit' the formation of an electrical field. Does this mean that given a electrical field $E$, one needs more energy to produce it in a medium with a higher $\epsilon$ ? Thanks in advance. </p> | 1,667 | [
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<p>The atoms in my table "stick together" to form a rectangle. Why? What makes them stick together?</p>
<p>I know about ionic/covalent bonding etc., but consider a sheet of pure iron. Just atoms of one element. The atoms still stick together to form a sheet instead of being "everywhere".But if I place ten balls on the ground beside each other they are completely independent of each other.</p>
<p>So what "connects" the atoms?</p> | 1,668 | [
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<p>Say you have a neutral rod, and you bring a positively charged rod beside it (call the side the charged rod is brought near side A and the other side side B). The electrons from the side B will start moving towards side A and the positively charged nuclei in side A will start moving to side B. </p>
<p>After a large part of side A consists of only of electrons, the electrons would start repelling each other and the movement of charges will stop. But at this point a very large portion of side A consists of electrons and a large part of side B consists only of positively charged nuclei. The system is at equilibrium.</p>
<p>But the rod has now been mostly split into a electron side and a positively charged nuclei side, however it doesn't look different at all? Why? Shouldn't a rod made of atoms look and be completely different than a rod that is mostly made up of electrons on one side and nuclei on the other?</p>
<p>I know that electrons are very small, and since side A mostly consists of electrons, shouldn't it be almost invisible? </p> | 1,669 | [
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] |
<p>I would like to have a general interpretation of the coefficients of the stiffness matrix that appears in <a href="http://en.wikipedia.org/wiki/Finite_element_method">FEM</a>. For instance if we are solving a linear elasticity problem and we modelize the relation between a node $i$ and a node $j$ as a spring system, then $K_{i,j}$ (where $K$ is the stiffness matrix of the system) can be seen as the stiffness constant of the virtual spring between the two nodes. But does there exist a more general interpretation? Perhaps in terms of internal work? </p>
<p>Another similar question is: What could be an interpretation of the coefficients $M_{i,j}$ of the mass matrix $M$?</p> | 1,670 | [
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<p>I've seen the Kaluza-Klein metric presented in two different ways., cf. Refs. 1 and 2. </p>
<ol>
<li><p>In one, there is a constant as well as an additional scalar field introduced:
$$\tilde{g}_{AB}=\begin{pmatrix}
g_{\mu \nu}+k_1^2\phi^2 A_\mu A_\nu & k_1\phi^2 A_\mu \\
k_1\phi^2 A_\nu & \phi^2
\end{pmatrix}.$$</p></li>
<li><p>In the other, only a constant is introduced:</p></li>
</ol>
<p>$$\tilde{g}_{AB}=\begin{pmatrix}
g_{\mu \nu}+k_2A_\mu A_\nu & k_2A_\mu \\
k_2A_\nu & k_2
\end{pmatrix}.$$</p>
<p>Doesn't the second take care of any problems associated with an unobserved scalar field? Or is there some reason why the first is preferred?</p>
<p>References:</p>
<ol>
<li><p>William O. Straub, <em>Kaluza-Klein Theory,</em> Lecture notes, Pasadena, California, 2008. The pdf file is available <a href="http://math.arizona.edu/~vpiercey/KaluzaKlein.pdf" rel="nofollow">here</a>.</p></li>
<li><p>Victor I. Piercey, <em>Kaluza-Klein Gravity,</em> Lecture notes for PHYS 569, University of Arizona, 2008. The pdf file is available <a href="http://www.weylmann.com/kaluza.pdf" rel="nofollow">here</a>.</p></li>
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<p>Tensor equations are supposed to stay invariant in <em>form</em> wrt coordinate transformations where the metric is preserved. It is important to take note of the fact that invariance in form of the tensor equations is consistent with the fact that the individual components of the tensor may change on passing from one frame to another.[Incidentally,Preservation of the metric implies preservation of norm, angles etc.]</p>
<p>But in General Relativity the tensor equations (examples: the geodesic equation, Maxwell's equations in covariant form) are considered to be invariant in form when we pass from one manifold to another. The metric is not preserved in such situations.
Preservation of the value of the line element is consistent with the fact that $g_{\mu\nu}$ may be considered as a covariant tensor of second rank:</p>
<p>$ds'^2=ds^2$</p>
<p>${=>}g'_{\mu\nu}dx'^{\mu}dx'^{\nu}=g_{\alpha\beta}dx^{\alpha}dx^{\beta}$</p>
<p>${=>}g'_{\mu\nu}=g_{\alpha\beta}\frac{dx^{\alpha}}{dx'^{\mu}}\frac{dx^{\beta}}{dx'^{\nu}}$ </p>
<p>Rigorous Calculations:</p>
<p>$ds'^2=g'_{\mu\nu}dx'^\alpha dx'^\beta$</p>
<p>$=g'_{\mu\nu}\frac{\partial x'^\mu}{\partial x^\alpha}{d x^\alpha}\frac{\partial x'^\nu}{\partial x^\beta}{d x^\beta}$</p>
<p>$=g'_{\mu\nu}\frac{\partial x'^\mu}{\partial x^\alpha}\frac{\partial x'^\nu}{\partial x^\beta}{d x^\alpha}{d x^\beta}$</p>
<p>$=>g_{\alpha\beta}=\frac{\partial x'^\mu}{\partial x^\alpha}\frac{\partial x'^\nu}{\partial x^\beta}g'_{\mu\nu}$</p>
<p>Therefore $g_{\mu\nu}$ is a covariant tensor of rank two.</p>
<p>But in the above proof we have assumed the value of $ds^2$ as invariant wrt to our transformation.This not true when different types of manifolds are in consideration.</p>
<p>Non-conservation of the value of $ds^2$ will result in dismissing $g_{\mu\nu}$ as a second rank tensor of covariant type. This will be the situation if we pass from one manifold to another.*It is important to emphasize the fact that the problem will remain even if when we pass from an arbitrary manifold to flat spacetime in particular to the local inertial frame.*Differential considerations are not improving matters as indicated in the above calculation. The very concept of a tensor gets upset by considering different/distinct manifold. </p>
<p>What is the mathematical foundation for the invariance of form of the tensor equations in such applications where we consider different/distinct manifolds?</p> | 1,672 | [
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<p>My general premise is that I want to investigate the transformations between two distinct sets of vertices on n-dimensional manifolds and then find applications to theoretical physics by:</p>
<ul>
<li><p>Minimalizing the change in the fundamental shape of the vertices (I want a convex polytope to stay a convex polytope).</p></li>
<li><p>Describing the dynamics of the transformation. (Possible applications to theoretical physics)</p></li>
<li><p>Finding constraint conditions on what can and cannot occur when the space in which the transformation is occurring is not "regular". This includes not-everywhere-differentiable manifolds, singularities, and other pathological cases.</p></li>
</ul>
<p>Well, what an ambitious project. But in any case I have made some progress for the n-dimensional case with n-vertices where the space is "regular" and I haven't introduced advanced mathematics yet. I'm mostly playing around in configuration space for vertices in euclidean space or something similar.</p>
<p>I have defined a functional in configuration space that gives the length of all possible paths in configuration space between two distinct sets of vertices in $\mathbb{R^n}$ as follows:</p>
<p>$$T = \int\limits_{\lambda_{1}}^{\lambda_{2}}
\sqrt{\sum_{I=1}^{n} \sum_{i=1}^{d}
\left(\frac{d}{d\lambda}\left(\sum_{j=1}^{d} s(\lambda)R_{j}^{i}(\lambda)(q_{I}^{j}(\lambda) + a^{j}(\lambda))\right)\right)^2} d\lambda$$</p>
<p>Where $\lambda_{1}$ and $\lambda_{2}$ correspond to the start-point and end-point of the transformation. The members of the gauge group are $s$ (dilatation), $R$ (rotation), and $a$ (translation). $q_{I}^{j}$ corresponds to a point in configuration space representing $I$ vertices and it lives in $j$ dimensions. ($i$ is also a dimension index)</p>
<p>You can think of the equation as finding the straight line between two points (in configuration space) given all of the varied paths when you solve for constraint conditions in the Euler-Lagrange equation (recall):
$$\frac{\partial{f}}{\partial{x}} - \frac{d}{d \lambda}
\left(\frac{\partial{f}}{\partial{\dot{x}}}\right) = 0$$
This is represented in the figure below when the space has no curvature.</p>
<p><img src="http://i.stack.imgur.com/JFxCH.png" alt="Transformative Norm Equation Figure"></p>
<p>Hopefully, you have a general idea about what I'm talking about, this is at about page 5 in the paper where this is introduced, there is a lot of background that I'm not going to discuss. I've constructed an explicit example with tetrahedra where a tetrahedron is approximated to another tetrahedron in a minimalized manner. Here's a figure I made:
<img src="http://i.stack.imgur.com/xvn4u.png" alt="Tetrahedral Approximation"> that may help you understand what I'm trying to work with... in any case I'll ask my questions now.</p>
<p>-</p>
<p><em>[Optional for the intrigued reader]</em>:</p>
<p>How can I begin to generalize this geodesic equation so that it can comment on spaces with different metrics, or irregular behaviour? As an example of what I am trying to reach for, I would be interested in seeing how a set of vertices could geodesically traverse a singularity in a smooth 2-manifold, and if I could define a general functional for any behaviour on any manifold (I would attempt to restrict it to 2-dimensions to start) eventually. There are two main types of singularities I would be interested in investigating, the first would be the "black hole type" in which a diffeomorphism in the manifold causes "a hole at the bottom". This means that there would be an exponential increase in curvature around the vicinity of the singularity, and I want to find out how a set of vertices deals with those imposed conditions. The other would be for a 2-manifold with no curvature and I know that for a particular configuration of vertices, with the geodesic (based on the equation above) lying along a certain line. If I create a hole (simply remove a point in the same way you can create a hole in a single-variable piecewise defined real function), what happens to the geodesic? How can I encode this information, so it either pushes right through the hole somehow, or defines a new path around it?</p>
<p>-</p>
<p><strong>Main Question:</strong>
For a functional (assuming I could derive one) that gives the path from a distinct set of vertices to another in an n-dimensional smooth manifold with shape preserving characteristics.
When a singularity is introduced into the manifold, will the geodesic equation:</p>
<ul>
<li><p>Somehow push through the singularity/ignore it is present.</p></li>
<li><p>Define an alternate path around the singularity (this is what I hope, so I can investigate further with applications to black hole dynamics)</p></li>
<li><p>Become undefined and nothing useful results</p></li>
</ul> | 1,673 | [
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<p>I am looking for some references which discuss Fourier transform methods in GR. Specifically supposing you have a metric $g_{\mu \nu}(x)$ and its Fourier transform $\tilde{g}_{\mu \nu}(k)$, what does this tell you about the Fourier transform of the inverse metric $\tilde{g}^{\mu \nu}(k)$ or the Riemann tensor $\tilde{R}^{\mu}{}_{\nu \rho \sigma}(k)$. There are some obvious identities you can derive and I am looking for a references which discusses these and says if they are useful or not. </p>
<p>An example of what I mean is the following identity:</p>
<p>$g^{\mu \alpha}(x)g_{\alpha \nu}(x) = \delta^{\mu}_{\nu} \implies (\tilde{g}^{\mu \alpha} \ast \tilde{g}_{\alpha \nu})(k) = \delta^{\mu}_{\nu} \delta^{4}(k)$. (This can be made sensible on a compact manifold or for metrics which are asymptotically flat etc.)</p> | 1,674 | [
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<p>A symmetry is anomalous when the path-integral measure does not respect it. One way this manifests itself is in the inability to regularize certain diagrams containing fermion loops in a way compatible with the symmetry. Specifically, it seems that the effect is completely determined by studying 1-loop diagrams. Can someone give a heuristic explanation as to why this is the case? And is there a more rigorous derivation that "I just can't find any good way to regularize this thing."?</p>
<p>An alternative approach, due to to Fujikawa, is to study the path integral of the fermions in an instanton background. Then one sees that the zero modes are not balanced with respect to their transformation under the symmetries, leading to an anomalous transformation of the measure under this symmetry. Specifically, the violation is proportional to the instanton number, and thus one finds the non-conservation of the current is proportional to the instanton density. This is also found by the perturbative method above.</p>
<p>My question, which is a little heuristic, is how is it that the effect seems perturbative (and exact at 1-loop) on the one hand, and yet related to instantons, which are non-perturbative, on the other?</p> | 1,675 | [
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<p>What lessons do we have from string theory regarding the fate of singularities in general relativity?</p>
<p>What happens to black hole singularities? What happens to cosmological singularities?</p>
<p>Which points of view on string theory yielded results in this respect? String field theory? AdS/CFT? Matrix theory? I suppose perturbative string theory is not applicable in the vicinity of singularities.</p> | 1,676 | [
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<p>Consider the following scenario: On earth, I pulse a laser focused at a far away mirror such that the time it takes for the light to reflect off the mirror and arrive back at me is at least a few seconds. Immediately after the laser pulse, I place a second mirror where the reflected beam will arrive. In the photon's frame, it would appear that the universe is moving around passed the photon at the speed of light. Furthermore, time on the moving universe would be infinitely dilated. Thus, the photon would never see the second mirror being raised. So how does it reflect off of it?</p>
<hr>
<p>EDIT: To be clear, I don't think moving frame velocity being equal to the speed of light here is necessary. I've written a similar example to my question below:</p>
<p>Consider an electron accelerated to 0.9c and then released in some arbitrary direction. After some amount of time, let the electron encounter an infinitely strong potential such that it reflects back to its release point. Meanwhile, an experimentalist calculates when they expect the electron to arrive and turns on an electric field right before it arrives that will curve the electron's trajectory so that it collides into a detector. In the electron's frame, the universe's time is dilated so that it never sees the electric field being turned on. Thus, how does it interact with the field and get detected?</p> | 1,677 | [
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<p>A few months ago <a href="http://physics.stackexchange.com/q/78442/25794">I asked about phonons</a>. I got some very good answers but I still have difficulty getting an intuition for phonons, while somehow photons, which in many ways are similar and which I realize I hardly understand anything about, seem more accessible to intuition.</p>
<p>In <a href="http://physics.stackexchange.com/a/18564/25794">Ron Maimon's answer</a> to the question "What exactly is a quantum of light?" he asserts that </p>
<blockquote>
<p>A quantum of light of wavelength λ is the minimum amount of energy which can be stored in an electromagnetic wave at that wavelength</p>
</blockquote>
<p>and </p>
<blockquote>
<p>the classical wave is a superposition of a large number of photons</p>
</blockquote>
<p>Translating this to vibrations in a crystal lattice, could we say that a phonon is the minimal amount of energy which can be stored in an lattice vibration in a given mode and that a classical vibration is a superposition of a large number of phonons?</p>
<p>I hope I am correct when I say that the electromagnetic field can interact with matter through the absorption of a photon, and it is this interaction that makes the photon into something particle-like. Do we have the same for phonon-interactions? I.e. that when a crystal vibration interacts with matter it does so by the creation/destruction of whole phonons at a time, which may also get absorbed at more or less precise locations, e.g. the energy of a single phonon is absorbed by a localized electron. </p>
<p>Finally I would like to understand how phonon exchange can effectively establish an attractive force between electrons, but I cannot say I have any intuition for how photons mediate the electromagnetic force either. I am afraid that for the moment this is beyond the scope of my background.</p> | 1,678 | [
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<p>In quantum mechanics, we talk about (1) vectors, (2) states, and (3) ensembles (e.g., a beam in a particle accelerator). Suppose we want to translate this into mathematical definitions. If I'd never heard of the von Neumann density matrix, I'd approach this problem as follows. Two vectors can represent the same state if they differ only by a phase, so we should define states as equivalence classes of vectors that differ by a phase. However, I would not see any reason to go a second step and define a further level of equivalence-classing, in which a hydrogen atom in its ground state is considered to be equivalent to a beam of hydrogen atoms in their ground states.</p>
<p>Von Neumann is obviously a lot smarter than I am, and his notion of a density matrix appears to be universally accepted as the right way to describe a state. We use the same density matrix to describe one hydrogen atom or a beam of them. Can anyone offer any insight into why there seems to be no useful notion of state that works the way I'd have thought, rather than the way von Neumann did it?</p>
<p>Does it matter whether we're talking about classical QM or QFT? Do we not want to distinguish states from ensembles because in QFT particles can be created and annihilated, so fixing the particle number is not really what we want to do in defining the notion of a state?</p>
<p>Related: <a href="http://mathoverflow.net/q/117125/21349">http://mathoverflow.net/q/117125/21349</a></p> | 1,679 | [
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<p>I recently came across two nice papers on the foundations of quantum mechancis, Aaronson 2004 and Hardy 2001. Aaronson makes the statement, which was new to me, that nonlinearity in QM leads to superluminal signaling (as well as the solvability of hard problems in computer science by a nonlinear quantum computer). Can anyone offer an argument with crayons for why this should be so?</p>
<p>It seems strange to me that a principle so fundamental and important can be violated simply by having some nonlinearity. When it comes to mechanical waves, we're used to thinking of a linear wave equation as an approximation that is <em>always</em> violated at some level. Does even the teensiest bit of nonlinearity in QM bring causality to its knees, or can the damage be limited in some sense?</p>
<p>Does all of this have any implications for quantum gravity -- e.g., does it help to explain why it's hard to make a theory of quantum gravity, since it's not obvious that quantum gravity can be unitary and linear?</p>
<blockquote>
<p>S. Aaronson, "Is Quantum Mechanics An Island In Theoryspace?," 2004, <a href="http://arxiv.org/abs/quant-ph/0401062">arXiv:quant-ph/0401062</a>.</p>
<p>L. Hardy, "Quantum theory from five reasonable axioms," 2001, <a href="http://arxiv.org/abs/quant-ph/0101012">arXiv:quant-ph/0101012</a>.</p>
</blockquote> | 1,680 | [
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<p>Einstein postulated that the speed of light in free space
is the same for all observers, regardless of their motion
relative to the light source, where we may think of an
observer as an <em>imaginary entity</em> with a sophisticated set
of measurement devices, <em>at rest with respect to itself</em>,
that perfectly records the positions and times of all
events in space and time. </p>
<p>It is really exciting to understand Einsteins thoughts. I have a doubt with respect Einsteins above postulate, </p>
<ul>
<li>Is there any significance in saying an <em>observer</em> as an <em>imaginary entity</em>? I thought light could be called as some thing imaginary, rather than calling observer as an imaginary entity. </li>
<li>What does it mean <em>one being at rest with respect to one self</em>? I think it mean that, observer is not under motion due to its own virtue, but the observer may be under motion, if considered inside a moving frame of reference. I don't know whether I am right with respect to this, if I have misunderstood, please explain. </li>
<li>Why is <em>speed of light considered same for all observers (imaginary entity which is at rest with respect to itself) regardless of their motion relative to the light source</em>? I thought, if we consider observer inside a moving frame of reference, speed of light should be different for the observer, due to relative motion with respect to light. </li>
</ul> | 1,681 | [
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<p>We can see diffraction of light if we allow light to pass through a slit, but why doesn't diffraction occur if we obstruct light using some other object, say a block? Why are shadows formed? Why doesn't light diffract around the obstruction as it does around the slit?</p> | 1,682 | [
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<p>I have a friend who has just show me his medical prescription for hyperopia (farsightedness) correction and he needs glasses with 4,25 diopters for that, which seemed to be weird for me because I had learned, from the mirror equation, that the maximum correction possible for hyperopia is 4 diopters:</p>
<p>$$
\frac{1}{f} = \frac{1}{p} + \frac{1}{p'}
$$</p>
<p>If we have $0.25m$ for the normal eye distant point and more than $0.25m$ for the farsighted eye distant point (negative sign, because it's a virtual image), then we would have:</p>
<p>$$
\frac{1}{f} = \frac{1}{0.25} + \frac{1}{p'} = 4 - \frac{1}{|p'|} \in\quad ]0,4[, \quad\text{since}\quad |p'| \geq 0.25m \quad\text{and}\quad p'<0
$$</p>
<p>I did some google search and find out that, indeed, <a href="http://www.essilor.com/EN/EYEHEALTH/VISIONDEFECTS/Pages/Hyperopia.aspx" rel="nofollow">hyperopia can reach values even greater, such as 20 diopters</a>, but I can't find pages where doctors explain that with equations or physics teachers explain how things really work in ophthalmology.</p>
<p>Either I am doing some terrible mistake, or doctors are doing some terrible mistake, or this equation just don't apply to hyperopia at all... Which one is true?</p> | 1,683 | [
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