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<p>Consider the operator $$T=pq^3+q^3p=-i\frac{d}{dq}q^3-iq^3\frac{d}{dq}$$</p>
<p>defined to act on the Hilbert Space $H=L^2(\mathbb{R},dq)$ with the common dense domain $S(\mathbb{R})$. Here $S(\mathbb{R})$ denotes Schwartz space. </p>
<p>How would I show that this operator is hermitian? </p>
<p>I understand the procedure of using the scalar products, i.e. $\forall f,g\in S(\mathbb{R})$ $$<f,Tg>=<T^*f,g>=<Tf,g>$$</p>
<p>But, I am unsure if I could just apply complex conjuagtion right to $T$ to test the operator. It seems if I do this that the operator is not hermitian as the minus sign switches to positive. This leads me to believe this method is flawed and a valid test of the operator being hermitian. </p>
<p>The generalization of this operator to $$T_n=pq^n+q^np$$ and its importance as an example of the subtle difference between hermitian and self-adjoint matrices defined on infinite dimensional Hilbert Spaces $H$ and usage with deficiency indices led me bring the question up.</p> | 1,876 | [
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<p>I have read from many sources that the Fraunhoffer diffraction pattern is the Fourier transform of the grating. But I feel like these explanations do not take into account the curvature of the initial beam. </p>
<p>Let's say my grating is defined by $G(x)$. I have a beam $E(x)$ incident on the grating and want to know what my image looks like in the far-field.</p>
<p>The normal technique for modeling diffraction seems to be to just take a fourier transform: $\mathscr{F} (G(x))$. How do I interpret this in spatial variables? I can see that the answer makes sense in the Fourier domain, but how can I map this again onto the spatial domain without taking a trivial $\mathscr{F}^{-1}$ back to my original function?</p>
<p>Also what if my initial beam has some arbitrary curvature. This method does not utilize any information about the input beam. I've thought of the following variations:
$$
\mathscr{F}^{-1} \left[ \mathscr{F}\left[G(x)\right]\times \mathscr{F}\left[E(x)\right] \right]$$.
and
$$\mathscr{F}^{-1} \left[ \mathscr{F}\left[G(x) \times E(x)\right] \right]$$.</p>
<p>I am having some numerical issues in visualizing the results so I can't confirm my answers. I'd appreciate any explanation of whether these variations are sensible or suggestions of what books might be useful. I've browsed through Goodman and a couple of other derivations of diffraction patterns but found nothing that seems to talk about arbitrary initial beams.</p> | 1,877 | [
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<blockquote>
<p><strong>Q:</strong>
A $0.150\text{ kg}$ glider is moving to the right ($+x$) on a frictionless, horizontal air track with a speed of $0.80\text{ m/s}$. It has an elastic collision with a $0.300\text{ kg}$ glider moving to the left ($-x$) with a speed of $2.20\text{ m/s}$.</p>
<p><strong>a.)</strong> What is the initial momentum of each glider? Express the momentum in terms of unit vectors.</p>
<p><strong>b.)</strong> Use the relative velocity formula to find $v_{2f}$ in terms of $v_{1f}$.</p>
<p><strong>c.)</strong> Use the relative velocity result to solve conservation of momentum to find the velocity (magnitude and direction) of each glider after the collision. </p>
</blockquote>
<p>I've figured part (a) using the definition of momentum: $p=mv$: </p>
<p><strong>1st glider</strong>: $$p_1 = m_1 v_1 = (0.15\text{ kg})(+0.8\hat{x}\text{ m/s}) = +0.12\hat{x}\text{ kg m/s}$$</p>
<p><strong>2nd glider</strong>: $$p_2 = m_2 v_2 = (0.3\text{ kg})(-2.20\hat{x}\text{ m/s}) = -0.66\hat{x}\text{ kg m/s}$$</p>
<p>Parts (b) and (c) are what have me confused at the moment. I'm not positive I have the equations for relative velocity right nor how to solve for $v_{2f}$ in terms of $v_{1f}$. My book lists this an equation that can be gotten from manipulation of a kinetic energy equation: $v_{1i} – v_{2i} = -(v_{1f} - v_{2f})$. Is this the relative velocity formula? would just isolating $v_{2f}$ in this equation be solving for $v_{2f}$ in terms of $v_{1f}$?</p>
<p>Any help appreciated. thanks!</p> | 1,878 | [
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<p>I'm trying to understand the way that Hamilton's equations have been written in <a href="http://pubs.rsc.org/en/content/articlelanding/2010/fd/b902479b" rel="nofollow">this paper</a>. It looks very similar to the usual vector/matrix form of Hamilton's equations, but there is a difference.</p>
<p>$$\frac{{\bf dZ}(t)}{dt} ~=~ J \frac{\partial H({\bf Z(t)})}{\partial z},
\qquad\qquad{\bf Z(0)} ~=~ z,$$</p>
<p>where $J$ is the block matrix $((0,1),(-1,0))$. $\bf{Z}(t)$ represents the point in phase space (positions and momenta). The part I don't understand is that the derivative of $H$ is taken with respect to the initial value (lowercase $z$) of ${\bf Z(t)}$. How does this form follow from the usual way of writing Hamilton's equations in vector form?</p>
<p>Note that my question is not about the vector notation for Hamilton's equations, which can be found in any introductory textbook on classical mechanics. I am specifically asking about the derivatives with respect to $z$, the initial values of ${\bf Z(t)}$.</p>
<p>Edit: I could not find an arxiv version of the paper, but it seems to be available as the first publication on <a href="http://www.uam.es/personal_pdi/ciencias/rdelgado/publications.html" rel="nofollow">one of the authors' websites</a>, under the section "Coarse-graining with proper dynamics." The equation in question is the first one in section II of the paper.</p> | 1,879 | [
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<p>This image from wikipedia, explains that there occurs a potential drop across a <a href="http://en.wikipedia.org/wiki/P%E2%80%93n_junction" rel="nofollow">pn semiconductor junction</a>, and an electric field confined to the depletion region.<img src="http://i.stack.imgur.com/7VaN3.png" alt="enter image description here"></p>
<p>I already know the reason for the existence of this drop and the calculation of the difference but I have two questions regarding this drop. </p>
<p><strong>1) If the <a href="http://en.wikipedia.org/wiki/Doping_%28semiconductor%29" rel="nofollow">n and p doped</a> regions are externally connected using a perfectly conducting wire, why will not any current flow?</strong><br>
Connecting the two regions should equalize their potentials (since wire is a resistance less) conductor, and therefore the gradient at the junction is destroyed resulting in a <a href="http://en.wikipedia.org/wiki/Diffusion_current" rel="nofollow">diffusion current</a> which is obviously against conservation of energy as the semiconductor has non-zero resistivity. Where will extra potential drops will be created so that Kirchoff's voltage rule holds without any current and the built-in potential difference $V_o$ of the juction persists? </p>
<p><strong>2)</strong> (probably naive) If an external Bias is applied, of say $|V|<|V_o|$, then the pn potential difference across the junction will just reduce by that amount ($|V|$). Assuming that the external voltage source is ideal without any resistance, <strong>what will be the potential drops which would sum to zero in this case? Will drop due to the resistance of semiconductor play a par in it? If yes, then there cannot be any ideal resistance-less semiconductor junction, can there?</strong></p> | 1,880 | [
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<p>For example, let us examine the case of quantum (discrete) fourier transform.</p>
<p>There are $2^N$ samples. How do we initialize these $2^N$ samples into $N$ qubits? I have a hard time understanding this. </p> | 1,881 | [
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<p>Based on the previous <a href="http://physics.stackexchange.com/questions/80294/is-the-su2-flux-defined-in-the-context-of-projective-symmetry-grouppsg-an">question</a> and the comment in it, imagine two different mean-field Hamiltonians $H=\sum(\psi_i^\dagger\chi_{ij}\psi_j+H.c.)$ and $H'=\sum(\psi_i^\dagger\chi_{ij}'\psi_j+H.c.)$, we say that <em>$H$ and $H'$ are gauge equivalent if they have the same eigenvalues and the same projected eigenspaces</em>. And the Wilson loop $W(C)$ can be defined as the trace of matrix-product $P(C)$(see the notations <a href="http://physics.stackexchange.com/questions/80294/is-the-su2-flux-defined-in-the-context-of-projective-symmetry-grouppsg-an">here</a> ). Now my questions are:</p>
<p>(1)"$H$ and $H'$ are gauge equivalent" if and only if "$W(C)=W'(C)$ for all loops $C$ on the 2D lattice ". Is this true? How to prove or disprove it?</p>
<p>(2)If the system is on a 2D torus, is $W(L)$ always a <em>positive real number</em> ? Which means that the 'total flux'(the phase of $W(L)$) through the torus is quantized as $2\pi\times integer$, where $L$ is the boundary of the 2D lattice.</p>
<p>(3)If the Hamiltonian contains extra terms, say $H=\sum(\psi_i^\dagger\chi_{ij}\psi_j+\psi_i^T\eta_{ij}\psi_j+H.c.+\psi_i^\dagger h_i\psi_i)$, is the Wilson loop still defined as $W(C)=tr(P(C))$?</p>
<p>Thanks a lot.</p> | 1,882 | [
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<p>I am stuck in this problem-</p>
<p><img src="http://i.stack.imgur.com/CBC4d.png" alt="The Problem"></p>
<p>I need to find the velocity of <a href="http://en.wikipedia.org/wiki/Efflux" rel="nofollow">efflux</a> at the hole of the container. [We can assume that the area of the hole is negligible in comparison with the base area of the container].</p>
<p>Here's my approach</p>
<p>Velocity of liquid at the upper-surface = $v_2$</p>
<p>Velocity of efflux (velocity of water at the hole, right?) = $v_1$</p>
<p>Using Bernoulli's equation for the surface and the hole - </p>
<p>$$
P_{atm} + \rho_2 g (h_1 + h _2) + \frac{1}{2}\rho_2 v_2^2 = P_{atm} + \rho_1 g h_1 + \rho_2 g h_2 + \frac{1}{2}\rho_1 v_1^2 \\
\implies \rho_2 g h_1 + \rho_2 g h_2 - \rho_1 g h_1 - \rho_2 g h_2 = \frac{1}{2}(\rho_1 v_1^2 - \rho_2 v_2^2) \\
\implies \frac{1}{2}(\rho_1 v_1^2 - \rho_2 v_2^2) = g h_1 (\rho_2 - \rho_1)
$$</p>
<p>Now, let area of the base be $A_2$ and that of the hole be $A_1$</p>
<p>then, using equation of continuity,</p>
<p>$$
A_1 v_1 = A_2 v_2 \\
\implies v_2 = \frac{A_1}{A_2} v_1 \\
\implies v_2 \approx 0 (\because {A_1 << A_2})
$$</p>
<p>Using this value in the previous Bernoulli's relation</p>
<p>$$
\frac{1}{2}(\rho_1 v_1^2) = g h_1 (\rho_2 - \rho_1) \\
\implies \frac{1}{2} \rho_1 v_1^2 = g h_1 (\rho_2 - \rho_1) \\
\implies v_1 = \sqrt {\frac {2gh_1(\rho_2 - \rho_1)}{\rho_1}}
$$</p>
<p>Which is not the correct answer.</p>
<p><a href="http://chat.stackexchange.com/transcript/message/12205725#12205725">I did get a correct answer in the chat room, but it was using a different method.</a></p>
<p>What's wrong with my method?</p> | 1,883 | [
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<p>I recently read R.P Feynman's <em>QED:A Strange Theory of Light and Matter.</em> It is believed that time travel to the past is not possible. Then why is particles going backward in time considered in the book while drawing Feynman diagrams?</p> | 1,884 | [
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<p>Let me start by apologizing if this question seems pedantic and say that I'm not very familiar with physics in general, as I'm a math major instead.</p>
<p>Anyway, say a body changes from temperature $T_1$ to $T_2$, with $T_2 \ge T_1$.</p>
<p>Then the change in temperature is
$$\Delta T = T_2 - T_1$$</p>
<p>Now, it's clear that if $\Delta T = x\text{K}$ then $\Delta T = x \text{°C}$, with $x \ge 0$.</p>
<p>But it's also clear that $x \text{K} \ne x \text{°C}$, which leads to a contradiction.</p>
<p>Then I don't really understand why are units typically written in $\Delta T$? I suppose it could be to illustrate the units used to measure $T_1$ and $T_2$, but is it really necessary?</p> | 1,885 | [
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<p>I really don't understand with the linearity conditions I have where this comes from.</p> | 1,886 | [
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<p>What are <a href="http://www.google.com/search?q=Ramsey+interaction" rel="nofollow">Ramsey interactions</a>? I am researching atomic clocks and am not sure why the atoms need to be exposed twice to an electromagnetic field in order to cause excitation.</p> | 1,887 | [
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<p>Dose anyone has a clue what <a href="http://www.google.com/search?q=symmetric+fission" rel="nofollow">Symmetric Fission</a> is?</p>
<p>I couldn't find any explanation on what is it on internet.</p> | 1,888 | [
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] |
<p>Can <a href="http://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensate" rel="nofollow">Bose-Einstein condensate</a> be written as non-linear wave equation (in terms of mean field approximation theory)?
the equation is: <img src="http://i.stack.imgur.com/X8kEs.png" alt="enter image description here"></p>
<p>source: <a href="http://xxx.tau.ac.il/abs/1308.2288" rel="nofollow">http://xxx.tau.ac.il/abs/1308.2288</a></p>
<blockquote>
<p>What I do understand by the Bose-Einstein condensate is, it is the
state of atoms at very low temperatues and at the moment the atoms
forget their previous identity. Therefore all the atoms stay in the
same quantum state.</p>
</blockquote>
<p>Am I right? </p> | 1,889 | [
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<p>I understand that an event, in a four dimensional space-time, produces a light cone. As time increases the cones gets larger on either side of the event (past and future). For example the if the sun where to "go out" it would take 8 minuets for the earth to be affected by it simply because it takes approximately 8 minuets for light from the sun to reach the earth due to its location in the future light cone of the event (the sun being the event). </p>
<p>Einstein made a suggestion that space-time is warped (vs flat) by the distribution of mass and energy and that bodies (like earth, jupiter, etc.) are meant to follow straight paths but cannot in a warped space (or appear not to because space is warped). However light supposedly follows these warped paths (called geodesics) as well even tho light is energy. </p>
<p>How can space bend light? and why does light have to follow a specific path which is warped by space? Things with mass (like water) have to go around things (like rocks), but light can go through certain things or expand, but it doesn't move to the side like water does to a rock. Does it?</p> | 1,890 | [
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<p>This probably includes a little of biology, but I believe it's mainly physics, so I hope it's ok to ask here:</p>
<p>Imagine 2 persons: Person A weights 120 pounds and person B weights 180 pounds.</p>
<p>Imagine they both fall from a big height:<br>
a) landing on their knees (a good jump)<br>
b) falling on their back (a bad fall)</p>
<p>For both of those persons, in both situations, for which of them the fall is easier to handle?</p>
<p>I reckon:</p>
<ol>
<li>when falling down, the speed is not relative to weight, therefore both those persons will hit the ground at the same speed</li>
<li>but, I assume, as the heavier person will create bigger force to the ground, the ground will hit him back with the same force, therefore a heavier person should get a bigger blow when hitting the ground</li>
<li>now, this one I'm not sure about, but I assume that a heavier person (considering it's not an unhealthy, fat person, but a healthy one, with muscles, etc) is able to withstand a greater force/blow</li>
</ol>
<p>Assuming the last assumption is correct, how are assumptions 2 and 3 related? Is there anything I missed?</p>
<p>Overall, which person can withstand falls from greater heights: a lighter one, or a heavier one?</p> | 1,891 | [
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<p>The formula used in Gyrochronology that relates a star's Period of Rotation-Mass-Age is empirical? </p>
<p>This news <a href="http://kepler.nasa.gov/news/nasakeplernews/index.cfm?fuseaction=ShowNews&NewsID=129" rel="nofollow">How to Learn a Star’s True Age</a> </p>
<blockquote>
<p>"“A star’s rotation slows down
steadily with time, like a top
spinning on a table, and can be used
as a clock to determine its age,"</p>
</blockquote>
<p>pointed to this paper <a href="http://arxiv.org/abs/1104.2912" rel="nofollow">THE KEPLER CLUSTER STUDY: STELLAR ROTATION IN NGC6811</a><br>
and later I found <a href="http://www.lowell.edu/media/content/release_supplements/gyro_background.pdf" rel="nofollow">this one (gyro_background)</a> with original work. </p>
<p>But I couldnt find a justification for the formula. Is the Period proportional to $age^{{1/2}}$ only an empirical result? </p>
<p>It seems to me that the formula is a data fit and not a direct result of a calculation of the stellar intrinsic dynamics. The rate of mass loss by radiation must have a 'word to say' in the formula.<br>
Any help is welcome. </p> | 1,892 | [
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<p>For the purpose of this question, let's restrict ourselves to BKL singularities. BKL cosmologies are homogeneous Bianchi type XIII and IV cosmologies which exhibit oscillatory chaotic behavior, although that's not relevant to this question. Most generic singularities can be approximated locally by a BKL solution. The volume of a BKL universe decreases linearly with time as the singularity approaches. If the matter falling toward the singularity has a nonzero entropy, and the second law of thermodynamics is satisfied, the entropy density will increase without limit as the singularity approaches. Can the entropy density exceed the inverse Planck volume? Being inside a black hole, the holographic bound does not apply. If entropy densities beyond the inverse Planck volume are forbidden, is the second law violated?</p> | 1,893 | [
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<p>I remember once getting new Teflon (non-stick) cookware; however, when I tried this new cookware on my induction cooker the <em>cookware did not heat up</em>. My regular steel cookware worked just fine on the induction cooker, both before and after trying the Teflon cookware. I've heard from at least two other people that they've noticed the same thing with Teflon cookware and induction cookers (at least one tried the same exact brand and style of cookware as myself). And so, I wonder:</p>
<p>Did my Teflon cookware not heat up with the induction cooker because of the Teflon or the process it went through to be Teflon coated? In other words, does Teflon cookware in general not work with induction cookers? Or is it possible that my Teflon cookware just happened to not be made of a ferrous material and thus the induction cooker had no effect on it, with or without the Teflon? </p>
<p>If it is the case that Teflon in no way affects the performance of the cookware on an induction cooker, is it possible that a majority (or all) of Teflon cookware is made of non-ferrous material (perhaps because it is easier to get the Teflon to stick to aluminum than is to stick to steel)? In this way Teflon cookware would not work on induction cookers but not because of the Teflon.</p>
<p>I unfortunately no longer live in the same apartment as the induction cooker or the Teflon cookware so I cannot verify whether the cookware is ferrous or not myself.</p> | 1,894 | [
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<p>I'm sorry if this is somewhat a dumb question.</p>
<p>First: "Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as <em>linear transformations of vector spaces</em>"</p>
<p>I know little about particle physics, but to what I know, physicists only deal with the groups of (linear) symmetric operators acting on vector space of states.</p>
<p>So in fact physicists are dealing with the <em>italic</em> part of the representation theory. Why would them bring it in? What is a significance of the action "representing an element of a group as linear transformation" in the work of physicists, who are already dealing with groups of linear transformations?</p> | 1,895 | [
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<p>I'm using the SOFTSUSY package to generate the sparticle spectrum at the EW scale. One of the input parameters is the ratio of the up and down-type Higgs vevs commonly known as $\tan\beta$. The $\mu$ parameter is computed as an output by constraining the Z-boson mass to be compatible with experiment. I was wondering if there a quick way to figure out the value of the $B\mu$ parameter from $\mu$ and $\tan\beta$. Even an approximate analytic expression would be nice to have.</p> | 1,896 | [
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<p>I was taught today that the Electromagnetic wave Theory is unable to explain black body radiation. The example that was given to me: When a metal is heated, it emits different frequencies of light as it gets hotter. If electromagnetic wave theory was correct, it would not be so, the frequency (color) of light would remain the same, but only the intensity will change. I don't understand why this is so. </p>
<p>My logic:
Electromagnetic waves occur when a charged body oscillates in a electric and magnetic field. If the metal is provided with more energy (in the form of heat) won't the charged body vibrate faster, thus changing the frequency of the light emitted?</p> | 1,897 | [
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<p>From the <a href="https://en.wikipedia.org/wiki/Gross%E2%80%93Pitaevskii_equation" rel="nofollow">Gross-Pitaevskii equation</a>
\begin{equation}i\hbar\frac{\partial\psi}{\partial t}=\left(-\frac{\hbar^2}{2m}\nabla^2+V+g|\psi|^2\right)\psi\end{equation}
using the variational relation
\begin{equation}i\hbar\frac{\partial\psi}{\partial t}=\frac{\partial\varepsilon}{\partial \psi^*}\end{equation}
we find the energy density
\begin{equation}\varepsilon=\frac{\hbar^2}{2m}|\nabla\psi|^2+V|\psi|^2+\frac{g}{2}|\psi|^4\end{equation}
The energy would be $E=\int d^3r \varepsilon$
and this is a prime integral of the motion, meaning it is a conserved quantity.</p>
<p>My questions are:</p>
<p>1) How do we get the variational relation?</p>
<p>2)How can we prove that $E$ is a conserved quantity?</p> | 1,898 | [
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<p>I've been looking deeply into many bibliographic references without finding the answer. I would be interested in knowing the numerical value of the critical 2d XY spin model on triangular lattice. Being derived analytically (if possible) or from Monte-Carlo doesn't matter.</p> | 1,899 | [
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<p><img src="http://i.stack.imgur.com/jKBIc.png" alt="enter image description here"></p>
<p>I have trouble interpreting this illustration. I see why <strong>r</strong> (position) and <strong>a</strong> (acceleration) are the way they are, but what happened to <strong>v</strong>? Why is it smaller than its coordinates? Is this another error in my textbook?</p> | 1,900 | [
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<p>I'll break this down to two related questions:</p>
<p>With a fission bomb, Uranium or Plutonium atoms are split by a high energy neutron, thus releasing energy (and more neutrons). Where does the energy come from? Most books I've ever come across simply state e=mc2 and leave it at that. Is it that matter (say a proton or neutron) is actually converted into energy? That would obviously affect the elements produced by the fission since there would be less matter at the end. It would also indicate exactly how energy could be released per atom given the fixed weight of a subatomic particle. I remember hearing once that the energy released is actually the binding energy that previously held the nucleus together.</p>
<p>With a fusion bomb, two hydrogen isotopes are pushed together to form helium and release energy - same question: where does this energy come from? Is matter actually converted or are we talking about something else here?</p>
<p>Sorry is this is rather basic - I haven't done physics since high school.</p> | 1,901 | [
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<p>What is the exact meaning of the word <a href="http://en.wikipedia.org/wiki/Vacuum">vacuum</a>? Is it just a state of very low pressure or is it nothingness (as in there is nothing)? Also, when we say space is vacuum - it must be referring to pressure as space has light travelling (which means photons) besides the big masses of comets, planets, stars.</p> | 1,902 | [
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<p>The following formula has been given in 't Hooft's black holes notes ($|\Omega \rangle$ is the vacuum state of Minkowski space, O is a operator):</p>
<p>$$\langle \Omega| O|\Omega \rangle = \sum_{n \ge 0} \langle n | O | n \rangle e^{-2 \pi n \omega}(1-e^{-2 \pi \omega})=Tr(O \rho_{\Omega})$$</p>
<p><strong>How does this mean that the radiation is thermal and follows Plancks black body law?</strong></p>
<ol>
<li><p><a href="https://www.gl.ciw.edu/static/users/bmilitzer/diss/node13.html" rel="nofollow">Here</a>, I read that $\langle O \rangle = \frac{1}{Z}\sum_n e^{-\beta E_n}\langle n |O|n\rangle$. How is this sum the same as that in the above expression?</p></li>
<li><p>How is the density matrix related to the law of black body radiation? How can I derive Planck's law from the expectation value of O in the first expression?</p></li>
</ol> | 1,903 | [
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<p>What is the difference and similarities between Stueckelberg mechanism and Higgs mechanism?
They both make the gauge field massive. Is the Stueckelberg mechanism a special case about U(1) gauge fields of Higgs mechanism? Does there exist Spontaneously symmetry breaking in Stueckelberg mechanism?</p> | 1,904 | [
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<p>A giant set of bar bells floating in space (like two identical sized planets connected by a long rod) would have a centre of mass midway between the two on the connecting rod. But surely it would have two centres of gravity, one at each end? If you were standing on one of the "bells" or planets, and threw a rock in the air, it wouldn't fly to the middle of the rod, surely?
And if I'm correct, then say, a big wobbly jelly shaped planet would also have multiple points of gravity.
We have to except a sphere on which the centre of mass and gravity are the same.</p>
<p>My interest in dark matter was brought about by a friend who explained that the observed mass was calculated with reference to the centres of galaxies - but in what sense can a galaxy have a centre if the above confusions come into play? Isn't a galaxy like a set of interconnected barbells? Is there really a "centre" for gravitational calcuations?</p> | 1,905 | [
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<p>What would be the general form of Lagrange Equation when instead of a scalar field we have a vector potential?
has anyone derived the klein gordon equation for a corresponding vector potential Lagrangian? </p> | 1,906 | [
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<p>I am quite confused by both these terms. I would like to know what's the exact difference between both these terms and which one is more accurate.</p> | 1,907 | [
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] |
<p>What makes the two 'color-neutral' gluons
$(r\bar r−b\bar b)/\sqrt2$ and
$(r\bar r+b\bar b −2g\bar g )/\sqrt6$ different from the pure $r\bar r +b\bar b +g\bar g $ ?</p>
<p>Why don't they result in long range (photon-like) interactions?</p> | 296 | [
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<p>Suppose you have a solid ball on a horizontal table.</p>
<ol>
<li>What is the direction of friction force when the ball I pushed horizontally and <strong>starts rolling</strong>?</li>
<li>Why is the direction of friction as it is? </li>
<li><p>Which forces acts at the contact point between delta time t0 to t1? (If we divide friction force in sub forces)</p>
<p>V=1Vx m/s</p>
<p>Fx=?</p></li>
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<p>Given the equation
$$F_{\text{net}} = ma$$</p>
<p>Does this not imply that if the net force on a certain object is positive, its acceleration will also be positive, and theoretically this object would accelerate forever to an infinite velocity?</p>
<p>i.e
envision a block on a surface. If a person were to apply a force (push) on this block that exceeds friction, ad continuously applies this force, this block should technically accelerate infinitely? </p> | 1,909 | [
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<p>Without any a priori knowledge of the mass, speed, distance, and size of local celestial bodies (aside from Earth's size), what can I calculate and how from my "backyard" through observation?</p>
<h3>Edit:</h3>
<p>My goal is to create a high-school level physics project that students can conduct over the course of a year/term as both a physics problem mixed with a bit of astrophysics history. Core concepts would include observation (within reason). This isn't meant to be extremely precise work (within an order of magnitude would be excellent), but to demonstrate the methods.</p>
<p>Given a bunch of knowns, it's trivial to find out one unknown, but I am unsure of how to start from square one like the early physicists and astronomers did in times past.</p> | 1,910 | [
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<p>The title sums it pretty much. Are all <a href="http://en.wikipedia.org/wiki/Diffeomorphism" rel="nofollow">diffeomorphism</a> transformations also <a href="http://en.wikipedia.org/wiki/Conformal_map" rel="nofollow">conformal transformations</a>?</p>
<p>If the answer is that they are not, what are called the set of diffeomorphisms that are not conformal?</p>
<p>General Relativity is invariant under diffeomorphisms, but it certainly is not invariant under conformal transformations, if conformal transformations where a subgroup of diff, you would have a contradiction. Or I am overlooking something important?</p> | 1,911 | [
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<p>I am a math teacher and I have to teach a topic called "Bruchterme" and "Bruchgleichungen" in german (I don't know the english word for it). For example </p>
<p>$$
\frac{x^2 - 3}{(x - 2)x^2} + \frac{4}{x} + 2
$$</p>
<p>is a "Bruchterm" and </p>
<p>$$
\frac{4x}{2x -3} = 4 - \frac{2x}{x-1}
$$</p>
<p>is a "Bruchgleichung".</p>
<p>Students have to learn, how to determine for which $x$ the term or equation is defined (i.e. the "singularities") and how to solve such equations. </p>
<p>Now my problem is that most textbooks about this offer little to no interesting applications of this, especially no applications of determining for which $x$ the term is defined.</p>
<p>Now I am looking for interesting examples from classical physics or engineering for this type of problem. Especially examples where singularities occur and are physically "interesting" in some way. Though it is for high school level, I don't want to restrict the question to trivial examples, but to classical physics.</p>
<p>The only physical applications I have in mind are the following: </p>
<p>Two parallel resistors $\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$<br>
Same type of equation: $\frac{1}{f} = \frac{1}{b} + \frac{1}{g}$ for the thin lens equation (where $b$ and $g$ are the distances from the image to the lens resp. from the object to the lense)</p>
<p>However here the denominators are rather trivial and I don't see why it would be interesting to determine for which value for example of $b$ (if the other values are fixed) the equation is not defined. </p>
<p>Gravitational force ($\propto \frac{1}{r^2}$) here one might discuss that the force goes to infinity for $r \to 0$. However the point of view of the curriculum is not to discuss such limits but just to look if one is allowed to plug in certain values for the variable or not (in a sense that dividing by 0 is not allowed or not defined without connection to limits). </p>
<ol>
<li><p>Are there some interesting examples from physics where students see that it is worth to learn how to solve such equations (from type and perhaps complexity as in my example above)</p></li>
<li><p>Are there examples from physics where students see that it is worth to learn to determine for which values the term or equation is defined?</p></li>
</ol>
<p><sub><strong>Edit (too long for a comment)</strong></sub></p>
<p><sub>
@Danu: I am teaching in the less known part of the german education system, called "Berufliche Schulen" more specifically I am at a school which focus is on technology, engineering and science. The system is a bit complicated with classes on very different levels. To make it short I have the motivation problem described above in a low level class and in a high level class (the level of the second one is equivalent to a German "Gymnasium" or even a bit higher in mathematics and physics) but the students are generally very interested in technology, engineering, physics and computer science (most of them want to take up a profession in this fields. Ranging from electricians or lab technician in the lower level classes up to enegineers or physicists in the higher level classes). Students are from 15 to 18 years old. Even in the lower level classes they are very interested in nontrivial physics even if they cannot grasp it conceptually or mathematically. So for motivation purposes some nontrivial, but for the students interesting examples would be good.
</sub></p> | 1,912 | [
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<p><strong>Explain me these projections please</strong></p>
<p><strong>Context:</strong> I was reading a paper (<a href="http://pra.aps.org/abstract/PRA/v68/i5/e052307" rel="nofollow">Phys. Rev. A 68, 052307</a>) which involved <strong>mesoscopic coherent states of light</strong>. There, in order to calculate the uncertainty of a single <strong>Poincaré angle</strong>(see polarization parametrization, for both polarization elipse and Poincaré sphere) <a href="http://en.wikipedia.org/wiki/Polarization_%28waves%29#Parameterization" rel="nofollow">http://en.wikipedia.org/wiki/Polarization_%28waves%29#Parameterization</a>) due to <em>shot noise</em> (by calculating $|\langle \Psi_k|\Psi_a\rangle|^2$ where $\Psi_k$ and $\Psi_a$ are two mesoscopic coherent pulses with the same amplitude and on a same great circle on the poincaré sphere)</p>
<p>I got found the following representation presented as a "manifold of two state in cartesian $(x,y)$ at $45^o$ from horizontal", where $(\Theta_k, \Phi_k)$ are Poincaré angles and $\gamma, \delta$ are <strong>projections</strong> on axes $x,y$, and $\alpha$ is the coherent amplitude:</p>
<p>$|\Psi(\Theta_k, \Phi_k)\rangle=|\alpha \gamma(\Theta_k, \Phi_k) \rangle \otimes |\alpha \delta(\Theta_k, \Phi_k) \rangle$</p>
<p>$\gamma = (1-i)e^{i \frac{\Phi_k}{2}} cos\frac{\Theta}{2} +(1+i)e^{i \frac{-\Phi_k}{2}}sin\frac{\Theta}{2} $</p>
<p>$\delta = (1+i)e^{i \frac{\Phi_k}{2}} cos\frac{\Theta}{2} +(1-i)e^{i \frac{-\Phi_k}{2}}sin\frac{\Theta}{2} $</p>
<p>And I don't understand. Thinking 'classicaly', writing with the polarization elipse (parameters $\Psi$ and $\chi$) I got</p>
<p>$\vec{E(t)}=|E|e^{i\omega t}[cos\chi,\pm i sin\chi]R_\Psi$</p>
<p>Where $R_\Psi$ is a rotation of $\Psi$ and the $\pm i$ account for clockwise and counter clockwise, but its different. I've put this here just for a base of what I expect the answer to cover. I've had many ideas of 'mappings' but none of them were good.</p> | 1,913 | [
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<p>If two rocks were tied together with a tight, absurdly long, non-elastic rope, and placed on planets at either end of Earth's observable universe - or beyond - What would happen?</p>
<ol>
<li>Is the structural integrity of the rope enough to "overcome" the expansion of the universe all along the rope, causing the rope to break, or the rocks to "slide off" the planets? Or will the rope expand?</li>
<li>If the rocks slide off the planets, and the planets were far enough apart, why would the rope not be able to move away from either planet faster than the speed of light?</li>
</ol> | 1,914 | [
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<p>A car moves on a plane road in east-west direction.At what latitude of earth should it move so that induced EMF in the axle connecting it's wheels is maximum?
A. At the poles
B. At the equator
C. At a latitude inclined at 45 degree to the equator</p> | 1,915 | [
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<p>Can someone tell me in layman's language how the $(1/2,1/2)$ represents a vector field and $(0,1/2)$ or $(1/2,0)$ represents spinors and $(0,0)$ represents scalar field. Please don't be pedantic on mathematics part. I didn't take a course on group theory yet. Give me physical arguments why this is true? I have come across this in QFT course that I am currently enrolled in.</p> | 1,916 | [
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<p>The coefficient of kinetic friction can be calculated using $\mu_k = F_k/F_n$. A change in angle does not affect the normal force, but doesn't a change in angle affect the friction? As in if the angle is steeper, wouldn't the object have a greater propensity to move down the slope, thus there will be less friction?</p> | 1,917 | [
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<p>In cold weathers it is suggested to put a humidifier since the air gets too dry. I wonder how the humidity affects how much time is needed to get the air at a temperature of 20 Celsius degrees? I mean suppose you have a cold room and you want to heat the air, Will the process be slower or faster in relation to variations in humidity?</p>
<p>Since the water tends to keep its temperature I feel the process will be slower with higher humidity, but I am not sure. </p> | 1,918 | [
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<p>As the title says. It is common sense that sharp things cut, but how do they work at the atomical level? </p> | 1,919 | [
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<p>I'm looking to create a rain harvesting system. I have a 275 gallon IBC tote that is 48" x 40" x 46". I have an adapter for a 3/4 garden hose at the bottom of the IBC tote. I'm trying to figure out three things:</p>
<ol>
<li><p>What is the pressure at the bottom of the tote, assuming that the tote is full?</p></li>
<li><p>Would the pressure from the column of water in the tote be able to reach an 8' tall planter 40' away via the hose?</p></li>
<li><p>How much would the pressure increase per foot that I elevated the tank?</p></li>
</ol> | 1,920 | [
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<p>Why is it that the direction of an area vector should be always along the normal drawn to the surface? Can't it also be some other angles with the plane?</p> | 1,921 | [
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<p>Is dark matter a candidate to fill void left by <a href="http://en.wikipedia.org/wiki/Luminiferous_aether" rel="nofollow">luminiferous ether</a> as a medium for light travel?</p> | 1,922 | [
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<p>'In the answers to one of the questions based on rotation of a disc in my physics book the answer includes the statement 'As we know that the velocity of outermost point on a rotating disc is double the velocity of center of mass'. But, I didn't know that and why is it like that?</p>
<p>My thinking: I know that as we go away from the center of mass the tangential velocity of particles of the disc increases according to $v=wr$. But, how does that result in the above result? </p> | 1,923 | [
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<p>What is the relationship between two eigenfunctions of the time-independent Schrödinger Equation (in one spatial dimension) if they both have the same eigenvalue?</p> | 1,924 | [
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<p>I read in a QFT book that local gauge symmetry implies causality. Could someone please explain that statement and why it's true?</p>
<p>Thank you.</p> | 1,925 | [
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<p>I'm currently struggling with the expression of operators in second quantization. I did an exercise in which I had to consider a fermion in a central potential $V(\vec{r})$ and show that the matrix elements of $V(\vec{r})$ and $V_{ij} a_i a^{\dagger}_j$ were identical. I think I get this right. Yet, when I try to do the same with two fermions in a central potential (not considering any interaction between them), I end up completely lost.</p>
<p>On the one hand (first quantization), I have</p>
<p>\begin{equation}
<a b ∣ V(r) ∣ cd > = \frac{1}{2} \int \int d^3 r_1 d^3 r_2 \Big(\phi_a^{\ast}(r_1)\phi_b^{\ast}(r_2) - \phi_a^{\ast}(r_2) \phi_b^{\ast}(r_1) \Big) [V(r_1) + V(r_2)] \Big(\phi_c^{\ast}(r_1) \phi_d^{\ast}(r_2) - \phi_c^{\ast}(r_2) \phi_d^{\ast}(r_1)\Big)
\end{equation}</p>
<p>(I should be able to further develop and simplify this expression, but I'm stuck with eight terms with no simplification on the horizon) and on the other hand (second quantization), I have</p>
<p>\begin{equation}<a b ∣ \sum_{ij} V_{ij} a_i a_j^{\dagger} ∣ cd > = < ab ∣ V_{ac} a_a a_c^{\dagger} + V_{ad} a_a a_d^{\dagger} + V_{bc} a_b a_c^{\dagger} + V_{bd} a_b a_d^{\dagger} ∣ cd>\end{equation}
I tried applying the anti-commutation relationship to this expression so as to simplify it, but unsuccessfully so far.</p>
<p>I am more or less sure I'm missing something terribly obvious and I should feel ashamed about it. I would gladly appreciate some fresh insight on this problem. </p> | 1,926 | [
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<p>In <a href="http://www.feynmanlectures.info/docroot/II_28.html#Ch28">Volume II Chapter 28</a> of the Feymann Lectures on Physics, Feynman discusses the infamous 4/3 problem of classical electromagnetism. Suppose you have a charged particle of radius $a$ and charge $q$ (uniformly distributed on the surface). If you integrate the energy density of the electromagnetic field over all space outside the particle, you'll get the total electromagnetic energy, which is an expression proportional to $c^2$. The energy divided by $c^2$ is what we usually call the mass, so if we calculate the "electromagnetic mass" in this manner we'll get $m = \frac{1}{2}\frac{1}{4\pi\epsilon_0}\frac{q^2}{ac^2}$. If, on the other hand, you took the momentum density of the electromagnetic field and integrated it over all space outside the particle, you'd get the total electromagnetic momentum, which turns out (for $v<<c$) to be proportional to the velocity of the particle. The constant of proportionality of momentum and velocity is what we call mass, so if we calculated the electromagnetic mass in this way we would get $m = \frac{2}{3}\frac{1}{4\pi\epsilon_0}\frac{q^2}{ac^2}$, which is $\frac{4}{3}$ times the value we got before! That is the 4/3 problem.</p>
<p>Feynman claims that this fundamental issue remains when we move to quantum electrodynamics. Was he right, and if so has the situation changed since the 1960's when he was writing? I've seen claims on the Internet (I don't have the links) that the 4/3 problem is still there in QED, but instead of $\frac{4}{3}$ the coefficient is something closer to 1. Is that true, and if so what's the coefficient? All of this is of course related to issues of self-energy and renormalization.</p>
<p>Any help would be greatly appreciated.</p> | 1,927 | [
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<p>I have a set of <a href="http://www.google.com/search?q=PLIF+images" rel="nofollow">PLIF images</a> of a passive scalar advected in a turbulent flow. I'm wondering if it's possible to estimate the integral length-scale based on the images of the passive scalar, and if so, how would I go about it. Any references to journal articles about this are welcome.</p>
<p>It seems that a 2D cross-correlation of the images would help, with defined peaks at various length-scales, but then you'd have to filter out a lot of noise.</p> | 1,928 | [
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<p>Almost every solid state physics textbook says crystal momentum is not really physical momentum. For example, phonons always carry crystal momentum but they do not cause a translation of the sample at all. </p>
<p>However, I learned that in indirect-band-gap semiconductors, we need phonons to provide the crystal momentum transfer to make happen electron transitions between the top of the valance band and the bottom of the conduction band. Along with absorbing or emitting photons, of course. </p>
<p>Photons do carry physical momentum. For the purpose of momentum conservation, it seems that phonons do carry physical momentum as well.</p>
<p>How can we explain this?</p>
<p>========================================</p>
<p>To put it more specifically, I drew a graph to tell the story:</p>
<p><img src="http://i.stack.imgur.com/syVah.png" alt=""></p>
<p>K (capital) is crystal momentum.</p>
<p>For such transition, photon provides most of the energy transfer (and a little momentum transfer hk, k in lower case), phonon provides most of the momentum transfer (and a little energy).</p>
<p>Similar graphs can be found in most solid state physics textbooks. The picture tells me, either the photon participating in the transition carries crystal momentum, which value is equal to physical moemntum hk, or the crystal momentum itself is a kind of physical momentum.</p>
<p>However, one can prove that a phonon does not carry physical momentum (here I quote Kittel's "Introduction to Solid State Physics"):</p>
<p><img src="http://i.stack.imgur.com/AohrD.jpg" alt=""></p>
<p>So, how do we explain the momentum transfer in the electron transition aforementioned?</p> | 1,929 | [
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<p>There's a conventional wisdom that the best way to minimize the force impact of a punch to the head is to lean into it, rather than away from it. </p>
<p>Is it true? If so, why?</p>
<p>EDIT: Hard to search for where I got this CW, but heres <a href="http://www.xomba.com/how_to_take_a_blow_to_the_head" rel="nofollow">one</a>, and <a href="http://www.wikihow.com/Take-a-Punch" rel="nofollow">another</a>. The reason it seems counter-intuitive is that I'd think if you move in the direction that a force is going to collide into you with, the collision would theoretically be softer. You see that when you catch a baseball barehanded; it hurts much more when you move towards the ball, rather than away from the ball, as it hits your hand. </p> | 1,930 | [
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<p>Does an analog of the Solovay-Kitaev theorem exist for quantum operations, a generalization of quantum gates that also includes all completely positive maps?</p> | 1,931 | [
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<p>I have a swimming pool of 5300Liters.
I have a heating element from a washing machine taking 1min17s to heat a bucket (10L) of water from 22 degrees to 30 degrees celcius.</p>
<p>the water in my pool is 19 degrees celcius.</p>
<p>how long would it take to warm my pool to 30 degrees?
How long would it take to warm it to 25 degrees?</p>
<p>is there any more information you need?</p> | 1,932 | [
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<p>I have a box with $x,y,z$ all ranging from 0 to $l$. It has $V(x)$=0 inside and =$\infty$ outside. By extending the 1D Schrodinger equation, I have that the allowed energy eigenvalues are $\hbar^2\pi^2\over2ml^2$$(n_1^2+n_2^2+n_3^2)$. What is the degeneracy of the 1st excited energy level? By "1st excited energy level" does that mean 1 of the $n_k$'s, say $n_1$, =2 while $n_2=n_3=1$? Or does it mean all 3 dimensions are in their 1st excited state -- $n_k=2$ $ \forall k\in${$1,2,3$}? Also, how does one find the degeneracy? I am guessing that it is 3? Thanks for any help.</p> | 1,933 | [
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<p>I know that surface tension plays a key role in the formation of a bubble. I guess a bubble contains air inside it. Now how is it so that a soap bubble contains air both inside it and outside it?</p>
<p>I will be grateful to an answer donor if he explains me the various stages of bubble formation. In brief "How to make a bubble from a glass of water, what are the various processes that take place during the bubble formation"??</p> | 1,934 | [
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<p>In class of Newton's laws of motion, it was explained that Newton's second law is valid only in inertial frames. Teacher give us a example by considering a lift which is going downwards with acceleration$=a$ then a man watching the lift from outside will write $mg-t=ma$ (where $m$ is mass of the lift and $t$ is upward force on the lift) while a man inside the lift will write $mg-t=0$ (for him there is no acceleration of lift) both equations will give us different value of $t$ so we have to have a frame specified for work ie inertial frame but nearly after 5 months I have a problem. In the observation of the man inside lift acceleration due to gravity is stated as $g$ but I think that it should have value $g-a$ (if a stone fall on Earth with acceleration $g$ then inside the lift it should fall with acceleration $g-a$) and then by substituting this value for acceleration due to gravity in equation of man inside lift we will get the same value of $t$ as we got in equations of man outside the lift. So if both frames give same value of $t$ then why we prefer one frame after other.</p> | 1,935 | [
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<p>In the absence of nonconservative forces such as friction and air resistance, the total mechanical energy in a closed system is conserved. This is why that when I toss an object directly upwards, the kinetic energy $K = (1/2)mv^2$ is transformed into potential energy as it increases in height with potential energy $U = mgh$</p>
<p>Because of the conservation due to the energies being transformed, we can express this relationship between the two energies as $K_i + U_i = K_f + U_f$</p>
<p>The question I was asked was to use these equations to find the maximum height $h_{max}$ to which the object will rise, as expressed in terms in $v$ and $g$.</p>
<p>I was able to solve this by saying that at this max height, the velocity and therefore the kinetic energy will be at <em>zero</em>. So I am able to say that $K_i + 0 = 0 + U_f$ or simply $K_i = U_f$</p>
<p>Mass cancels and we are left with $\frac{v^2}{2g} = h_{max}$</p>
<p>This was easy enough, and it is likely that my misunderstanding is simply a mathematical one, but I am at a loss when asked</p>
<blockquote>
<p>At what height $h$ above the ground does the projectile have a speed of $0.5v$</p>
</blockquote>
<p>How do you approach this problem?</p> | 1,936 | [
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<p>Regarding the future applications of terahertz technology, I thought we already knew the entire electromagnetic spectrum. In fact we do not, terahertz lies between microwaves and infrared radiation. The future applications of this is astonishing. For example, faster communications and next generation scanning and imaging technologies. The TSA will pretty much be out of a job, because T-Ray(terahertz) scanners will automatically detect chemical weapons and bombs. I know this because they leave a trace on the terahertz spectrum. The downside with communications is terahertz is absorbed easily by the Earth's atmosphere, so the communications cannot be too high in the sky. What I am asking is, are there any other frequencies on the electromagnetic spectrum we might discover? I was taught in school, that there are radio waves, microwaves, infrared, visible, ultra violent, x-rays, and gamma rays. Now the books have to be rewritten because of the existence of terahertz or t-rays. Remember it lies right between microwaves and infrared radiation. Another question is what about sound? So far there is infra sound, sound, and ultrasound. Do you think in the future we will discover other types of sound? Sorry for the length of my question, as you can see I know a lot about science and technology.</p> | 1,937 | [
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<p>There is a mass $m$ in a potential such that </p>
<p>$$
V(r) = \left\{
\begin{array}{lr}
0, & a \leq r \leq b\\
\infty, & \text{everywhere else}
\end{array}
\right.
$$
I am looking to find the solution $u(r)$
to the radial equation $$ -\frac{\hbar^2}{2m} u^{\prime \prime}(r) + \bigg( V(r) + \frac{\hbar^2}{2m} \frac{l(l+1)}{r^2}\bigg) u(r) = E\, u(r)$$
in the case $l=0$.</p>
<p>$\textbf{Progress so far}$</p>
<p>Looking at the region with zero potential, and letting $l=0$, I define $$ k \equiv \frac{\sqrt{2mE}}{{\hbar}} $$ so that I have the second order differential equation $$ u^{\prime \prime}(r) = - k^2 u(r)$$
which has the solution
$$u(r) = A \sin(kr) + B \cos( kr) $$
The potential is such that the boundary conditions are
$$ u(a) = u(b) = 0 $$
$$ A \sin(ka) + B \cos(ka) = A \sin(kb) + B \cos(kb) $$
It seems that, since, for instance, $\sin(kb) = \sin(ka)$, $$kb = ka + 2\pi n$$But this must be wrong because, for instance, $$\sin\bigg (\frac{2 \pi n a}{b - a} \bigg ) \neq 0 .$$
Could someone tell me where I went wrong in finding $k?$ </p> | 1,938 | [
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<p>How light years are measured. Once i remember the distance between earth and moon are measured by light which travels and comeback and by the delay the distance is calculated.. But how light years are calculated</p> | 1,939 | [
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<p>Consider this reverse biased diode : </p>
<p><img src="http://i.stack.imgur.com/z9dZo.png" alt="enter image description here"></p>
<p>I read that no or very small current flows in reverse biased diode as depletion layers get widened and huge resistance is offered so no electrons can cross it. But, why the electrons or holes need to cross the depletion layer? In the diagram above, the positive charges (holes) are moving towards left and the current due to electrons is also in left, so won't the circuit be completed?</p> | 1,940 | [
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<p>I've heard occasional mentions of the term "bootstraps" in connection with the S Matrix. I believe it applies to an old approach that was tried in the 1960s, whereby - well I'm not sure - but it sounds like they tried to compute the S Matrix without the interaction picture/perturbation theory approach that we currently use.</p>
<p>I'm aware that the approach was abandoned, but my question is: how was it envisaged to work ? What was the input to the calculation supposed to be and how did the calculation proceed ?</p>
<p>I know it's got something to do with analyticity properties in terms of the momenta, but that's all I know....</p>
<p>For example, from the wikipedia article:</p>
<blockquote>
<p>Chew and followers believed that it would be possible to use crossing symmetry and Regge behavior to formulate a consistent S-matrix for infinitely many particle types. The Regge hypothesis would determine the spectrum, crossing and analyticity would determine the scattering amplitude--- the forces, while unitarity would determine the self-consistent quantum corrections in a way analogous to including loops.</p>
</blockquote>
<p>For example - I can't really understand how you would hope to compute the scattering amplitude just given crossing symmetry and assuming analyticity</p> | 1,941 | [
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<p>I am following Griffiths' Introduction to Quantum Mechanics, as well as an online lecture that follows a different book, and both sources give different equations for the general solution of the 1-D Schrodinger equation for a free particle. </p>
<p>Griffiths has it as: $$\Psi(x,t)=\frac{1}{\sqrt {2\pi}}\int_{-\infty}^{+\infty}\phi(k) e^{ i\left(kx-\frac{hk^2}{2m}t\right)}dk\tag1$$</p>
<p>The online lecture has it as:$$\Psi(x,t)=\frac{1}{\sqrt {2\pi\hbar}}\int_{-\infty}^{+\infty}a(p) e^{ \frac{i}{\hbar}\left(px-\frac{p^2}{2m}t\right)}dp\tag2$$</p>
<p>The most glaring addition to the second equation is the $\hbar$ under the square root sign. Given this, I can't see how they're both equivalent. </p>
<p>The link to the online lecture is <a href="http://www.youtube.com/watch?feature=player_detailpage&v=xm-LMpkqSUQ#t=1273s" rel="nofollow">http://www.youtube.com/watch?feature=player_detailpage&v=xm-LMpkqSUQ#t=1273s</a></p> | 1,942 | [
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<p>Can any one please let me know what is the general procedure to construct the momentum operator under some coordinate transformation? For example, I understand that if </p>
<p>$${\bf{r}}\rightarrow{\bf{r'}}=a{\bf{r}}$$ </p>
<p>then the momentum operator will change as: </p>
<p>$$\frac{\partial}{\partial{\bf{r}}}\rightarrow\frac{1}{a}\frac{\partial}{\partial{\bf{r}}}.$$</p>
<p>However, what if the factor $a$ itself if a function of ${\bf{r}}$. My understanding is that in such a case, the differential $d{\bf{r'}}$ may be written as $$d{\bf{r'}}=a\ d{\bf{r}}+{\bf{r}}\ da({\bf{r}}).$$
Then, is there any procedure to calculate how would the momentum operator transform?</p> | 1,943 | [
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<p>Consider a ideal conductor in free space.For all purposes here,the zero of the potential is taken at infinity.</p>
<p>Suppose I give a charge $Q$ to the conductor.As a result,the conductor will have a potential $V$. The question is can we say that $$Q=CV$$ where $C$ is a constant which depends only on <strong>the shape and size on the conductor and not on the charge Q</strong>?</p>
<ol>
<li><p>If yes, how shall we prove it mathematically.</p></li>
<li><p>Also then how can we find the proportionality constant explicitly given the shape and size of the conductor?</p></li>
</ol> | 1,944 | [
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<p>So, I was watching various sci channel shows, and they touch on how extremely massive stars live only 100k years, vs the sun which lives ~10 billion years, and dwarf stars live some unspecified time longer.</p>
<p>So, lets say you have a galactic civilation, worried about "heat death", and in order to avoid it, they decide to take apart a number of large stars, making them into a cluster of dwarf stars, so the star will take much longer to burn out...</p>
<p>First, can a group of dwarf stars orbit a common center, close enough that we can considering their combined luminosity as a unit? maybe if we can get them all within the orbital distance of mercury from the common center? Not sure how stable orbits would be once your talking a dozen or more objects?</p>
<p>Second, how much luminosity loss is there? Say you have a combined cluster of 20 dwarf stars, each 1/20 of the mass of the sun, and all within a sphere of mercury's orbit? How much lower would the combined luminosity be -- and so, how much closer would earth have to be in order to maintain liquid water? I'd guess you'd still have some extreme seasons as you made closest approch to any one dwarf.</p>
<p>Third, if we know how close earth would have to get -- could there be a stable orbit around such a cluster, at that range?</p>
<p>Finally, if you can do all that, do you really get much added time? Assuming the source star was halfway thru it's main sequence when split into the cluster of dwarfs, so I guess you'd be estimating the remaining lifetime of a dwarf star (each 1/20 of a stellar mass), which is halfway thru its main sequence?</p>
<p>Edit: Since this is such a multi-stage question: if we know we can refute something right away (ie: perhaps you cannot have a dozen dwarf stars in any stable orbit unless they are very far apart), that could end the issue right away.</p> | 1,945 | [
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<p>How can we see living cells underneath it?</p> | 1,946 | [
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<p>I realize the situation where a laser beam moves vertically in a moving vehicle<img src="http://i.stack.imgur.com/qoDgm.gif" alt="time dilation proof"></p>
<p>but what if the laser beam was a normal ball If we do the same steps of the proof considering that the velocity of the ball is not absolute and will have different velocities in different reference frames there will be no time dilation
what is wrong in my understanding because according to special relativity there should be time dilation whether the event is a laser beam bouncing or a ball.</p> | 1,947 | [
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<p>In classical thermodynamics, <a href="http://en.wikipedia.org/wiki/Thermodynamic_equilibrium" rel="nofollow">equilibrium conditions</a> means maximum entropy for a closed state. However, people always talk about equilibrium for open systems as well. How can one say that an open system has reached equilibrium with out negating the definition.</p> | 1,948 | [
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<p>I know that electric charge is lorentz invariant quantity and I can easily think of experiment to check that. Is a though experiment that can prove that also?</p> | 1,949 | [
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<p>Suppose my system involves:</p>
<p>1) A mounted wheel with some outward flap</p>
<p>2) A bullet already in motion</p>
<p>Initially the net angular momentum is 0 and the net kinetic energy is just that of the speeding bullet.</p>
<p>The bullet hits the flap, causing the wheel to turn, and continues on (slightly slower).</p>
<p>Now the net angular momentum of the system is > 0 and the net kinetic energy is lower.</p>
<p>1) Is energy being converted into angular momentum here (so net energy is conserved)?</p>
<p>2) How is the net angular momentum of this system being conserved with the net amount before/after has changed?</p> | 1,950 | [
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<p>When I close one eye and put the tip of my finger near my open eye, it seems as if the light from the background image bends around my finger slightly, warping the image near the edges of my blurry finger tip.</p>
<p>What causes this? Is it the heat from my finger that bends the light? Or the minuscule gravity that the mass in my finger exerts? (I don't think so.) Is this some kind of diffraction?</p>
<p><img src="http://i.stack.imgur.com/e32lY.jpg" alt="Light bending around my finger"></p>
<p>To reproduce: put your finger about 5 cm from your open eye, look through the fuzzy edge of your finger and focus at something farther away. Move your finger gradually through your view and you'll see the background image shift as your finger moves.</p>
<hr>
<p>For all the people asking, I made another photo. This time the backdrop is a grid I have on my screen (due to lack of grid paper). You see the grid deform ever so slightly near the top of my finger. Here's the setup:</p>
<p><img src="http://i.stack.imgur.com/t2ioO.png" alt="Setup"><img src="http://i.stack.imgur.com/biR3w.jpg" alt="Finger on a grid"></p>
<p>Note that these distances are arbitrary. It worked just as well with my finger closer to the camera, but this happens to be the situation that I measured.</p>
<hr>
<p>Here are some photos of the side of a 2 mm thick flat opaque plastic object, at different aperture sizes. Especially notice hoe the grid fails to line up in the bottom two photos.</p>
<p><img src="http://i.stack.imgur.com/33jVJ.jpg" alt="Object photographed from the side"></p> | 1,951 | [
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<p>The Hamiltonian of Bose-Hubbard model reads as
$$H=-J\sum\limits_{<i,j>}b_i^{\dagger}b_j+h.c.+\frac{U}{2}\sum\limits_{i}n_i(n_i-1)-\mu\sum\limits_in_i~~~~~~~~~(1)$$ </p>
<p>For this we plot phase diagram in ( $J/U$, $\mu/U$ ) space. </p>
<p>Same way if I want to plot phase diagram of Hamiltonian which looks like
$$H=-J\sum\limits_{<i,j>}b_i^{\dagger}b_j+h.c.+\frac{U}{2}\sum\limits_{i}n_i(n_i-1)~~~~~~~~~(2)$$ </p>
<p>How to get phase diagram of such hamiltonian? I am solving this model Numerically by Exact Diagonalisation. </p> | 1,952 | [
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<p>In <a href="http://en.wikipedia.org/wiki/Gordon_Kane" rel="nofollow">Gordon Kane</a>'s <a href="http://www.goodreads.com/book/show/15843083-supersymmetry-and-beyond" rel="nofollow"><em>Supersymmetry and Beyond</em></a> (p. 118), he states:</p>
<blockquote>
<p>String theory has to be formulated in nine space dimensions or it is not a consistent mathematical theory. There doesn't seem to be a simple way to explain "Why nine?"</p>
</blockquote>
<p>Which, rather unexpectedly, is followed immediately by this seemingly simple explanation:</p>
<blockquote>
<p>What happens is that if theories to describe nature and to include gravity in the description are formulated in $d$ space dimensions, they lead to results that include terms that are infinite, but the terms that are infinite are multiplied by a factor $(d-9)$, and drop out only for a factor $d=9$.</p>
</blockquote>
<p>This leads me to think</p>
<ol>
<li>that perhaps this explanation is not as simple as I perceive it to be, or</li>
<li>that it perhaps misses out on subtleties and therefore perhaps is not <em>quite</em> correct, or</li>
<li>it's not really considered an <em>explanation</em> by Kane because it doesn't explain why some terms are multiplied by $(d-9)$.</li>
</ol>
<p>My guess is option 3, but I'd like to be sure. So, my main question is: <strong>Is the quoted "explanation" fully correct?</strong></p> | 1,953 | [
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<p>A proton accelerated with electric field gives off E.M. radiation and therefore should lose mass. Larmor's formula gives us a value for the power emitted (varies as acceleration squared). However, as the proton picks up speed, it also gains mass. Now, say I set up an immense electric field which provides an immense acceleration to the proton. In the initial moments of motion, even though its acceleration is extremely high, its velocity is low. In those moments, does the proton lose mass faster than it gains?</p> | 1,954 | [
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<p>As I checked, the energy-momentum tensor defined as ${T^\mu}_\nu=\frac{\partial {\cal L}}{\partial(\partial_\mu \phi)}\partial_\nu \phi-{\cal L}{\delta^\mu}_\nu$ at the solution $\phi$ of equation of motion(Euler-Lagrange equation) satisfies automatically the conservation law: $\partial_\mu{T^\mu}_\nu=0$, without any reference to the translation symmetry under $x^\mu\rightarrow x^\mu-a^\mu$.<br>
So, what is the need of this symmetry?<br>
Or, could there be something wrong with my calculation or conceptual issues?</p> | 1,955 | [
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<p>Yes the title is an odd question, so I will provide a little background! Recently the sheer amount of padlocks on the Pont d'Art in Paris caused a part of its railing to fall off. Similar padlocks have been found on the Eiffel tower, so naturally the first question that came to my head was: "How many padlocks would have to be put on one side of the Eiffel tower before <em>it</em> falls over?"</p>
<p><strong>Assumptions</strong></p>
<ol>
<li>The centre of mass of the Eiffel tower for the $x$ axis is directly through the middle of the tower</li>
<li>The $y$ position of the centre of mass doesn't really matter too much.</li>
<li>The tower will fall when it's centre of mass is just over the edge of the base</li>
<li>The tower doesn't bend, it just falls over</li>
<li>The number of padlocks $n$ is sufficiently large that the volume of padlocks will bulge over the edge of the base (therefore fulfilling assumption 3).</li>
<li>The padlocks and the tower are particles</li>
<li>The tower is not bolted to the ground, and is free to fall over</li>
</ol>
<p><strong>Variables</strong></p>
<p>Half the width of base of tower $d \approx 50$m</p>
<p>Mass of tower $M \approx 1 \times 10^7$kg</p>
<p>Mass of padlock $m \approx 2.83 \times 10^{-2}$kg</p>
<p>Volume of padlock $V \approx 1.41 \times 10^{-5}$m$^3$</p>
<p><strong>Method</strong></p>
<p>The volume of the sphere of padlocks (of radius $x$) must be equal to the volume of $n$ padlocks:</p>
<p>$\frac{4\pi x^3}{3} = nV$</p>
<p>The centre of mass $x$ of the padlocks (as they form a clump) needs to be greater than $d$ hence:</p>
<p>$x = \sqrt[3]{\frac{3nV}{4\pi}} \geq d$</p>
<p>and if we consider the centre of mass of the Eiffel tower to be at the origin, we can make the following equation:</p>
<p>$(M + mn)d = (nm)x \Rightarrow (M - nm)d = (nm)\sqrt[3]{\frac{3nV}{4\pi}}$</p>
<p>Solving the resulting quartic for $n$ (and subbing back into $x$) gives:</p>
<p>$n = 3.82 \times 10^{10}$ padlocks </p>
<p>$x = 50.5 \geq d \therefore$ condition 3 is fulfilled</p>
<p>Is this method providing anywhere near a reasonable estimation for the number of padlocks required to topple the Eiffel tower (given assumptions 1 & 4 at least), and how can it be improved?</p>
<p>(p.s. not sure on what the tags should be!)</p> | 1,956 | [
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<p>So this has been really bugging me over the past few days (and forgive me if the answer is so simple). Let's say we're observing the Sombrero galaxy. </p>
<p>It is about 29 million light years away and 50 thousand light yyears in diameter.
So we should be observing the "front" of it at what it looked like 29 million years ago, and the "back" of it 29.05 million years ago. </p>
<p>Why doesn't this extra distance change the galaxy's shape? If, for example, the galaxy was moving directly away from us in a straight line (not that it is), wouldn't the galaxy be compressed? Hope this makes sense.</p> | 1,957 | [
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<p>I am doing a scattering simulation of a gaussian wave packet on a finite square well. I have solved numerically the Schroedinger equation and I know the values of the wave function after the scattering process before and after the square well.
To be more clear, this is the situation after the scattering process: (the wave function is not normalized, the biggest packet is moving towards left, the smallest towards right)<img src="http://i.stack.imgur.com/vcbmQ.png" alt="enter image description here"></p>
<p>I know that the tasmission coefficient is defined as:</p>
<p>$T=\frac{\vec{j}_{trasmitted}}{\vec{j}_{incident}}$</p>
<p>where the flux $\vec{j}$ is defined as:</p>
<p>$\vec{j}(x)=\psi^*(x)\frac{\hat{p}}{2m}\psi(x)-\psi(x)\frac{\hat{p}}{2m}\psi(x)^*$</p>
<p>Now, for an unormalizable function like $C*e^{ikx}$ I am able to determine T, it is easy because the flux is constant.
But in this case the flux is not constant before and after the square well.</p>
<p>Maybe the probability to find the particle after the square well (and so the area under the part of the normalized $|\psi|^2$ that crossed the barrier) is the correct value of the transmission coefficient?</p>
<p>If my hypotesis is correct, could you give me some references where it is expalined?</p> | 1,958 | [
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<p><em>This question related to <a href="http://physics.stackexchange.com/questions/51554/why-are-magnetic-lines-of-force-invisible">Why are magnetic lines of force invisible?</a> and is motivated by a comment of @BlackbodyBlacklight, based on that, the illustrating example may depend on that linked question as context to be clearly understandable.</em></p>
<p>A remote magnetic field, in the sense that it is not at the location of measurement, could influence the location of measurement in some (possibly indirect) way that allows to derive information about it's structure.</p>
<p>This is comparable to deriving information about a remote temperature profile based on properties of the local electromagnetic field, like when <em>using a camera</em>, or just <em>seeing something glow</em>.</p>
<p>It might well turn out that it is fundamentally impossible to derive information about a remote magnetic field, (given some sensible constraints).<br>
In this case, an Answer should ideally explain why that is the case.</p>
<p>What is described above is roughly comparable to human perception, which was the context where the question came up originally. Therefore, I will illustrate my initial ideas in that context in the section below:
<hr>
<em>Establishing the context for the question (The biological aspects referred to are part of the <strong>illustration</strong>, <strong>not</strong> directly related to the question):</em></p>
<p>The motivating idea was: <em>"We can not see magnetic fields, but that may be because it was not important during evolution to acquire this capability."</em></p>
<p>Could it be possible, in principle, to "see" magnetic fields?</p>
<p>Now, if it would have been helpful during evolution - what kind of
perception is possible purely <strong>from the physical side</strong> of the question - assuming "perfect evolution".</p>
<p>The linked question asks about seeing magnetic field lines - so could something like eyes for <em>seeing field lines</em> have evolved?<br>
I <strong>assume not</strong>, so we do not need to go into details whether to see them on surfaces, as lines at a fixed distance, etc. (Feel free to make creative assumptions as needed regarding how to "see")</p>
<p>What did evolve, in some birds and bacteria, is perception of the field of Earth in terms of <strong>direction of the local(!) field</strong> lines - something like "feeling north and south".</p>
<p><hr>
<em>The actual question, related to physics of magnetic fields, in comparison to phenomena for which human perception exists:</em></p>
<p>What are the <strong>physical constraints</strong>? </p>
<p>Seeing a magnetic field like a fourth base color would not work - there is no radiation.</p>
<p>Something similar to spacial sound perception? Which would mean to measure from a finite set of "local" reference points to collect measurements on a given remote locatioin.</p>
<p>Anything better than measuring a local field vector is certainly interesting. </p> | 1,959 | [
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<p>Consider the following schematics of a Bainbridge mass spectrometer</p>
<p><img src="http://i.stack.imgur.com/UDrwC.png" alt=""></p>
<p><sub>
(Source: <a href="http://www.schoolphysics.co.uk/age16-19/Atomic%20physics/Atomic%20structure%20and%20ions/text/Mass_spectrometer/images/1.png" rel="nofollow">http://www.schoolphysics.co.uk/age16-19/Atomic%20physics/Atomic%20structure%20and%20ions/text/Mass_spectrometer/images/1.png</a>)
</sub></p>
<p>Suppose the gas contains two sorts of atoms (for example two isotopes of one element) with different masses. The electric force between the electrodes of in the gas chamber which accelerates them is $qE$, where $E$ is the electric field and $q$ the charge of the ions. For simplicity consider the case that you have one ion of sort A and one of type B each with equal charge q, but different masses $m_A$ and $m_B$ with $m_A > m_B$. Then the velocity of the $A$ before entering the Wien-filter will be smaller than hat of atom $B$ (because of it's mass). Furthermore suppose that the Wien filter only lets pass particles with velocity $v_A$, which is by assumtion the same as the velocity of our atom $A$. Because $B$ is faster than $A$, ions of type $A$ will pass the Wien-filter, but ions of type $B$ will not. </p>
<p>But then the magnetic field after the Wien-filter would be useless, because only ions of type $A$ would pass the filter. </p>
<p>So I guess that there must be another reason why the velocity distributions of type $A$ and $B$ atoms overlap after leaving the ion source such that both types $A$ and $B$ could reach the magnetic field after the Wien-filter. </p>
<p>Why is this the case? How can one quantitatively estimate how large the difference of $m_A$ and $m_B$ may be such that the velocity distributions of $A$ and $B$ overlap? Can you give me a quantitaive example of $A$ and $B$ and the concrete velocity distributions from experiments?</p>
<p>I am also looking for good references where those questions are discussed.</p> | 1,960 | [
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<p>This may sound dumb, but I have been observing that, If my mobile phone is placed on the wire of my hands free (earphones) and if my mobile vibrates, I could listen the vibration from the ear plugs. I first thought that it is just my imagination, but every time my mobile is placed on my earphones wire i could totally listen the mobile vibrating through the earplugs, even when I am listening to loud music, I could still totally hear the vibration clearly.</p>
<p>What is the phenomenon behind this ?</p> | 1,961 | [
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<p>For example how radio signals of a base transceiver station (BTS) penetrate through buildings?</p> | 1,962 | [
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<p>Image we have an ultra-high intensity, ultra low frequency laser, with wattage on the order of terawatts and a wavelength on the order of a lightsecond. We rotate it that the electric field component is oriented on the $\hat z$ axis, then fire it at a macroscopic block with a positive electric charge. Because of the low frequency the block will experience an electric that doesn't immediately change direction, and because of the high intensity the field will be very strong. So from this naive understanding of classical physics, the block will briefly levitate.</p>
<p>Except this blatantly contradicts both QM (Compton scattering) and multiple macroscopic experiments (like solar sails), which both say that the block will be pushed in the direction of the laser. What assumptions in the original problem are missing/wrong?</p> | 1,963 | [
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<p>For example We know glass when rubbed by silk will become positively charged while the silk will be charged negative. </p>
<p>What exactly makes glass appropriate for losing electrons in that experiment? (</p> | 1,964 | [
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<p>I was in a physics group, then a student (or a professor, I don't know) posted this-</p>
<blockquote>
<p>Mohammad Shafiq Khan > Physics > The space-time concept including the formula $E=mc^2$ are proved baseless in the published paper "Experimental & Theoretical Evidences of Fallacy of Space-time Concept and Actual State of Existence of the Physical Universe" available at <a href="http://www.indjst.org/index.php/indjst/issue/view/2885" rel="nofollow">http://www.indjst.org/index.php/indjst/issue/view/2885</a> on the same premises on which Einstein had derived it.</p>
</blockquote>
<p>I went to the link, only I don't have a device which has Adobe plug-in. I just want to ask, does this article have any value? Was Einstein wrong about relativity?</p> | 1,965 | [
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<p>As I understand it, you can't see beyond the Cosmic Microwave Background Radiation because the plasma of the early universe was opaque to electromagnetic radiation. What if you had a "neutrino telescope" with sufficient resolution? Would you be able to observe the primordial universe from before recombination?</p> | 1,966 | [
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<p>It is a long time question for me. </p>
<p>For me, it seems that WKB is applicable for a given $E$ if and only if $\hbar$ is sufficiently small. Or in other words, WKB is applicable if and only if the quantum number is large enough. </p>
<p>Is this understanding right?</p>
<p>I would take the exactness of WKB for the harmonic oscillator as purely accidental. </p> | 1,967 | [
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<p>It is very well-known that for bosonic operators a Gauge transformation can always be associated with it
$$a\rightarrow e^{i\phi}a.$$
Obviously this is a Unitary transformation. Something like
$$a^{\prime}=\mathcal{U}^{\dagger}a\mathcal{U}$$</p>
<p>I want to know what is $\mathcal{U}$?</p> | 1,968 | [
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<p>Are there any exactly solvable models in statistical mechanics that are known to have critical exponents different from those in mean-field theory, apart from the two-dimensional Ising model? I wonder about this because most easily solved models are either mean-field or do not exhibit a phase transition (Ising chain).</p> | 1,969 | [
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<p>I am reading at the moment the paper <a href="http://arxiv.org/abs/1401.5203" rel="nofollow">http://arxiv.org/abs/1401.5203</a> and try to reproduce the results. One result is the proximity correction $S_{\Sigma}$ to the system</p>
<p>$$
e^{-S_{\Sigma}} =\frac{\int\prod_{j\sigma\omega}D\Psi_{j\sigma}\left(\omega\right)D\Psi_{j\sigma}^{\dagger}\left(\omega\right)e^{-S_{T} - S_{S}}}{\int\prod_{j\sigma\omega}D\Psi_{j\sigma}\left(\omega\right)D\Psi_{j\sigma}^{\dagger}\left(\omega\right)e^{-S_{S}}}\text{,}
$$</p>
<p>where $S_{S}$ is the action for the superconductor and $S_{T} = \sum_{\omega}\sum_{\sigma}\left(-\gamma\Psi_{j=1,\sigma}^{\dagger}\psi_{D,\sigma} -\gamma\psi_{D,\sigma}^{\dagger}\Psi_{j=1,\sigma}\right)$ is the tunneling Hamiltonian between superconductor and the System.</p>
<p>After a Bogoliubov transformation the tunneling terms becomes</p>
<p>$$
S_{T} = \sum_{w}\frac{2}{\pi}\int_{0}^{\pi}dq-\gamma\sin\left(q\right)\left[\left(u_{q}\phi^{\dagger}_{+}\left(q,\omega\right) - v_{q}\phi_{-}\left(q,-\omega\right)\right)\psi_{D\uparrow}\left(\omega\right) + \left(v_{q}\phi_{+}\left(q,\omega\right) + u_{q}\phi_{-}^{\dagger}\left(q,-\omega\right)\right)\psi_{D\downarrow}\left(-\omega\right) + \text{ h.c.}\right]
$$</p>
<p>With</p>
<p>$$
\prod_{j\sigma\omega}D\Psi_{j\sigma}\left(\omega\right)D\Psi_{j\sigma}^{\dagger}\left(q,\omega\right)\to\prod_{q\tau\omega}D\phi_{\tau}\left(q,\omega\right)D\phi_{\tau}^{\dagger}\left(\omega\right)
$$</p>
<p>the final result for $S_{\Sigma}$ should be</p>
<p>$$
S_{\Sigma} = \sum_{\omega}\gamma^{2}\frac{2}{\pi}\int_{0}^{\pi}dq\sin^{2}\left(q\right)\left[\left(\frac{v_{q}^{2}}{i\omega + E_{q}} - \frac{u_{q}^{2}}{-i\omega + E_{q}}\right)\sum_{\sigma =\uparrow,\downarrow}\psi_{D\sigma}^{\dagger}\psi_{D\sigma} + \frac{2u_{q}v_{q}E_{q}}{\omega^{2} + E_{q}^{2}}\left(\psi_{D\downarrow}\left(-\omega\right)\psi_{D\uparrow}\left(\omega\right) + \text{ h.c.}\right)\right]
$$</p>
<p>However, I can not reproduce this final result. I watch in the Books by Altland & Simons and Negele & Orland and find the Gaussian integral</p>
<p>$$
\int d\eta^{\dagger}d\eta e^{-\eta^{\dagger}H\eta + \psi^{\dagger}\eta + \psi\eta^{\dagger}} = det\left(H\right)e^{\psi^{\dagger}H^{-1}\psi}
$$</p>
<p>which sould be solved my problem. But I do not known how I get from this integral the result which I want. </p>
<p><strong>Edit:</strong> My problem is that if I write for $S_{T}$ each term and try to integrate out the quasiparticle operators I do not obtain the same result (without the factors in front):</p>
<p>$$
S_{T} \sim \sum_{w}\left[\left(u_{q}\phi^{\dagger}_{+}\left(q,\omega\right) - v_{q}\phi_{-}\left(q,-\omega\right)\right)\psi_{D\uparrow}\left(\omega\right) + \left(v_{q}\phi_{+}\left(q,\omega\right) + u_{q}\phi_{-}^{\dagger}\left(q,-\omega\right)\right)\psi_{D\downarrow}\left(-\omega\right) + \text{ h.c.}\right] = u_{q}\left(\phi^{\dagger}_{+}\left(q,\omega\right)\psi_{D\uparrow}\left(\omega\right) + \psi^{\dagger}_{D\uparrow}\left(\omega\right)\phi_{+}\left(q,\omega\right) + \phi^{\dagger}_{-}\left(q,-\omega\right)\psi_{D\downarrow}\left(-\omega\right) + \psi^{\dagger}_{D\downarrow}\left(-\omega\right)\phi_{-}\left(q,-\omega\right)\right) + v_{q}\left(\phi_{+}\left(q,\omega\right)\psi_{D\downarrow}\left(-\omega\right) + \psi^{\dagger}_{D\downarrow}\left(-\omega\right)\phi^{\dagger}_{+}\left(q,\omega\right) - \phi_{-}\left(q,-\omega\right)\psi_{D\uparrow}\left(\omega\right) - \psi^{\dagger}_{D\uparrow}\left(\omega\right)\phi^{\dagger}_{-}\left(q,-\omega\right)\right)\text{.}
$$</p>
<p>I know that</p>
<p>$$
\int\prod_{j\sigma\omega}D\Psi_{j\sigma}\left(\omega\right)D\Psi_{j\sigma}^{\dagger}\left(\omega\right)e^{-S_{S}} \sim i\omega + E_{q}
$$</p>
<p>However, with this integral I have trouble:</p>
<p>$$
\int\prod_{j\sigma\omega}D\Psi_{j\sigma}\left(\omega\right)D\Psi_{j\sigma}^{\dagger}\left(\omega\right)e^{-S_{T} - S_{S}}
$$</p>
<p>The terms with the $u_{q}$ are the integral which I know</p>
<p>$$
\int d\eta^{\dagger}d\eta e^{-\eta^{\dagger}H\eta + \psi^{\dagger}\eta + \psi\eta^{\dagger}} = det\left(H\right)e^{\psi^{\dagger}H^{-1}\psi}
$$</p>
<p>But how I can solve this with the $v_{q}$'s?</p>
<p><strong>Edit2:</strong> I see my Substitution was wrong. Now I obtain</p>
<p>$$
\frac{1}{-i\omega + E_{q}}\left(\left(v_{q}\psi_{\downarrow}\left(-\omega\right) + u_{q}\psi_{\uparrow}^{\dagger}\left(\omega\right)\right)\left(v_{q}\psi_{\downarrow}^{\dagger}\left(-\omega\right) + u_{q}\psi_{\uparrow}\left(\omega\right)\right) + \left(u_{q}\psi_{\downarrow}^{\dagger}\left(-\omega\right) - v_{q}\psi_{\uparrow}^{\dagger}\left(\omega\right)\right)\left(u_{q}\psi_{\downarrow}\left(-\omega\right) - v_{q}\psi_{\uparrow}^{\dagger}\left(\omega\right)\right)\right) = \frac{1}{-i\omega + E_{q}}\left(\left(u_{q}^{2} - v_{q}^{2}\right)\left(\psi_{D\uparrow}^{\dagger}\left(\omega\right)\psi_{D\uparrow}\left(\omega\right) + \psi_{D\downarrow}^{\dagger}\left(-\omega\right)\psi_{D\downarrow}\left(-\omega\right)\right) +2u_{q}v_{q}\left(\psi_{D\uparrow}^{\dagger}\left(\omega\right)\psi_{D\downarrow}^{\dagger}\left(-\omega\right) + \psi_{D\downarrow}\left(-\omega\right)\psi_{D\uparrow}\left(\omega\right)\right) - 2v_{q}^{2}\right)\text{,}
$$</p>
<p>which is very nice but not the same as in the paper. What I overlook at the moment?</p> | 1,970 | [
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<p>In my textbook, as a preliminary to Faraday's law of induction, magnetic flux is defined over a closed loop as </p>
<p>$$\Phi_B = \oint \vec{B}\cdot d\vec{A}$$</p>
<p>Then it draws a parallel with electric flux and says: "as in electric flux, $d\vec{A}$ is a vector of magnitude $dA$ that is perpendicular to a differential area $dA$". But the electric flux is through a closed <em>surface</em>, and the direction of $d\vec{A}$ is defined as coming out of the surface. The book does not mention anything about how direction is the direction of area vector is defined in case of magnetic flux. Also, there is not necessarily a current in the loop so we can't assign a direction using that? </p>
<p>I wasn't able to find an explicit clarification anywhere else on the web. </p> | 1,971 | [
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<p>I know there are many examples of metals with a negative
dielectric constant at optical frequencies. But what about insulators?</p> | 1,972 | [
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<p>I have a problem to understand what does exactly mean (definition, how is induced, in which direction works) "flaring stress" in reference to the thread relief of internally/externally threaded hydraulic accumulator body. I found that phrase in EN 14359 (very long and complicated formula) and in ASME VIII Div. 1 (it is just mentioned there), but I have no any more information about it. Google is just giving answers about some type of migraine.</p> | 1,973 | [
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<p>I am actually working with Green-Schwarz anomaly cancellation mechanism in which I have came across a strange formula which relates trace in the adjoint representation (Tr) to trace in fundamental representation (tr).For the special case of $SO(n)$, the relation is </p>
<p>$Tr(e^{iF})= \frac{1}{2}(tre^{iF})^2-\frac{1}{2}(tre^{2iF})$</p>
<p>This relation can be found in 'String theory and M theory' of Becker, Becker and Schwarz Chapter 5. They say that this is a result which follows from Chern character factorization property (I guess something similar to how Chern character can be represented by a product of Chern characters defined on two vector bundles, when the whole character is evaluated on the product of vector bundles but I am not sure because I have never came across such term).</p>
<p>Can any one tell me how to relate the traces in one representation to the other because I have never seen it before in any Group theoretic context.</p>
<p>Chapter 13 of GSW also contains some information on it but that is not much useful. Thanks for any help.</p> | 1,974 | [
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