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<p>Ok, so on Science channel was a special about Hawking/Susskind debating black holes, which can somehow remove information from the universe.</p>
<p>A) In stars, fusion converts 4 hydrogen into 1 helium, with a small fraction of the mass being converted to energy. </p>
<p>This happens all the time, and nobody is that amazed about it anymore. So, I have to assume that science is ok with the idea of fusion happening, and so, converting matter into energy doesn't destroy the information of the matter.</p>
<p>Let's take that as Assuption 1:<br>
** 1: Converting matter to energy is OK.</p>
<p>B) In empty space, quantum fluctuations of the energy levels are presumed to allow "pair creation" where a particle and its anti-particle can be created out of the really tiny amounts of energy that exist in empty space. For quantum fluctutations, if I understand it, the uncertaincy principle actually dictates that the amount of energy can fluctuate, allowing a pair to be created even if there "shouldn't be enough energy", because the amount of energy in a small space is uncertain.</p>
<p>The tiny amount of energy becomes a particle + anti-particle, for a tiny instant, and then they fall back into each other, and are converted back to the same tiny amount of energy.</p>
<p>Anyways, ignoring the fact that I'm fuzzy on all the requirements for pair creation -- it is generally accepted as part of quantum physics that it can happen. </p>
<p>So, again, I have to assume that science is ok with the idea of pair creation. And so, converting energy into matter, must also not destroy the information value of the energy, and that information value must remain even after the 2 particles annihilate and go back to being energy.</p>
<p>Let's take that as Assumptions 2 + 3:
** 2: Converting energy into matter is OK.
** 3: Pair creation, doesn't mess up anything.</p>
<p>C) Ok, now black holes. My (weak) understanding of Hawking's idea on how black holes emit energy is that pair creation can happen close to the black hole. Since the position of energy is uncertain at the quantum level, some of the energy that "should be" inside the black hole's event horizon, can be treated as uncertainly outside the black hole, and then that extra energy can cause pair creation. If the black hole then pulls in both particles, then nothing special happens -- the mass/energy stays inside the black hole. </p>
<p>But if one or both of the particles moves away from the black hole, and somehow escapes, that can cause the energy to leave the black hole. This is assumed to be a very unlikely event, but still within possibility.</p>
<p>Anyways, that's my (very rough) understanding of Hawking's theory on how black holes can loose mass over time, when in theory "nothing can escape a black hole". Let's assume Hawking's correct...</p>
<p>If a black hole continues to loose mass over cosmic time, it will eventually disappear, and the problem is stated that when it vanishes: All of the information which the black hole had formerly consumned, will be lost. </p>
<p>Which contradicts the Law of Conservation of Information. At least, that's how the Science channel presented it.</p>
<p>So, my idea/question:</p>
<p>** Every time the black hole looses mass, it's through the processes of energy->matter, or matter->energy->matter, involved in pair creation. Based on assumptions 1, 2 + 3 above, none of these should violate the Law of Conservation of Information. It seems that the very slow process of loosing mass, should also be a very slow process of "leaking" information back into the rest of the universe.</p>
<pre><code>*** So, WHAT's the problem? ***
</code></pre>
<p>By my understanding: By the time the black hole vanishes, it's OK -- because all of it's information should also have leaked back into the rest of the universe, and no information will be lost.</p>
<p>OK, so please tell me why this is wrong. It seems intuituve to me, but really smart people have spent years looking into this. So, am I missing something?</p>
<p>Thanks!
-Chuck</p>
<p>ps. Also, if my statement of fusion, pair creation, and/or hawking radition is wrong -- BUT NOT IN A WAY THAT MATERIALLY AFFECTS THE QUESITON -- please overlook that and just focus on the question.</p> | g9544 | [
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<p>In an infinite plane elastic plate of thickness $d$, it is shown that the modes of oscillation corresponding to a fixed time-frequency $\omega$ have wave-numbers given by solutions of the Rayleigh-Lamb equations</p>
<p>$$\frac{\tan (pd)}{\tan (qd)}=-\left[\frac{4k^2pq}{\left(k^2-q^2\right)^2}\right]^{\pm 1}$$</p>
<p>where the +1 exponent corresponds to symmetric modes and the -1 exponent to antisymmetric modes, and</p>
<p>$$p^2=\frac{\omega ^2}{c_L^2}-k^2$$</p>
<p>$$q^2=\frac{\omega ^2}{c_T^2}-k^2$$</p>
<p>where $c_L^2=\frac{2\mu +\lambda }{\rho }$ is the longitudinal wave-speed and $c_T^2=\frac{\mu }{\rho }$ is the transverse wave-speed ($\mu$ and $\lambda$ being the Lame constants). From this equation one obtains certain dispersion curves, relating the wave-number of each mode to the frequency. My question is: For each value of the frequency, the Rayliegh-Lamb equations give a discrete number of wave-numbers. How do you decide which wave-number belongs to which mode? That is, if I have a set of wave-numbers corresponding to one value of the frequency, say $\omega_1$, and then I get a new set of wave-numbers corresponding to a different frequency, say $\omega_2$, then how do I identify wave-numbers corresponding to different frequencies as belonging to the same mode?</p>
<p>I hope I've explained myself adequately. You can see that my question can also be framed for many other trascendental equations with multiple roots that show up in mathematical physics. I chose this particular problem to make my point. One might think that it is just a matter of ordering the wave-numbers in increasing order. But I've seen dispersion curves in which the curves for different modes intersect, so an ordering criterion is not applicable (not sure if this happens for the Rayleigh-Lamb modes, though)</p>
<p>Thanks. And sorry for the brick of text.</p> | g9545 | [
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<p>I'm trying to understand how the <a href="http://en.wikipedia.org/wiki/Gell-Mann%E2%80%93Nishijima_formula" rel="nofollow">Gell-Mann–Nishijima relation</a> has been derived:
\begin{equation}
Q = I_3 + \frac{Y}{2}
\end{equation}
where $Q$ is the electric charge of the quarks, $I_3$ is the isospin quantum number and $Y$ is the hypercharge given by:
\begin{equation}
Y = B + S
\end{equation}
where $B$ is the baryon number and $S$ is the strangeness number. </p>
<p>Most books (I have looked at) discuss the Gell-Mann–Nishijima in relation to the approximate global $\mathrm{SU(3)}$ flavour symmetry that is associated with the up-,down- and strange-quark at high enough energies. But I have yet to fully understand the connection between the Gell-Mann–Nishijima and the $\mathrm{SU(3)}$ flavour symmetry. </p>
<p>Can the Gell-Mann–Nishijima relation somehow been derived or has it simply been postulated by noticing the relation between $Q$, $I_3$ and $Y$? If it can be derived, then I would be very grateful if someone can give a brief outline of how it is derived.</p> | g9546 | [
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<p>I am trying to recreate the results of an article (membrane simulation) and I have the following line:</p>
<blockquote>
<p>Both particles have the same soft radius,</p>
<p>$U_{rep} (r)/\epsilon = \text{exp}\left\{ -20 (r/\sigma -1 )\right \}$</p>
</blockquote>
<p>With $\epsilon$ being the energy scale, $\sigma$ being the length scale, and $r$ is the distance between two particles. This is a two body interactions. If I calculate the interactions between two particles, should I put $U_{rep}$ or half of it into each particle? </p> | g9547 | [
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<p>I want to have a general solution for calculating orbits. Wikipedia says how to put object on circle orbit. We must give it speed $v_1 = \sqrt{g(h_o +R)} $ Where $h_o$ - is the orbit height, and $R$ - Earth radius.</p>
<p>But how to calculate the speed that the object must have, if we want to place it in an elliptical orbit?</p>
<p>For example: </p>
<p>Having massive massive spherical object with mass $M$ and radius $R$. </p>
<p>which speed $V$ we must to give to this object to put it on elliptical orbit with eccentricity $\epsilon = E$ , and perihelium $P$?</p> | g311 | [
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<p>In a Young’s double slit experiment 10 fringes are observed in a given segment of the screen when light of wavelength 500.0 nm is used. When the experiment is performed in a medium of refractive index 1.2, the number of fringes observed in the same segment remains 10. The wavelength of light used in this case is? </p>
<p>(Also provide some explanations)</p>
<p>(But do look at my Answer & say whether it is correct or not.)</p>
<p>My Explanations: </p>
<p>Number and position of fringes formed in young’s double slit experiment is only dependant on the frequency of light used. Here, the number of fringes in both cases remains same because light of same frequency is used in two different mediums. To compare the wavelengths, we should only look at the change in speed and wavelength caused by the new medium, as compared to vacuum.
As given in the question, the number of fringes in a particular segment doesn’t change, so fringe width (β) remains same.
Therefore,</p>
<p><img src="http://i.stack.imgur.com/Dh1NJ.jpg" alt="enter image description here"></p>
<p>Where β = fringe width</p>
<p>When a ray of light passing through vacuum with wavelength λ1 is passed through a medium of refractive index µ, the new wavelength λ2 is given by:</p>
<p><img src="http://i.stack.imgur.com/o5Zmp.jpg" alt="enter image description here"></p>
<p>(which can be proved by the fact that velocity of light in a medium is inversely proportional to refractive index. Since frequency of light is constant (depends on the source only), refractive index is also inversely proportional to wavelength.)
In the question, µ = 1.2, λ1 = 500nm. </p>
<p>Hence, λ2 = (500) / (1.2) = 416.6667 nm.</p> | g9548 | [
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<p>If <strong>yes</strong>, then how does this accord with relativity: the laws of physics are the same in all reference frames? We can move from a reference frame in which the photon has near zero energy density, to a reference frame where it has near infinite energy density.</p>
<p>If <strong>no</strong>, why are there no gravitational effects for a photon with an arbitrarily high energy density? For some reference frames, a photon can have more energy that the mass of the sun. How are these reference frames consistent with each other?</p> | g9549 | [
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<p>I am working on the Hydrogen atom and I was trying to calculate $\frac{d<r>}{dt}$ using $$\frac{d<r>}{dt} = \frac{i}{\hbar} <[\hat{H} , \hat{r}]>.$$ Here $r = \sqrt(x^2 + y^2 + z^2)$ and $H = \frac{p^2}{2m} + V$ where $p^2 = -\hbar^2 \nabla^2 $. Now according to Ehrenfest's theorem should behave classically and give me some equivalent of velocity, and indeed I do get something but it does't resemble velocity: $\frac{-\hbar^2}{2m} (2\nabla r \nabla f + f \nabla^2 r)$ </p>
<p>where $f$ is a test function.</p>
<p>Steps:
$[H ,r]f = [\frac{p^2}{2m} + V , r]f = \frac{p^2(rf)}{2m} + Vrf - \frac{rp^2(f)}{2m} - rVf = \frac{1}{2m}[p^2 ,r]f = \frac{1}{2m}[-\hbar^2\nabla^2 , r]f = \frac{-\hbar^2}{2m}[\nabla^2 , r]f $$= \frac{-\hbar^2}{2m} (\nabla^2(rf) - r\nabla^2(f)) = \frac{-\hbar^2}{2m} (\nabla r\nabla f + r\nabla^2f + \nabla f \nabla r + f\nabla^2 r - r\nabla^2f) = \frac{-\hbar^2}{2m} (2\nabla r \nabla f + f \nabla^2 r) $</p>
<p>Am I doing something wrong?</p> | g9550 | [
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<p>Firstly I think shades of this question have appeared elsewhere (like <a href="http://physics.stackexchange.com/questions/10088/why-does-gravity-need-to-be-quantised">here</a>, or <a href="http://physics.stackexchange.com/questions/57228/why-is-gravity-such-a-unique-force">here</a>). Hopefully mine is a slightly different take on it. If I'm just being thick please correct me.</p>
<p>We always hear about the force of gravity being the odd-one-out of the four forces. And this argument, whenever it's presented in popular science at least, always hinges on the relative strength of the forces. Or for a more in depth picture <a href="http://physics.stackexchange.com/questions/57228/why-is-gravity-such-a-unique-force">this excellent thread</a>. But, having had a single, brief semester studying general relativity, I'm struggling to see how it is viewed as a force at all.</p>
<p>A force, as I understand it, involves the interaction of matter particles with each other via a field. An energy quantisation of the field is the force carrying particle of the field.</p>
<p>In the case of gravity though, particles don't interact with one another in this way. General relativity describes how space-time is distorted by energy. So what looked to everyone before Einstein like two orbiting celestial bodies, bound by some long distance force was actually two lumps of energy distorting space-time enough to make their paths through 3D space elliptical. </p>
<p>Yet theorists are still very concerned with "uniting the 4 forces". Even though that pesky 4<sup>th</sup> force has been well described by distortions in space time. Is there a reason for this that is understandable to a recent physics graduate like myself? </p>
<p>My main points of confusion:</p>
<ul>
<li>Why is gravity still viewed as a force?</li>
<li>Is the interaction of particles with space time the force-like interaction?</li>
<li>Is space-time <em>the</em> force field?</li>
<li>If particles not experiencing EM/weak/strong forces merely follow straight lines in higher-dimensional space (what I understand geodesics to be) then how can there be a 4<sup>th</sup> force acting on them? </li>
</ul>
<p>Thanks to anyone who can help shed some light on this for me!</p> | g9551 | [
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<p>There is a large disk of glass sitting outside, it is pretty thick and wide. Via different sensors hooked up to a controller, I can measure the air temperature, glass temperature (on the surface), and air humidity. Since the disk is fairly large, the temperature difference between it and the air could reach a few degrees-- due to thermal inertia and the fact that air temperature can change rapidly after sunset and sunrise.</p>
<p>Is there a formula or some empirical data that will tell me when the conditions are right for the disk to start getting covered in dew?</p>
<p>Would air flow across the glass surface matter? If it does, assume the air is nearly static (except for convection) - there's no wind. Also, humidity such as fog or rain drops falling from the sky can't reach the disk. All I care about is humidity in the air around the disk condensing on the glass surface.</p> | g9552 | [
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<p>Let us define a 4D rotation by using two unit quaternions: $$\mathring{q}_l=\frac{a+ib+jc+kd}{\left|a+ib+jc+kd\right|}$$ and $$\mathring{q}_r=\frac{e+ib+jc+kd}{\left|e+ib+jc+kd\right|}.$$ They differ only by the real term (the denominators are only for normalization). And let us consider two cases: the $a,b,c,d,e \in \mathbb{R}$ in the first case and $a,b,c,d,e \in \mathbb{C}$ in the second case - that would be biquaternions. Now let us consider a rotation of an arbitrary (non-unit) quaternion $\mathring{q}$ so that: </p>
<p>$$\mathring{q}_{rotated}=\mathring{q}_l\,\mathring{q}\,\mathring{q}_r^* $$ </p>
<p>(note the conjugate of $\mathring{q}_r$). I can represent this 4D rotation using a $4 \times 4$ rotation matrix, and thus I can find eigenvalues and eigenvectors of this matrix. The question is - what is the interpretation of eigenvectors of such rotation?</p>
<p>In 3D rotations the invariant vector (a vector that is not being rotated) is the rotation axis, and the eigenvalue has to be 1 (because it is a rotation, so no particular interpretation here). In 4D rotations I know that there are two planes around which the rotation occurs, but are those planes the invariant of the rotation? If yes, then how can I describe this plane as an eigenvector quaternion - would such quaternion be a "normal vector" of this plane, analogous to 3D space where normal vector of a plane is just a 3D vector?</p>
<p>How is that different between <a href="http://en.wikipedia.org/wiki/Quaternion" rel="nofollow">quaternions</a> and <a href="http://en.wikipedia.org/wiki/Biquaternion" rel="nofollow">biquaternions</a>?</p>
<p>Please note that since $e \neq a$ the normalized quaternion $\mathring{q}_r$ will be very different from $\mathring{q}_l$ - all the imaginary components of unit quaternion (those created by $b,c,d$) will be in fact different, thanks to normalization.</p> | g9553 | [
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<p>In physics total energy is extremal for the static solutions of equation of motions.</p>
<p>Can anyone explain this sentence to me?</p> | g9554 | [
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<p>The short story is, that I have to calculate some transport coefficients, but using the the MB distribution as my distribution function.
What I currently need to solve is:</p>
<p>${{\mathcal{L}}^{\,\left( \alpha \right)}}=\frac{1}{2}\frac{{\bar{N}}}{V}{{\left( \frac{{{h}^{2}}}{2\pi m{{k}_{B}}T} \right)}^{3/2}}\frac{1}{{{k}_{B}}T}\int{{{e}^{-\varepsilon /{{k}_{B}}T}}}{{\left( \varepsilon -\mu \right)}^{\alpha }}{{\varepsilon }^{3/2}}d\varepsilon$</p>
<p>As it shows, it's not that the integral is hard, it's more what I use for $\mu$ that is my concern. According to my teacher I need to use:</p>
<p>${{e}^{-\mu /{{k}_{B}}T}}=\frac{1}{2}\frac{V}{{\bar{N}}}{{\left( \frac{2\pi m{{k}_{B}}T}{{{h}^{2}}} \right)}^{3/2}}$</p>
<p>But I can't see how I can use this instead of $\mu$ without solving for $\mu$. And then I would just end up with a ln() term, which I really can't imagine doing anything good for me. But maybe I'm mistaken ?</p>
<p>Any hint would be appreciated :)</p> | g9555 | [
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<p>I am trying to reproduce a calculation by <a href="http://dx.doi.org/10.1080/00218469508014354" rel="nofollow">Carre et al. (1995)</a> in which they calculate the shape of a droplet on an incline.</p>
<p>My issue is in the derivation of the potential energy (essentially the center of mass) of the droplet. The droplet is defined in the coordinate system as shown below, with the droplet base being circular. The function for $z$ is known, but not important for now.</p>
<blockquote>
<p><img src="http://i.stack.imgur.com/18JOI.png" alt="coordinate system for droplet on an incline"></p>
</blockquote>
<p>The potential energy is defined with respect to the origin of the coordinate system. </p>
<p>The function given in the paper (in Eq. 5) is
$$E_p=\int_{-\pi}^{\pi}\int_0^{r_0} \rho g z r \left(\frac{z}{2}\cos\alpha-r\sin\alpha\cos\phi\right) dr\,d\phi$$</p>
<p>I cannot figure out how they get to this. I understand that there should be a $\cos\alpha$ in there due to the inclination. Moreover, I would say the potential energy should be $$E_p=\int_{-\pi}^{\pi}\int_0^{r_0}\int \rho g z dz\,rdr\,d\phi $$ and since I am going to apply the equation in an euler-lagrange like scheme I only need to integrate the $z$ component. However, I am having some doubts here, because the height of the center of mass above the origin would be $h=z \cos\alpha$.</p>
<p>Apart from those 'vague' statements above I can't get to the answer. Could someone please explain (i) whether my equation for the potential energy is correct (or whether I need something in terms of $h$?) and (ii) how I could go about doing the integration.</p> | g9556 | [
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<p>There is something I don't really understand about flashes of lightning.</p>
<p>When a flash occurs, how come electricity be transferred at the speed of light since electricity's displacement is very slow ?</p>
<p>It is not that I don't understand anything about that. I've read wikipedia and the others. But on that particular point there's something I really don't get ...</p>
<p>But the question could be for lighting as for any charge strike between two points of course.</p> | g9557 | [
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] |
<p>I am trying to understand the workings of a CMOS image sensor. I understand that increasing wavelength results in an increased penetration depth in the silicon often used in <a href="http://en.wikipedia.org/wiki/CMOS" rel="nofollow">CMOS</a> image sensors. What I am in need of assistance in understanding is how the wavelengths (corresponding to the primary colours, red green and blue) relate to the vertical structure of the CMOS sensor.</p>
<p>After reading several articles and performing some experiments, I am aware that CMOS sensors are indeed sensitive to UV wavelengths, but I am trying to understand how I was able to get a reddish-pink response using an incident UV source.</p> | g9558 | [
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<p>When we use the quantum fourier transform, for a function, the output is entangled, so if a measurement is made on the output, the result may not be that of the function that one wanted in the first place. For a trivial example, consider addition of two polynomials, using DFT, or
even FFT, this is straight forward, but if we consider the same setting in the quantum case, where FFT, or DFTs are subsituted with Quantum gates, how can the result of the addition two
polynomials be accomplished. </p> | g9559 | [
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<p>Given a fixed shell with the mass of $M$ and a radius $R$ ,
what would be the metric tensor for $r<R$?
I do know that using Birkhoff Theorem the metric for $r>R$ should be schwarzschild.
I'm not sure how to solve $G_{\mu\nu}=0$ for the inner part, and I'm not sure if I can demand continuity at $r=R$.</p> | g50 | [
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<p>Inspired by a similar post, I wanted to know what video games, past or present, most accurately employ the laws of physics. </p>
<p>I believe this is an appropriate question for this site, since the users are far more qualified to answer it than anyone on typical gaming sites, who think they know a lot more about physics than they really do. In my limited experience, most games claiming "realistic physics" are anything but.</p> | g9560 | [
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<p>Let us consider the functional integral:
\begin{equation}
\int \mathcal{D} A e^{iS[A]}
\end{equation}
where $S[A]$ is the action for $U(1)$ gauge field and
\begin{equation}
\mathcal{D}A\equiv \mathcal{D}A_0 \mathcal{D}A_1 \mathcal{D}A_2 \mathcal{D}A_3; \\
\mathcal{D}A_i = \prod_x dA_i(x).
\end{equation}</p>
<p>Now I have two questions:</p>
<p><strong>1. How to show that the integration measure $\mathcal{D} A $ is invariant under the gauge transformation:</strong>
\begin{equation}
A_\mu (x) \to A_\mu (x) + \frac{1}{e}\partial_\mu \alpha(x)
\end{equation}
<strong>2. How to show that the integration measure $\mathcal{D} A $ is invariant under the Lorentz transformation?</strong></p> | g9561 | [
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<p>This is a question motivated by <a href="http://en.wikipedia.org/wiki/Angry_Birds" rel="nofollow">Angry Birds</a>. </p>
<p>When playing the game, I notice that if the initial velocity is constant, the way to land a bird furthest away from the launching point is by launching the bird at 45 degrees from the ground. </p>
<p>Is it possible to obtain a derivation and a proof of this?</p> | g9562 | [
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<p>I know a lot about Theoretical Physicists from <a href="http://en.wikipedia.org/wiki/Sheldon_Cooper" rel="nofollow">Sheldon Cooper</a> of <a href="http://en.wikipedia.org/wiki/The_Big_Bang_Theory" rel="nofollow">Big Bang Theory</a>. I know about parallel universes (and I wonder if one codingtales in one of the parallel universes believes if parallel universes do not exist).</p>
<p>I'm sorry, but this is just for fun.
But I've real questions:</p>
<ul>
<li>Are all the equations that appear on the board and Sheldon tries to solve true?</li>
<li>How is a Rocket Scientist different from a Theoretical Physicist?</li>
<li>I figure out some errors in the definition of a "Screw" that Sheldon gives. He says he is on an inclined plane wrapped helically around its axis. As far as I know there are no inclined planes with axes. Can you please explain it to me?</li>
<li>Any other suggestion that would help me understand Big Bang Theory better would be appreciated.</li>
</ul>
<p>(And please forgive the tags. I didn't find "Theoretical-Physics" tag. I have to bluntly use some other alternative.)</p> | g9563 | [
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<p>Related: <a href="http://physics.stackexchange.com/questions/18117/is-a-white-object-always-white">Is a white object always white?</a></p>
<p>If you are standing in front of a glass window during the day, you can see your dim white t-shirt’s reflection in the window. The reflection is dim because only 4% of the light is reflected (assuming you’re perpendicular to the window) while the rest is transmitted through the glass. If a white object is placed in front of this glass pane, it’s obvious that white light is reflected as white and not colored. Why is this? </p>
<p>I vaguely remember that visible light that is reflected penetrates glass about λ/2, so that means that blue light penetrates the least while red light penetrates more. If I shine white light onto a glass window (assume RGB color model), I have to account for all three colored scatterers in roughly equal amounts of intensity to produce white. Here is what I think.</p>
<p>• Blue light penetrates shorter distances into the glass window’s atomic layers and recruits less atomic scatterers that back scatter more intensity blue light. I assume Rayleigh scattering is valid since wavelengths of light are much greater than the atomic spacing’s in glass, therefore $I ∝ 1/λ^4$.</p>
<p>• Green light penetrates deeper into the glass window’s atomic layers and recruits more atomic scatterers that back scatter more “less intensity” green light according to $I ∝ 1/λ^4$?. So the larger number of green atomic scatterers scattered less intense green light but there are more green atomic scatterers.</p>
<p>• In a similar fashion, red light would recruit an even larger number of red atomic scatterers (but less intense than green or blue), however, the increased number of red atomic scatterers compensates for a more overall intense red intensity. </p>
<p>In summary, there are roughly equal amounts of blue, green and red light reflected such that a glass window reflects white objects white. Here is what I am asking: is this correct way to view this? I was unable to find this anywhere online or on . I would greatly appreciate feedback, especially if this is not correct. </p> | g9564 | [
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<p>Suppose we have a container divided into equal halves. Right half is fixed at temperature $T$, volume $\frac{V}{2}$. </p>
<p>Initially it has pressure $P_0$, a hole of area $A$ is opened between them. I have to show that the pressure in the right half is:</p>
<p>$$ p = \frac{p_0}{2} \left[ 1 + exp \left(-\langle v \rangle\frac{A}{V}t \right) \right] $$ where $\langle v \rangle$ is average speed of molecules in right half.</p>
<p>I got an answer that's very very close, but off by a factor of $2$ in the exponential!</p>
<p><strong>Attempt</strong></p>
<p>$$ dN = -A\phi_{(t)} dt $$</p>
<p>Using flux $\phi = \frac{1}{4} n\langle v \rangle $,</p>
<p>$$dN = - \frac{A}{4} n \langle v \rangle dt = -\frac{A}{4}\left( \frac{N}{V/2} \right) \langle v \rangle dt = -\frac{NA}{2V} \langle v \rangle dt $$</p>
<p>The graph looks something like this: <img src="http://i.stack.imgur.com/KisWM.png" alt="enter image description here"></p>
<p>We solve for the dotted graph, then translate upwards by $\frac{N_0}{2}$.</p>
<p>Integrating, and translating upwards i get: $ p = \frac{p_0}{2} \left[ 1 + exp \left(-\langle v \rangle\frac{A}{2V}t \right) \right] $</p>
<p>Not sure why I'm missing a factor of $2$..</p> | g9565 | [
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<p>OK, so $\ce{Fe}$ is the most 'stable element'. As such, why do all elements above it not decay into $\ce{Fe}$? In all cases, would it not lead to an increase in binding energy and therefore energy been released, meaning it is energetically feasible, and should happen spontaneously (given enough time)? </p> | g9566 | [
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<p><strong>Disclaimer</strong>: I am not asking whether or not the events that occur in Anathem are possible. I understand that this book is a work of speculative science fiction and this site has no interest in it as such. I'm asking about the physics of a <strong>particular</strong> plot device.</p>
<p>In Anathem, Neal Stephenson states that the configuration of hadrons within atomic nuclei created during (stellar) nucleosynthesis is dependent on the fundamental constants of the universe (non derivable constants such as the speed of light and the cosmological constant), and that this configuration affects the electrical properties of the elements, and thus their chemistry.</p>
<p>In the book, physicists have created machines that can generate energy densities on the order of the big bang (or so). The idea is, in the beginning of the universe, the fundamental constants were unfixed and ambiguous. As the universe cooled down, the constants became fixed, the forces separated, and the universe was born. By creating high energy densities, the values of the constants can be fudged, allowing nucleosynthesis that generates nuclei with hadronic configuration different from ordinary matter. With that different configuration comes different electrical properties and thus different chemistry. This last feature prevents (or interferes with) 'newmatter' or matter from parallel universes with different constants from interacting chemically with matter of the universe the book is set in. These physicists tweaked the constants until they generated 'newmatter' with desirable chemical properties.</p>
<p>I have two questions. First, do the constants of nature affect nucleosynthesis in any way that measurably changes the electrical properties of the orbitals? Second, is there any veracity to the claim that the fundamental physical constants were 'unfixed' during the first instants of the big bang and that they can be locally unfixed by high enough energy densities?</p> | g9567 | [
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<p>I'm revisiting the elementary algorithms of renormalization that are taught in a classroom setting and find that the procedure taught to students is as follows:</p>
<ol>
<li>Write down the bare Lagrangian: $\mathcal{L}_0$</li>
<li>Renormalize the field strengths: $\phi_0\rightarrow Z^{1/2}\phi_r$</li>
<li>Renormalize coupling constants: <em>e.g.</em> $Z^2\lambda_0\rightarrow Z_\lambda \mu^{2\epsilon}\lambda_R$</li>
<li>For perturbation theory rewrite $Z=1+\delta$, where $\delta$'s are counterterms.</li>
</ol>
<p>I have thought long and hard about steps 2 and 3 (especially in the context of operator renormalization), and have come to the following conclusions which I would like verified.</p>
<p>In <strong>step 3</strong>, the renormalization of coupling constants is actually the renormalization of the operator product multiplying the coupling constant. And, in <strong>step 2</strong>, when they renormalize the field strength, it is not the renormalization of a single operator; they are actually renormalizing another composite operator: the kinetic term $\partial_\mu\phi\,\partial^\mu\phi$. The renormalization of a single operator $\phi$ actually corresponds to tadpoles, the one-point function.</p>
<p>Am I interpreting this correctly?</p> | g9568 | [
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<p>Can some one please explain to me why electricity flowing though a copper coil generates a magnetic field or where I could possibly find that information? Are there other materials that produce a magnetic field when a current is run through them in a different shape? Thanks!</p> | g9569 | [
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<p>Consider the typical problem, "You have a conducting sphere of charge $Q$ and a point charge q a certain distance away, what is the force on the point charge?". The solution is a simple application of the method of images, but in Griffiths part of the argument relies on saying
$$Q_{conductor} = \sum_i q_{image}$$
Or, that the image charges in the conductor add to the total charge. I don't really see how this is a mathematically valid statement. I was thinking it may come from the Neumann boundary conditions and integrating Poisson's eq, but I am not sure.</p>
<p>For reference consider Griffiths Problem 3.8 which deals with the $Q=0$ case.</p>
<p>An explanation would really be helpful.</p> | g9570 | [
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<p>Hydrogen is the lightest element, so it's cable of lifting the most weight in out atmosphere (probably not the best terminology there, but you get the picture)</p>
<p>Would hot hydrogen (in the same sense as hot air) be able to lift even more mass?
Would a higher or lower density of hydrogen in a ballon lift more?
If you could have a balloon which had nothing in it (it was a vacuum inside) would that lift more than a hydrogen balloon?</p>
<p>Basically what are the physics to balloons and lifting?</p>
<p>(really not sure what to tag this, so if someone else could that'd be great)</p> | g978 | [
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<p>The change of entropy is defined $$\Delta S = \int \frac{dQ_\mathrm{rev}}{T}.$$ If a system is isolated the heat transfer between the system and the surroundings is zero ($dQ = 0$), thus $\Delta S = 0$.</p>
<p>However, it is commonly stated that the entropy of an isolated system can increase. How is this possible, given the above definition of entropy?</p> | g9571 | [
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<p>I was in Solid State Physics lecture yesterday and we BRIEFLY went over what causes phonons to collide with one another. Things such as crystal imperfections, boundaries, Temperature, but I was wondering:</p>
<p>Do Optical mode phonons interact differently than acoustic ones? Also, is there a way I can quantify this?</p> | g9572 | [
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<p>As I see it, as light comes through approximately same angular area of the lens, it should have less aberrations, and at the same time, it should still have high θ -> diffraction limit should stay as without this diaphragm.</p>
<p>What am I missing?</p> | g9573 | [
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<p>I'm looking for a list of immiscible (and miscible) liquids.</p>
<p>I currently developing a game that will use the property of differents liquid, some of them will mix, other will not (positioning in the recipient is important so density is).</p>
<p>I'm searching but not found anything very relevant.</p>
<p>Of course I know water, oil (and mercury). But I'm looking for more than two (at least 8 differents elements in three or four groups).</p>
<p>I can always invent them but I'd like to keep things real as far as I can. </p>
<p>The elements must be at the liquid state in normal conditions.</p> | g9574 | [
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<p>This may sound as a mathematical question, but it should be very familiar to physicists. I am trying to perform an expansion of the function $$f(x) = \sum_{n=1}^{\infty} \frac{K_2(nx)}{n^2 x^2},$$ for $x \ll 1$. Here, $K_2(x)$ is the modified Bessel function of the second kind. This series is a result of solving the integral $$f(x) = \frac{1}{3}\int_1^\infty \frac{(t^2-1)^{3/2}}{\mathrm{e}^{xt}-1}\mathrm{d}t.$$ The result should be $$f(x) \approx \frac{\pi^4}{45 x^4} - \frac{\pi^2}{12 x^2}+\frac{\pi}{6x}-\frac{1}{32}\left( \frac{3}{2}-2\gamma+2\ln4\pi-\ln x^2\right)+\mathcal{O}(x^2),$$ where $\gamma$ is the Euler-Mascheroni constant. It agrees numerically with $f(x)$ for small $x$. However, by using the series expansion of the Bessel function $$K_2(nx) = \frac{2}{n^2x^2}-\frac{1}{2}+\frac{1}{2}\sum_{k=0}^\infty \left[\psi(k+1)+\psi(k+3)-\ln\frac{n^2x^2}{4}\right]\frac{\left(\frac{n^2 x^2}{4}\right)^{k+1}}{k!(k+2)!},$$ with $\psi(x)$ being the digamma function and using the zeta regularization for summation over $n$, I am able to reproduce all the terms except $\frac{\pi}{6x}$. I.e., my result is $$f(x) = \frac{2\zeta(4)}{x^4} - \frac{\zeta(2)}{2x^2} + \frac{1}{8}\sum_{k=0}^\infty \left[\left(\psi(k+1)+\psi(k+3)-\ln\frac{x^2}{4}\right)\zeta(-2k) + 2 \zeta'(-2k)\right]\frac{\left(\frac{x^2}{4}\right)^{k}}{k!(k+2)!}.$$ It seems very strange that the $\frac{\pi}{6x}$ term should appear in the expansion since only even powers of $x$ appear in $K_2(nx)$. But, numerically, it is certainly there. How did I miss it?</p>
<p>I think that the missing term indicates that the zeta regularization isn't used properly. The term $\frac{\pi}{6x}$ appears right in the middle, separating the convergent sums $\zeta(4)$ and $\zeta(2)$ from the (regularized) divergent sums $\zeta(-2k)$ and $\zeta'(-2k)$. So, the missing term may be the price to pay for using the zeta regularization. Unfortunately, I don't know enough formal math to, so to speak, "repair the damage". Any help on the subject is more than welcome.</p> | g9575 | [
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<p>The dynamic viscosity of air is $1.78\cdot 10^{-5}Pas = 1.78\cdot 10^{-5}\frac{kg}{ms}$. How do I convert this to the minute scale? I see a reason for multiplying the value with 60 (because of the $\frac{kg}{ms}$) or dividing it by 60 (because of the $Pa s$).</p>
<p>Background: I need the value in a numerical CFD simulation, where I have all my timesteps in the minute scale.</p> | g9576 | [
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] |
<p>What do the supercharges in <a href="http://en.wikipedia.org/wiki/Supersymmetry_algebra">extended supersymmetry</a> do?</p>
<p>In $N=1$ supersymmetry there are a certain number of fermions and and equal number of bosons. You can transform all fermions to the bosons (and vice versa) in a 1 to 1 fashion using a single supercharge, $Q$.</p>
<p>So what happens when you have, for example, $N=2$ supersymmetry with 8 supercharges? Since $Q$ is a generator of supersymmetry transformations, is it a linear combination of these supercharges that act on the particles? In which case could one particle be acted on by two separate linear combinations of $Q$? Or is it strictly one linear combination of $Q$ per fermion/boson?</p>
<p>Also, what does $N$ mean physically? What difference does $N=2$ have to $N=1$ other than more supercharges? Or is that the only difference between the two theories?</p> | g9577 | [
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<p>The other day we derived Kepler's third law.</p>
<p>$$
\left( \frac{T_1}{T_2} \right)^2 = \left( \frac{r_1}{r_2} \right) ^3
$$</p>
<p>In order to derive this, you can look at a given planet that revolves around the sun. If you assume that the sun is way heavier than the planet and that the planet moves on a circle, you can simply state the force on the Planet by the sun:</p>
<p>$$
F_G = G \frac{mM}{r^2}
$$</p>
<p>Where $F_G$ is the force on the planet, $G$ is the gravitational constant, $m$ is the mass of the planet, $M$ the mass of the sun and $r$ the distance of the planet to the sun.</p>
<p>Our tutor said that you now have to have a centrifugal force $F_c$ that is equal in absolute value to $F_G$, but opposite in direction so that the sum of the forces would be zero. The absolute value of this $F_c$ would then be equal to $F_G$:</p>
<p><img src="http://wstaw.org/m/2011/11/15/m9.png" alt=""></p>
<p>Then you plug in the formulas for the respective forces:</p>
<p>$$ |\vec{F_c}| = |\vec{F_G}| $$
$$ m \frac{v^2}{r} = G \frac{mM}{r^2} $$
$$ \frac{v^2}{r} = G \frac{M}{r^2} $$
$$ v^2 = G \frac{M}{r} $$
$$ v = \sqrt{G \frac{M}{r}} $$</p>
<p>Where $v$ is the tangential velocity of the planet.</p>
<p>Together with $v=\frac{2\pi r}{T}$, where $T$ is the revolution period, you get:</p>
<p>$$ \frac{2\pi r}{T} = \sqrt{G \frac{M}{r}} $$
$$ \frac{4\pi^2 r^2}{T^2} = G \frac{M}{r} $$
$$ \frac{r^3}{T^2} = G \frac{M}{4\pi^2} $$</p>
<p>Since the right part is constant for each solar system, you can form a ratio and optain Kepler's law. So this approach does work.</p>
<hr>
<p>I think that this view is only valid if you construct it from a rotating observation point, i. e. standing on the planet and facing the sun at all times. Our tutor said that this is from a static (with respect to the sun) observation point.</p>
<p>In the rotating system, you do have the centrifugal force, but it is not a real force in this sense. So by saying that you have a centrifugal force, I think that you are in this rotating reference frame. And in the rotating frame, the planet is not moving at all. And by not moving in this rotating frame, it does revolve around the sun in the static frame. Since the planet is not moving, there must not be any force on it. Constructing a force equilibrium is the way to achieve this.</p>
<p>But our tutor told me that there is a force equilibrium in the static reference frame as well. I say that if you have an equilibrium there, there is no net force, therefore the planet does only move along a straight line in a static frame. I was then told that although there is no net force, the forces are still there.</p>
<p>At that point, I think that this is converting dynamics into statics by adding opposing forces so that everything cancels out.</p>
<p>Say we <em>do</em> have a net force (gravity) on the planet.</p>
<p><img src="http://wstaw.org/m/2011/11/15/m10.png" alt=""></p>
<p>This force is something like this, if the planet has rotated $\theta$ around the sun:</p>
<p>$$
\vec{F_G} \propto
\left(
\begin{matrix}
-\cos(\theta) \\
-\sin(\theta)
\end{matrix}
\right)
$$</p>
<p>If you integrate this, you will get the velocity and the position of the planet.</p>
<p>$$
\int \vec{F_G} \, \mathrm d\theta \propto
\left(
\begin{matrix}
-\sin(\theta) \\
\cos(\theta)
\end{matrix}
\right)
$$</p>
<p>This seems very good, since it goes along the circle, so this is the tangential velocity.</p>
<p>$$
\iint \vec{F_G} \, \mathrm d\theta \propto
\left(
\begin{matrix}
\cos(\theta) \\
\sin(\theta)
\end{matrix}
\right)
$$</p>
<p>And that is simply the position around the unit circle.</p>
<p>You can substitute $\theta$ with $\omega t$ if you wish to express the angle with time.</p>
<p>So I claim that you need a net force in order for the circular trajectory to appear at all. My tutor told me that if the force is too high (too weak), the planet will spiral into (away from) the sun, and therefore the equilibrium has to be there. That makes no sense to me. What makes sense to me is that if the trajectory is deflected too much (not enough) to form a circle, there is no circle.</p>
<hr>
<p>If you have no force, integration turns out quite different:</p>
<p>$$
F = 0
$$</p>
<p>$$
\int F \, \mathrm dt = \underbrace{0t}_{a=0} + \underbrace{c_1}_{v_0}
$$</p>
<p>$$
\iint F \, \mathrm dt = \underbrace{0t^2}_{a=0} + \underbrace{c_1t}_{v_0} + \underbrace{c_2}_{d_0}
$$</p>
<p>The latter is your $d = v_0 t + d_0$, which models a linear, unaccelerated movement. In the rotating system, $v_0$ is just zero, and it works out. In the static system, that would mean that the planet either remains still or moves on a straight line -- which is not what is really happening.</p>
<hr>
<p>So I claim that you must <em>not</em> have a force equilibrium for the earth to rotate -- that is the basic disagreement we have.</p>
<p>Who is right? Or are we both right, just looking differently at the same problem?</p> | g9578 | [
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<p>So this appears in a problem which looks simple enough in its context; It's something like this:</p>
<blockquote>
<p><em>Two discs, <strong>A</strong> and <strong>B</strong>, are mounted coaxially on a vertical axle. The discs have moments of inertia $I$ and $2I$ respectively about the common axis. Disc A is imparted an initial angular velocity $2\omega$ using the entire potential energy of a spring compressed by a distance $x_1$. Disc B is imparted an angular velocity $\omega$ by the same spring compressed by a distance $x_2$. Both the discs rotate in the clockwise direction.</em></p>
<p><em>When disc B was brought in contact with A, they acquire a common angular velocity in time $t$. The average frictional torque by the other during this period is: <strong>?</strong></em></p>
</blockquote>
<p>The answer ($2I\omega/3t$) was obtained by applying angular momentum conservation and is exactly what confused me. <strong>How can we apply angular momentum conservation when friction is present?</strong></p> | g9579 | [
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<p>The Hamiltonian for an electron of mass $m$ and charge $e$ in an exterior electromagnetic field is
$$H=\frac{1}{2m}(p-(e/c)A)^2+e\varphi.$$
The corresponding (via canonical quantization) quantum mechanical Hamiltonian is not invariant under Gauge transformations (for $A$ and $\varphi$). What's the physical meaning of this? And what is the physical meaning of the fact that the classical hamiltonian is not invariant under gauge transformations?</p> | g9580 | [
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<p>In <a href="http://www.itp.uzh.ch/lectures/jetzer/GR2013/GR.pdf" rel="nofollow">these lecture notes</a> the static isotropic metric is treated as follows (p. 71):</p>
<p>Take a spherically symmetric, bounded, static distribution of matter, then we will have a spherically symmetric metric which is asymptotically the Minkowski metric. It has the form (in spherical coordinates):
$$ds^2=B(r)c^2dt^2-A(r)dr^2-C(r)r^2(d\theta^2+\sin^2\theta d\phi^2)$$
And then it goes on eliminating $C$ and expanding $A$ and $B$ in powers of $\frac{1}{r}$. No explanations are given on why we can assume that form for the metric. Could someone explain why, please?</p>
<p>Personally, I would rather assume the form (in cartesian coordinates):
$$ds^2=f(r)dt^2-g(r)(dx^2+dy^2+dz^2)$$
which would certainly give a spherically symmetric metric, and <em>then</em> change to spherical coordinates, obtaining something looking like:
$$ds^2=f(r)dt^2-g(r)(dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2)$$
which looks substantially different from the above. Is this approach wrong? Why?</p>
<p>By the way, don't be afraid of getting technical. I have a pretty good mathematical basis on the subject (a course of one year on differential geometry).</p> | g9581 | [
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<p>For an harmonic oscillator we can write the Hamiltonian eigenvalues in the basis of the amplitude eigenvalues : for example the ground state is a gaussian : $⟨x|0⟩=a.e^{-b.x^{2}}$. </p>
<p>I was wondering if we can do that for the energy states of a field : can we calculate $⟨a|n⟩$ with |a⟩ an eigenvalue of the field operator $\phi$ (let's say a Klein-Gordon field) ?</p>
<p>I failed to carry out the ladder operator method, but I think we should be able to calculate (or at least have implicit equation of) any operator eigenvalue in any other operator eigenvalue basis, or is there something obvious that I missed ? </p>
<p>Because we always play to write state in all the basis we can : spin on position, energy on spin etc ..., but I have never seen in classes or in books energy states of a field written in the field operator basis.</p>
<p>EDIT : I'll look in the reference given by Peter : "Weinberg I" and try to explain how to do.</p>
<p>EDIT : I didn't find the answer in the given reference :(</p> | g9582 | [
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<p>Two inertial frames K and k’ are considered. They are in relative uniform motion along the x-x’ direction with relative speed =v.
In the frame K’ we have a cuboidal piece of dielectric [at rest wrt K’]with a flat face perpendicular to the x-x’ direction, that is, this particular face is parallel to the y-z direction.The dielectric is homogeneous and isotropic within itself.
We now consider Maxwell’s equations [in a medium] wrt the dielectric in the rest frame of the dielectric,ie,K’.
If these equations are transformed, they should retain their form in K[according to the first postulate of SR]. But the individual values of the variables may change</p>
<p>With this information we proceed into the paradox.</p>
<p>Speed of light in the dielectric as observed from k’=nc [n is a positive fraction less than 1]</p>
<p>Relative speed between the frames, v=cn’[n’ is also a positive fraction less than 1]</p>
<p>For normal incidence:
Speed of light in the dielectric as observed from K[From Velocity-Addition Rule of SR]:
$v=\frac{nc+n’c}{1+nn’}$ ------------------- (1)</p>
<p>For oblique ray inside the medium at $\theta$ degrees degrees with respect to the x’ axis in the K’ frame:</p>
<p>$v’_x=ncCos (\theta)$</p>
<p>$v’_y=nc Sin(\theta)$</p>
<p>$v’_z=0$</p>
<p>[$v'_z$ has been taken to be zero for the convenience of calculations]</p>
<p>Observations from K:</p>
<p>$v_x=\frac{nc Cos(\theta)+n’c}{1+nn’Cos (\theta)}$</p>
<p>$v_y=\frac{nc Sin(\theta)}{1+nn’Cos(\theta)}\sqrt{1-n’^2}$</p>
<p>$v_z=0$
Therefore,</p>
<p>$v=\sqrt{[\frac{nc Cos(\theta)+n’c}{1+nn’Cos (\theta)}]^2+[\frac{nc Sin(\theta)}
{1+nn’Cos(\theta)}\sqrt{1-n’^2}]^2}$ ------------------- (2)</p>
<p>The results from (1) and (2) are not identical , though from the invariant Maxwell’s equations[in a medium] we understand that the speed of light should be the same in all directions inside the dielectric as observed from K.
What would be your answer to this paradox.
[My assessment:This paradoxical situation arises from the fact that we have applied SR in an incorrect context.It has been applied in an anisotropic and inhomogeneous configuration.
You could of course have a different assessment]
[The dielectric within itself is homogeneous and isotropic. But the overall space being considered is not homogeneous and isotropic] </p>
<p>If Maxwell's equations change their form wrt the Lorentz transformations,Gauss Law,Div B=0 etc may change if a piece of dielectric is loaded into a moving train!</p> | g9583 | [
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<p>So the wave moves like. a wave, it moves up and down, up and down. But how do photons move? Do they follow the same path or do they just go straight forward without oscillating?</p> | g9584 | [
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<p>Let's say I have a cylindrical piston filled with air and fitted with an electrical resistor. The resistor has a current going across it and a voltage for a certain amount of time. My system would be the air inside the piston.</p>
<p>In the first law of thermodynamics: Q - W = E,</p>
<p>Would the power from the resistor for a designated amount of time perform a heat transfer or a work transfer on the air?</p> | g9585 | [
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<p>Not sure if this is perfectly a physics question:</p>
<p>A rope is wrapped around a cylinder of diameter D. The cylinder is slightly eccentric so that the distance between the axis of the cylinder and the axis of rotation is e. As the cylinder rotates, it pulls the rope. At t=0, the far end of the rope is at position 0.</p>
<p>What is the function of the rope position with respect to the angle of rotation of the cylinder? (This is not a homework question BTW)</p>
<p><img src="http://i.stack.imgur.com/no8wz.png" alt="enter image description here"></p> | g9586 | [
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<p>For example; is $$\partial_{\rho}\partial_{\sigma}h_{\mu\nu} - \partial_{\sigma}\partial_{\rho}h_{\mu\nu}=0$$ correct?</p> | g9587 | [
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<p>First I must let you know that I don't have much understanding of neither GR nor quantum mechanics, and therefore this question.</p>
<p>I've mentally pictured Newtonian physics, GR and quantum mechanics all somewhat competing (in terms of use case) physics models. For one situation Newtonian physics model may be sufficient, but for another (e.g. interplanetary calculations) general relativity is needed. And then there's quantum mechanics.</p>
<p>Is it possible that the exact same exercise can be solved in Newtonian physics, GR and quantum mechanics? Under "solved" I mean that it can be calculated according to the model (Newtonian, GR or QM) but the answer would come different and hence illustrating the need of why and for what GR was required and the same for QM.</p>
<p>Basically what I mean is that is there any problem that could be solved with different physical models? For example if I would have a problem A, then by applying a solution based on Newtonian physics (NP), GR or QM, I would get different results, e.g:</p>
<pre><code>NP(A) = x
GR(A) = y
QM(A) = z
</code></pre>
<p>If it's possible, can someone please give an concrete example?</p> | g9588 | [
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<p>A free particle moves along geodesics, one form being</p>
<p>\begin{split}
\ddot x^\mu &= -\Gamma^{\mu}_{\sigma \rho} \dot x^\sigma \dot x^\rho \\
&= -\frac{1}{2}g^{\mu \nu}(\partial_\sigma g_{\rho \nu} + \partial_\rho g_{\sigma \nu} - \partial_\nu g_{\rho \sigma}) \dot x^\sigma \dot x^\rho
\end{split}</p>
<p>Applying the lowering operator, we have</p>
<p>\begin{split}
\ddot x_\mu &= g_{\mu \alpha} \ddot x^\alpha \\
&= -\frac{1}{2}\delta^\mu_\nu(\partial_\sigma g_{\rho \nu} + \partial_\rho g_{\sigma \nu} - \partial_\nu g_{\rho \sigma}) \dot x^\sigma \dot x^\rho
\end{split}
Hence
$$\tag{1}
\ddot x_{\mu} = -\frac{1}{2}(\partial_\sigma g_{\rho \mu} + \partial_\rho g_{\sigma \mu} - \partial_\mu g_{\rho \sigma}) \dot x^\sigma \dot x^\rho
$$</p>
<p>Equivalently, using the intrinsic derivative, </p>
<p>$$
\frac{Dt_\mu}{D\tau} = 0 $$ for a free particle, so $$
\frac{dt_\mu}{d\tau} = \Gamma^\sigma_{\mu \rho} \dot x_\sigma \dot x^\rho
$$ </p>
<p>or </p>
<p>$$
\ddot x_\mu = \frac{1}{2}g^{\sigma \nu}(\partial_\mu g_{\rho \nu} + \partial_\rho g_{\mu \nu} - \partial_\nu g_{\rho \mu}) \dot x_\sigma \dot x^\rho
$$
Hence
$$\tag{2}
\ddot x_\mu= \frac{1}{2}\partial_\mu g_{\sigma \rho}\dot x^\sigma \dot x^\rho
$$
(The summation on $\sigma$ and $\rho$ in the last two terms cancels)</p>
<p>Clearly (1) $\neq$ (2). Does anyone know where my reasoning has gone astray?</p>
<p>Much Appreciated</p> | g9589 | [
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<p>Is there a software (open source preferred) where I would input something like "Ingoing: a fermion $(p1, s1)$ and a photon $(p2, s2)$. Output: A fermion $(k1, r1)$ and a photon $(k2, r2)$" and I would could then get each diagram (that is, each term in the Dyson series) up to whichever order I choose, also showing terms that evaluate to zero (just to see that they are there)?</p>
<p>I have tried to compute the above diagram to 2nd order and I'm just baffled with how many details there are to collect, this seems like the perfect job for a computer, but I can't find anything that would do such a symbolic calculation (hopefully showing the steps along the way) and not just the final result. </p> | g9590 | [
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<p>I can not do the following problem:</p>
<p>Prove the identity:</p>
<p>$$\left( \frac{\partial x}{\partial y} \right)_{z}\left( \frac{\partial y}{\partial z} \right)_{x}\left( \frac{\partial z}{\partial x} \right)_{y}=-1$$</p>
<p>State the properties that must be $x=x(y,z)$, $y=y(x,z)$, $z=(x,y)$.</p>
<p>The truth is I do not know how to start, and they do not know how to interpret the functions $x$, $y$, $z$. Any help or explanation will be the most grateful.</p> | g9591 | [
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<p>Say I have a laser. If I spin the laser so that the beam sweeps in an arc along a very distant object, could that dot travel faster than the speed of light?</p>
<p>In Diagram form:
<img src="http://i.stack.imgur.com/tqv8C.jpg" alt="laser beam thought experiment"></p> | g312 | [
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<p>A candle is floating in a liquid placed in a container. The container is a cylinder of diameter $D$, and the candle is of width $d$. ($D>d$) The height of the liquid from the bottom of the container is $p$, and the height of the candle flame from the bottom is $h$. The density of the candle material is $p_c$ and that of the liquid is $p_l$.</p>
<p>If the length of the candle changes by $\Delta L$, find the change in the level of the liquid $\Delta p$, and the change in height of the flame $\Delta h$.</p>
<p>My attempt:</p>
<p>I'm trying to use Archimedes principle. Suppose $x$ is the height of the submerged candle when its length is $L$. Then, balancing gravitational and buoyant forces, $$p_c \pi \left(\frac{d}{2}\right)^2 L g = p_l \pi \left(\frac{d}{2}\right)^2 x g $$</p>
<p>So $$x = \frac{p_c}{p_l}L$$</p>
<p>Now suppose the length of the candle changes by $\Delta L$, causing the liquid height to change by $\Delta h$. I wrote: $$p_c \pi \left(\frac{d}{2}\right)^2 (L+\Delta L) g = p_l \pi \left(\frac{d}{2}\right)^2 (x+\Delta p) g$$</p>
<p>Which gives $$\Delta p = \frac{p_c}{p_l}\Delta L$$</p>
<p>Here I assumed that the change in liquid level would contribute to additional buoyant force. However, I'm not getting the right answer, which involves $D$ as well. So I'm not sure how to use Archimedes Principle for the required case. Any suggestions?</p> | g9592 | [
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<p>So I am wondering what kind of two dimensional Fermi surface is called quasi one dimensional, what is its character? Also, when there are orbital hybridization taking place in lattice site, what are characters of hybridized Fermi surface?</p> | g9593 | [
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<p>I'm studying the introduction of flavour degrees of freedom in the AdS/CFT correspondence and now I'm supposed to calculate the mass spectrum of mesons in the <a href="http://www.google.com/search?as_q=coulomb+branch" rel="nofollow">Coulomb branch</a>. I have searched the concept but I always find very long and complex explanations. Could anyone explain it in a direct way, pointing out some physical intuition?</p> | g9594 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/8659/invariant-spacetime-distance-circular-motion">Invariant spacetime - distance - Circular Motion</a> </p>
</blockquote>
<p>This is a question that I thought up a few years ago when I was taking mechanics. I asked the professor but didn't really get a straight answer. </p>
<p>Imagine a record spinning at relativistic speeds. In the lab frame, the circumference of the record should decrease according to the Lorentz contraction. However the radius of the record should remain fixed since it is orthogonal to the direction of motion. So does the shape of the record appear distorted since the circumference is smaller but the radius is the same? What would it look like?</p>
<p>I have a feeling the answer is obvious and/or well known since it would seem to be similar to the situation of relativistic particles traveling around a circular accelerator but I can't think of how it can be solved.</p> | g306 | [
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<p>I have difficulty understanding the following vector calculus example. Text can be found <a href="http://www.nhn.ou.edu/~shaferry/41832005_files/l9.pdf" rel="nofollow">here</a>. It is the 5th Q&A -- starting with equation (31.1035).It concerns finding the vector potential of a current loop.</p>
<p>I get the argument up to $$\vec A=\frac{\mu I}{4\pi r^3}\oint \vec r'\cdot \vec r \,\,\,d\vec l'$$.
But I don't understand why $$\oint \vec r'\cdot \vec r \,\,\,d\vec l'=\left(\int d\vec a\right)\times \vec r$$</p>
<p><em><strong>[ADDED]</em></strong>: Definitions of the terms:</p>
<p>This describes a current loop, with a current $I$ flowing in it.</p>
<p>$\vec r$ is the position vector of an arbitrary point in space. $\vec r'$ is a position vector of a point on the loop. $d\vec l'$ is an infinitesimal line element along the loop.</p>
<p>$\vec A(\vec r)$ is the vector potential, given by $$\frac{\mu I}{4\pi}\oint \frac{d\vec l'}{|\vec r'-\vec r|}$$
The above expression is obtained by Taylor expanding this definition.</p>
<hr>
<p>The first thing I tried is the vector triple product identity
$$(\vec r'\cdot \vec r )\,\,\,d\vec l'=\vec r\times(d\vec l'\times \vec r')+(\vec r\cdot d\vec l')\vec r'$$
The first term looks like it's doing the right sort of thing, but I don't see why the second term vanishes (or fits in) ...Maybe I have missed out something quite obvious?</p> | g9595 | [
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<p>In a nutshell, Noether's theorem states that for every continuous symmetry a corresponding conserved quantity exists.</p>
<p>Now, the Hamiltonian equations of motion (let's talk about a classical system here) are invarinat under addition of any constant to the Hamiltonian function</p>
<p>$$H\rightarrow H+ \text{const.}$$</p>
<p>Is there a corresponding conserved quantity?</p> | g9596 | [
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<p>The action</p>
<p>$$S=\int L \;\mathrm{d}t$$</p>
<p>is an important physical quantity. But can it be understood more intuitively? The Hamiltonian corresponds to the energy, whereas the action has dimension of energy × time, the same as angular momentum.</p>
<p>I've heard the action being described as a measure of <em>change</em>, although I don't know how this description can be justified.</p> | g9597 | [
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<p>I've had this conceptual problem with Faraday's law and inductance for a while now.</p>
<p>Take the example of a simple current loop with increasing area in a constant field (as in <a href="http://physics.stackexchange.com/a/1790/12255">this answer</a>). So Faraday's law states that the increasing flux (due to the increasing area) causes an EMF and hence a current. Due to the minus sign in Faraday's law, or by Lenz's law the direction of the current is such that the magnetic field it creates opposes the external field.</p>
<p>Now why do we never consider the magnetic field created by the induced current, when calculating the change in flux? All workings I have seen always calculate the flux from $\underline{\mathbf B} \cdot \underline{\mathbf A}(t)$. Why is $\underline{\mathbf B}$ not adjusted by the induced magnetic field? Is it just <em>that</em> small that we can neglect it unconditionally?</p>
<p>I have the same problem with self-inductance in AC circuits (although, maybe if I understood the above problem, this would become apparent to me as well). Say we start from current $I=0 \text{A}$. Then the EMF in the circuit increases (but is still very low), which increases $I$, which in turn creates an increasing $\underline{\mathbf B}$ inside the coil. Wouldn't the induced counter-EMF be much greater than the external EMF applied to the circuit? And if so, how come there is current moving in the first place, if the slightest increase in EMF causes a counter-EMF which acts to stop the current?</p>
<p>Is it just that I am looking at idealised situations or that the magnitudes of the external and induced effects differ greatly? Or do I have a conceptual misunderstanding about how (self-)inductance works?</p> | g9598 | [
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<p>I've had a wild idea which I can not discuss at length in this forum, but it comes down to the following problem:</p>
<p>A sphere of radius R=~10μm and mass m=~10-16Kgr is travelling towards the earth at v = ~10^8m/sec. The sphere carries a charge Q and intersects the earth's magnetic field perpendicularly, at a distance r, such that it enters an elliptic orbit. Given an electron wave function of ~5eV, is there any possible configuration of Q and r that would
allow the sphere to end up in a stable geo-stationary orbit?</p>
<p>Would releasing charges at particular points of each rotation make any difference?</p>
<p>Even though I did get a physics BS 13 years ago, I took a different direction and the calculations required are way beyond my current capabilities. If anyone finds the problem interesting, I'll be glad to hear from you.</p> | g9599 | [
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<p>Wigner crystal and FQH effect are both due to strong electron-electron interaction under magnetic field. As we know, Landau's symmetry-breaking cannot be used to describe FQH state. But can it be used to describe Wigner crystal state and which symmetries are broken in Wigner crystal?</p> | g9600 | [
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<p><em>Note:</em> This is a rewrite of the original question, which was titled <a href="http://physics.stackexchange.com/revisions/20511/2">What would time be for 2D beings?</a></p>
<hr>
<p>In my current, non-physicist's understanding, every instant of our three‑dimensional world is just another 'slice' of a four‑dimensional body. I don't mean that as an analogy, but quite literally... Obviously, it would not be a straight 'slice', it would still be bent and curved by gravity, speed and other relativistic factors. Is this wrong?</p>
<p>Also, both 'spacial' and 'temporal' dimensions are — in my mind — fundamentally the same thing, given different names because we experience them differently because of our nature. I've had people explicitly say in the comments that this is wrong and that time and space are not the same 'type' of dimension. I'd like to understand what are the fundamental differences.</p>
<hr>
<p><em>In my original question</em> I used these two assumptions of mine (that space is a slice of time and that all dimensions are fundamentally the same) to make an analogy.</p>
<p>I noted that taking two-dimensional slices of a three-dimensional body — just as a slice of four‑dimensional time is three‑dimensional space — and displaying them is rapid succession looks like a bunch of matter that is changing over time (like in this brain scan below). </p>
<p><img src="http://i.stack.imgur.com/rezJs.gif" alt="Brain scan"></p>
<p>Assuming all dimensions, temporal or spatial, are fundamentally the same, would that mean that for a hypothetical two‑dimensional being time would be the thrid dimension, not that fourth?</p>
<p>The same question in a more general form: for any $n$‑dimensional being, would time for it be the $n+1$ dimension?</p>
<p>In particular, what would time be for a being living in a hypothetical fifth dimension?</p> | g9601 | [
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] |
<p>How do we estimate the energy released from knocking on a wooden table or a door?</p> | g9602 | [
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] |
<p>I am learning about circular motion and not quite sure how to approach this particular problem. Any help would be greatly appreciated!</p>
<blockquote>
<p>A particle moves along a circular path over a horizontal $x$-$y$ coordinate system, at constant speed. At time $t_1 = 5$ s, it is at point ($4.00$ m, $8.00$ m) with velocity ($1.60$ m/s)$\hat{\mathbf{j}}$ and acceleration in the positive $x$ direction. At time $t_2 = 8$ s, it has velocity ($-1.6$ m/s)$\hat{\mathbf{i}}$ and acceleration in the positive $y$ direction. What are the coordinates of the center of the circular path?</p>
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<p>When you boil a liquid the KE stays the same (as temp is the same), so where does the extra energy go? I know the extra energy put in breaks bonds between the molecules/ atoms, but once it has broken the bonds where does the energy go?</p>
<p>And where does the potential energy in a liquid go- as (ideal) gases don't have PE? I've heard it's turned to latent heat, but what is this, how is it stored?</p> | g9604 | [
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<p>Let's say we have two systems A and B. Each system is described by a density matrix $\rho_A$ and $\rho_B$. I'm wondering about the formal notation to write down the expectation value of an operator that takes us from system A to system B or vice versa.</p>
<p>Normally the expectation value of an operator M acting on a system X is just:</p>
<p>$ <M> = Tr(\rho_X M)$.</p>
<p>But how would I write this for two systems? Would it be a tensor product of the two density matrices: $\rho_A \otimes \rho_B \rightarrow$ $ <M> = Tr(\rho_A \otimes \rho_B M)$ ?</p>
<p>If I would use state vectors I would just write: $<M_{ij}> = <i|M|j>$ and get my expectation value for a specific M or for all M if I do that for all possible states (in this case pure states for simplification).
Any idea how to express this in the density matrix trace formalism? How would it look for pure and mixed states?</p> | g9605 | [
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<p>I have to find the Lagrangian for a system.
In the point of interest I have come up with the following position coordinates:</p>
<p>$$x = Rcos(\omega t)+\ell sin(\phi)$$
and
$$y = Rsin(\omega t)-\ell cos(\phi)$$
Now, since I want to find the Lagrangian I need to take the derivative with respect to $t$ of both of them (To get the $\dot{x}$ and $\dot{y}$), and plug it into the formula for kinetic energy. Although it seems easy, the result I should get I'm a little confused about.</p>
<p>In my opinion only the first term in each equation is the only ones that is dependent of $t$, but I know that the answer should be:</p>
<p>$$\dot{x} = -R \omega sin(\omega t)+\ell cos(\phi)\dot{\phi}$$
and
$$\dot{y} = R \omega cos(\omega t)+\ell sin(\phi)\dot{\phi}$$</p>
<p>So my question is, mathematically, how do I justify, that I just take the derivative of the second term in each equation, as it was dependent of $t$ (Which is obviously is), and the just put a $\dot{\phi}$ outside?</p> | g9606 | [
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<p>I posted this in the mathematical forums. Maybe you will help me. I found an hard article <a href="http://prola.aps.org/abstract/PR/v105/i3/p776_1" rel="nofollow">http://prola.aps.org/abstract/PR/v105/i3/p776_1</a> of yang huang and luttinger. The authors begins with the sum: $$\frac{{E_2 }}{{E_0 }} = \frac{{16\pi ^2 a^2 \lambda ^2 }}{{V^2 }}\sum\limits_{} {'\frac{{\langle n_\alpha \rangle \langle n_\gamma \rangle \langle n_\lambda \rangle }}{{\frac{1}{2}\left( {k_\alpha ^2 + k_\beta ^2 - k_\gamma ^2 - k_\lambda ^2 } \right)}}\delta \left( {\vec k{}_\alpha + \vec k{}_\beta - \vec k{}_\gamma - \vec k{}_\lambda } \right)} $$ which represents the second order perturbation term of the energy of a bose gas. ${\langle n_\alpha \rangle }$ is: $$\langle n_\alpha \rangle = \sum\limits_{n = 0}^\infty {n\left( {ze^{ - \beta \varepsilon _\alpha } } \right)} ^n /\sum\limits_{n = 0}^\infty {\left( {ze^{ - \beta \varepsilon _\alpha } } \right)} ^n = \frac{{ze^{ - \beta \varepsilon _\alpha } }}{{1 - ze^{ - \beta \varepsilon _\alpha } }} $$ In the sum the terms with a vanishing denominator are omitted. There is also the restriction ${\vec k{}_\alpha \ne \vec k{}_\beta }\ \ $ and ${\vec k{}_\gamma \ne \vec k{}_\lambda }\ \ $. Now the passage that is unclear to me is the passage to the integral: $$\frac{{E_2 }}{{E_0 }} = \frac{{16\pi a^2 \lambda ^2 }}{{V^2 }}\left( {\frac{{4V^3 }}{{\pi ^3 \lambda ^5 }}} \right)\sum\limits_{i,j,k}^\infty {\frac{{z^{j + k + l} }}{{\left( {j + k + l} \right)^{1/2} \left( {j - k} \right)l}}} \frac{{\partial J}}{{\partial u}} $$ where: $$J = \int\limits_0^\infty {\int\limits_0^\infty {dqdq\frac{{\cosh \left( {upq} \right)}}{{q^2 - p^2 }}} } e^{ - vq^2 - wp^2 } $$ and: $$u = \frac{{\hbar ^2 \beta }}{{2m}}\left( {\frac{{2\left( {j - k} \right)l}}{{j + k + l}}} \right) $$ $$v = \frac{{\hbar ^2 \beta }}{{2m}}\left( {\frac{{\left( {j + k} \right)l}}{{j + k + l}}} \right) $$ $$w = \frac{{\hbar ^2 \beta }}{{2m}}\left( {\frac{{\left( {j + k} \right)l + 4jk}}{{j + k + l}}} \right) $$ $\frac{{\partial J}}{{\partial u}}$ is calculated here: <a href="http://math.stackexchange.com/questions/63534/difficult-integral">http://math.stackexchange.com/questions/63534/difficult-integral</a>
Maybe we don't have to do thermal average of n and we need : $$n = \frac{1}{{z^{ - 1} e^{\varepsilon /kT} - 1}} = \sum\limits_{n = 1}^\infty {\left( {ze^{ - \varepsilon /kT} } \right)^l } $$ so: $$\sum\limits_{} {'... = \sum\limits_{i,j,k = 1}^\infty {\sum\limits_{} ' } } z^{ijk} \frac{{e^{ - \frac{{\hbar ^2 k_a^2j }}{{2m}}} e^{ - \frac{{\hbar ^2 k_\gamma ^2k }}{{2m}}} e^{ - \frac{{\hbar ^2 k_\lambda ^2l }}{{2m}}} }}{{\frac{1}{2}\left( {k_a^2 + k_b^2 - k_\gamma ^2 - k_\lambda ^2 } \right)}} $$ This explain the factor $z^{ijk}$ and the sum. How can we continue?Now we have to transform the sum' in a integral. The density of state depends of $kdk$ so i think we have to evaluate: $$\int\limits_0^\infty {\frac{{e^{ - \frac{{\hbar ^2 k_a^2j }}{{2m}}} e^{ - \frac{{\hbar ^2 k_\gamma ^2k }}{{2m}}} e^{ - \frac{{\hbar ^2 k_\lambda ^2l }}{{2m}}} }}{{\frac{1}{2}\left( {k_a^2 + k_b^2 - k_\gamma ^2 - k_\lambda ^2 } \right)}}k_a k_b k_\gamma k_\lambda } dk_a dk_b dk_\gamma dk_\lambda $$ We have to evaluate two integrals, i think. Do you have any suggestion?</p> | g9607 | [
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<p>I have action as shown below
$$S=\int \mathrm{d}t \int \mathrm{d}x^3 \bar\psi\left(i\partial_t\psi +\frac1{2m}\bar\nabla^2\psi-V(x)\psi\right)$$ </p>
<p>How do I manipulate it to obtain the Lagrangian density in Standard form
$$\mathcal{L}\left(\{\varphi_r\},\{\partial_\mu\varphi_r\} \right)$$ </p> | g9608 | [
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] |
<p>how electronic ink display retains the particles on top of micro capsule when
charge on electrode is released or when device is powered off?</p>
<p>Doesnt that make particles loose on the dielectric fluid on capsule after volatge on electrode is removed?</p>
<p>How e ink maintains image even when device is powered off?</p>
<p>I have understood basics of E ink , couldnt understand this part</p> | g9609 | [
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<p>I have a question on the reformulation of the Klein Gordon equation in terms of Killing fields.
Suppose we have a static spacetime with timelike Killingfield $\xi^{\mu}$
(e.g. Schwarzschild).
Then the wave eq. ($\nabla_{\mu}\nabla^{\mu}-m^{2})\phi=0$
can be rewritten in the following form
$$
\partial_{t}^{2}\phi=VD^{i}(VD_{i})\phi-V^{2}m^{2}\phi
$$</p>
<p>with $V^{2}=-\xi_{\mu}\xi^{\mu}$
and $D_{a}$
being the spatial covariant derivative.</p>
<p>Can anybody tell me how this comes about?</p> | g9610 | [
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<p>Inspired by <a href="http://arxiv.org/abs/1401.5761" rel="nofollow">Stephen Hawking</a> I recently tripped upon an idea of what is really inside a black hole.</p>
<p>I thought if NASA (or any other space agency) could send a super protected camera into a black hole, then we could see what's inside black hole.</p>
<p>Is this even possible?</p> | g9611 | [
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<p>A recent paper by the CDMS collaboration (<a href="http://prl.aps.org/abstract/PRL/v111/i25/e251301">PRL here</a> and <a href="http://cdms.berkeley.edu/CDMSII_Si_DM_Results.pdf">free text here</a>) makes this statement in the abstract:</p>
<blockquote>
<p>This blind analysis of 140.2 kg day of data taken between July 2007 and September 2008 revealed three WIMP-candidate events...The probability that the known backgrounds would produce three or more events in the signal region is 5.4%.</p>
</blockquote>
<p>I am not part of the particle physics communities or the dark matter communities so my questions are:</p>
<ol>
<li>Is this convincing evidence for the existence of WIMPs?</li>
<li>What does this say, if anything, about the overall composition of dark matter? I.E. is this convincing evidence that a large fraction of dark matter is from WIMPs?</li>
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<p><img src="http://i.stack.imgur.com/MEo7X.jpg" alt="enter image description here">
<img src="http://i.stack.imgur.com/U1dyL.png" alt="enter image description here"></p>
<p>My interpretation: When we have no angular momentum, the potential well looks like this, my question is: How do you find the point where the wavefunction penetrates its classical forbidden region, i.e. quantum tunnelling effect? The answer is $r>2a_0$ where $a_0$ is the Bohr's radius. </p>
<p>My answer: It seems you have to equate the Coulomb potential with the energy of the ground state of the hydrogen atom.</p>
<p>And what would the wavefunction intersect? Does it pass through $(0,0)$? Can we view the infinite deep well, as a infinite barrier, by imagining we shift the zero reference level down to infinity?</p>
<p>I'm quite confused by the idea of having negative energy. Does it mean I need infinite energy to get out of the well? Just like an infinite square well?</p> | g9613 | [
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<p>Suppose you accelerate a body to very near the speed of light $c$ where $v = c - \epsilon$. Although this would take an enormous energy, is it possible the last arbitrarily small velocity needed -- $\epsilon$ -- could be overcome with a minor bump in velocity due to the <a href="http://en.wikipedia.org/wiki/Uncertainty_principle" rel="nofollow">uncertainty principle</a>?</p> | g9614 | [
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<p>It appears that velocity is a quantity of motion meaning that all objects can have assigned to them a particular velocity. Through the application of forces (ex: gravity, E&m) we measure changes in velocity, which is called acceleration but here is the crux:</p>
<p>Once the force ceases to be applied the body will stop accelerating.... Meaning that while forces can be applied there is no actual quantity of force</p>
<p>There exists a quantity called momentum (mv) such that a body of mass m with momentum Q will move indefinitely velocity Q/m unless acted on by a force (m*a)</p>
<p>There theoretically could be this thing called 2nd momentum (ma) such that a body of mass m with 2nd momentum Q would accelerate indefinitely unless acted upon by a 2nd order force (mass*jerk) </p>
<p>The reason the above ^ does not occur is because for it to occur would require a constantly increasing amount of energy... ie infinite energy.</p>
<p>But we don't actually mean energy, we mean the quantity 1/2 * mv^2 would grow arbitrarily large for this to occur... </p>
<p>In an abstract mathematical world where it is not 1/2 mv^2 that is conserved but rather the quantity 1/2 m*a^2 that is conserved objects could be in a state of uniform acceleration indefinitely and still satisfy conservation of 2nd order momentum... </p>
<p>But obviously our universe IS NOT that abstract mathematical world of the 2nd kind... It is of the first... Why? What dictated that momentum (mv) became the quantity of motion and not (ma) or for that matter (mass *jerk) or why not fractional derivatives of velocity?</p> | g9615 | [
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] |
<p>What's the physical difference between the quantities $\left\langle v_{i}v_{j}\right\rangle $ and $\left\langle v_{i}\right\rangle \left\langle v_{j}\right\rangle $? </p>
<p>Where $\left\langle v_{i}\right\rangle$ is the mean value of the $v_{i}$ velocity component with $i=1,2,3$. </p>
<p>(Same story with $v_{j}$.)</p>
<p>This is in the context of stars' velocities in galaxies.</p> | g9616 | [
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<p>A pion is made out of a pair of up and/or down quarks. A neutron or proton is three up or down quarks. So naively I'd expect a pion to be about 2/3 the mass of a nucleon.</p>
<p>In fact it's less than 1/6 the mass. Why is it so much lighter? Is there some reason why a lot less energy is contained in the gluon field of a pion than in the gluon field of a nucleon?</p> | g9617 | [
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<p>We tried to measure background radiation using a geiger counter for a experiment at school. The meter showed $0.12$-$0.21$ microSv/h during the day averaging at about $0.14$ mcSv/h.</p>
<p>As we tried to see ways how to shield incoming radiotion nothing seemed to work. Taking a cue from nuclear power plant we put the meter into a watertigt glasscontainer and sank it in the gym pool so that the container had a good three feet of water all around. The measurement did not budge at all. We tried all settings fast/slow/auto. The geiger meter should be okay as it had recently been calibrated.</p>
<p>Can someone explain. </p> | g9618 | [
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] |
<p>So I am told that pressure = Force per Area --> F/A..</p>
<p>When considering the units of Force I find that force = kg * m/s^2</p>
<p>When considering the units of Area I find that area = m^2</p>
<p>Thus the units of Pressure = kg/s^2 *1/m = d^2(mass)/dt^2 * 1/m </p>
<p>If I interpret this correctly that means that if one had a long glass tube </p>
<p>(picture below):</p>
<p>=========================================================================</p>
<p>=========================================================================</p>
<p>And we considered a segment of the tube </p>
<pre><code> this is the segment:
</code></pre>
<p>===============|===================================|=====================</p>
<p>===============|===================================|=====================</p>
<p>and we assumed that there was matter flowing through the entire tube:</p>
<p>===============|===================================|=====================</p>
<p><strong>* * *</strong> * * <strong><em>*</em>**</strong> * * <strong><em>*</em>*</strong> * * * * * * <strong><em>*</em>*</strong> * * * * * * ** * * <strong><em>*</em>**</strong> *</p>
<p>===============|===================================|=====================</p>
<p>Then at any given moment: the pressure in this particular segment could be given by:</p>
<p>The change in the rate of mass flow/length of segment:</p>
<p>= d^2(mass in segment)/dt^2 *1/length</p>
<p>Is this correct?</p> | g9619 | [
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<p>Consider the following problem taken from a problem booklet. My questions are:</p>
<ol>
<li>What is displacement vector?</li>
<li>And how to determine the direction of displacement vector at a certain point?</li>
<li>Where is the position with zero displacement vector?</li>
</ol>
<p><img src="http://i.stack.imgur.com/XuGiu.png" alt="enter image description here"></p> | g9620 | [
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<p>I'm trying to write down the bath Hamiltonian for a system of dimers and trimers. Imagine each of the monomers in the excited state can interact with several phonons with given frequencies. The bath Hamiltonian has the following form
$\sum\limits_{n = 1}^N {\sum\limits_{j = 1}^M {{\omega _{nj}}} } a_{nj}^\dagger {a_{nj}}$. Here N is the number of monomers and M is the number of vibrational modes. I don't understand how the matrix of vibrational modes look like for a trimer with lets say four modes. It looks like I will have a 3-by-4 matrix, where all the rows are the same since I have the same modes for all the monomers. This doesn't make sense to me, since I'm supposed to diagonalize this matrix. Another ambiguity arises when I'm trying to figure out what values the number operator takes for each element of this matrix, since it looks to me it has to be 1 for each element. Imagine, you have two or three monomers which are all coupled to the same modes.</p>
<p>I will appreciate your comments! </p> | g9621 | [
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<p>Why <a href="http://en.wikipedia.org/wiki/Perpetual_motion" rel="nofollow">perpetual motion</a> wouldn't be possible if we are so technological advanced?</p>
<p>It is just a thing that I was wondering for too long. I mean, we are able to create so powerful permanent magnets, like neodymium magnets which can store an massive quantity of energy for a long period of time, but we are not able to use that energy at all. Why?</p> | g9622 | [
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<p>I was going through <a href="http://arxiv.org/abs/quant-ph/0001106" rel="nofollow">arXiv:quant-ph/0001106v1</a>, the first paper by Farhi on adiabatic quantum computation.
Equation 2.24 says, $$\tilde{H}(s) = (1-s)H_B + sH_P$$ which means the adiabatic evolution starts from the ground state of $H_B$ and slowly evolves until it arrives at the ground state of $H_P$. In section 3.1, the one qubit example has the adiabatic Hamiltonian as $$\tilde{H}(s) = \begin{pmatrix}
s & \epsilon (1-s) \\
\epsilon (1-s) & 1-s
\end{pmatrix}$$
I don't see how the plot of Figure 1 is drawn. In figure 1, Farhi plotted eigenvalues of the Hamiltonian for s while the range for s was 0 to 1. The Hamiltonian is supposed to evolve according to the Schrodinger equation (eq 2.1), $$i \frac{d}{dt} |\Psi (t) \rangle = H(t) |\Psi (t) \rangle$$
Was this evolution solved to draw the plot? Or did Farhi derive the formula for eigenvalues in terms of s using just matrix math and plotted accordingly?</p> | g9623 | [
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<p>I am given the following equations.
$$\frac{\partial ^2\tilde \psi(x,\omega)}{\partial x^2}+k^2(x,\omega)\tilde\psi(x,\omega)=0 $$
Where
$$k^2 =\frac{2m\omega}{\hbar}$$
And then
$$\frac{\partial ^2\tilde E(x,\omega)}{\partial x^2}+\frac{n^2(x)\omega^2}{c^2}\tilde E(x,\omega)=0$$
We have the boundary conditions:
$$\tilde\psi(\pm a/2)=\tilde E(\pm a/2)=0$$
I am asked for the allowed values of $\omega$ and $k$ and then the normalized mode functions. </p>
<p>I did this in the same way we treat the Infinite Well in QM. I took that the solution was $Ae^{ikx}+Be^{-ikx}$ and then found each solution was of the same form, which was $\sqrt{2/a}\sin(kx)$ for both. But then the allowed values of k and $\omega$ were different depending on the k values above. Am I doing this correctly?</p> | g9624 | [
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<p>My questions are:</p>
<ol>
<li>How does a black body absorb photons? </li>
<li>Why does a black body absorb the most photons of all objects (e.g. those with another color)? </li>
<li>Are there any relationship between the electron and a black body?</li>
</ol> | g9625 | [
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<p>Does the inverse square law have anything to do with conservative behavior of the central forces? </p> | g9626 | [
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<p>The book I am reading takes the unjustified step $$e^{-\frac{i}{\hbar}\vec{p}\cdot\vec{r}}f(\vec{p}) = f(\frac{\hbar}{i}\vec{\nabla})e^{-\frac{i}{\hbar}\vec{p}\cdot\vec{r}}$$</p>
<p>and similarly, he uses elsewhere $$e^{-\frac{i}{\hbar}\vec{p}\cdot\vec{r}}g(\vec{r}) = g(i\hbar\vec{\nabla}_p)e^{-\frac{i}{\hbar}\vec{p}\cdot\vec{r}}$$</p>
<p>I have a gut feeling this follows somehow from the commutation relation $[r^i,p_j] = i\hbar\delta^i_j$ but I cannot justify this. </p>
<p>Anyhow, the book has not introduced the standard commutation relations, and is working only with the fact that $\phi(\vec{p},t)$ and $\psi(\vec{r},t)$ are Fourier transforms of each other, and $f(\vec{p})$ and $g(\vec{r})$ are analytic. </p>
<p>Could someone give me a hint?</p> | g9627 | [
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<p>I know that by Heisenberg's Uncertainty Principle that it is not possible to know the exact values of position and momentum of a particle simultaneously, but can we know the exact values of momentum and velocity of a particle simultaneously? I would think the answer would be no because even if we were 100% certain of the particle's position, we would be completely unsure of the particle's momentum, thus making us also completely unsure of the particle's velocity. Does anyone have any insight into this?</p> | g9628 | [
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<p>I'm a little confused.. When I write the equation of the circuit, do I have to keep the signs of the induced emf and self-inductance opposite?</p>
<p>I'll try to explain better my doubt:
Consider a coil that has resistance R and inductance L. Running a current will give rise to both an induced emf and self-inductance, and the situation is:</p>
<p>induced emf= -k </p>
<p>self-inductance= $-L \frac{dI}{dt}$</p>
<p>Now, my question is: the equation of the circuit is given by:</p>
<p>$-k -L \frac{dI}{dt}=RI$</p>
<p>or by </p>
<p>$k -L \frac{dI}{dt}=RI$ ?</p> | g9629 | [
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<p>Dynamic structure factor is the spatial and temporal Fourier transform of Van Hoves time dependent pair correlation function. It is written as</p>
<p>$$ S(k,\omega)= \frac{1}{2\pi}\int F(k,t)\exp(i\omega t)dt $$</p>
<p>$F(k,t)$ is intermediate scattering function.
My question is how did we use spatial and temporal Fourier Transform of Van Hoves to get dynamic structure factor? and what does spatial and temporal Fourier transform mean?</p> | g9630 | [
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<p>Connect the positive terminal of a battery to a piece of p-doped semiconductor, say, silicon doped with boron. Will the terminal pull electrons out of the doped silicon, or equivalently, inject holes into it?</p>
<p>The atomic structure of p-doped semiconductors certainly <em>accepts</em> electrons freely, but I feel like it should be stingier than an insulator when it comes to <em>donating</em> electrons. As an example of an insulator, undoped silicon is unwilling to give away electrons, or accept holes, because that would rip gaps in the stable crystalline bonding. P-type silicon should behave similarly here; thus, I would guess that the p-type block remains neutral, unlike a copper block which would take on the same charge as the terminal. However, if it were connected to a negative terminal, it would accept enough electrons to basically saturate it and give it a negative charge. Is this understanding correct? Or could the p-type somehow donate electrons/accept holes because of some kind of instability associated with doping?</p>
<p>EDIT: Admittedly, this is a rather petty question, and I doubt that anyone has ever bothered to check the answer in a published experiment. Still, knowing that the p-type stays neutral would confirm most people's (or at least my) model of why doped semiconductors conduct. Their conductivity possesses a kind of asymmetry, unlike that of copper, which gives and takes electrons equally freely.</p> | g9631 | [
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<p>In inflationary theory, many papers start off by making the slow-roll approximation, on which many things depend. This approximation is usually presented by requiring that two 'slow-roll parameters' are small:
$$\epsilon_V\equiv\frac{1}{16\pi G}\left(\frac{V'}{V}\right)^2 \ll 1$$
$$|\eta_V|\equiv \frac{1}{8\pi G}\left(\frac{V''}{V}\right)\ll 1$$
We then have
$$H^2=\frac{8\pi G}{3}V$$
$$3H\dot{\phi}=-V'$$
Now, the first of these two conditions is reasonably easily derived:
\begin{align*}
\frac{\ddot{a}}{a}&\gg 0\\
\dot{H}+H^2 &\gg 0\\
-\frac{\dot{H}}{H^2}&\ll 1\\
\frac{1}{16\pi G}\left(\frac{V'}{V}\right)^2=\epsilon_V&\ll 1 \hspace{1cm}\text{used slow roll approx.}
\end{align*}</p>
<p>However, I'm not sure how to find the second one (note that I am <em>not</em> asking for an intuitive explanation, I understand what the second parameter represents, I just wanna know how to derive it). Could someone tell me how to derive it (or what the original premise, analogous to $\ddot{a}/a\gg 0$ for $\epsilon_V$, is)?</p> | g9632 | [
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<p>I'm new to physics and am just going through some of the free online classes at World Science U, and after watching this video on the nature of the speed of light and its constancy, a question came to mind about photons. (Video: <a href="http://www.youtube.com/watch?v=fDm1PQgIq7U">YouTube Video</a>, <a href="http://www.worldscienceu.com/courses/1/elements/5H4am8">World Science U course</a>) </p>
<p>I know that photons don't have mass, but what happens when photons — even the photons from distant stars — reach us? Are we merely observing the occurrence of photons moving through space relative to us, or are we really being "bathed" in photons? I know that when I observe rain, I can both observe it from a distance but could also be immersed in it as well if in the path of that rain. But with distant starlight, are we just observing it or are the photons actually reaching and penetrating the earth around us? If they are penetrating, does science tell us what is actually happening on an atomic or sub-atomic level?</p> | g9633 | [
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<p>given a Hamiltonian and the semiclassical WKB partition function in units $ \hbar =2m=1 $</p>
<p>$ \Theta (t) = \frac{1}{2\pi} \iint dx dp exp(-tp^{2}-tV(x)) $</p>
<p>can i use this Theta function to evaluate the Funntional determinant ??</p>
<p>$ \zeta (s,z^{2}) \Gamma (s)= \int_{0}^{\infty}dt\Theta (t) exp(-tz^{2})$</p>
<p>$ \zeta (s,z^{2})= \sum_{n=0}^{\infty}(z^{2}+E_{n})^{-s}$</p>
<p>the functional determinant is evaluated as $ det(H+z^{2})=exp(- \partial _{s} \zeta (0,z^{2}))$</p>
<p>so we need only the Theta function which is $ \Theta (t)= \sum_{n}exp(-tE_{n})$ i have simply approximated this sum by an integral over phase space involving the Hamiltonian</p> | g9634 | [
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<p>Put the classical electron-emitting double-slit apparatus in a sealed box. At each slit there's a counter to check whether an electron has passed it or not, so had the apparatus been left in open, we would have observed dots pile after the two slits on the screen with no interference pattern. But now the apparatus is running in the sealed box, and the box is so designed that 1) an outside observer can't know anything inside the box unless he opens it. 2) the box destroys the counters just before you open it, leaving no information of their past recordings. My question is, will you see interference pattern after you open the box? Why or why not?</p>
<p>To quote Feynman:"You do add the amplitudes for the different indistinguishable alternatives inside the experiment... At the end of the process you may say that you 'don't want to look at the photon.' That's your bussiness... Nature does not know what you are looking at, and she behaves the way she's going to behave whether you bother to take down the data or not" Does this comment apply here? But which slit the electrons have passed does seem indistinguishable to an outside observer during or after the experiment. </p> | g9635 | [
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<p>I'm supposed to calculate the capacitance of an unknown capacitor in series, but I'm not sure exactly which equation to use.</p>
<p>I know the voltage across the resistor (Vr), voltage across the capacitor (Vc), the total voltage (V), the resistance value of the resistor (R), the frequency (f), and since I was using an oscilloscope, the change in time between Vr and V.
I know there's an equation that deals with these values (it includes natural number e), but I can't find it, and even if I had it I'm not exactly sure how I would rederive it to solve for the capacitance. How would I attempt this problem?</p> | g9636 | [
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<p>What's the difference between particle's helicity, spinor's helicity and chirality ?</p>
<p>I read a saying that right-handed spinor correspond to the left-handed positron, right-handed spinor correspond to the right-handed positron. </p>
<p>Why the particle's helicity is different from the spinor's? </p> | g9637 | [
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<p>In the late nineties Bender has started a research program on what is called PT symmetric QM, or non hermitian QM, in which he has shown that if the hamiltonian enjoys a PT symmetry then the spectrum is real even if the hamiltonian was not hermitian. Now a search for "PT symmetric" on the arxiv will return over 400 papers.</p>
<p>This relaxation of the hermiticity of the hamiltonian has led to new forms of hamiltonians that have been rejected before because they were non hermitian. It turns out that such PT symmetric hamiltonians can describe real physical systems.</p>
<p>Since this PT symmetric QM cannot be considered a generalization to conventional quantum mechanics I suppose in which observables are mathematically represented by hermitian operators, I find it intriguing that something like this can describe real physical systems. So how can one understand PT symmetric QM from first principles? I mean how does it fit in the grand picture, why it works? </p> | g9638 | [
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