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<p>I am a bachelor physics student and as an assignment we had to perform measurements on an FT spectroscopy setup. </p>
<p><strong>Context.</strong></p>
<p>Our setup consisted of a Michelson interferometer through which the light of two sources namely a HeNe-laser and a sodium lamp, passed at the same time. The light of the interference pattern was split into two. One of the beams was passed through a polarizer to filter out the laser light (partially). Both beams were sent to their respective detectors which measured the intensity in a single spot of the interference pattern. </p>
<p>One of the mirrors of the interferometer was attached to a moving stage which was moving at a constant velocity during the measurement. </p>
<p>Here is a drawing of the setup:</p>
<p><img src="http://i.stack.imgur.com/fbS8c.png" alt="Fourier transform spectroscopy setup"></p>
<p>The detectors measured the intensity during the time the stage was moving. We therefore received intensity vs. time as data. Detector 1 measured the intensity of the sodium light + the weakened laser light. Detector 0 measured the intensity of both the laser light and the sodium light (the latter was negligible since the laser light was so much stronger than the sodium light). </p>
<p>The data from detector 0 was used for calibration. We assumed the laser light consists of a single wavelength of light. We measured however that there were varying frequencies in the intensity. These variations must therefore be caused by flaws in the setup (like faulty calibration or inconsistency in the speed of the stage). </p>
<p>The variations were corrected by moving the data points along the time axis such that a nice sinusoid remained. The frequency of that sinusoid was chosen to be the average frequency of the original data (I am not sure if this is true. We were not told the exact details of the data correction procedure.) To prepare the data for Fast Fourier Transform the data points were moved along the sinusoid such that they were spaced equally horizontally. </p>
<p>Here is an illustration of the procedure:</p>
<p><img src="http://i.stack.imgur.com/zMMug.png" alt="Data correction procedure"></p>
<p>Because the variations caused by flaws in the setup were the same in the signals from both detectors (since they were measuring at the same time) the corrections of the data from detector 0 were applied to the data from detector 1 as well. </p>
<p>The data from detector 1 was Fourier transformed (using the FFT algorithm). The frequencies found from this were scaled down by a factor of $\frac{2v}{c}$, where $v$ was the speed of the stage. (I am leaving out the derivation.) We scaled the frequencies back up such that the frequency of the laser peak matched its optical frequency and turned those into wavelengths. </p>
<p>This gave the following spectrum:</p>
<p><img src="http://i.stack.imgur.com/XYq15.png" alt="Sodium spectrum"></p>
<p>Now onto the main problem. How do I determine the spectral resolution of this setup? Spectral resolution is the smallest difference in wavelength the spectrometer can detect at a certain wavelength. </p>
<p>The time for measurements has long passed so it's no longer possible to make new measurements. </p>
<p><strong>Question.</strong></p>
<p>We were told that because the laser peak is sharpest in our spectrum its width gives us the resolution. But I do not understand how these are related. Can someone explain that? If the width of the laser peak does not give the resolution, what would and why? </p>
<p>What I am also interested in are the factors that influence the resolution. </p>
<p><strong>Attempt.</strong></p>
<p>I was thinking about a hypothetical case where the detectors could measure infinitely fast producing a continuous signal. The setup would also have a special laser with a variable wavelength instead of a sodium lamp. </p>
<p>By collecting the data from detector 0 which is measuring the intensity of both lasers and applying a continuous Fourier transform (we are skipping the data correction procedure) a spectrum is obtained. </p>
<p>Because of the variations in the frequencies each laser will produce a normal distribution (a Gaussian) in the spectrum. The spectrum then consists of a sum of two Gaussians. </p>
<p>One method of finding out how many peaks there are in a spectrum is counting the maxima. If the Gaussians are far apart we can see two maxima. But when two Gaussians get closer and closer together (the distance between their maxima decreases), at some point their sum will only have a single maximum before the tops of the two Gaussians have overlapped. At this point the two peaks can no longer be distinguished and our method will tell us there is only one peak. </p>
<p>For the following plot we took the sum of two identical Gaussians $e^{-(x-a)^2}+e^{-(x+a)^2}$. The two Gaussians are a distance $2a$ apart. By taking the derivative with respect to $x$ and setting it equal to $0$, the extrema can be found. In the plot the $x$-coordinate of the extremas are plotted against $a$. <a href="http://www.wolframalpha.com/input/?i=d/dx%28e%5E%28-%28x%2ba%29%5E2%29%2be%5E%28-%28x-a%29%5E2%29%29=0" rel="nofollow">WolframAlpha plot</a> </p>
<p>The minimum distance that the Gaussians have to be apart such that two maxima can be seen could be called the resolution. This distance is a function of the amplitudes and standard deviations of the two Gaussians which are themselves determined by the quality of the setup. </p>
<p>To find the resolution of this hypothetical setup one could decrease the difference in wavelength between the two lasers till only one maximum is visible and repeat this for all wavelengths. </p>
<p>However, this is just a hypothetical setup and not the real one. In the real one, the detectors measure a finite amount of points per second, there is data correction, FFT and a finite amount of point per nanometer in the spectrum and it's not clear how the resolution is determined.</p>
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<p>I am part of a school project, <a href="http://projectstratos.com/" rel="nofollow">Project Stratos</a> to send a balloon to the edge of space (the closer side :P) and was wondering how you would work out the accent rate of a large balloon (roughly 1m^3 of helium with 100g of mass) and the size of it as it increases its Altitude. I am creating a live map (that will be based on predictions rather than its actual location) and want to know the speed it will float up into the atmosphere. Currently we are assuming the ascent rate will be about 5m/s but I doubt that is very accurate and would this speed increase as it gets higher?</p>
<p>Edit: I would also quite like to know the burst height of the balloon.</p>
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<p>Suppose, we have the most simple <a href="http://en.wikipedia.org/wiki/Double_pendulum" rel="nofollow">double pendulum</a>:</p>
<ul>
<li>Both masses are equal.</li>
<li>Both limbs are equal.</li>
<li>No friction.</li>
<li>No driver.</li>
<li>Arbitrary initial conditions (no restriction to low energies)</li>
</ul>
<p>Does this pendulum have transients for any initial conditions or is it immediately on its respective attractor?</p>
<p>I have seen several time series that suggest that there are no transients, however, I could not find any general statement on this.</p>
<hr>
<p>I here say that a system has no transients, if its trajectory comes arbitrarily close in phase space to its initial state again, i.e., for a given initial state $x(0)$ and for all $ε>0$, there is a $T>0$ such that $\left | x(T) - x(0) \right | < ε$.</p>
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<p>I've done a little research but I can't seem to find the answers to these questions in regards to a spinning space station simulating gravity.</p>
<p>1) If we imagine the space station as a hollow doughnut and I'm floating in the center of the tube while the station is stopped what happens when the station starts? How are forces translated to my body that push me towards the floor or would I just remain floating there while the station flies by?</p>
<p>2) Does the spinning station rely on friction to simulate gravity? If the inner surface of the station were frictionless would I still move along with the station or would the station simply fly by me?</p>
<p>Thanks!</p>
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<p>Ok, as I understand the expansion of the initial singularity was caused by quantum fluctuations like the ones predicted by the Heisenberg Uncertainty Principle. But how can these fluctuations occur within a singularity? And how can they cause the expansion of that singularity? Or do I have a terrible misunderstanding of the theory? </p>
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<p>Why is it concluded that the cosmos is expanding at an ever increasing rate when looking further out and therefore further back in time?</p>
<p>Surely, if the time were reversed to play forwards, the conclusion would be that the expansion of the universe is actually deaccelerating, and the need to postulate 'Dark Energy' would be avoided.</p>
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<p>Consider the following thought experiment:</p>
<p>Imagine an object of a certain mass density which allows it to float in water. </p>
<p>Now if this object is viewed from a moving frame with high speed, it will look Lorentz contracted hence its density will look greater than the rest density. </p>
<p>So from the rest frame it floats but from the moving frame it sinks!</p>
<p>What am I missing?</p>
| 7,321 |
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] |
<p>In this <a href="http://physics.stackexchange.com/questions/18595/is-decoherence-even-possible-in-anti-de-sitter-space">link</a>, one of the answers contains the statement</p>
<blockquote>
<p>If you examine the space-time near a finite area quantum black hole,
you will see an approximate AdS space.</p>
</blockquote>
<p>Presumably "approximate" means this is only true to some order in the distance from the horizon ? Could someone outline where the result comes from, or provide a reference.</p>
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<p>Barring aside considerations such as heat dissipation, capacitance, etc... (aka any sort of technological issue) what is the fastest speed of a processor?</p>
<p>I am told that at distances of 1 planck length $L =1.616$ x $ 10^{-35}$ m</p>
<p>Heisenberg's uncertainty principle hovers around an error factor of 1. </p>
<p>Does that suggest then CPU speeds (in this case measured via period) theoretically cannot exceed something along the same order of magnitude of:</p>
<p>$L/c = 5.387$ x $10^{42}$ calculations per second? (pretending a single cycle of a lightwave bouncing across a single planck length of distance amounts to an entire calculation)</p>
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<p>I've been looking at the <a href="http://arxiv.org/abs/1303.5076" rel="nofollow">Planck 2013 cosmological parameters</a> paper, trying to update my toy cosmology simulator with the most recent data. Most of the interesting values such as $H_0$, $\Omega_m$, and $\Omega_\Lambda$ can be found in Table 2 on page 12, but the one thing I didn't find was an estimate of the energy density of radiation. Can this be derived from some other parameters in these data?</p>
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<p>Suppose that I am inside of a moving train. I have a coin in my hand and I am standing still. If I toss this coin straight up, it will fall back into my hand. </p>
<p>Now, suppose that I am on that moving train's roof. Now if I toss the coin straight up, the coin will not land in my hand, rather it will fall toward the place where the train was when I tossed it. </p>
<p>Why is there a difference?</p>
| 7,325 |
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<p>The relation I was referring to in the title is</p>
<p>$${\Lambda_a}^b= \eta_{ac} {L^c}_d \eta^{db}$$</p>
<p>where ${\Lambda_a}^b$ is the inverse transpose of $L$, the <a href="http://en.wikipedia.org/wiki/Lorentz_transformation" rel="nofollow">Lorentz transformation</a>.</p>
<p>I was wondering about the significance of such an equation? What does it imply in general relativity?</p>
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] |
<p>In the Lattice Boltzmann method we require (based on mass conservation) that the sum of the distribution functions for a node is equal to the density, i.e.</p>
<p>$$
\sum_i f_i = \rho
$$</p>
<p>But what units do $f_i$ have that make their sum equal to the density, which has units of $kg/m^3$?</p>
<p>I guess it means that $f_i$ must have same units as the density, but how can I interpret $f_i$ in that case?</p>
| 7,327 |
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<p>If you push massive object it accelerates slowly compared to object which is of same size but less density/mass.</p>
<p>However, all objects do Accelerate at the same rate.</p>
<p><code>F=ma</code>....according to this if gravity is a force than spherical ball made of cotton shouldn't also accelerate toward earth with acceleration of 9.8 m/s2 just as the ball made of steel would also do at the same acceleration.</p>
<p>But they both do. All i can think of is that force experienced by object increases or decreases linearly according to the mass an object has, is that true. Force object experiences changes according to its mass. </p>
<p>I guess the whole confusion could be that if gravity wasn't called a force, in my mind force is something that has magnitude which is same regardless of the mass of object it's touching. Word forcefield would make more sense even though i won't have any idea what does that really mean. </p>
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<p>Could a device with very thin columns of glass or something that attracts water more be used to pull water up and then release it to drive generator and perhaps add vacuum. </p>
<p>I have been wondering about this on and off for like 8 years.</p>
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<p>Let's say we have an object lying on a slope, the weight of the object can be divided into two components one horizontal and one vertical. My question is does the weight act only in these two directions or does it spread evenly between the the both components?</p>
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<p>Quite a technical question!</p>
<p>I have measured the Point Spread Function of 100nm fluorescent breads with my Olympus scanning head. I'm two-photon exciting the beads with a wavelength of 800nm and focused in the sample with a 100x with N.A.: 1.4</p>
<p>The theory suggests me the resolution of such a system to be:
$$d=\frac{0.7\cdot\lambda}{N.A.}\approx 400nm$$
</p>
<p>Now that value should be equal (with at most a 18% correction) to the FWHM of the 2D gaussian I obtain on the image.</p>
<p>But from my analysis of the images I obtain a FWHM close to 2um. Now surely the formula is for an <em>ideal</em> optical setup but a factor five seems to me quite strange! Is that possible to obtain such a result, in the case of <strong>very not ideal</strong> optical elements, or should I look for some sort of problem with the acquisition sw that tells me the pixel dimension of the images? </p>
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<blockquote>
<p>A gold nucleus is 460 fm (1 fm = 10^(-15) m) from a proton, which initially is at rest. When the proton is released, it speeds away because of the repulsion that it experiences due to the charge on the gold nucleus. What is the proton's speed a large distance (assume to be infinity) from the gold nucleus? (Assume the gold nucleus remains stationary.)</p>
</blockquote>
<p>My solution: suppose the gold nucleus' charge is $Q$, and the proton's charge is $q$, then $\int_{R}^{\infty} {KQq\over (r^2)}=\frac{1}{2}mv^2$, where $R=460~\text{fm}$. </p>
<p>That's my solution.</p>
<p>But the question doesn't tell me the mass of the proton and the quantity of charge for the gold nucleus. How can I solve it? I am confused!</p>
| 7,332 |
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<p>In classical mechanics, we had Newton's law of gravity $F \propto \frac{Mm}{r^2}$. Because of this, all laws of classical electrostatics applied to classical gravity if we assumed that all charges attracted each other due to Coulomb's law being analogous. We can "tweak" classical electrostatics to fit gravity.</p>
<p>In modern physics, does the reverse work? Can we "tweak" General Relativity to accurately describe electrostatics or even electromagnetism?</p>
| 247 |
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<p>Gravity is inversely proportional to the <em>distance</em> between objects.</p>
<p>Do we use Euclidean distance or the invariant interval for that distance?</p>
<p>Using the invariant interval makes everything a bit more complex: An object that is one lightyear away, is also one year "younger" when we observe it. Does that temporal distance contribute, so that when we calculate trajectories and physical phenomena on a distance - they are different from what's observed locally?</p>
<p>Does this affect observed things such as the trajectory of two equally distant (from us) objects and rotational velocity of distant objects?</p>
<p>An example:</p>
<p>When we look towards the center of the milky way, we're looking about 27 000 years back in time; i.e. the temporal distance is 27000 years, and spatial distance is 27 kly</p>
<p>When we look at the center of the andromeda galaxy, it's about 25 Mly away. When we look at another object in the andromeda galaxy, which is also 25 Mly away - the temporal distance between those objects, <em>relative to us</em> is 0.</p>
<p>So; <em>is the relative temporal distance between objects and the observer incorporated in Einsteins field equations?</em></p>
| 7,333 |
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<p>Suppose initially I have a uniform magnetic field.</p>
<p>Now, in this formerly-uniform magnetic field, I place an un-magnetised cylindrical body of iron into this field of known dimension, with the axis of rotation aligned with the (formerly uniform) magnetic field.</p>
<p>How would one go about mathematically describing the magnetic field? How does this solution change as the cylinder is rotated about a different axis?</p>
<hr>
<p><strong>Note:</strong> This isn't a homework problem, but I'm guessing that there is a classic solution in ferromagnetism or magnetostatics to this problem. I'm not quite sure where to start looking - it's a bit beyond my freshman physics textbook that was gathering dust on my shelf. My vector calculus is a bit rusty, but I can probably work it out :)</p>
<p>The actual shape isn't too important, so if an object other than a cylinder helps illustrate the solution, then I am happy to use it.</p>
| 7,334 |
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<p>Specifically, it is a one-dimensional wave equation for waves on a string with a non-constant cross-section, i. e. $$S(x)=S_1+S_2 \cos{2x}; \qquad c(x)=\sqrt{F/\rho\, S(x)}.$$ Separating the variables yields the Mathieu differential equation $$\frac{d^2y}{dx^2}+[a-2q\cos (2x) ]y=0,$$ $a$ and $q$ are parameters of the string. One form of the solution is the Floquet solution $$F(a,q,x) = \exp(i \mu \,x) \, P(a,q,x),$$ where $P(a,q,x)$ is a periodic function and $\mu$ is a complex exponent (another form of the solution are the Mathieu sine and cosine, which are odd and even functions respectively, if that is of any use). The temporal part of the solution is sinusoidal. My question is how to get the dispersion relation $\omega(k)$ from the equation.</p>
<p>With the normal wave equation one would plug the straight wave solution $e^{i(kx-\omega t)}$ into the equation, which would yield the dispersion relation $\omega=c\,k$.</p>
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<p>I'm not the brightest person and I was wondering if it was possible, in the same way that it's possible to make ink that's visible under UV light, if it's possible to make ink that requires a distinct combination of colours of light to work. If there are different coloured UV lights would it be possible to make an ink (or buy one) that reacts only when all three are present and overlapping?</p>
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<p>Speed of sound is related to the derivative of pressure with respect to density:</p>
<p>$v_s=\sqrt{(\frac{\partial P}{\partial \rho})_S}$</p>
<p>where S tells us the derivative must be taken while keeping entropy constant.</p>
<p>For an ideal Bose gas, $P$ and $\rho$ can be expressed in terms of Bose-Einstein functions:</p>
<p>$g_v(z) = \frac{1}{\Gamma(v)}\int \frac{x^{v-1}}{z^{-1}e^x -1}$</p>
<p>where in the given context $z=e^{\frac{\mu}{T}}$ and $\lambda = \lambda (T)$ is the thermodynamic wavelength</p>
<p>Those expressions are:</p>
<p>$ P = \frac{T}{\lambda^3} g_{\frac{5}{2}}(z)$</p>
<p>$ \rho = \frac{m}{\lambda^3} g_{\frac{3}{2}}(z)$</p>
<p>Having all this in my hands, I still fail to perform the differentiation. Could you please hint what should my first step be?</p>
| 7,337 |
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<p>I'm referring to <em>"Path integral approach to birth-death processes on a lattice"</em>, L. Peliti, J. Physique 46, 1469-1483 (1985), available at: <a href="http://people.na.infn.it/~peliti/path.pdf">http://people.na.infn.it/~peliti/path.pdf</a></p>
<p>The article is about a reformulation of the master equation for a Markov process in terms of the path-integral formalism. However my question is mainly about Quantum Mechanics.</p>
<p>The author defines a Hilbert space $\mathcal{H}$, an orthogonal basis of which is given by $\mid n \rangle$, $n \in \mathbb{N}$, with:</p>
<p>$$ \langle n \mid m \rangle = n! \delta_{n,m} $$</p>
<p>The creation/annihilation operators are defined on $\mathcal{H}$ as follows:</p>
<p>$$ a \mid n \rangle = n \mid n - 1 \rangle $$
$$ \pi \mid n \rangle = \mid n + 1 \rangle $$</p>
<p>and they are easily to be seen each other's hermitean conjugates, according to the scalar product just defined.</p>
<p>The conventions are a little bit different from Quantum Mechanics, but this is not really relevant for my question. The author implies that it is possible to rewrite every operator $O: \mathcal{H} \rightarrow \mathcal{H}$ only in terms (sums of products) of creation/annihilation operators.</p>
<p>I cannot demonstrate this assertion. I have tried taking the matrix elements of a generic operator $O$, and demonstrating that everything can be rewritten in terms of $a$ and $\pi$ but actually this is not working.</p>
<p>Thank you in advance for any replies.</p>
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<p>I know developments in physics and mathematics often nourish each other, and at different points in history, physics and math have taken turns leading each other. An example of a physicist "inventing" the required mathematics is Newton and his calculus. At other times, the mathematics have been discovered as a curiosity, and the physicists come along to show where it is used in nature. </p>
<p>My question is: In the current state of affairs, in which fields does the physics lead mathematics?</p>
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<p>Okay, first off, I know that there is a very similar problem about statically indeterminate trusses <a href="http://physics.stackexchange.com/questions/44310/solving-a-statically-indeterminate-truss">here</a>, but my problem was a bit different and I didn't know how to adapt the answer for that problem to mine.</p>
<p><img src="http://i.stack.imgur.com/Ucm9F.png" alt="enter image description here"></p>
<p>So my problem is this: I'm trying to solve for the displacement at node 7 using the energy method, and then I need to solve it using the Finite Element Method. Since the Energy Method seemed the easiest, I went for that first. Through the sum of moments at node 7 and at node 1, I found out that the vertical reaction force at 1 and 3 are the same, and their horizontal reaction forces are equal but in opposite directions, thus cancelling out. 2 doesn't have a horizontal reaction force. Thus I found out that this was statically indeterminate, since now I had $P = 2\cdot(R_{1y}) + (R_{2y})$.</p>
<p>So my next step was to use the energy method from the question I linked to earlier, in order to solve the forces in each member. However, being the idiot that I am, I don't know how to solve the forces in the members to begin with. I don't know which forces point where. I know that if the forces at the ends of a member point away, it's tension. Otherwise, it's compression. But since I know that the reaction forces at 1, 2, and 3 point up, that would mean that the force in the members at 1, 2 and 3 must point down, which means that it is in compression....at least in my head it does. In short, could someone help me get this so that I can go on with my Energy method equation? </p>
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<p>I have a doubt, say when a body moves in a circular loop, there are basically two types of acceleration acting on the object. One is the linear acceleration which is basically tangent to the circle, and there is an acceleration that acts inward which is basically centripetal acceleration. These two acceleration vectors are perpendicular to each other and they have a net acceleration vector pointing inward. My question is since there is a net acceleration due to these two acceleration vectors then the net force must also be acting in that direction. Then why does the object not move in the direction of that net force rather than moving along the circular path. Please understand that i am not talking about the centripetal force, rather i am taking about the net resultant force(centripetal + tangential force).
I feel that along the tangential direction of my circular path, the force would be basically the force of my engine which is driving my car. This force kind of propels me forward in the linear direction, and during the circular turn, the friction force basically would provide me the centripetal force and prevent me from flying off my turn. So, what would be the meaning of the resultant of these two forces would imply in my real world because the resultant of these two force would be parallel to the X-axis and would act inward. I am basically moving the circular arc due to the combined effects of these two forces. Please I am standard 10 student, so my physics concepts are not really very advanced. Can someone explain this to me in a little conceptual manner.</p>
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<p>I had to do a problem, and part of it was to find the mechanical energy of satellite orbiting around mars, and I had all of the information I needed. I thought the total mechanical energy would be the kinetic energy+ the potential energy, or $KE+PE$. However, I had the answer sheet and it said that I had to do $KE-PE$, because when you integrate $Gm_1m_2/r^2$ you get a negative sign. I can see why it works out mathematically, but I don't understand why you are actually losing energy when orbiting in a gravitational field.</p>
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<p>The equation for the photon propagator is straightforward
$$
D_{ij} = \langle 0 |T \{ A_{i}(x')A_{j}(x) \}|0 \rangle
$$
However, $A_{i}(x)$ is gauge-dependent and therefore unphysical (in the arguable sense). Then, since the propagator is dependent on the vector potential, the propagator is unphysical. Sadly, my whole understanding of what amplitudes mean may be skewed, but I would assume the probability amplitude for a photon to propagate between $x$ and $x'$ is something we would want to be gauge-<em>independent</em>.</p>
<p>Edit:</p>
<p>I guess I wasn't clear enough. By computing the probability amplitude for a process, we obtain a complex number that when multiplied by it's complex conjugate we obtain a probability for such a process to occur (when normalized). Here, the physical process is propagation, and the probability is $|\langle 0 |T \{ A_{i}(x')A_{j}(x) \}|0 \rangle|^2$. However, this probability is gauge dependent, and hence, the usual physical interpretation of $|\langle 0 |T \{ A_{i}(x')A_{j}(x) \}|0 \rangle|^2$ is questionable to me. Where has my interpretation gone astray?</p>
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<p>These ideas/questions probably represent a lack of understanding on my part,
but here they are:</p>
<p>1) Cosmologists talk about the increasing speed of expansion of the universe and talk of dark energy as the cause. But, I keep thinking that the farther out we observe, is farther back in time and the light we observe is from a time billions of years ago. We don't know what they look like or how fast they are moving now. Who's to say they haven't slowed down? How can we know now?
So, how can we say that the universe is increasing in it's rate of expansion? All we can say for sure is that it was expanding at that rate.</p>
<p>Another thought- on black holes:</p>
<p>2) If Black holes where not composed of highly dense matter, why would there be different "sizes" of Black holes. if all black holes were collapsed to a "singularity", there should be no difference in "size" (the diameter of the light free area). Therefore, I have trouble with the singularity concept and think they are just another form of dense matter that just happens to have enough gravity to hold back even photons.</p>
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<p>I'm having a bit of a slow day, and can't see how to do the following integral in Peskin and Schroeder (page 107 for anyone with the book). We've derived in the centre of mass frame the integral over 2 body phase space</p>
<p>$$\int d\Pi_2 =\int \frac{dp_1p_1^2d\Omega}{(2\pi)^3)2E_12E_2}(2\pi)\delta(E_{cm}-E_1-E_2)$$</p>
<p>where $E_1^2=p_1^2+m_1^2,\ E_2=p_1^2+m_2^2$ and $E_{cm}$ is the total initial energy.</p>
<p>Apparently one can then "integrate over the final delta function" to get</p>
<p>$$\int d\Pi_2=\int d\Omega\frac{p_1^2}{16\pi^2E_1E_2}(\frac{p_1}{E_1}+\frac{p_1}{E_2})^{-1}$$</p>
<p>How is this done? I suppose it's some kind of clever change of variables or trick but I can't see it! Apologies if I'm missing something easy! To be honest I'm just puzzled as to why we're left with any factors of $p_1$ after doing the integral, because it's just an integration variable, isn't it?</p>
<p>A clear argument would be very much appreciated, as would some intuition as to why I'm misunderstanding the notation! Many thanks in advace.</p>
| 7,345 |
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] |
<p>Criteria for thermodynamical stability is the convex of entropy. But for black hole entropy is non-additive. </p>
| 7,346 |
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<p>Suppose someone tells me what constellation the Sun, Moon, Mercury,
Venus, Mars, Jupiter, and Saturn were in when they were born, and the
information is astronomically accurate (ie, not astrological signs).</p>
<p>Could I use this information to estimate their date of birth
(including year) to a reasonable approximation? My thoughts:</p>
<ul>
<li><p>12 constellations, 7 planets (incl Sun/Moon) yields 35,831,808
possibilities, and we know the moon changes constellations daily. If
the planets were in purely random constellations every day, it would
take 98104 years for all possible combinations. Using this, and
additional information that the person was born in the last 200
years, it would seem easy to find their birth date.</p></li>
<li><p>Of course, planets aren't in random constellations. Additionally,
the birthday paradox suggests there will be many repeats despite the
35,831,808 combinations. </p></li>
</ul>
<p>However, it still seems like you could do a fairly good job of
approximating someone's birthday with this information? </p>
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<p>Is the structure of the periodic table fundamental enough that other forms of intelligent life would likely structure their own listing of the elements in the same way (i.e. are there other ways of arranging the periodic table (in 2D) which would be equally useful, or is the current structure the most efficient way to convey the properties of the elements)? </p>
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<p>It makes intuitive sense to me for the apparent frequency of a sound as modified by the Doppler effect to be based entirely on the speed at which the observer and the source get closer or farther apart, regardless of whether the source or the observer is moving, based on the principle of relativity. However, the general equation I receive from every source $$f'=f((v±v_o)/(v∓v_s))$$ does not have this property. For example, if the speed of sound in the medium is 2 units and the observer and source are coming closer at a rate of 1 unit, then if we take the observer to be moving then we get $$f'=f(3/2)$$ but if we take the source to be moving then we get $$f'=f(2/1)$$ What is the difference between these two cases?</p>
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<p>What causes humans to emit infra red radiation and why don't we also emit other types of electromagnetic radiation such as ultra violet or microwave? Sunlight contains UV and our body takes it in, but when we give off radiation, do we not radiate it back? </p>
<p>I know that the constant motion of molecules embodies heat, and that when an electron is excited by colliding with another, it jumps to a higher energy level and falls back again, releasing energy in the form of photons. My question is, then, does this energy 'in the form of photons' carry heat? Or is heat another form of energy release that is separate from EM radiation?</p>
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<p>I wonder if there is a certain amount of weight that I could carry in a rolling bag to help prevent me from being knocked over by the wind. I have balance problems and some lack of muscle strength in my legs and sometimes it feels like strong wind gusts here in Boston, MA may knock me over, especially when I walk in a wind tunnel between buildings. I have a rolling bag that I pull along that weighs 6 lbs. when it has a pair of shoes in it. I also fill it with kids' books and toys for work but they don't usually weigh very much. The handle extends 20" above the bag and I can keep a good grip on it. Would the bag have to weigh more than I do in order to be of any help? Thanks for any ideas. Donna</p>
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<p>What is the <strong>physical meaning</strong> of the kinetic term in the classical scalar field Lagrangian $$\mathcal{L}_{kin}~=~\frac{1}{2}(\partial_\mu\phi)(\partial^\mu\phi)~?$$ It gives how does the field change from one space-time point to the other. Right? What is the significance of the relative minus sign between the square of the time gradient and square of the space gradient? </p>
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<p>So few days ago I came across this new tool for smart phones <a href="http://www.consumerphysics.com/myscio/" rel="nofollow">http://www.consumerphysics.com/myscio/</a>. It says it uses NIR spectroscopy, which I don't know lot about. My question is: <strong>Is it possible for such a small device (73x25x16.5 mm) to be accurate? What are the size limitations of this technology?</strong> </p>
<p>They claim that scio will be able to recognize molecules, which concentration is above 1%. Is it possible? How deep under surface can NIR spectroscopy go? What are the material limitations of this technology?</p>
<p>I know I have lot of questions, but I am really interested in this topic. If anyone could suggest me some literature about this as well, I will be glad.</p>
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<p>The following functional derivative holds:
\begin{align}
\frac{\delta q(t)}{\delta q(t')} ~=~ \delta(t-t')
\end{align}
and
\begin{align}
\frac{\delta \dot{q}(t)}{\delta q(t')} ~=~ \delta'(t-t')
\end{align}
where $'$ is $d/dt$.</p>
<p>Question: What is
\begin{align}
\frac{\delta q(t)}{\delta \dot{q}(t')}?
\end{align}
I'm asking this because in QFT, the lecturer defined the canonical momentum field to $\phi$ by
\begin{align}
\pi(x,t) ~:=~ \frac{\delta L(t)}{\delta \dot{\phi}(x,t)},
\end{align}
where $L$ is the Lagrangian, a functional of the field: $L[\phi,\partial_\mu \phi] = \int d^d x \mathcal{L}(\phi,\partial_\mu \phi)$.</p>
<p>I know I should get
\begin{align}
\pi(x,t) ~=~ \frac{\partial \mathcal{L(x,t)}}{\partial \dot{\phi}(x,t)}.
\end{align}
(Note it's now a partial derivative with respect to the Lagrangian density.) But doing it I get:
\begin{align}
\delta L = \int d^dx \frac{\partial \mathcal{L}}{\partial \phi} \delta \phi + \frac{\partial \mathcal{L}}{\partial \partial_\mu \phi} \delta\partial_\mu \phi.
\end{align}
So somehow we ignore the first term $\int d^dx \frac{\partial \mathcal{L}}{\partial \phi} \delta \phi$! Why is that?</p>
<p>It can't be that we are treating $\delta \phi$ and $\delta \dot{\phi}$ as independent because if I were to take the functional derivative w.r.t. $\phi(x')$, I would have to move the dot from $\delta \dot{\phi}$ over to $\frac{\partial \mathcal{L}}{\partial \dot{\phi}}$ which will give me </p>
<p>$$\int d^dx (\frac{\partial \mathcal{L}}{\partial \phi} - \partial_\mu\frac{\partial \mathcal{L}}{\partial \partial_\mu \phi}) \delta \phi$$ </p>
<p>i.e. the functional derivative gives the Euler-Lagrange equations.</p>
<p>So how to I take the functional derivative of a functional w.r.t to the derivative of a function?</p>
| 7,354 |
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<p>I've read that the faster you travel in space, the slower you travel in time.
And when you reach the speed of light (which we won't be able to) time will stand still.
So when light travels at the speed of light, it doesn't move in time, so no time will have past when the light arrives at it's "destination". Right?</p>
| 195 |
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<p>If the second and third matter families (so-called generations: muon, muon neutrino, tau, tau neutrino, ...) didn't exist, would that affect how the universe runs? Are they optional or required?</p>
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] |
<p>Today, I was tutoring and explained the space-time. I explained how one can convert North-South into West-East by rotating, and how you can convert time into space with velocity.</p>
<p>Below the Energy-Momentum stuff the book had some problems. One was the following:</p>
<blockquote>
<p>Given an event at (1 Lightsecond, 2 s), give it's coordinates in a frame of reference that is moving with 0.6 of speed of light to it.</p>
</blockquote>
<p>This the diagram I came up with, in blue there is the event. (Not really to scale.)</p>
<p><img src="http://wstaw.org/m/2011/11/11/m8.png" alt=""></p>
<p>Then I projected the point onto the orange axes, and I get smaller values on each. This makes sense to me, as a moving observer would see thing smaller and slower, resulting in less values in either measurement.</p>
<p>But how do I get the orange lines? I cannot see how I get this angle. If I got them, I could tell you the speed of the reference frame, but the speed is given here …</p>
<p>And did I do the projection of the speed right?</p>
| 7,356 |
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<p>According to <a href="http://en.wikipedia.org/wiki/Density_matrix#Pure_and_mixed_states" rel="nofollow">Wikipedia</a>, if a system has $50\%$ chance to be in state $\left|\psi_1\right>$ and $50\%$ to be in state $\left|\psi_2\right>$, then this is a mixed state.</p>
<p>Now consider state $\left|\Psi\right>=(\left|\psi_1\right>+\left|\psi_2\right>)/\sqrt2$, which is a superposition. Let $\left|\psi_i\right>$ be eigenstates of Hamiltonian. Then measurements of energy will give $50\%$ chance of it being $E_1$ and $50\%$ of being $E_2$. But this then corresponds to the definition above of mixed state! At the same time superposition is defined to be a pure state.</p>
<p>So, what is the mistake here? What is real difference from mixed state and superposition of pure states?</p>
| 7,357 |
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<p>I'm curious what will happen if we 'drive' a car with (almost) the speed of light, and all of sudden we switch the car headlights on? Will the car headlights ray have double speed (speed of 600.000km/h) or that ray will never be seen?</p>
| 109 |
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] |
<p>Why is the Dirac operator considered so important - in both physics and (pure) mathematics?</p>
| 7,358 |
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<p>Given the energy momentum tensor in E&M:
$T_{\mu\nu} = -F_{\mu\alpha} g^{\alpha \beta} F_{\beta \nu} +\frac{1}{4} g_{\mu \nu} F_{\sigma \alpha} g^{\alpha \beta} F_{\beta \rho} g^{\rho \sigma}$
I want to show it is homogenous and isotropic in Robertson-Walker spacetime. How would you go about performing a rotation and a translation to show that $T_{\mu \nu}$ is left invariant?</p>
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<p>I have been trying to understand Einstein's principle of relativity for quite some time. I have at least I believe, understood the notion that time is relative and can actually 'change' relative to some frame of reference close to c. My question is, time is actually a variable that we invented based on how we thought about it; it's the progression of events, life and existence itself. So when, we say in Twin Paradox, one of the twin when he comes back to Earth is younger, how can that be possible? Again, it is true time would slow for him during his travel in space but that's because of the way we defined time and consequently its limitations close to c. How can that affect his physical structure (aging process)? Time should be independent of his personal physique. </p>
| 117 |
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] |
<p>This is a very trivial question, but I can't seem to reason it out, again, as to why charges gather at points and edges.</p>
| 7,360 |
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<p>Today in an engineering thermodynamics lecture, the professor gave an example of a gas doing work. We had a cylinder full of helium at a pressure of something like 200 kPa absolute and the valve was opened so that the gas would flow out against the atmospheric pressure until the pressures were equal. Also the cylinder was assumed to be in thermal equilibrium with its surroundings so the temperature of the gas was equal to the temperature of the ambient air. However, the way he calculated the work perturbed me. He said that this was an isobaric process because the gas was expanding against a constant atmospheric pressure. I was under the assumption that an isobaric process means that the working fluid stays at constant pressure throughout the process which is not the case in this expansion. And in this case, the gas pressure is dropping as it leaves the cylinder.</p>
<p>The professor then proceeded to calculate the work as $W = P_\text{atm}\Delta V$. But I didn't think that was right and that simple.</p>
<p>I should also note that the system was modeled as a closed system, so the gas inside the cylinder and the escaping gas were the system.</p>
<p>If this is not an isobaric process why is the work calculated with a constant pressure?</p>
<p>Am I correct, or is the professor?</p>
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] |
<p>Every once in a while, we get a question asking for a book or other educational reference on a particular topic at a particular level. This is a meta-question that collects all those links together. If you're looking for book recommendations, this is probably the place to start.</p>
<p>All the questions linked below, as well as others which deal with more specialized books, can be found under the tag <a href="/questions/tagged/resource-recommendation" class="post-tag" title="show questions tagged 'resource-recommendation'" rel="tag">resource-recommendation</a> (formerly <a href="/questions/tagged/books" class="post-tag" title="show questions tagged 'books'" rel="tag">books</a>).</p>
<p>If you find a question that should be added to the list, edit it into the existing list answer rather than creating your own answer for it. See <a href="http://meta.physics.stackexchange.com/a/4698/22494">our policy</a> on resource recommendations for details.</p>
<p><strong>Related Meta:</strong> <a href="http://meta.physics.stackexchange.com/q/5307/23119">Do we need/want an overarching books question?</a> </p>
| 98 |
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<p>Many of us may remember the problems from thermodynamics where we had to determine if a pretend air conditioning unit supplier's claims violated the 2nd law. I want to pose a similar question regarding a new "green" Air Conditioning (AC) technology.</p>
<p>An ice energy storage unit for AC works by making ice during the night and using the latent heat (mostly) to cool the building during the day. The major motive for doing this is to move the energy demand from the peak energy use times to the low-demand times at night when it is cheaper. The cost of electricity is not relevant to this question. The claims of a vendor of this technology, <strong>Ice Bear Energy Storage System</strong>, include the following:</p>
<blockquote>
<p><a href="http://www.ice-energy.com/ice-bear-energy-storage-system" rel="nofollow">Bottom line: The Ice Bear system reduces total net energy consumption for most buildings under virtually all operating conditions and installations.</a></p>
</blockquote>
<p><strong>Question:</strong> Is this claim physically plausible, given reasonable assumptions about climate? There are some example cities below, I imagine that considering an average location or just the extremes would be sufficient.</p>
<p>Here are a few things to note, but these are certainly not comprehensive. Creativity may be important for identifying all the factors at play.</p>
<ul>
<li>The temperature of ice is $0^{\circ} C$ ($32^{\circ} F$)</li>
<li>Room temperature is $22.2^{\circ} C$ ($72^{\circ} F$)</li>
<li>The outside ambient temperature is lower at night than during the day. Exactly how much depends on the location and <a href="http://en.wikipedia.org/wiki/Diurnal_temperature_variation" rel="nofollow">humidity particularly.</a> Locations mentioned on their website include <a href="http://www.ice-energy.com/case-studies" rel="nofollow">Glendale, CA; Howell, NJ; Las Vegas, NV; Fort Collins, CO; Orange County, CA; and Redding, CA.</a></li>
<li>The ice storage duration is 6 hours and since these are multiple MW products, I'll say it's fair to assume almost perfect insulation unless someone wants to challenge this.</li>
<li>You <em>can</em> obtain useful work while exchanging the heat from the office building to the ice. I believe this would be important considering the wording of "total net energy consumption".</li>
</ul>
<h1>Temperatures</h1>
<p>I have <a href="http://www.bestplaces.net/climate/" rel="nofollow">looked up</a> the temperatures for most of the mentioned cities. For some I couldn't get data for I used a nearby city instead.</p>
<ul>
<li><strong>(City/State) (night-day difference) (seasonal difference) (hottest month high)</strong></li>
<li>Redding CA, 27.4, 44, 99</li>
<li>Las Vegas NV, 26.5, 48, 104</li>
<li>Denver CO, 27.8, 45, 88</li>
<li>Lakehurts NJ, 27.8, 45, 88</li>
<li>Santa Anna CA, 20.5, 18, 83</li>
</ul>
<p>These seem to be fairly consistent for the night-day difference in particular. I hope that simplifies the problem. It seems clear that running at night "buys" you about 27.5 degrees F. Now, the absolute temperatures as well as the seasonal variation will affect the efficiency, but to a much smaller degree.</p>
<p>My conflict: Can a $27.5$ degree difference really make up for a $72-32=40$ degree penalty on the thermal cycle? <em>Really</em>? It's not a trivial problem, but I am skeptical. I will certainly send them an email letting them know they violated the 2nd law depending on the outcome here.</p>
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] |
<p>What is meant by a cyclic rotation of a matrix, specifically in proving that j*k-k*j=i*l
where j,k,l are cyclic rotations of the Pauli spin matrices sigma x, sigma y, and sigma z</p>
| 7,363 |
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<p>The Kolmogorov theory of turbulence indicates an energy cascade in turbulence. Is there a corresponding version of relativistic fluid?</p>
| 7,364 |
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<p>Suppose you do a Stern-Gerlach experiment with atoms in a $j=1$ state. There would be three separate beams ($m_j = -1, 0, 1$) coming out of the apperatus. But what would be the relative distribution of atoms in these beams?</p>
<p>On first thought I would say that the distribution should be $\frac{1}{3}$, $\frac{1}{3}$, $\frac{1}{3}$. But considering the "classical" distribution of the projections of the angular momentum onto one axis I could imagine that there could be more atoms in the $m_j = 0$ state. This would still be compatible with symmetry considerations.</p>
<p>To state the question differently: how does the density matrix of an unpolarized beam of $j=1$ atoms look like?</p>
| 7,365 |
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<p>If we know a function $f(\phi, \theta)$ in $\mathbb R^3$ only over a convex surface (which for simplicity let's assume a sphere of radius $r$), is there any measure for the degree of anisotropy over this surface? That is, given two functions, $f_1, f_2$ is there any objective measure $A(f)$ where we can make qualitative comparisons like $A(f_1) > A(f_2)$?</p>
<p>Thoughts:</p>
<ul>
<li>For an electrostatic field produced by a centered monopole we should get $A(f_{\text{monopole}})=0$.</li>
<li>For sufficiently large $r$, the field produced by a possibly non-centered monopole versus that of a possibly non-centered dipole should be $A(f_{\text{dipole}}) > A(f_{\text{monopole}})$.</li>
<li>An expansion in spherical multipole moments seems like a good start, but how would you compare the two expansions?</li>
<li>Would the comparison generalize to a non-spherical but concave surface provided you normalized for the surface area?</li>
<li>This may be related to the calculation of the degree of anisotropy of the <a href="http://en.wikipedia.org/wiki/Cosmic_microwave_background" rel="nofollow">CMB</a>, but I'm not familiar with how they define anisotropy.</li>
</ul>
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<p>I am trying to understand <a href="http://en.wikipedia.org/wiki/Coherent_states">coherent states</a>. As far as I could find there are three equivalent definitions and in general many sources start from a different one, still I fail to see their equivalence. I restate the definitions and their equivalences given in the Wikipedia page:</p>
<ol>
<li>Eigenstate of annihilation operator: $$a|\alpha \rangle =\alpha|\alpha \rangle$$</li>
<li>Displacement operator of the vacuum: $$ |\alpha \rangle =e^{\alpha a^{\dagger}-\alpha^{*} a}|0 \rangle $$</li>
<li>State of minimal Uncertainty: $$ \Delta X= \Delta P = \frac{1}{\sqrt{2}} $$</li>
</ol>
<p>I fail to see how they are the same!
Can someone please explain how to derive these from one another?</p>
| 339 |
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<blockquote>
<p>A 9.5 kg block (A) is traveling in the positive x-direction with a speed of 3.0 m/s. At some point, it explodes and breaks into two pieces. After the explosion, block C, which is 6.0 kg, moves off in the negative y-direction, and block B, which is 3.5 kg, moves off at an angle 35° with respect to the x axis. </p>
</blockquote>
<p><img src="http://i.stack.imgur.com/KZk8n.jpg" alt="sorry for the crappy recreation"></p>
<p>The two parts of this question that I am stuck on are:</p>
<blockquote>
<p><strong>a)</strong> find the momentum of the 2-block combination after the explosion and write it in vector notation</p>
</blockquote>
<ul>
<li>Here I know that momentum is conserved, so Pa = Pbc = 28.5 kg*m/s</li>
<li>I am confused about how to separate the x and y components of the momentum after the explosion: can I use trigonometry with the mass of block B as the hypotenuse?</li>
</ul>
<blockquote>
<p><strong>b)</strong> Find the speeds of blocks B and C after the explosion</p>
</blockquote>
<ul>
<li>Since momentum is conserved, and Mass(A) = Mass(B)+ Mass(C), I know that V(A)=V(B)+V(C), but I don't understand how to split them (I know it's not in half...)</li>
</ul>
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<p>Do the full inhomogeneous Maxwell equations obey parity (P) and time reversal (T) symmetry separately or only the full CPT symmetry?</p>
<p>I believe the homogeneous Maxwell equations obey parity and time reversal symmetry separately - is that right? The homogeneous Maxwell equations reduce to a wave equation in which space and time appear as second order derivatives.</p>
<p>Here's my effort at answering my first question.</p>
<p>In the first inhomogeneous equations one has:</p>
<p>$$\mathbf{\nabla . E} = \frac{\rho}{\epsilon_0}$$</p>
<p>If I change the sign of x,y,z, in $\mathbf{\nabla . E}$ then I must change the sign of $\rho$ to balance it.</p>
<p>In the third equation one has:</p>
<p>$$ \mathbf{\nabla \times E} = - \frac{\partial \mathbf{B}}{\partial t}$$</p>
<p>As the sign of x,y,z has changed then the sign of t must change to balance it.</p>
<p>Thus we must have CPT symmetry. This can be checked in the last equation:</p>
<p>$$\mathbf{\nabla \times B} = \mu_0 \mathbf{j} + \frac{1}{c^2}\frac{\partial \mathbf{E}}{\partial t}$$</p>
<p>The sign of x,y,z in $\mathbf{\nabla \times B}$ has changed. This is balanced by a sign change of t in $\frac{1}{c^2}\frac{\partial \mathbf{E}}{\partial t}$. The sign of $\mathbf{j}$ has changed because the charge has changed sign.</p>
<p>Thus the inhomogeneous Maxwell equations obey CPT symmetry rather than C,P or T alone.</p>
<p>Is this reasoning right?</p>
| 7,368 |
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] |
<p>Recently I am reading Di Francesco's book Volume I on Conformal Field Theory. In order to reduce the number of fields in a correlator, the calculation of Operator Algebra is extremely important. To do this, equation (6.175), which is derived from (6.174), is necessary. The author writes in the book below equation 6.174:
"Applying this relation (here the author refers to eq.6.174) on the r.h.s of Eq.(6.169), we find:"</p>
<p><img src="http://i.stack.imgur.com/HDFai.png" alt="enter image description here"></p>
<p>But I still can not understand how 6.174 is applied to deduce this equation presented above. Nor can I understand why the holomorphic dimension h appears on the right hand side of this equation. Since this is very important for the discussion on the calculation of the Operator Algebra, I can not go on without an understanding of this.</p>
| 7,369 |
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<p>If an electron falls into a black hole. How can the Heisenberg uncertainty principle hold? The electron has fallen into the singularity now so it has a well defined position which means that it doesn't have a well defined momentum? Furthermore, the electron can't have a well defined position in space because space eigenkets are unphysical. It's momentum must certainly cause it to move.</p>
<p>Another question, Can one calculate the amount of new mass(the relativistic mass) that the black hole acquire after quantum particles fall in the singularity?</p>
<p>Doesn't this mean that the electron can't be described by a wave packet at the singularity of the black hole? If we want quantum mechanics to be applicable inside a black hole,the wavefunction should leak outside right ?</p>
| 7,370 |
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<p>Who can provide me some elegant solution for </p>
<p>$$\left[\hat{a}^{M},\hat{a}^{\dagger N}\right]\qquad\text{with} \qquad\left[\hat{a},\hat{a}^{\dagger}\right]~=~1$$</p>
<p>other than brute force calculation?</p>
<p>================================================</p>
<p>O lala!!! Thanks for @Prathyush and @Qmechanic !!! I got the same result with Qmechanic... I think Prathyush's suggestion should be equivalent to the my suggestion of the correspondence up to a canonical transformation. Here is my calculation (I was not confident to post it...) </p>
<p>$\begin{array}{c}
\mbox{representation of }\left(\hat{a},\hat{a}^{\dagger}\right)\mbox{ on polynomial space }span\left\{ \frac{x^{n}}{\sqrt{n!}}\right\} _{n\ge0}\\
\hat{a}\left[f\left(x\right)\right]=\frac{d}{dx}f\left(x\right)\;;\;\hat{a}^{\dagger}\left[f\left(x\right)\right]=xf\left(x\right)\;;\;\left[\hat{a},\hat{a}^{\dagger}\right]\left[f\left(x\right)\right]=id\left[f\left(x\right)\right]\\
\left|0\right\rangle \sim 1\;;\;\left|n\right\rangle \sim x^{n}/\sqrt{n!}
\end{array}$</p>
<p>$\begin{array}{c}
\mbox{calculate the normal ordering }\left[\hat{a}^{M},\hat{a}^{\dagger}{}^{N}\right]\mbox{:}\\
\sim\left[\frac{d^{M}}{dx^{M}},x^{N}\right]=\frac{d^{M}}{dx^{M}}\left(x^{N}\star\right)-x^{N}\frac{d^{M}}{dx^{M}}\left(\star\right)\\
\sim\left\{ \overset{min\left\{ M,N\right\} }{\underset{k=0}{\sum}}\frac{N!}{\left(N-k\right)!}C_{M}^{k}\left(\hat{a}^{\dagger}\right)^{N-k}\left(\hat{a}\right)^{M-k}\right\} -\left(\hat{a}^{\dagger}\right)^{N}\left(\hat{a}\right)^{M}\\
\end{array}$</p>
<p>======================
One comment on 02-12-2012:
The representation I was using is actually related to Bergmann representation with the inner product for Hilbert space (polynomials) being:</p>
<p>$\left\langle f\left(x\right),g\left(x\right)\right\rangle :=\int dxe^{-x^{2}}\overline{f\left(x\right)}g\left(x\right)\,,x\in\mathbb{R}\,,\, f,g\in\mathbb{C}\left[x\right]$</p>
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<p>Let's say a 10N cube (in air, on Earth) rests flat on a scale at the bottom of a tank of water, and the scale reads 8N, so there is 2N of buoyant force on the cube. How does the buoyant force manifest itself on the cube? I assume the force is normal to the surface of the scale, but if there isn't any water under the cube, how does the water push <em>up</em> on the cube?</p>
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<p>Let $ H = \frac{-h^2}{2m}\frac{\partial^2 }{\partial x^2}$. I want to find the matrix elements of $H$ in position basis. It is written like this:</p>
<p>$\langle x \mid H \mid x' \rangle = \frac{-h^2}{2m}\frac{\partial^2}{\partial x^2} \delta(x -x')$.</p>
<p>How do we get this? are we allowed to do $\langle x | \frac{\partial^2}{\partial x^2} \mid x' \rangle = \frac{\partial^2}{\partial x^2} \langle x \mid x' \rangle$? Why? It seems some thing similar is done above. </p>
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<p>What experiments prove the greenhouse effect? I've read a little on <a href="http://en.wikipedia.org/wiki/Greenhouse_effect" rel="nofollow">wikipedia</a> but I've got only some names but not their experiments.</p>
| 710 |
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] |
<p>I know the equations, but I am having difficulties to get the difference between this two concepts, binding energy and adsorption energy.</p>
| 7,374 |
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] |
<blockquote>
<p>Can a liquid insulator be electrically charged by touching a charged
conductor?</p>
</blockquote>
<p>I understand that solid insulator will only be charge on the surface where it is touch, but the case is different from liquid which it circulates, so I imagine that slowly liquid insulator will be fully charge due to circulation.</p>
| 7,375 |
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] |
<p>What is the temperature of objects in Low Earth Orbit?</p>
<p>Consider LEO to be 600km to 800km.</p>
| 7,376 |
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<p>My mind is drawing a blank right now. In systems with spin and orbital angular momentum, I know that rotational invariance implies that $[H, \mathbf{J}]=0$ where $\mathbf J=\mathbf L+\mathbf S$. But I don't see why rotationally invariant systems should necessarily commute with the <em>square</em>-magnitude operators:</p>
<p>$$[H,\mathbf{L}^2]=0\qquad\text{and}\qquad[H,\mathbf{S}^2]=0\,.$$</p>
<p>But the funny thing is, I am unable to come up with model Hamiltonians that are rotationally invariant, but do not commute with $\mathbf{L}^2$ and $\mathbf{S}^2$. So my question is: is there an easy algebraic reason for this commutation relation? or are there systems that do not conserve these operators? In the latter case, I need help coming up with examples.</p>
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<p>What is the material for optical focusing that produces the less diffraction losses?</p>
<p>Suppose one have a sequence of serial optical elements $R$ that keep refocusing a beam of collimated light, each a distance $D$. Now suppose that the wavelength $\lambda$, the optical element and the distance are such that the $\epsilon$ is the amount of radiation that falls outside the next optical element, which i'll take to be small. </p>
<p>I'm concerned with other losses in this system, for instance, the optical material can have some intrinsic diffusion, or it absorbs a portion of the radiation</p>
<p>What are the main losses that an optical element can have of this kind, and which are the best refocusing optical elements that minimise those losses?</p>
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<p>I've heard and read that our solar system lies near to the peripheral region of the Galaxy. Then accordingly we would have a greater probability of sustaining to eventual gulping down by the super-massive black hole. But how long ?</p>
| 405 |
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] |
<p>Can the statement be regarded as true? That every particle, or element in the universe can be regarded as a combination of black hole and white hole in variable proportion.</p>
| 7,379 |
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<p>Can someone explain these 3 concepts into a unified framework.</p>
<ol>
<li><p>Randomness : Randomness as seen in a coin toss, where the system follows known and deterministic (at the length and scale and precision of the experiment) physics but, the complexity allows us to treat the system as random. (i.e we can always run an exact physical simulation to predict the coin's fall)</p></li>
<li><p>Chaotic randomness: randomness as seen in weather or fluid flow, where the system is highly sensitive to initial conditions and as a result no matter what the precision of measurement/simulation is, we cannot predict the out come. </p></li>
<li><p>Quantum randomness: The inherent probability distributions that we study in QM, where the electron's position probablity distribution is set by QM. </p></li>
</ol>
<p>how do the concepts of randomness in coin toss, randomness in chaos and randomness in QM fit into a unified picture. Can someone please elaborate ? </p>
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<p>When doing physics with two-level systems and introducing rotations, a term that appears quite often is the rotation of a pauli matrix by another one:</p>
<p>$e^{- i \sigma_i \theta/2} \sigma_k e^{i \sigma_i \theta/2}$</p>
<p>The way I know to evaluate this is by using</p>
<p>$\exp(i \sigma_i \theta/2) = I \cos \frac{\theta}{2} + i \sigma_i \sin \frac{\theta}{2}$</p>
<p>and expanding the previous equation. If $i = j$ it's trivial, but in the other cases it get's quite tedious. Is there an easier way to do that (or some shortcut)?</p>
| 7,381 |
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<p>The trajectory of a cannonball fired from the origin with initial speed $v_0$ at an angle $\theta$ is given by
$$y = x\tan\theta - \frac{g}{2v_0^2 \cos^2\theta}x^2.$$
For fixed $v_0$, at what angle $\theta$ should the canon ball be fired so that it will hit a target located at $(x_0,y_0)$? What is the area of the region in which the target can be located such that it is possible to hit the target?</p>
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] |
<p>We know that the uncertainty product of general states is bounded by the inequality described by Heisenberg's uncertainty relation. And the ground state of the quantum harmonic oscillator falls under the minimum uncertainty product ($\frac{\hbar}{2}$). Do the excited states of QHO have also the minimum uncertainty product or higher?</p>
| 7,383 |
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<p>I'm bit confused about 'black' as a color. As per my knowledge, it is not given in visible color spectra like other colors for example red, violet etc. Also I'm confused with definition of color--does it mean the reflection of respective wavelengths from an object perceived by human eyes or latest electronic sensors determines the color of that object? This is okay for normal people to know that this this color is this this. But how will you tell that some object has Red color to a blind person, how that person can think of that? I can define a bicycle as a vehicle with two wheels, a handle, a seat--this way, a blind person can imagine that vehicle, but tell me the definition of color so as blind person can imagine about specific color considering he/she is blind since birth. Also one more question regarding black--what is the color of an apple if it is placed in a dark room--to eyes, it seems blindly blackish (oh, what the hell this black is again, and why we feel blind in dark places although eyes are open, huh?), but if we focus a torch on that apple with a daylight like color, it looks with original color, why so?</p>
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<p>im trying to develop a flexible robotic arm filled with ferro fluid. Different flexible motions can be performed using this concept.</p>
| 7,384 |
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<p>I am trying to solve the following problem that involves motion in a straight line. I am not quite sure how to go about it though. I know it will involve formulas and derivatives but which specific ones would I use?</p>
<blockquote>
<p>An electron has a constant acceleration of 2.6 m/s^2. At a certain
instant its velocity is 12.1 m/s.</p>
<p>What was its velocity 2.5 s earlier? What was its velocity 2.5 s
later?</p>
</blockquote>
| 7,385 |
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] |
<p>Why do we have two different types of phonons, <a href="http://en.wikipedia.org/wiki/Phonon#Acoustic_and_optical_phonons" rel="nofollow">optical and acoustical</a>? Why are they called optical phonons? </p>
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<p>From what I understand, it seems like you can only "add" beams together. You can use a beam combiner, basically using a beam splitter in reverse, to combine two beams. In homodyne detection, you use a local oscillator beam to mix with the signal beam so you can decipher the difference in phase change which would give you information about the system your signal beam is probing. Homodyne detection can give information regarding the position of an atom.</p>
<p>Say I wanted to detect the relative distance of two atoms. Each atom/ion is stuck inside an electric harmonic potential. The location of the harmonic potentials are known, (if this is possible) Now, let's say I turn off the harmonic potential trap and at the same time I use the homodyne measurement technique to determine the position of both of the atoms. This means shining a laser to detect the atom's position. (The light is absorbed and re-emitted?, or reflected?) In any case, light that comes out of the atom/ion cavity is related to the atom's position/momentum and it is combined with a local oscillator beam to determine the position and momentum of the atom. The light hits photodetectors which then integrate the signals together producing a photocurrent. The analysis of this photocurrent will allow one to figure out the atom's position relative to the center of the harmonic trap.</p>
<p>To figure out the relative position of the atoms from each other, I'll need to "subtract" the position of one atom with the other. If one atom's position is labeled x1 and the other is labeled x2, than wouldn't we simply do x2 - x1? If so, how is one to do this?! How does one "subtract" the position of one from the other using light beams?</p>
| 7,387 |
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] |
<p>How would you determine whether an object is at equilibrium or not?</p>
<p>What is the definition of <a href="http://en.wikipedia.org/wiki/Mechanical_equilibrium" rel="nofollow">equilibrium</a>?</p>
| 7,388 |
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<p>There are lots of devices that purport to measure the absolute value of sound pressure levels. <a href="http://rads.stackoverflow.com/amzn/click/B000A6V5FS" rel="nofollow">Here is an example</a>, <a href="http://rads.stackoverflow.com/amzn/click/B000I4ZVHU" rel="nofollow">here's another</a>, there's also this <a href="https://itunes.apple.com/ca/app/decibel-10th/id448155923?mt=8" rel="nofollow">iPhone App</a>. Putting such devices side-by-side in a sound-filled environment usually yields a discrepancy between the <a href="https://en.wikipedia.org/wiki/Sound_pressure_level" rel="nofollow">decibel</a> readings of each device. This inevitably leads to the question "Which reading is correct?"</p>
<p>If we were similarly measuring temperature with a collection of thermometers, there are a variety of naturally-occurring physical references against which to verify or calibrate the thermometers. For example, you could use <a href="https://en.wikipedia.org/wiki/Water_%28molecule%29#Triple_point" rel="nofollow">the triple point of water</a> or the <a href="https://en.wikipedia.org/wiki/Critical_point_%28thermodynamics%29" rel="nofollow">critical point</a> of another substance. </p>
<p>So, for thermometers, we have a naturally occurring phenomenon with a known temperature to which we can compare our thermometer's reading.</p>
<p>Is there a naturally occurring phenomenon with a known sound pressure level to which we can compare a sound pressure level meter's reading?</p>
| 7,389 |
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<p>If a container like a balloon but with constant volume is filled, is it possible to pack air molecules so closely together that they don't have any empty space between them? If so, what would this material look like?</p>
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] |
<p>The standard lore in QFT is that one must work with renormalised fields, mass, interaction etc. So we must work with "physical" or renormalised quantities and all our ignorance with respect to its relation with "bare quantities" is taken care by counter terms which (surprisingly!!) cancel infinities. My question is:</p>
<ol>
<li><p>Why do we care to regulate such field theories, when all physical quantities are rendered finite by the way of renormalisation . Why is there such song and dance about regulator and regularizing scheme(one which break certain symmetries and what not)? </p></li>
<li><p>The following is not very physical question- Is there a deeper understanding how and why counter terms seem to render all physical quantities finite. I guess a satisfactory answer to this question is a matter of taste, nevertheless what I am looking for is a constructive argument which a priori guarantees finite sensible values of measurable quantities.</p></li>
</ol>
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] |
<blockquote>
<p>A $150~\text{V}$ battery is connected across two parallel metal plates of area $28.5~\text{cm}^2$ and separation $0.00820~\text{m}$. A beam of alpha particles (charge $+2e$, mass $6.64 \times 10^{-27}~\text{kg}$) is accelerated from rest through a potential difference of $1.75~\text{kV}$ and enters the region between the plates perpendicular to the electric field. </p>
<p>What magnitude and direction of magnetic field are needed so that the alpha particles emerge undeflected from between the plates?</p>
</blockquote>
| 7,392 |
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<p>Cern recently stated that antimatter may be repelled by matter, much like the opposite effect of Gravity. So is it possible that antimatter is actually repelled to the edges of the Universe to create a sort of outer-shell, something that allows the expansion of the Universe into nothingness..</p>
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] |
<p>Problem:</p>
<p>An object is undergoing simple harmonic motion with period 1.2s and amplitude 0.6m. At $t=0$, the object is at $x=0$. How far is the object from the equilibrium position when $t=0.480$s?</p>
<p>Attempt at solution:</p>
<p>I used the displacement equation :</p>
<p>$$x(t)=A\cos(\omega t+\phi)$$</p>
<p>and also found out what the angular frequency $\omega$ is (5.2rad/s). Then I found that $\phi$ is 0. I plugged my results in the equation and when I looked at the solution they used the following equation:</p>
<p>$$x(t)=A\sin(\omega t)$$</p>
<p>Im not quite sure how I was supposed to know to use this equation instead of the cosine equation that is written on my equation sheet.</p>
<p>Thank you.</p>
| 7,394 |
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<p>what does string theory have to say about the trace anomaly in the expectation value of the stress energy tensor of massless quantum fields on a curved background and its interpretation as the emergence of an extra scalar degree of freedom in the semiclassical limit of GR + QFT?
Is it something that can be proved? that emerges naturally? Or that can be shown to be false?</p>
<p>The question has actually two quite distinct parts: 1) the presence of the anomaly and its form and 2) the possibility of interpreting it as the emergence of an extra degree of freedom at the classical level with its own EOM and a specific stress energy tensor satisfying the constraint Trace (T_mu_nu) = some function of the curvature.</p>
<p>Thanks a lot</p>
<p>NOTE ADDED
As a warm-up I will reduce the scope of the question.</p>
<p>String theory and the triangle anomaly in Q.E.D.?</p>
<p>Are there examples where, starting from "first principles" in string theory, one compactifies to 3+1 dimensions and reproduce the physical effects of the so called/chiral triangle anomaly.
When I say physical effects, I mean I don't care how it shows up, as long as it reproduces the final result in some approximation.
Than we can think of the fact that there are different kind of anomalies etc.</p>
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<p>I have a question about the restoring force in elastic band or rope which confusing me for a long time.</p>
<p>As I was told in high school physics, for an elastic band (or spring), if Hooke's law holds, we have $F = k\Delta x$. What's confusing me is: should F be the total force acting on the object or the restoring force only? Or I ask this way: is there anything called "restoring force" existing independently, just like gravity, friction or tension?</p>
<p>To my understanding, restoring force should be the total force which is pointing to the equilibrium point. For example, if we consider a bungee cord, we should always count the tension of the cord as well as the gravity so the restoring force at any time should be the total force of tension and gravity; hence, when we apply Hooke's law, we should always have $F$ being the total force, not just the tension. Is that correct?</p>
<p>This is pretty confusing to me because there use many terms in the book. Sometimes they said it is the tension in Hooke's law, sometimes they say the restoring force and sometimes the total force....</p>
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<p>I'm a statistician with a little training in physics and would just like to know the general consensus on a few things.</p>
<p>I'm reading a book by <a href="http://en.wikipedia.org/wiki/John_Moffat_%28physicist%29">John Moffat</a> which basically tries to state how GR makes failed predictions in certain situations. I know GR is well extremely tested, but I imagine all physicist are aware it doesn't always hold up.</p>
<p>The book tries to put forth modified theories of gravity that make do without the need of dark matter and dark energy to make GR match real world observations. (ie speed of galaxy rotations etc)</p>
<p>Are modified theories of gravity creditable?</p>
<p>Is dark energy/matter the 'ether' of the 20th/21st century? Is it likely scientists are looking for something that simply doesn't exist and there are unknown fundamental forces at work? What's the best evidence for it's existence other than observations based on the <a href="http://en.wikipedia.org/wiki/Bullet_Cluster">'bullet' cluster</a>?</p>
<p>Any insight would be great.
Cheers :)</p>
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<p>Recently I heard that both of "languages" of scattering theory (formal solution by method of classical scattering theory and its reworking by Feynman with his rules) predict virtual particles. But for first language intermediate states may be interpreted as particles which satisfy corresponding wave equations but don't satisfy 4-momentum conservation law; then, for second language intermediate states refer to functions which don't satisfy wave equation but satisfy 4-momentum conservation law. </p>
<p>May someone add some details to this text? As for me, there is so small information for understanding clearly. Perhaps my question contains some incorrect phrases, but I hope that it can be understood.</p>
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<p>I understand that changing electric field produces magnetic field and changing magnetic field produces electric field.</p>
<p>Are these produced magnetic and electric field produced due to one defined to be constant or variable?
If these are defined to be variable then do they continue to produce one another? By this I mean if changing electric field produces changing magnetic field, does this changing magnetic field produce a new electric field or the same one again?</p>
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] |
<p>It seems that the Bloch equations, or a suitable generalization thereof, are enough to phenomenologically model the measurement process in NMR. Has anyone attempted a fully quantum mechanical model of the NMR spin system and measurement coil together? </p>
<p>It seems to me that these coils are intentionally of low quality to ensure no back-action (more phenomenology, so far as I can tell) on the spin system. But perhaps one can engineer more efficient control or parameter estimation strategies with high quality coils and an accurate quantum mechanical model of the measurement interaction, no?</p>
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<p>That is perhaps a bit of an advertisement, but a couple of collaborators and myself just sent out a <a href="http://arxiv.org/abs/1110.2791v1" rel="nofollow">paper</a>, and one of the results there is a little bit surprising. We found (in section 6E) a fermionic local symmetry which closes to a tensor gauge symmetry and therefore does not imply (contrary to common lore) the presence of gravity. The construction is in the context of the six dimensional (2,0) theory, and uses in an essential way properties of chiral fermions in six dimensions, but one has to wonder if this is more general. </p>
<p>So my question is simple - does this ring a bell? are there any other known examples?</p>
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<p>I am currently studying generalized probabilistic theories. Let me roughly recall how such a theory looks like (you can skip this and go to "My question" if you are familiar with this).</p>
<p><strong>Recall</strong>: In a <em>generalized probabilistic theory</em> (GPT), the set of states is given by a convex subset $\Omega \subset V$ of a real vector space (in the special case of quantum theory, $\Omega$ is the set of density operators and $V$ is the vector space of Hermitian operators on a Hilbert space). A the statistics of a measurement are described by a set of <em>effects</em>. An effect is a linear functional $e \in V^*$ such that $0 \leq e(\omega) \leq 1$ for all $\omega \in \Omega$. Let the set of effects be denoted by $E(\Omega)$. The <em>unit effect</em> $u \in E(\Omega)$ is given by $u(\omega) = 1$ for all $\omega \in \Omega$ (in a GPT, the set of states $\Omega$ is always such that such a functional $u \in V^*$ exists). A measurement is a set $\{ e_1, \ldots, e_n \}$ of effects such that $\sum_{i=1}^n e_i = u$.</p>
<p>In order to describe composite systems in a GPT, the concept of tensor products is introduced (in what follows, a "tensor product" of state spaces $\Omega_A$ and $\Omega_B$ is a rule that tells you how to combine systems, and this rule does not have to coincide with what is usually called a tensor product in mathematics, whereas the tensor product $e_A \otimes e_B$ means the usual tensor product; I think this is a bad terminology, but it is very common in the theory of GPTs). A state $\omega^{AB} \in \Omega^{AB}$ of a system composed of two subsystems $A$ and $B$ has to satisfy
\begin{equation}
\text{normalization:} \quad (u^A \otimes u^B)(\omega^{AB}) = 1
\end{equation}
and
\begin{equation}
\text{positivity:} \quad (e_A \otimes e_B)(\omega^{AB}) \geq 0 \quad \forall e_A \in E(\Omega_A), \forall e_B \in E(\Omega_B),
\end{equation}
where $\otimes$ denotes the usual (in mathematical terminology) tensor product. These two requirements have to be fulfilled by every state of a composite system; they are the minimal restrictions for a composite system.</p>
<p>The <em>maximal tensor product</em> $\Omega_A \otimes_\text{max} \Omega_B$ of two systems is given by the set of all $\omega^{AB} \in V_A \otimes V_B$ that satisfy normalization and positivity (it's called maximal since minimal restrictions lead to a maximally large set of states).</p>
<p>The other extreme case is the <em>minimal tensor product</em> which is given by all convex combinations of product states $\omega_A \otimes \omega_B$, i.e. by all <em>mixtures</em> of product states.</p>
<p>There are also other possible ways to combine systems, i.e. other tensor products apart from the maximal and the minimal tensor product. For example, the "tensor product" in the quantum case (encompassing all density operators on $\mathcal{H}_A \otimes \mathcal{H}_B$) is neither the minimal nor the maximal tensor product.</p>
<p><strong>My question:</strong> I wonder how much one can infer about the structure of the state space $\Omega_A \otimes \Omega_B$ from the structure of the local state spaces $\Omega_A$, $\Omega_B$ when considering different kinds of tensor products. More precisely, I wonder whether one can relate the statements that the local states form a polytope and that the composite states form a polytope (are there tensor products such that one statement implies the other?). Are there tensor products such that the composite states form a polytope while the local states do not? Are there tensor products such that the composite states always form a polytope? I am interested in all kinds of arguments that make statements about sets of states being (non-)polytopic when arising from certain kinds of tensor products.</p>
<p>I appreciate any kind of argument or comment, etc.</p>
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<p>This question comes from reading <a href="http://theoreticalphysics.stackexchange.com/questions/278/which-letter-to-use-for-a-cft">Andre's question</a> where I wandered whether that question even makes sense physically. In mathematics, VOAs form a category, does this category as a whole have a physical interpretation? </p>
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<p>The gravitational acceleration on Earth is approximately $ 10 \mathrm{m}/\mathrm{s}^2 $. Compared to this, the tidal effect of the Moon's gravity gives a local variation in the acceleration of approximately $ 9 \cdot 10^{-7} \mathrm{m}/\mathrm{s}^2 $, that is, seven orders of magnitudes less. The level of the water can rise $ 1 \mathrm{m} $ during high tide, which is only three and a half orders of magnitude smaller than the depth of the ocean. </p>
<p>How can this small variation in gravity move so much water and cause such a strong effect on the seas? </p>
| 7,404 |
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<p>The single inpurity Anderson Hamiltonian is
$H=\sum_{\sigma}\epsilon_{\sigma}n_{d,\sigma}+Un_{d,\uparrow}n_{d,\downarrow}+\sum_{k,\sigma}\epsilon_{k}c_{k,\sigma}^{+}c_{k,\sigma}+\sum_{k,\sigma}(V_{k}c_{d,\sigma}^+c_{k,\sigma}+h.c.)$</p>
<p>where $n_{d,\sigma}$ is the occupation number of d electron and $\epsilon_d$ is the energy level of d electron, $U$ is the Hubbard interaction.</p>
<p>I know that there are several regimes of parameters $\epsilon_d, U, \Delta$ ($\Delta$ is the resonance width. I do not know why it is called resonance width and this is actually my question), in which the physics of anderson model is different. </p>
<p>For example, there are intermediate valence regime where $\epsilon_d$ and $\epsilon_d+U$ are comparable with $\Delta$. For another example, when $\epsilon_d-\epsilon_F>>\Delta$ and $\epsilon_d+U-\epsilon_F<<\Delta$, they are called empty orbit regime.</p>
<p>My questions are</p>
<ol>
<li><p>What is the resonance width? What is its physical meaning?</p></li>
<li><p>Why should we compare $\epsilon_d-\epsilon_F$ and $\epsilon_d+U-\epsilon_F$ with $\Delta$, rather than 0, when we distinguish different regimes of Anderson Model?</p></li>
</ol>
<p>Thanks very much!</p>
| 7,405 |
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