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<p>I'm wondering if it exists a composition law for the squeezing operation ? I guess so for geometric reason, since they are (generalized, and the phase is annoying of course) hyperbolic rotations of the annihilation $a$ and creation $a^{\dagger}$ operators of some bosonic modes.</p>
<p>I define the squeezing operator as $$ S\left(\zeta\right) = e^{\Sigma} \;\; ; \;\; \Sigma = \frac{\zeta^{\ast} aa -\zeta a^{\dagger}a^{\dagger}}{2}$$
for any complex parameter $\zeta$.</p>
<p><strong>I would like to know the rule for $S\left(\zeta_{1}\right)\cdot S\left(\zeta_{2}\right)$ ?</strong></p>
<p>For instance, we know that
$$D\left(\alpha\right)\cdot D\left(\beta\right)=D\left(\alpha + \beta\right) e^{\left(\alpha \beta^{\ast}-\beta\alpha^{\ast} \right)/2}$$
for the coherent / displacement operator $D\left(\alpha\right)=e^{\Delta}$ with $\Delta = \alpha^{\ast} a - \alpha a^{\dagger}$. </p>
<p>A reference for $S\left(\zeta_{1}\right)\cdot S\left(\zeta_{2}\right)$ would be sufficient. </p> | 1,308 | [
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<p>As an electric motor spins, the energy from the electricity is 'conducted' to the rotor by the magnetic fields. However, when the motor is stopped, the energy becomes heat and burns up to motor. What causes this heat to be formed? Is it purely generated by the current flowing through the wire or is the magnetic field form the permanent magnets involved somehow? Conversely, why is there less heat when the motor is spinning? I'm hoping this is not just because the motor contains a fan.</p> | 1,309 | [
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<p>Momentum is defined as
$$p = \gamma m_0 v$$
And here is another law
$$E^2=(m_0c^2)^2+(pc)^2$$
And <a href="http://www.pveducation.org/pvcdrom/properties-of-sunlight/energy-of-photon" rel="nofollow">this website</a> says the energy of a red photon is $1.9074 eV$. Also, light has a rest mass of $0$. The problem is that by the first equation implies momentum is $0$, and then the second equation would imply energy is $0$, and $0 \ne 1.9074 eV$. What is wrong?</p> | 1,310 | [
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<p>To my knowledge there are three types of acceleration when a body (e.g. a rod) is moving in a circle about an axis. These are:</p>
<ol>
<li><p>Angular acceleration : this is the rate of change of angular velocity.</p></li>
<li><p>Tangential acceleration : this is the linear acceleration of the system in a tangential direction to the circle and equals the radius times the angular acceleration.</p></li>
<li><p>Radial/centripetal acceleration : this is the linear acceleration of the system that is directed inwards towards the center of the circle.</p></li>
</ol>
<p>I also think there are two types of velocity:</p>
<ol>
<li><p>Linear velocity: this is its velocity in the tangential direction and is constantly changing</p></li>
<li><p>Angular velocity or angular frequency: this is the rate of change of angle.</p></li>
</ol>
<p>Is the above correct? And have I missed anything? </p> | 1,311 | [
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<p>When an EM wave diffracts, I can imagine that its EM field interacts with the charges in a certain obstacle thus inducing a wave behaviour on the charges of the matter that will interact with the EM of the photon.</p>
<p>However, I am having difficulty to make the analogy with matter waves. Since a matter wave is a result of a momentum, which implies kinetic energy, how does a matter wave is created in order to interfere with an electron with, for instance, 200 keV?</p>
<p>Since a single electron can be diffracted, this means that its wave must interfere with other induced matter waves, right? But wouldn't that imply the creation of high kinetic energy particles?</p> | 1,312 | [
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<ol>
<li><p>In electroweak theory, for the first generation of leptons, in the doublet
$$\psi_L=\begin{pmatrix}
\nu_{eL}\\
e_L
\end{pmatrix}$$
we assign a non-abelian charge $I=\frac{1}{2}$. Is this a Noether charge? </p></li>
<li><p>A $SU(2)$ transformation on the Lagrangian:
$$\mathcal{L}^{ferm}=i\bar\psi_R\gamma^\mu D_\mu\psi_R+i\bar\psi_L\gamma^\mu D_\mu\psi_L$$
leads to this conservation of this charge. Right? </p></li>
<li><p>When that corresponding charge operator acts on the one-particle (or multi-particle states) it gives us the corresponding charge. Right? </p></li>
<li><p>The weak hyper charge $Y$ appears in the $U(1)$ transformation: $e^{\beta(x) Y/2}$ and $I$ appears in the $SU(2)$ transformation $e^{i\vec\alpha(x)\cdot\vec I}$? One can define, Noether charge (operator) for both these transformations, for $U(1)$ case it will be related to $Y$ and for $SU(2)$ case it will be related to $I$. But do we have one-particle and many-particle states here? </p></li>
<li><p>On what states will these Noether charges act on?</p></li>
</ol> | 1,313 | [
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<p>Need help doing this simple differentiation.
Consider 4 d Euclidean(or Minkowskian) spacetime.
\begin{equation}
\partial_{\mu}\frac{(a-x)_\mu}{(a-x)^4}= ?
\end{equation}
where $a_\mu$ is a constant vector and the indices are summed over since one really doesn't need to bother about upper and lower indices in flat space.
Also $(a-x)^2=(a-x)_\mu(a-x)_\mu$. A simple minded calculation gives me the result $0$. But I think the answer may contain Dirac Delta function like the relation below in 3 dimension
\begin{equation}
{\bf{\nabla}} . \frac{\hat{r}}{r^2}=4\pi \delta ^3(r)
\end{equation}
A corollary question. What will be $\partial_{\mu}\frac{1}{(a-x)^2}$ ? Is it
$\frac{2(a-x)_\mu}{(a-x)^4}$ or does it also involve the Delta function?</p> | 1,314 | [
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<p>There are many decorative devices, that keep a small metal sphere levitating in air by attracting it to a top electromagnetic coil while monitoring their height by optical sensors.</p>
<p>There is also the 'levitron' where a passive permanent magnet spinning top is hoovering above some permanent magnets, it's orientation is controlled by gyro forces.</p>
<p>And there are bottom based electromagnetic devices, which keep a disk hoovering in air above them by rotating magnetic fields. Like this one: <a href="http://usables24.de/shop/article_GL02.001/Gravity-Lifter-Mini.html" rel="nofollow">http://usables24.de/shop/article_GL02.001/Gravity-Lifter-Mini.html</a></p>
<p>On the first look, it seems simple to exchange the rotation of the top by the rotation of the field. </p>
<p>But: A rotating field must employ the orientation and position control by some electrodynamic means I guess, as the disk is spinning quite slow in these devices, and thus is not stabilized by gyro forces. </p>
<p>What mechanisms are behind this? Maybe the 'levitron' is stabilzed by magnetic forces too, not by the gyro force alone? </p> | 1,315 | [
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<p>So I'm having problems with the double infinite potential well given by</p>
<p>$$V(x)= \left\{\begin{array}{ll}
\infty & -\infty < x < -a-b \\
0 & -a-b< x < -a \\
V_0 & -a < x < a \\
0 & a < x < a + b \\
\infty & a+b < x < \infty \\
\end{array}\right.$$</p>
<p>I have to use the fact that the potential well is symmetric about x=0. I have solved the Schrödinger equation in all the regions and end up with</p>
<p>$$ψ1=A\sin(k_1x)+B\cos(k_1x)$$
$$ψ2=Ce^{k_2x}+De^{−k_2x}$$
$$ψ3=E\sin(k_1x)+F\cos(k_1x)$$
where $k_1=\sqrt{\frac{2mE}{\hbar^2}}$ and $k_2=\sqrt{\frac{2m(V_0-E)}{\hbar^2}}$. I can use the symmetry argument and get C=D which means cosh for even and sinh for odd wave functions. I know that I'm meant to get $A\sin(k_1(a+b−x))$ between the $-a-b$ and $-a$ and between $a+b$ and $a$ but I'm not sure how. If I know how to do that it may help me normalise the wave function which is key at the moment. Many thanks in advance!</p> | 1,316 | [
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<p>I've heard that the big bang began in all places at once, but my physics teacher told my class that the universe was all once compressed into a single dense piece of matter. If my physics teacher is right, then did that dense piece of matter have the same mass as the entire universe now? And if it was so small and yet to dense, wouldn't it technically have been a black hole?</p> | 15 | [
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<p>By fundamental definition of a entangled system we can say that if we know the quantum state of one subsystem then we can describe the state of another subsystem.
A particle possess wave-particle duality. If one experiment verify the wave nature of particle then we can not see its particle behaviour in same exp and vice-versa.
Can we say that wave and particle behaviour are in some sort of entanglement state.</p> | 1,317 | [
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<p>Although there is no standard measure of entanglement, the GHZ states </p>
<p>$|GHZ\rangle=\frac{1}{\sqrt{2}}(|0\rangle^{\otimes n}+|1\rangle^{\otimes n})$</p>
<p>are often deemed as maximally entangled states of $n$ qubits. We know by now that even separable states can exhibit correlations that are impossible in the classical world. This makes me wonder: is there a similar characterization of <em>separable</em> states that exhibit maximum non-classical correlations?</p>
<p>For example, I imagine that states of the form</p>
<p>$\rho=(1/4)(|0\rangle\langle 0|\otimes|+\rangle\langle +|)+(|1\rangle\langle 1|\otimes|-\rangle\langle -|)+(|+\rangle\langle +|\otimes|1\rangle\langle 1|)+(|-\rangle\langle -|\otimes|0\rangle\langle 0|)$</p>
<p>are two-qubit separable states with maximum correlations. Is this true? Is there a general notion for $n$ qubits?</p> | 1,318 | [
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<p>Does the heat of vaporization of water depend strongly on the relative humidity of the gas into which it evaporates? </p>
<p>Some context: If we want to calculate the dew point of water, we find the temperature at which the <em>partial pressure</em> of the water lies on the liquid/vapor boundary of the water phase diagram. This is why water can evaporate from our bodies even though we do not heat it to anywhere near its boiling point.</p>
<p>The heat of vaporization should be pressure dependent (in addition to temperature dependent). Yet, when specifying the heat of vaporization, most references only specify the <em>total</em> ambient temperature, usually 1 atmosphere. Why is the total pressure used in this case instead of the partial pressure? And if the partial pressure is what matters after all, then wouldn't relative humidity be important when calculating heats of vaporization? Of course, relative humidity governs the rate and the total amount of evaporation, which is why we can't cool ourselves by sweating in humid weather, but that's not what my question is about.</p> | 1,319 | [
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<p>I have some knowledge about LC circuits, like I know Maxwell's equations, the differential equations of circuit oscillator and so on. I am armed with equations to solve physical problems involving inductors and their behavior.</p>
<p>But I always need to understand things as they really are. And I can't find anything on the Internet about how the inductor really works.</p>
<p>So the alternating current in the coil gives rise to a magnetic field in the coil and as I read the energy is stored in it in a magnetic field (but magnetic field cannot do work so I guess it is stored only for split second and it just changes to electric field instantly in a way opposing the current that created this effect and treating the magnetism as the relativistic effect of moving charge which in fact is nothing more how there could be energy stored in it?). </p>
<p>But how exactly does it happen?</p>
<p>So we have LC circuit. We put the electrons on one plate of a inductor. The electrons push the electrons on the other side away creating plus charges and the potential difference which drives the current what in fact makes the capacitor plate be more and more charged, when it is fully charged the current stops.</p>
<p>Now we have also the coil in our circuit. And when the capacitor is fully charged for some reason starts to make current in the other direction (i get that it can be reversed when there is emf and we change its direction but i dont get how it happens only with capacitor and coil). As it does so it moves the electrons through the coil creating magnetic field, which creates electric field in which energy is stored which can be then once more changed into charges moving to the capacitor. And then it repeats.</p>
<p>But how an electron on a microscopic level creates this magnetic field inside the coil, how is energy stored in such inductor, how it gets there, is it stolen from kinetic energy of electron it slows down and gives energy to the coil? How the magnetic field then holds the energy or the electric field that happanes just after? In a static electric field it happens because of plus and minus charges which create the field around but how it happens in that case?</p>
<p>I really lack of good explanation of that process and I can't really think of any electron behavior that would create such effects. Please explain it to me or point me to some readings so I can finally get how it works.</p> | 1,320 | [
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<p>We want to find the energy of a hydrogen atom ($Z=1$) in the ground state
$$
\psi_{100} = \frac{1}{\sqrt{\pi}}e^{-r}\ \ \ \ \ \ (\mbox{atomic units})
$$
with Hamiltonian
$$
H = -\frac{1}{2}\nabla^2-\frac{1}{r}
$$</p>
<p>Then</p>
<p>$$
\begin{align*}
\langle \psi_{100}|H|\psi_{100}\rangle &=\int_0^\infty \int_0^{2\pi} \int_0^\pi r^2\sin\theta\frac{1}{\sqrt{\pi}}e^{-r}\left(-\frac{1}{2}\frac{d^2}{dr^2}-\frac{1}{r}\right)\frac{1}{\sqrt{\pi}}e^{-r} d\theta d\phi dr \\
&=\int_0^\pi \sin\theta d\theta \int_0^{2\pi}d\phi \int_0^\infty \frac{r^2e^{-r}}{\sqrt{\pi}}\left(-\frac{1}{2\sqrt{\pi}}e^{-r}-\frac{1}{r\sqrt{\pi}}e^{-r}\right)dr \\
&= 4\pi \cdot \frac{1}{\pi}\int_0^\infty \left(-\frac{r^2}{2}e^{-2r}-re^{-2r}\right)dr \\
&= 4\left(-\frac{3}{8}\right) \\
&= -\frac{3}{2}
\end{align*}
$$</p>
<p>However, I've read everywhere that $E = -\frac{Z^2}{2n^2}$, and so for a hydrogen atom in the ground state we should have $E=-\frac{1}{2}$. So why am I getting $-\frac{3}{2}$? I've double-checked with Mathematica. </p> | 1,321 | [
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] |
<p>Some physical properties of water change in the presence of solutes: vapor pressure, boiling point, freezing point and osmotic pressure. In particular, these four properties are called <a href="http://en.wikipedia.org/wiki/Colligative_properties" rel="nofollow"><em>colligative properties</em></a> because they depend only on the ratio of concentrations of solute and solvent, and not on the nature of the solute.</p>
<p>I am interested in the viscosity of water. I have two questions:</p>
<ol>
<li><p>Does the viscosity of water change in the presence of solutes? </p></li>
<li><p>Does the change depend on the nature of the solute? On the concentration of the solute? How?</p></li>
</ol>
<p>Of course, if the answer to the first question is in the negative, the second question is nullified.</p> | 1,322 | [
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<p>I´m wondering how can one get to the definition of Functional Derivative found on most Quantum Field Theory books:</p>
<p>$$\frac{\delta F[f(x)]}{\delta f(y) } = \lim_{\epsilon \rightarrow 0} \frac{F[f(x)+\epsilon \delta(x-y)]-F[f(x)]}{\epsilon}$$</p>
<p>from the definitions of Functional Derivatives used by mathematicians (I´ve seen many claims that it is, in effect, the Fréchet derivative, but no proofs). The <a href="http://en.wikipedia.org/wiki/Functional_derivative" rel="nofollow">Wikipedia article</a> says it´s just a matter of using the delta function as “test function” but then goes on to say that it is nonsense.</p>
<p>Where does this $\delta(x-y)$ comes from?</p> | 1,323 | [
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<p>Many documentaries regarding the double slit experiment state that they only send a single photon through the slit. How is that achieved and can it really be ensured that it is a single photon?</p> | 1,324 | [
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<p>So, I have a camera crane that has a pivot leaving 2 distances, 0.5mts and 1.5mts respectively. Plus a small L-shaped piece of metal and the end of d2 (1.5) which holds a camera, which I suppose I have to add it to the force applied on that side. One additional variable is that there's a piece of metal attached to the fulcrum and then connected to the second distance (d2, 1.5) at the end, which leaves me the doubt of where to add that weight (which is 580grs, let's say d2*) to the force, or add it as another variable in the equation. The current equation that i'm using (which was without the second piece of weight on distance 2) is:
$$
w = \dfrac{f * d2}{d1}
$$</p>
<p><img src="http://i.stack.imgur.com/1P35v.png" alt="Lever example"></p>
<p>So i'm dubious if this is the right equation taking in count the variables I have explained above. Thanks!</p> | 1,325 | [
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<p>When considering tabulated standard values of heat of combustion (∆H° combustion) or any other similar value, the results are reported for species in their standard states (1 bar pressure, 25°C, etc). When something combusts, however, macro state variables such as pressure and temperature change. For the tabulated values, are the conditions kept at constant temperature and pressure for the duration of the reaction? Or is the reaction started at standard conditions and the temp/pressure/etc allowed to fluctuate as the system reaches equilibrium?</p> | 1,326 | [
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] |
<p>This question is regarding <a href="http://en.wikipedia.org/wiki/Noether%27s_theorem" rel="nofollow">Noether's Theorem</a> in general, but also in the application to an example. The example is:</p>
<blockquote>
<p><em>Find the conserved current for the Lagrangian</em>
$$L=\bar{\psi}(\frac{i}{2}\gamma^{\mu}\partial_{\mu}-m)\psi.$$</p>
</blockquote>
<p>Is my first step to find a transformation that leaves the action invariant? But isn't there more than one symmetry meaning this question has several answers?</p>
<p>From Peskin and Schroeder, it says we need to find</p>
<p>$$\partial_{\mu}j^{\mu}(x)=0 \; \text{ for } \; j^{\mu}(x)=\frac{\partial L}{\partial (\partial_{\mu}\psi)}\Delta\psi-J^{\mu}.$$</p>
<p>So I think I'm right in saying this $J^{\mu}$ is dependent on the transformation we're making? e.g. if it's just $\psi \rightarrow \psi +a$ then $J^{\mu}=0$. </p>
<p>The Lagrangian changes under the transformation of the field as</p>
<p>$$L \rightarrow L+\alpha \partial_{\mu}J^{\mu}.$$</p>
<p>But this doesn't help me because plugging $\psi \rightarrow \psi +\alpha \Delta \psi$ (an arbitrary transformation of the field $\psi$) into the Lagrangian at the top gives me (after no more than one line)</p>
<p>$$L \rightarrow L+\alpha \bar{\psi}(\frac{i}{2}\gamma^{\mu}\partial_{\mu}-m)\Delta \psi$$</p>
<p>and how do we use this to find $J^{\mu}$?</p>
<p>(Something I've assumed throughout all of this is that $\psi$ and $\bar{\psi}$ are treated completely separate and here we're only considering $\psi$. I hope I am right in doing this.)</p>
<p>I haven't really pin pointed a question here as my understanding breaks down at many points and when I think I finally understand it, I'm given a new Lagrangian and become stuck again. There must be some general procedure?</p> | 1,327 | [
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<p>A tank has 2000 litres of capacity (Ct). It fills at 5 l/s. It is filled with 800 l at time 0.</p>
<p>a) How many seconds will it need to fill half the tank.</p>
<p>So I did this:</p>
<p>X = time.</p>
<pre><code>5 l * X 2000L
---- = ------- - 800L
s 2
X = 200L
---------
5 L
---
s
X = 200
-------
5
---------
s
</code></pre>
<p>Eventually, I get to <code>X = 40 / s</code>, but I guess it should be <code>X = 40s</code>.</p>
<p>However, I fail to see what I should do to get the seconds multiplying the 40.</p>
<p>PS: Shouldnt I give the X a unit at the beggining, or is it ok to be just X?</p> | 1,328 | [
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<p>I'm doing some numerical Monte Carlo analysis on the 2 dimensional Ising model at the critical point. I was using the Metropolis 'single flip' evolution at first with success, though it suffers from critical slow down and makes studying large lattices unlikely possible. I'm now looking at cluster flip algorithms, specifically the Wolff algorithm.</p>
<p>I managed to implement it, and it looks to be working as it should (flipping a unique spin at $T = +\infty$, the whole lattice at $T=0$, matches the right energy density in the thermodynamic limit...) but I don't get the right behaviour for the two point $<\sigma_i\sigma_j>$ correlation function.</p>
<p>According to CFT it should behave like:</p>
<p>$$<\sigma_i\sigma_j> \;\propto \;\frac{1}{|i-j|^{\frac{1}{4}}}$$</p>
<p>I'm more and more convinced that it has to do with boundary conditions, I use non periodic free boundaries. The literature on the subject doesn't say much on this point.</p>
<p>Am I missing a subtlety (or an evidence) in this procedure, or in the use of this algorithm?</p> | 1,329 | [
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<p>I have a pulse profile (binned photon counts versus phase) of a star, and for each count rate I have its statistical error.</p>
<p>I want to calculate the so-called pulsed-fraction $P_{frac}=\frac{F_{max}-F_{min}}{F_{max}+F{min}}$, where $F_{max}$ and $F_{min}$ are the maximum and minimum count rates, respectively.</p>
<p>Is it possible to do propagation error to find the error for $P_{frac}$? If yes, how?</p> | 1,330 | [
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<ol>
<li><p>Can physical states be treated as information (strings over some alphabet)?</p></li>
<li><p>If (1) is true, isn't this a trivial conclusion that the universe can be simulated by a Turing machine or a cellular automaton or any other computational model, given that differential equations that determine the evolution of physical states (the bits in digital physics) are simulated by the transition function of the Turing machine?</p></li>
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<p>I have a basic enough question. Assume that one has one of those ideal see saws, i.e. the teeter-totter pivots on point source, the balance is of uniform mass etc.</p>
<p>Now assume that one places an object of mass m on one end. This will force the balance to tilt on side. Now suppose that one adds another mass of m on the other side. What happens? It seems that everyone agrees that the the balance would move to the stable equilibrium, that is the middle. What I don't understand is, what is the Force that moves the teeter-totter to the middle position?</p>
<p>Once the second mass has been placed on the other side, the two net forces acting on the pivot is the same. And since the distance is the same, the torque is the same. So what force is it that moves the balance to the middle? </p>
<p>And if no such force exists then every position of the balance is valid if the two weights are equal, then how do those mass balances work?</p> | 1,332 | [
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<p>Can somebody explain to me, when the following two equations (equations 2.48 and 2.50 <a href="http://physics.usask.ca/~hirose/p812/notes/Ch2.pdf" rel="nofollow">in this document</a>) are applicable and what $\Phi_s$ and $\Phi$ actually are? The thing is, I want to find general equations that determine the field produced by conducting spherical sphere in an external field and was wondering whether these are the equations I am looking for.</p>
<p>$$\Phi (r,\theta,\phi)=\sum_{l,m}\left(\frac{a}{r} \right )^{l+1}Y_{lm}(\theta,\phi)\oint\Phi_{s}(\theta',\phi')Y^*_{lm}(\theta',\phi')d\Omega',\,\,\,\,r>a$$</p>
<p>$$\Phi (r,\theta,\phi)=\sum_{l,m}\left(\frac{r}{a} \right )^{l}Y_{lm}(\theta,\phi)\oint\Phi_{s}(\theta',\phi')Y^*_{lm}(\theta',\phi')d\Omega',\,\,\,\,r<a$$</p>
<p>Or is it rather this equation (equation (17.3) in <a href="http://www.physics.udel.edu/~jim/PHYS809_10F/Class_Notes/Class_17.pdf" rel="nofollow">this document</a>), probably they are one and the same:</p>
<p>$$\Phi (\mathbf{x})=-\frac{1}{4\pi}\int_S \Phi(\mathbf{x}')(\hat n.\nabla')G_D(\mathbf{x},\mathbf{x}')dS'$$</p> | 1,333 | [
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<p>I am attempting to model the impact of a deformable body (clean, dry sand characterized as a continuous fluid) with flexible diaphragm (in this case a sheet of rubber). Assuming that this is a closed system (no sand escapes) do you think work-energy theorem is adequate for approximating the total force imparted on the fluid by the diaphragm? Disregarding heat loss, can I take the KE of the flow (assuming uniform flow), look at the total deflection of the barrier and get an approximation of the force? I guess this comes down to my lack of knowledge about work. I know the force on the flow from the barrier increases linearly with deflection, but does that matter if I am only concerned about the total force required to take the flow from it's initial velocity to rest?</p>
<p>Sorry for the long winded description. Any guidance would be appreciated.</p> | 1,334 | [
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<p>I've bumped into the study of the $SU(N)$ theory in the <a href="http://en.wikipedia.org/wiki/1/N_expansion" rel="nofollow">large-$N$ limit</a>.
I'm wondering in which way the study of this Yang-Mills theory, can give contribution to QCD with gauge group $SU(3)$, i.e. $N=3$ Any suggestion?</p> | 1,335 | [
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<p>The oil, vinegar and other liquids in homemade salad dressing separate into layers after sitting for a while, making the mixture become more organized as time evolves. Why doesn't this violate the 2nd law of thermodynamics?</p>
<p>I assume that the answer is that since the separation is due to gravity, the effect is due to an external force and so the system isn't closed which is necessary for the 2nd law, but I'm not sure.</p>
<p>If that's the answer, then what happens if I consider the whole system include the salad dressing, gravitational field and whatever mass is generating the gravitational field? The entropy of the salad dressing seems to decrease and so the entropy of some other component of this system must be increasing, but it's hard to see what this other component would be.</p> | 1,336 | [
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<p>In the context of <a href="http://en.wikipedia.org/wiki/Inflation_%28cosmology%29" rel="nofollow">Inflationary Cosmology</a>, it is postulated that there was a period of shrinking <a href="http://en.wikipedia.org/wiki/Hubble_radius" rel="nofollow">Hubble Sphere</a> radius $(aH)^{-1}$.</p>
<p>$$ \frac{d}{dt} (aH)^{-1} < 0 $$</p>
<p>Then the regions of the universe which appear to us to not be in causal contact can indeed have been in causal contact at some (conformal? physical? I'm not sure. My intuition would be physical) time in the past.</p>
<p>This solves the <a href="http://en.wikipedia.org/wiki/Horizon_problem" rel="nofollow">Horizon Problem</a>, namely that we observe a homogenous isotropic universe to one part in 10,000, yet it doesn't seem that all regions of the universe were in casual contact to achieve such a uniformity.</p>
<p>$\\$</p>
<p>Anyway, when this shrinking Hubble sphere radius $(aH)^{-1}$ is invoked, it has the result of pushing the Hot Big Bang singularity back to negative infinity conformal time.</p>
<p>It is this statement that I am having trouble grasping.</p>
<p>$\\$</p>
<p>Explicitly</p>
<p>$$ \chi_{PH}(\tau) = \tau - \tau_i = \int_{t_i}^{t} \frac{1}{a(t)} dt = \int_{a_i}^a (a H)^{-1} d\log(a) $$</p>
<p>where by Friedmann</p>
<p>$$ (aH)^{-1} \propto a^{\frac{1+3w}{2}} $$</p>
<p>giving</p>
<p>$$ \chi_{PH}(a(\tau)) = \tau - \tau_i \propto \frac{2}{1+3w} \left[ a(\tau)^{\frac{2}{1+3w}} - a_i^{\frac{2}{1+3w} } \right] $$ </p>
<p>So for a strong energy violating fluid such that</p>
<p>$$ 1 + 3w < 0 $$</p>
<p>this becomes</p>
<p>$$ \tau_i \propto \frac{2}{1+3w} a_i^{\frac{2}{1+3w}} \rightarrow -\infty $$</p>
<p>$\\$</p>
<p>I understand the maths behind it, and can't argue against that the initial conformal time goes to minus infitity, but I don't know how to think about that.</p>
<p>In a standard Hot Big Bang the singularity occurs at $\tau_i = 0$ conformal time. This seems fine to me.</p>
<p>I am aware that conformal time is unphysical, but unless the scale factor goes negative, then negative conformal time suggests to me negative physical time.</p>
<p>And a negative scale factor would bring a whole other conceptual problem!</p>
<p>Namely, they are directly related by</p>
<p>$$ d\tau = \frac{1}{a(t)} dt $$</p>
<p>$\\$</p>
<p>So my questions are;</p>
<p>What does it mean to have negative conformal time?</p>
<p>And what does it mean to have negative infinity conformal time?</p>
<p>$\\$</p>
<p>Thanks</p> | 1,337 | [
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<p>Is it possible to consider the regular polygons ($n$-gons) as deformed circles and use a <a href="http://www.google.com/#q=pseudo-polar+coordinate+system" rel="nofollow">pseudo-polar coordinate system</a> to calculate their moment of inertia over its center $O$. Inasmuch as I know (I am a solid mechanics beginner), we can find the area moment of inertia $J_O$ of the circle $C(O,R)$ considering a circular disk of thickness $dr$. </p>
<p>Can I apply the same to regular polygons, namely if their area is equal to: </p>
<p>$$A = \frac{nr^2}{2 \sin{(2 \pi / n})}$$ </p>
<p>then the differential area element $dA$ will be equal to the area of the outer polygon with radius $r+dr$ minus the area of the inside one of radius $r$:</p>
<p>$$dA = A(r+dr) - A(r) = \frac{n}{2 \sin{(2\pi/n})}dr (2r+dr)$$</p>
<p>so the moment of inertia at the center O of the solid polygon can be calculated as:</p>
<p>$$J_O = \int^R_0 r^2 dA = \int^R_0 r^2 \frac{n}{2 \sin{(2\pi/n})}dr (2r+dr)$$</p>
<p>where $R$ is the radius of the polygon (the distance between its center and its vertices).</p>
<p>Considering the above $J_O$ tends to be equal to the one of the perfect circle:</p>
<p>$$\lim_{n \to +\infty} J_O = \pi r^4 / 4$$</p>
<p>Are my claims right?</p> | 1,338 | [
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<p>In the US HEPA filters will remove 99.97% of particles sized 0.3 microns. In Europe HEPA filters do not have to meet this exact figure and there are different classes of HEPA filters e.g. h10(removes 85% at .3 microns, h11 removes 95% at .3 microns etc, h14 removes 99.97% etc.). </p>
<p>Suppose you have a vacuum cleaner that is h10(removes 85% of particles at .3 microns), when vacuuming 15% of .3 microns will pass through the filter. If we continue to vacuum the area these will go into the vacuum again and be filtered again. Therefore even though the filter performs less, is it correct to say eventually(with continued vacuuming) the filter will remove the remaining 15% so its not a big deal if you get one with 85% or 99.9% filtration so long as you are bothered to continue vacuuming?</p>
<p>Thanks</p> | 1,339 | [
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<p>In <a href="http://iopscience.iop.org/1742-5468/2005/05/P05012" rel="nofollow">Spin-glass theory for pedestrians</a> by Castellani and Cavagna, the initial formula used to introduce the replica trick is written as:</p>
<p>$$\overline{\log Z}=\lim_{n\rightarrow0}\frac{1}{n}\log\overline{Z^{n}}\qquad(1)$$</p>
<p>where the overbar denotes average over quenched disorder. I don't know how to prove this formula.</p>
<p>In other treatments I have seen of the replica method (<a href="http://en.wikipedia.org/wiki/Replica_trick" rel="nofollow">wiki</a>, for example), one starts from:</p>
<p>$$\log Z = \lim_{n\rightarrow 0} \frac{Z^n-1}{n}\qquad(2)$$</p>
<p>which I understand. How are (2) and (1) connected? What's the proof of (1)?</p> | 1,340 | [
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<p>I have an electric field and a certain charged particle in it that has a certain potential energy associated with it. Where does the energy go if I remove the field?</p> | 1,341 | [
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<p>It is well known that the orbits of Ganymede, Europa and Io are in a 4:2:1 resonance. Most online sources (including but not limited to <a href="http://en.wikipedia.org/wiki/Galilean_moons" rel="nofollow">Wikipedia</a>) say that such an <a href="http://en.wikipedia.org/wiki/Orbital_resonance" rel="nofollow">orbital resonance</a>, along with the 3:2 resonance, is "stable and self-correcting", but fail to explain why this is so.</p>
<p>The textbook <em>Fundamental Astronomy</em> says that this phenomenon is due to "tidal forces" but does not elaborate further. I presume it refers to the tidal deceleration which causes the orbits of the moons to evolve outwards, which, other sources say, caused the moons to eventually enter into resonance, but this also does not explain why the resonance is stable.</p>
<p>I am aware of a similar question <a href="http://physics.stackexchange.com/q/90517/">here</a> but I'm more interested in stability rather than instability in this case.</p>
<p>In short, why is the orbital resonance of the Galilean moons stable, and how is it different from other cases of orbital resonance that are unstable? I don't mind (and would prefer) if the answer is mathematical in nature.</p> | 1,342 | [
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<p>How can we know that a black hole exists if it attracts both matter and light, and to prove that it exists would need to observe it?</p>
<p>It's a very common question but I don't know the answer.</p> | 16 | [
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<p>A fluid in a rotating bucket will take on a parabolic shape (for example of some simple derivations of this result see <a href="http://en.wikipedia.org/wiki/Bucket_argument" rel="nofollow">http://en.wikipedia.org/wiki/Bucket_argument</a>). The assumptions that play into the derivations that I've seen do not take into consideration a Coriolis effect. Suppose one did not know that the earth was spinning. Is it possible to measure the angular frequency of the earth using a bucket of fluid by considering how the Coriolis effect modified the curvature of the paraboloid in question? </p>
<p>Perhaps it's possible to optically measure the curvature of the paraboloid setup by the spinning process and from that determine the angular frequency of the planets rotation. I am concerned though that the effect of the rotation of the earth is so minuscule on the parabolic shape that it would be impossible to physically measure it. </p> | 1,343 | [
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<p>Why would the blood boil? Is it because of the temperature or pressure? Because I really can't figure it out. I thought space didn't have a temperature above freezing unless close to a star or the sun. But if it is the pressure how would that work?</p> | 1,344 | [
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<p>If you drop a glass cup on the ground, it will break and shatter into pieces. This happens all the time and is consistent with quantum mechanics. But it never happens that a shattered glass cup rearranges itself from the ground into someone's hand as a whole glass cup, even though this is also consistent with quantum mechanics. We see from this example that not everything that is consistent with quantum mechanics is possible.</p>
<p>As far as I know, scalable quantum computing has never been demonstrated either backwards in time or forwards in time. So <em>a fortiori</em>, I would think that this would be good enough evidence to suggest that scalable quantum computing is impossible. Yet, some physicists believe that scalable quantum computing is still possible. Why? </p> | 611 | [
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<p>I have a question that's been bothering me for years.</p>
<p>Given a rod of uniform mass distribution with total mass $M$ and length $L$ that lies on a horizontal table (with one end fixed to the table around which the rod is free to rotate in the horizontal plane, and a force F applied perpendicular to the rod at the other end), how do you solve for the motion of the rod (and the internal forces) using only Newton's laws and the assumption that the rod is a rotating rigid body? By that I mean only using the most basic conception of Newton's laws and the system's constraints, without the ideas of torque and moment of inertia, energy and momentum, and even without the idea that the net force on the rod gives the acceleration of the center of mass--so only using Newton's laws for point particles, or in this case infinitesimal $dm$ sections of the rod.</p>
<p>I've tried to solve this by breaking the rod down into these small $dm$ components and using an idea I've seen (at least I think I've seen) where you set $F(x+dx)-F(x)=dm(a)$ and then are able to find $F'(x)$ and integrate and then use the boundary conditions on the force. I did this for both tangential and radial components, with radial acceleration equal to $x(\omega(t))^2$ and tangential acceleration equal to $\omega\ '(t)x$, but was unable to obtain the right answer. I used the force at one end of the rod as a boundary condition (is this correct?), but was unable to even solve for the force at the pivot, let alone the angular velocity as a function of time, and have no idea if this technique is even valid. I feel like at a certain point it might be that my force equation switches sign--as the net force that accelerates the infinitesimal mass starts coming from the inward instead of outward side.</p>
<p>I'd also be interested to know more generally how to solve for internal forces and motion of a rigid body using only these most basic assumptions, such as for a free uniform square on a horizontal table with a force applied perpendicularly to one side into one corner.</p> | 1,345 | [
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<p>Is the Laplacian operator, $\nabla^{2}$, a <a href="http://en.wikipedia.org/wiki/Self-adjoint_operator" rel="nofollow">Hermitian operator</a>? </p>
<p>Alternatively: is the matrix representation of the Laplacian Hermitian?</p>
<p>i.e.</p>
<p>$$\langle \nabla^{2} x | y \rangle = \langle x | \nabla^{2} y \rangle$$</p>
<p>I believe that $\nabla^{2}$ is Hermitian (if it was not, then the Hamiltonian in the time-independent Schroedinger equation would not be Hermitian), but I do not know how one would demonstrate that this is the case.</p>
<p>More broadly, how would one determine whether a general operator is Hermitian?
One could calculate every element in a matrix representation of the operator to see whether the matrix is equal to it's conjugate transpose, but this would neither efficient or general. </p>
<p>It is my understanding that Hermiticity is a property that does not depend on the matrix representation of the operator. I feel that there should be a general way to test the Hermiticity of an operator without evaluating matrix elements in a particular matrix representation.</p>
<p>Apologies if this question is poorly posed. I am not sure if I need to be more specific with the definitions of "Hermitian" and "Laplacian". Feel free to request clarification.</p> | 1,346 | [
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<p>Say you have two white noise signals with different variation amplitudes A1 and A2 as shown in this beautiful Excel graph:</p>
<p><img src="http://i.stack.imgur.com/6ASjL.png" alt="White noise signals" title="White noise"></p>
<p>Ignoring the DC offset as it's been represented here, how do you relate the amplitudes A1 and A2 to the magnitude of the Fourier coefficients after a Fourier transform (as shown in the diagram below)?</p>
<p><img src="http://i.stack.imgur.com/VeXgm.png" alt="Fourier transforms of white noise signals" title="Fourier transforms"></p>
<p>In other words, is it possible to relate A1 to Mag1 and A2 to Mag2? Can this even be done analytically or will it require a bit of simulation?</p>
<p>Any help is much appreciated! </p> | 1,347 | [
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<p>For the oscillation of a torsion pendulum (a mechanical motion), the time period is given by<br>
$T=2\pi\sqrt{\frac{I}{C}}$ which is a result of the angular acceleration $\alpha=\frac{d^2\theta}{dt^2}=-(\frac{C}{I})\theta$ where $C$ is the restoring <a href="http://en.wikipedia.org/wiki/Couple_%28mechanics%29" rel="nofollow">couple</a> of the string.</p>
<p>Do we relate $T=\frac{2\pi}{\omega}$ and $\alpha$ for finding the time period of torsion pendulum? I can't really understand how they arrived at $2\pi\sqrt{\frac{I}{C}}$ ?</p> | 1,348 | [
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<p>I'm dealing with a problem here and even that I'm trying to solve it i can't</p>
<p>It says:</p>
<blockquote>
<p><em>In what figures the voltmeter and ampere-meter are wrong positioned?</em></p>
</blockquote>
<p><img src="http://i.stack.imgur.com/fKKPJ.jpg" alt="enter image description here"></p>
<p>I think that all the the others are correct except the second one.</p>
<p>Can anyone help me?</p> | 1,349 | [
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<p>I've got to calculate the thermionic emission through a diode, so I need to use <a href="http://en.wikipedia.org/wiki/Thermionic_emission#Richardson.27s_Law" rel="nofollow">Richardson's Law</a>. However, one thing's got me confused - according to the Wikipedia page:</p>
<p>$$J = A_GT^2e^\frac{-W}{kt}$$</p>
<p>I could live with that, but for $A_G$. Apparently, I'm not the only one; Wikipedia's a bit cryptic about what $A_G$ is, mentioning that physicists have struggled with it for decades, "but there is agreement that $A_G$ must be written in the form:"</p>
<p>$$A_G = \lambda_RA_0$$</p>
<p>"Where $\lambda_R$ is a material-specific correction factor that is typically of order 0.5, and $A_0$ is a universal constant."</p>
<p>That's the last real mention of $\lambda_R$, and the only reference for it is in French.</p>
<p>So, what is $\lambda_R$, really? How can I figure out its numerical value so that I can actually use Richardson's Law?</p> | 1,350 | [
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<p>To me, adiabatic processes are idealisation. What do people mean with statements such as: "turning off the coupling constant (in QED say) adiabatically"?</p> | 1,351 | [
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<p>I have often heard it said (by professional cosmologists) that if the universe is infinite, then there necessarily exist infinitely many copies of me repeated throughout.</p>
<p>The reasoning seems to be that any finite volume of space can contain at most finitely many fundamental particles existing in finitely many configurations. From here it is implied that all possible (permissible) configurations must both exist and occur infinitely often.</p>
<p>This does not sound like a sound argument to me, so I have to believe that I have misunderstood the argument. </p>
<p>Is this unsettling implication true in an infinite universe, and if so how is it justified.</p> | 1,352 | [
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<p>Consider a rollercoaster that goes down a slope:</p>
<p><img src="http://i.stack.imgur.com/94VBI.png" alt="rollercoaster"></p>
<p>At the higher level it has speed $v_0$, then it goes down a slope and at the end it has speed $v_0 + \Delta v$. The carriage is not powered and has negligible friction. </p>
<p>So by conservation of energy we should, after eliminating mass $m$ that appears in all terms, have</p>
<p>$$
\frac{1}{2}v_0^2 + gh = \frac{1}{2}(v_0 + \Delta v)^2 \\
\frac{1}{2}v_0^2 + gh = \frac{1}{2}v_0^2 + v_0\Delta v + \frac{1}{2}\Delta v^2 \\
gh = \frac{1}{2}\Delta v^2 + v_0\Delta v
$$</p>
<p>(left side is energy at the beginning, right side energy at the end).</p>
<p>But what if we consider the same situation in a reference frame moving left with speed $v_0$. Then we should get just</p>
<p>$$ gh = \frac{1}{2}\Delta v^2 $$</p>
<p>These differ by a $v_0\Delta v$ term, so solving for $\Delta v$ gives different result. But obviously the speeds should be the same. So where is the error?</p> | 1,353 | [
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<p>I am confronted with the following scaling equation (in Laplace domain if that matters):</p>
<p>$\tilde\psi(s) = a \tilde\psi(s/b)$</p>
<p>I know the answer is such that $\tilde\psi(s) \sim s^{\beta} K(s)$ where $\beta = \ln a/\ln b$ and I don't really care what K is. I'm not sure if the functions $\tilde\psi(s)$ are need to solve this but I do know them if they are necessary.
How is this type of equation solved?</p> | 1,354 | [
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<p>M. Mezard, G. Parisi and coworkers have written about replica symmetry and its breaking in spin glasses, structural glasses, and hard computational problems.</p>
<p>I am just getting acquainted with this literature. Where are the best places to start understanding replica symmetry?</p>
<p>Can anyone here explain the concept?</p> | 1,355 | [
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<p>I have searched for an answer to this question on physics SE but I have not seen a question in which it is addressed properly. Please let me know if there is an answer already.</p>
<p>My question briefly is, is the uncertainty principle a technical difficulty in measurement OR is it an intrinsic concept in Quantum Mechanics irrelevant of any measurement??</p>
<p>Everyone knows the thought experiment of measuring the position of an electron. One can detect electron's position by hitting it with a photon, due to Compton scattering the collision of the photon with electron will change electron's momentum. This experiment is used to explain uncertainty principle to layman, but it is over simplified, isn't it? It also gives an impression that if there was a better suited experimental method the uncertainty principle becomes irrelevant. </p>
<p>I personally think it is intrinsic as it arises from the non-zero commutator of position and momentum operators irrespective of the measurement process. Am I right?</p>
<p><strong>EDIT:</strong> My question is similar to certain extent to <a href="http://physics.stackexchange.com/questions/54184/is-the-uncertainty-principle-just-saying-something-about-what-an-observer-can-kn">this question</a> and <a href="http://physics.stackexchange.com/questions/24068/isnt-the-uncertainty-principle-just-non-fundamental-limitations-in-our-current">this question</a>. The answers there are nice but they focus on explaining basics of quantum mechanics more than they comment on the technical difficulty part. In the answers of question 2 there are statements like "So, it's not a knowledge limit" and "you're sort of correct when you say it's an observational limit" without further comments</p>
<p>To summarize, assume hypothetically we managed to find a way in the future where we can have a look at an electron without disturbing it by measurement or causing its wave function to collapse, would the uncertainty principle still hold in such a case?? Why/Why not? </p> | 1,356 | [
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<p>Mass and Energy can warp space-time around them, but that doesn't answer what space-time is, what is space?</p> | 1,357 | [
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<p>I'm doing experiments that require using a big, thin, contrast grid, for example 1 black pixel and then 1 white pixel alternately. </p>
<p>I've met with a few DLP projectors and in each of them I had to manually adjust <strong>clock adjustment/phase adjustment.</strong> If I didn't do so, image was flickering or halos were present around contours.</p>
<p>I have read few manuals looking for description of this setting, but none of them explained why it happens. </p>
<p>Computer and projector are connected through VGA, and of course, the resolution is set properly to meet the projector's.</p>
<p>Do you think you have an explanation for this?</p>
<p>It's exactly what happens here, behind the grid: <a href="http://www.lagom.nl/lcd-test/clock_phase.php#clockcalib" rel="nofollow">http://www.lagom.nl/lcd-test/clock_phase.php#clockcalib</a>
But why?</p> | 1,358 | [
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<p>The problem that got me thinking goes like this:- </p>
<blockquote>
<p><em>Find $dp/dx$ where $p$ is the probability of finding a body at a random instant of time undergoing linear shm according to $x=a\sin(\omega t)$. Plot the probability versus displacement graph. $x$=Displacement from mean.</em> </p>
</blockquote>
<p>My work: </p>
<p>$$v=dx/dt=\omega \sqrt{a^2-x^2}$$ </p>
<p>Probability of finding within $x$ and $x+dx$ is $dt/T$ where dt is the time it spends there and T$$ is the total period.<br>
Therefore </p>
<p>$$dp=dt/T=\frac{dx}{\pi \sqrt{a^2-x^2}}$$</p>
<p>because $t=2\pi /\omega$ and the factor 2 is to account for the fact that it spends time twice in one oscillation. The answer matches the answer and also the condition that integration $-a$ to $a$ of $dp =1$.</p>
<p>But when i try to find p as a function of x to plot the graph I get </p>
<p>$$p=\frac{1}{\pi}\arcsin(x/a)+C.$$ </p>
<p>But then I get stuck as there is no way to find $C$ (except the fact that for $C=0$ the probability at the mean position is $0$ and hence $C$ cannot equal 0) which I know of. So how can I get a restraint on $C$ to find its value and hence to properly graph it with the condition that the probability from $-a$ to $a$ be 1?</p> | 1,359 | [
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<p>I am confused on <a href="http://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation" rel="nofollow">bra-ket notation</a> in quantum mechanics. My professor says that a ket is an eigenfunction of some operator. However, for some time now I thought a ket could represent a general wavefunction. Obviously any wavefunction should be able to be expressed as a linear combination of eigenfunctions (and thus a linear combination of kets based on my professor's definition), but I've always thought that a ket could represent any wavefunction, not just an eigenfunction. Which of these two viewpoints is right, or, if they are both right, how are they both right?</p> | 1,360 | [
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<p>So I am dealing with the following hamiltonian, and the following perturbation:
$$H=-\mu B_0\sigma_z$$
$$V=\mu B_1(\cos(\omega t)\hat x-\sin(\omega t)\hat y)\cdot{\bf \sigma}$$
I am asked for the probability that the spin will go from ground state at time 0 to the excited state at time t using first order time dependent perturbation theory.</p>
<p>I know that the ground state is $\begin{pmatrix}1\\0 \end{pmatrix}$ with energy $-\mu B_0$. This means the excited state is $\begin{pmatrix}0\\1 \end{pmatrix}$ with energy$+\mu B_0$. So the probability I want is:</p>
<p>$$P=|(0 1)U^{(1)}(t,0)\begin{pmatrix}0\\1\end{pmatrix}|^2$$
Where I have
$$U^{(1)}=\frac{1}{i\hbar}\int^t_0 dt_1e^{-iH_f(t-t_1)/\hbar} V(t_1) e^{-iH_it_1/\hbar}$$
So I have 2 questions. The first is if this is the correct expression I want to use for finding the probability. And secondly, how do I treat the matrices? Seeing as they are a constant, can I just pull them out and then integrate the remaining pieces?</p> | 1,361 | [
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<p>I don't fully understand what would happen if we could travel at the speed of light. But I saw somewhere here that it would mean events happen out of order. But why is this a problem. It is said that cause has to happen before the effect, but why does this have to be linear? </p>
<p>And why is it that the speed of light is the maximum? I take it that light (photons?) are massless, so why can't they travel faster?</p> | 137 | [
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<p>I'm a computer scientist that likes to read about math and physics occasionally. A local author at a nearby aviation center brought bernoulli's flight equations into question.</p>
<p>It was clear enough logic, but I didn't understand all the math involved. He basically said that the lift equations don't account for why a plane can have lift while it's still upside down and took some sample data on his small aircraft to show.</p>
<p>Anyhow, I didn't know what to make of it, my only point of view and my first acquaintance with flight physics.</p>
<p>How much effect does Bernoulli's equations play in flight, especially when an aircraft is <strong>upside down</strong>?</p>
<p>He suggested that <strong>Newton</strong> did some early work on lift equations, is this true?</p>
<p>What is a <strong>sufficient</strong> way to view Bernoulli's lift equations?</p> | 17 | [
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<p>I'm taking a solid state course, and is currently on the subject of dielectrics. In one of the sections, concerning "Impurities in Dielectrics" the books says:</p>
<p><em>"Impurities can also be used to make insulators conductive, exactly like doped semiconductors. Donor or acceptor levels have to lie close to the conduction band and valence band, respectively, in order to give rise to a finite density of electrons and holes and thereby conductivity. The important advantage over a usual semiconductor material is that high-temperature applications are possible. In a narrow-gap semiconductor, high temperatures are a problem because of the exponentially increasing number of intrinsic carriers. In a doped insulator, this is not an issue for practically relevant temperatures"</em></p>
<p>Now, I'm not getting why more carriers are a bad thing ? I see, that if the gap is too small (Semiconductor), and the temperature is high, the you should be able to excite electrons into the conduction band, instead of just the doped bands, right ? But why is that a bad thing ?
I see that high-temperature applications can be bad, since you have to get high temperatures, which is not always easy and great, but if you already are at that temperature, what is the problem ?</p> | 1,362 | [
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<p>As far as I know, since the earth is spinning, centrifugal force should be throwing you off the earth. However, gravity counteracts that by being a stronger force. So if you stood an object with the exact mass and size of earth, but it wasn't spinning, would you perceive gravity to be stronger?</p>
<p>If this is true, wouldn't an object outside of the atmosphere not in orbit of the earth have greater acceleration than an object in orbit?</p> | 18 | [
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<p>I have just begun studying quantum field theory and am following the book by Peskin and Schroeder for that.
So while quantising the Klein Gordon field, we Fourier expand the field and then work only in the momentum space. What is the need for this expansion?</p> | 1,363 | [
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<p>First thing I've been wondering is how the gravitational field is emitted. Matter emits gravitational waves, and I guess that those waves travel at around the speed of light. If that's not the case, please direct me to something that explains that.<br>
For now I'll assume that this emits a waves that travels at the speed of light.</p>
<p>Now those waves are emitted constantly without the apparent need of energy. Just the mass being there is enough to emit those waves, and the mass itself isn't consumed by the emission of gravitational waves. Again that's my assumption, please tell me if I'm wrong.</p>
<p>Now if we would be able to modulate the mass of an object, we would be able to modulate the gravitational waves this object emits. Hence we would be able to transmit information through those waves, if of course we'd have a device that detects gravitational waves with enough sensitivity.</p>
<p>Those waves would be transmitted at no cost, because the waves are emitted by the mass. The only cost of this transmission would be the "mass modulator", which has yet to be invented and which would require energy. However the actual transmission doesn't require energy, and the gravitational waves are harmless, unlike the EM ones.</p>
<p>So here comes my question (as in the title): would it be possible somehow to use the gravitational waves to transmit information?</p> | 876 | [
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<p>Recently I read an interesting article about <a href="http://en.wikipedia.org/wiki/Negative_temperature" rel="nofollow">negative temperature</a>. I was puzzled because I thought before that <a href="http://en.wikipedia.org/wiki/Temperature" rel="nofollow">temperature</a> has definite meaning in thermodynamics: it tells about how fast atoms jiggle. Now if temperature can be negative, this means temperature has wider meaning...</p>
<p>I wonder if there is any deep physical meaning of temperature, not mathematical one. </p> | 1,364 | [
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<p>I have been wondering about the axiom of choice and how it relates to physics. In particular, I was wondering how many (if any) experimentally-verified physical theories require axiom of choice (or well-ordering) and if any theories actually require constructability. As a math student, I have always been told the axiom of choice is invoked because of the beautiful results that transpire from its assumption. Do any mainstream physical theories require AoC or constructability, and if so, how do they require AoC or constructability? </p> | 1,365 | [
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<p>Well this is pretty noobish question and I am not sure how to ask.<br />
When We talk we don't talk in an uniform frequency. Then how can one measure frequency of ones sound/voice ? <br />
I am asking this cause several times I've heard people to say this is his/her frequency of voice. Is this a vague term ? or it has any real meaning ?</p> | 1,366 | [
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<p>I've tried to find the moment of inertia of a cylinder rotating about an axis parallel to its base (i.e about the 'End diameter') as one can see <a href="http://hyperphysics.phy-astr.gsu.edu/hbase/icyl.html" rel="nofollow">here</a> . But when I checked my results with different references ,I've found that it's incorrect!.I need a help to figure out where I did it wrong. </p>
<p>since $$I=\int\limits x^2.dm$$ </p>
<p>$$dm = \rho.dv$$
where</p>
<p>$$dv= r.d\theta . dr .dh$$
&</p>
<p>$$\rho=\frac{M}{\pi R^2 H}$$</p>
<p>$M$:cylinder total mass.
$H$:total height of cylinder.
$R$:the radius of the cylinder.</p>
<p>The distance of each infinitesimal element from the axis of rotation would be : </p>
<p>$$\sqrt{r^2+h^2}$$ </p>
<p>Therfore, </p>
<p>$$I= \frac{2M}{ R^2 H}\int\limits_{0}^{R} \int\limits_{0}^{H}\, r(r^2+h^2)dh\,dr$$ </p>
<p>and this gives a result of : $$\frac{M~H^2}{3}+\frac{M~R^2}{2}$$</p>
<p>Which is obviously wrong since the second term should be multiplied by a factor of $1/2$. </p> | 1,367 | [
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<p>The partition function of a dipole in an electric field is a well-known problem, analytical solvable (nice integral, can be calculated with pen and paper), for example in the Langevin treatment of paramagnetism.</p>
<p>I'm right now looking at a classical quadrupole in an electric field. This looks very different, and I have not been able to solve it by myself. Before I try harder and reinvent the (broken) wheel, would anyone know if it's been addressed, and possibly solved, before?</p>
<hr>
<p><strong>The way I see it:</strong> I consider a quadrupole $\mathcal{Q}$ fixed in space but with free orientation. It interacts with an electric field $\mathbf{E}$, whose derivatives form the tensor $\mathcal{E}$ ($\mathcal{E}_{ij} = \frac{\partial E_i}{\partial x_j}$). The potential energy is thus:</p>
<p>$$ U = \frac{1}{3} \mathcal{E}:\mathcal{Q} $$</p>
<p>and the partition function I want to calculate is:</p>
<p>$$ Z = \int \exp\left(-U/kT\right) $$</p>
<p>where the integral is taken over all orientations of the quadrupole with respect to the field, i.e. a 3D rotation matrix determined by three angle $(\theta,\phi,\xi)$. I've started doing this integral, but neither my brain nor Mathematica can express it in a nice way.</p>
<p><em>I don't know if I'm missing a way to solve this in this fashion, or if it would be better attacked from another angle…</em></p> | 1,368 | [
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<p>spin-0 is massive or massless?
How does we separate the massive and massless degrees of freedom for spin-2?
What is the partially massive?</p> | 1,369 | [
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<h2>Definitions / Background</h2>
<p>In LTE, Kirchoff's law for radiation holds:</p>
<p>$$ \frac{j_{\nu}}{\alpha_{\nu}} = B_{\nu} (T) $$</p>
<p>where $j_{\nu}$ is the specific radiative emissivity, $\alpha_{\nu}$ is the monochromatic radiative absorption, and $B_{\nu} (T)$ is the Planck function evaluated at the temperature $T$. </p>
<p>Consider a gas of two-level atoms with energies $E_u$ and $E_l$, with $E_u > E_l$, statistical weights $g_u$ and $g_l$, and number densities $n_u$ and $n_l$. The transition between these states has Einstein coefficients $A_{ul}$, $B_{ul}$ and $B_{lu}$ that we can use to write the emissivity and absoprtion of the transition:</p>
<p>$$ j_{\nu} = \frac{h \nu}{4 \pi} n_u A_{ul} \psi({\nu})$$
$$ \alpha_{\nu} = \frac{h \nu}{4 \pi} [ n_l B_{lu} \phi({\nu}) - n_u B_{ul} \chi({\nu})]$$</p>
<p>where $\psi$, $\phi$, and $\chi$ are line profile functions accounting for line-broadening mechanisms such as thermal motion. </p>
<p>Then, making use of the standard relations between the Einstein coefficients, we have
$$ \frac{j_{\nu}}{\alpha_{\nu}} = \frac{2 h \nu^3}{c^2} \frac{\frac{\psi}{\phi}}{\frac{g_u n_l}{g_l n_u} - \frac{\chi}{\phi}}$$</p>
<h2>Question</h2>
<p>I want to understand how the right-hand side of the last equation simplifies to the Planck function under conditions of LTE, over the entire width of the line, without assuming that the line is narrow.</p>
<p>A standard discussion of this topic, as found in e.g. the astrophysics textbook by Rybicki and Lightman, does not seem to achieve this. In my reading, they proceed by making the following observations and/or assumptions:</p>
<p>1) There is complete redistribution of frequency between absorption and all types of emission, so that $\psi = \chi = \phi$.</p>
<p>2) In LTE at temperature T, the fraction $ g_u n_l / (g_l n_u)$ is equal to $\exp[h \nu_0/(kT)]$, where $\nu_0 = (E_u - E_l)/h$. </p>
<p>If those are true, then we have</p>
<p>$$ \frac{j_{\nu}}{\alpha_{\nu}} = \frac{2 h \nu^3}{c^2} \frac{1}{\exp[h \nu_0/(kT)] - 1}$$</p>
<p>But this does not have the correct exponential term in the denominator to match the Planck function, because $\nu_0$ is constant (frequency-independent). Or from another perspective, the problem is that $g_u n_l / (g_l n_u)$ is frequency-independent.</p>
<p>So, what is going on? Do we have to break either of the assumptions/observations 1 or 2 above? If so, how? If not, then does Kirchoff's law simply not apply on a frequency-by-frequency basis over the width of a broad line, although it still might apply in a line-averaged sense? Is there some other possibility or detail I have overlooked?</p> | 1,370 | [
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<p>A purely phase-modulated signal has no power modulation. This is obvious enough if you look at the time series, but I'd like to "see" it in the frequency domain.</p>
<p>In physical terms, if we take a laser beam and apply phase modulation via an electro-optic modulator and then send this beam to a photodiode, we will see only a DC term on the photodiode. I would like to work out how the sidebands created by phase modulation manage to conspire to eliminate any beat frequencies.</p>
<p>The operation of phase modulation at a frequency $\Omega$ consists of multiplication by $\exp\{i\Gamma\sin\Omega t\}$ which is easily seen to have unit amplitude at all times. Via the Jacobi-Anger identity, this can be written in terms of sidebands: </p>
<p>$$\exp\{i\Gamma\sin\Omega t\} = \sum_{n=-\infty}^\infty J_n(\Gamma) \exp\{i n \Omega t\}$$</p>
<p>where $\exp\{i n \Omega t\}$ is interpreted as creating a sideband at a frequency of $n\Omega$ Hz away from the carrier whose amplitude is given by the nth Bessel function $J_n(\Gamma)$ where $\Gamma$ is the modulation depth.</p>
<p>A photodiode will see the modulus squared of this signal. We can write down the result by grouping terms by like powers of $\exp\{i\Omega t\}$.</p>
<p>It's easy to see that the energy is conserved at DC due to the relation $ 1 = \sum_{n=-\infty}^\infty |J_n(x)|^2 $.</p>
<p>Likewise, it's easy to see that there is no signal at frequency $1\Omega$ on the photodiode, because the beat of the $n$th upper sideband with the $(n-1)$st upper sideband always cancels with the beat of the nth lower sideband with the $(n-1)$st lower side band, because of the property of the Bessel functions that $J_{-n}(x)=(-1)^n J_n(x)$.</p>
<p>However, the same sort of nice cancellation does not appear for the $2\Omega$ signal, because the bessel functions always appear in products with two bessel functions of the same parity, i.e. $\cdots + J_{-3}J_{-1} + J_{-2}J_0 + J_{-1}J_1 + J_0 J_2 + J_1 J_3 + \cdots$.</p>
<p><strong>How can we show that there is no beat signal at frequencies $2\Omega$ (and higher)?</strong></p> | 1,371 | [
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<p>Suppose i try to derive the most generic Dirac-like equation (that is, as factors of first-order expression in momenta and mass operator where we allow coefficients that are associative, don't necessarily commute to each other, but they commute with the space-time degrees of freedom)</p>
<p>Usually the Dirac equation is obtained by taking this expression:</p>
<p>$E^2 = P^2 + m^2$
with $E=i \hbar \partial_{t}$, $P=-i \hbar \partial_{x}$</p>
<p>and getting $E^2 = ( \beta m + \alpha_{i} P_{i} )^2$</p>
<p>and obtaining a satisfying algebra:</p>
<ul>
<li><p>$\beta \alpha_{i} + \alpha_{i} \beta = 0$</p></li>
<li><p>$\alpha_{i} \alpha_{j} + \alpha_{j} \alpha_{i} = 0$</p></li>
<li><p>$\beta^2 = \alpha_{i}^2 = 1$</p></li>
</ul>
<p>However what happens if i don't assume a squared expression and just try to factorize the operator; I note that the most generic expression satisfying the above would be this:</p>
<p>$$E^2 = ( \hat{\beta} m + \hat{\alpha_{i}} P_{i} ) ( \hat{\beta^{-1}} m + \hat{\alpha^{-1}_{i}} P_{i} )$$</p>
<p>where i use the hat to denote that these are more generic that the usual $\beta, \alpha_{i}$ that we deal with traditionally</p>
<p>This system satisfies the original expression if we demand the following algebra;</p>
<ul>
<li><p>$\hat{\beta} \hat{ \alpha^{-1}_{i}} + \hat{\alpha_{i} } \hat{\beta^{-1}} = 0$</p></li>
<li><p>$\hat{\alpha_{i}} \hat{ \alpha^{-1}_{j}} + \hat{\alpha_{j} } \hat{\alpha^{-1}_{i}} = 0$</p></li>
</ul>
<p>The question here is: It seems to me that there is a gauge freedom in specifying these quantities; <strong>what is the meaning of these gauge symmetries and what, if any, interesting invariants we should expected to get from them?</strong></p>
<p><strong>Note 1:</strong> if i assume the following representation of these hatted cliffords in terms of the well-known ones;</p>
<ul>
<li><p>$\hat{\beta} = \boldsymbol{L} \beta \boldsymbol{M} $</p></li>
<li><p>$\hat{\alpha_{i}} = \boldsymbol{L} \alpha_{i} \boldsymbol{M}$</p></li>
</ul>
<p>i notice that this satisfies the required algebra <em>(and makes the extra degrees of freedom explicit!)</em></p>
<p><strong>Note 2;</strong> it seems to me that the most generic solutions look like:</p>
<ul>
<li>$( \hat{\beta} m + \hat{\alpha_{i}} P_{i} ) \boldsymbol{u} = \lambda_{0} \boldsymbol{v} $</li>
<li>$( \hat{\beta^{-1}} m + \hat{\alpha^{-1}_{i}} P_{i} ) \boldsymbol{v} = \lambda_{1} \boldsymbol{u} $</li>
</ul>
<p><strong>is this just a rewrite of the left-hand and right-hand Weyl spinors, or is something different?</strong></p>
<p>I have purposely not assumed any space-time or spinorial dimensionality in the above, but feel free to make your answers about concrete dimensions</p> | 1,372 | [
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<p>Get up off the plane of the ecliptic by a couple of million miles. Look back at the Sun and watch the Earth's orbit in time-lapse for a few centuries. The orbit is an ellipse tilted at 23.5 degrees from the ecliptic. Hold station with the Sun as it rotates about the galactic center.</p>
<p>Question: Does that tilted ellipse -- Earth's orbit -- precess? Over millions of years, does the long axis rotate with the Sun's rotation in the galaxy? Or does it stay fixed in some larger frame of reference, such as the Local Group of galaxies? Or does it exhibit some more complex and subtle motion?</p>
<p>Does the long axis nutate?</p>
<p>Does the angle of the orbit to the ecliptic change in any periodic way?</p>
<p>Why? What are the forces in play?</p> | 1,373 | [
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<p>A positive pion is an up and an anti-down. A negative pion is a down and an anti-up. What's a pion with an electrical charge of 0?</p> | 1,374 | [
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<p>I am hoping that someone can clarify this for me.</p>
<p>With these equations. (Boltzmann Law) and radius of Radius of star how does surface temp scale with mass</p>
<blockquote>
<p>$R _x \approx R_\bigodot (\frac{M_x}{M_\bigodot})^.5$ </p>
<p>$L _x \approx L_\bigodot (\frac{M_x}{M_\bigodot})^3.5$ </p>
<p>$\frac{L}{\pi R^2} = \sigma T^4$</p>
<p>$T_\bigodot = 5800 K$ = Surface Temp of Sun</p>
</blockquote>
<p>What I have done is that I have substituted $R_x$ and $L_x$ in the 3rd equation and simplified it in terms of $T$ where my final expression is </p>
<blockquote>
<p>$T = (\frac{L_\bigodot M_x^2.5}{\pi\sigma M_\bigodot^2.5 R_\bigodot})^0.25$</p>
</blockquote>
<p>Am i done? What am I suppose to do next?
Thanks</p> | 1,375 | [
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<p>The title says it all but I will add some details. I believe that interference takes place only in waves which are parallel to each other. See the picture to understand what I mean by parallel.</p>
<p><img src="http://i.stack.imgur.com/omkAt.png" alt="enter image description here"></p>
<p>We see that in the image the interfering waves are parallel. If interference can happen in non-parallel waves then please give me an explanation but if it does not consider the double slit experiment. How are the waves parallel in the double slit experiment?</p>
<p><img src="http://i.stack.imgur.com/2GKvA.png" alt="enter image description here"></p>
<p>We see that in the double-slit experiment the waves are not parallel. Please answer accordingly as I have given both the situations.</p> | 1,376 | [
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<p>How do we calculate the expectation value for speed? I have heard that we must first calculate the expectaion value for kinetic energy. Someone please explain a bit what options do we have.</p> | 1,377 | [
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<p>In differential geometry and general relativity space is said to be flat if the Riemann tensor $R=0$. If the Ricci tensor on manifold $M$ is zero, it doesn't mean that the manifold itself is flat. So what's the geometrical meaning of <a href="http://en.wikipedia.org/wiki/Ricci_curvature" rel="nofollow">Ricci tensor</a> since it's been defined with the <a href="http://en.wikipedia.org/wiki/Riemann_curvature_tensor" rel="nofollow">Riemann tensor</a> as </p>
<p>$$\mathrm{Ric}_{ij}=\sum_a R^a_{iaj}?$$ </p> | 1,378 | [
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<p>Since we see the <a href="http://en.wikipedia.org/wiki/New_moon" rel="nofollow">new moon</a> at least once in a month when the Moon gets in between of the Sun and the Earth during the evening ( and as far as I know if this happens during the day, you'll get to see a <a href="http://en.wikipedia.org/wiki/Solar_eclipse" rel="nofollow">solar eclipse</a>. Why don't we get to see this often or in the day? </p>
<p>Does it mean that in some part of world there's a solar eclipse when we are seeing a new moon? I'm looking for a diagram or interactive way to understand this if possible as I'm not a native English speaker, but I'll try my best to do so. </p> | 110 | [
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<p>The formula for calculating the center of mass is</p>
<p>$$ r_{center} = \frac{m_1 \cdot r_1 + m_2 \cdot r_2}{m_1+m_2} $$</p>
<p>Why can't I use it to calculate the barycentre of two planets?</p>
<p><img src="http://i.stack.imgur.com/KVVK5.png" alt=""></p>
<p>I understand how to use ratios to approach this problem.</p>
<p>$$ \frac{r_1}{r_2} = \frac{m_1}{m_2} $$
$$ r_1 = \frac{m_2}{m_1+m_2}\cdot (d_{Sun->Earth}) $$</p>
<p>However, I do not understand why I can not use the $r_{center}$ formula.</p> | 1,379 | [
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<p><strong>Groundhog Day Update, 2014</strong></p>
<p>The simple and dumb way to ask my main question is this: If something like a neutron start goes sailing by at very close to the speed of light, say fast enough to double its total mass-energy, do you "feel" the energy it is carrying as gravity as it passes by?</p>
<p>I'm about 99% sure the answer is yes, but I'd sure appreciate some confirmation of that. I was surprised (and appreciative) at the complexity of the first two attempts at answers, but must confess I thought it was a simpler problem than that.</p>
<p>For one thing, frame equivalence should not an issue, since in terms of gravitational effects it doesn't matter whether it was you or the star that doubled in mass. And since the mass used to speed up the star cannot simply disappear, its gravitational pull has to go <em>somewhere</em>, so why not in the star itself?</p>
<p>I now gather that such a simple argument gets very, very messy when expressed in tensor form. Ah... oops, but again, thanks. And I <em>think</em> that's what John Rennie answered...</p>
<hr>
<p>Contingent on relativistic mass actually having gravitational effects, I think I can now answer my second question myself.</p>
<p>Imagine a self-contained cluster of masses within a compact region of space, all initially motionless relative to each other. At a large distances the gravitational behavior of the cluster will asymptotically approach that of a single large gravitational mass, one equal in magnitude to a simple sum of the individual masses in the cluster.</p>
<p>Next, have the cluster go crazy, all by itself and without any external stimulus. Huge parts of it are annihilated into pure energy, which in turn drives other parts outward at relativistic velocities. (If that sounds a bit outrageous, look up black hole jets sometime.) Since there are no external influences, the outbound parts must have momenta that sum to zero, with the simplest case being two equal-mass objects moving in opposite directions at the same velocities.</p>
<p>Now from a great enough distance, the cluster will continue to look asymptotically close to a single mass that is still equal to the sum of the original cluster masses. To that distant observer, the conversion of huge chunks of the cluster into pure momentum makes not a whit of difference: The cluster still has exactly the same mass-energy, with the only difference being that it is getting harder to approximate as a point.</p>
<p>So <em>of course</em> the few chunks of the original cluster that were never accelerated during the explosion will look a bit unique to the distant observer. In particular they will appear to have the lowest masses, since they did not acquire any of the converted mass energy during the explosion. Nothing profound, that... yet still interesting, especially as you broaden the argument to include larger and larger assemblages of mass-energy.</p>
<p>The black holes at the center of most (all?) galaxies would be examples of mass-gravity minimum frames, and their dual jets examples of entities with excess gravitation.</p>
<hr>
<p>Oh, and one other point: Will stars or mass (large jets?) moving at relativistic speed exhibit higher levels of gravitational lensing? I would assume so...</p>
<p>My original more specific and inadvertently homework-like version of the above thought experiment is below.</p>
<hr>
<p>Start with five large objects $\{m_a,m_b,m_c,m_d,m_e\}$ of equal mass $m$. Oddly, $m_b$ is made of antimatter.</p>
<p>$m_a$ remains unchanged at the center of mass of the group.</p>
<p>$m_b$ and $m_c$ are mutually annihilated. Their energy is used to launch $m_d$ and $m_e$ along ${\pm}x$ paths. The energy imparts to each a velocity as measured from $m_a$ of $(\sqrt{\frac{3}{4}})c$, which in turn gives $m_d$ and $m_e$ each a relativistic mass of $2m$, again as measured from $m_a$. The annihilated masses of $m_b$ and $m_c$ have in effect been "added" in the form of momentum to the rest masses of $m_d$ and $m_e$.</p>
<ol>
<li><p>In terms of <em>gravity</em>, where in the resulting three-body system do the masses of $m_b$ and $m_c$ reside?</p></li>
<li><p>If you answered "$m_d$ and $m_e$", what happened to frame invariance?</p></li>
</ol>
<hr>
<p><strong>Notes</strong></p>
<p>A related (but definitely different) question is:</p>
<p><a href="http://physics.stackexchange.com/q/51733/7670">Does the increase of (relativistic) mass, while flying near speed of light, has any impact on astronauts?</a></p>
<p>I could not find any exact matches, but I also cheerfully acknowledge that my question search skills are not as good as some on this group.</p> | 1,380 | [
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<p>I'm trying to diagonalize the Hamiltonian for a 1D wire with proximity-induced superconductivity. In the case without a superconductor it's all fine. However, with a superconductor I don't get the correct result for the energy spectrum of the Hamiltonian in</p>
<blockquote>
<p>Tudor D. Stanescu and Sumanta Tewari. “<a href="http://dx.doi.org/10.1088/0953-8984/25/23/233201" rel="nofollow">Majorana fermions in semiconductor nanowires: fundamentals, modeling, and experiment.</a>” Journal of Physics: Condensed Matter <strong>25</strong>, no. 23 (2013): 233201. <a href="http://arxiv.org/abs/1302.5433v2" rel="nofollow">arXiv:1302.5433</a> [cond-mat.supr-con].</p>
</blockquote>
<p>given by</p>
<p>$$
H = \eta_{k}\tau_{z} + B\sigma_{x} + \alpha k\sigma_{y}\tau_{z} + \Delta\tau_{x}
$$</p>
<p>Here $\sigma$ and $\tau$ are the Pauli matrices for the spin and particle-hole space.</p>
<p>Now the correct result is: $E^{2}_{k} = \Delta^{2} + \eta_{k}^{2} + B^{2} + \left(\alpha k\right)^{2} \pm 2\sqrt{B^{2}\Delta^{2} + \eta^{2}_{k}B^{2} + \eta^{2}_{k}\left(\alpha k\right)^{2}}$</p>
<p>Now, my problem is that I don't know how I can bring the Hamiltonian in the correct matrix form for the calculation of the eigenvalues. If I try it with the upper Hamiltonian I have completely wrong results for the energy spectrum. I believe my mistake is the interpretation of the Pauli matrices $\tau$ but I don't know how I can write the Hamiltonian in the form to get the correct eigenvalues.</p> | 1,381 | [
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<p>This is a question I have been asking myself for some time since the following technique is often used in the nonlinear dynamics community, but never managed to get an answer why it could be applied.</p>
<p>In symplectic mechanics, consider a dynamical system with no rigid body movements (all eigenvalues are non-zero, ie $\lambda_i\neq 0$). The centre manifold theorem states that the dynamics can be written in the following form:
$$ \begin{bmatrix} \dot{\mathbf{x}}_\mathrm{s} \\ \dot{\mathbf{x}}_\mathrm{c} \\ \dot{\mathbf{x}}_\mathrm{u} \end{bmatrix}
=\begin{bmatrix} [ A_\mathrm{s}] & & \\ & [A_\mathrm{c}] & \\& &[A_\mathrm{u}] \end{bmatrix}\begin{bmatrix} \mathbf{x_\mathrm{s}} \\ \mathbf{x_\mathrm{c}} \\ \mathbf{x_\mathrm{u}} \end{bmatrix} +\text{nonlinear terms}$$
and that there exists a <a href="http://en.wikipedia.org/wiki/Center_manifold" rel="nofollow">centre manifold</a> (also stable and unstable manifolds, but they are not of interest here) of dimension the number of lines of $A_\mathrm{u}$.</p>
<p>Because all eigenvalues corresponding to the <a href="http://www.scholarpedia.org/article/Center_manifold" rel="nofollow">centre manifold</a> ($\mathrm{Re}(\lambda_i)=0$) are purely imaginary and non-zero (no rigid body movements), $A_\mathrm{c}$ is composed of $2\times2$ blocks on the diagonal.</p>
<p>Now, we are searching for periodic orbits, so we focus and the centre manifold and we name the coordinates of $\mathbf{x}_\mathrm{u}$ such that $\mathbf{x}_\mathrm{u}=[u_1,v_1,\dots,u_n,v_n]^T$ (even dimension, as explained above).</p>
<p>In order to calculate the <em>nonlinear normal mode $i$</em>, many people write that any pair $(u_i,v_i)$ can be chosen as <em>master coordinates</em> and that all other pair $(u_j,v_j)$, $i\neq j$, can be written as:
$$\left\lbrace \begin{aligned} u_j &=f_j(u_i,v_i) \\ v_j&=g_j(u_i,v_i)\end{aligned}\right.$$</p>
<p><strong>Question:</strong> What ensures that choosing a pair of master coordinates is always possible? If not, could you give a counterexample?</p>
<p><strong>EDIT</strong> The presence of an internal resonance in the dynamical system constitutes a counterexample: the system above does not have any solutions, then. Yet, I don't really understand why is it so.</p> | 1,382 | [
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<p>How does:</p>
<p>$$L \cdot\frac{dL}{dt} = \frac{1}{2}\frac{d(L^2)}{dt}$$</p>
<p>where <strong>L</strong> is a vector (I dunno how to make it bold in the equation).
How do they reach to this right hand side equation?
And what is the difference between <strong>||L||</strong> and <strong>|L|</strong> ?</p> | 1,383 | [
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<p>Given von Neumann equation
$$\frac{d}{dt} \rho(t) = -i [H, \rho(t)] = -i e^{-iHt}[H, \rho(0)]e^{iHt}.$$</p>
<p>If we know that $[H, \rho(0)] \neq 0$, how do we prove in details the solution of von Neumann equation will never be stabilized, i.e. $$\lim_{t\to\infty}\frac{d}{dt} \rho(t) \neq 0~?$$ </p> | 1,384 | [
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<p>Could anyone give me a brief explanation why multilayer OLEDS as more efficient than single layer OLEDs? What other advantages exist for multilayer OLEDs over single layer OLEDs?</p> | 1,385 | [
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<p>The gas mileage of my vehicle tends to improve the more I have been driving on the interstate on that tank of gas - if I go through a tank of gas without at any point driving on the interstate, I will typically get 24-26 MPG, but when I have driven almost exclusively on the interstate, I will typically get 26-29 MPG. </p>
<p>What confuses me is that, when traveling on the interstate, I am driving great distances at higher RPMs, which I would expect to correlate to more gas usage and thus worse gas mileage. Why does the inverse seem to be true?</p> | 1,386 | [
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] |
<p>I learned how to sweat a pipe today from my father. If you're not familiar with the process, <a href="http://www.popularmechanics.com/home/skills/4213319" rel="nofollow">this</a> might help.</p>
<p>One thing that jumps out at me is this line (from the above link, as well as my father's explanation)</p>
<blockquote>
<p>Solder, which melts at low temperatures, wicks into a joint by capillary action and bonds with copper at the molecular level.</p>
</blockquote>
<p>This doesn't seem to be quite right to me. I always thought of capillary action as something like putting a piece of paper vertically into a dye and watching the dye rise up the paper. I also didn't think the solder bonded with the copper at the molecular level, just that it melted and filled in the gaps really well. <a href="http://en.wikipedia.org/wiki/Capillary_action" rel="nofollow">Wikipedia</a> seems to agree with my definition of capillary action</p>
<blockquote>
<p>Capillary action ... is the ability of a liquid to flow in narrow spaces without the assistance of, and in opposition to, external forces like gravity.</p>
</blockquote>
<p>When sweating a pipe you place the solder above the pipe, letting it drip down (ie with the assistance of gravity) which makes it <em>not</em> capillary action... right?</p>
<p>Is there something I'm just not understanding here?</p> | 1,387 | [
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<p>Does an object's kinetic energy increase, decrease, or stay constant when it reaches <a href="http://en.wikipedia.org/wiki/Terminal_velocity" rel="nofollow">terminal velocity</a> while falling? </p> | 1,388 | [
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<p>Length of a dipole antenna according to antenna theory should be lambda/2 for best reception. I am just curious about the outcome when length of dipole antenna >> lambda. Impedance will be zero in such a case and there will be no signal received. Is it correct?</p> | 1,389 | [
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<p>I have seen the term topological charge defined in an abstract mathematical way as a essentially a labeling scheme for particles which follows certain rules. However I am left guessing when trying to explain what physical properties of a system lead to the need to introduce this new type of "charge." </p>
<p>If it makes any difference, I'm interested in studying the statistical properties of Quantum Hall systems (and in particular anyonic interferometry), where the different topological charges contribute to the total number of quantum states. </p> | 1,390 | [
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<p>If I have a complex voltage $V_z$ is real voltage $V$ (i.e. the voltage used in the normal ohms law and the voltage we normally talk about) always given by $V=Re(V_z)$. Does the same apply to current? and if it is not the case how do we find $V$ from $V_z$?</p> | 1,391 | [
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<p>Assume you are sliding down a steep hill. Is it safer to go down with your back against the hill or the front of your body against the hill? I assumed that since the surface area of your body which is touching the hill is about the the same regardless of which way you are going down, it won't make a difference. Is this the correct answer?</p> | 1,392 | [
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<p>This question is entirely on tensorial notation in Wald's General Relativity. When specifying the properties of the Riemann tensor on pg39, he states:</p>
<p>$R_{[abc]}^{\quad \ \ \ d} = 0$</p>
<p>and</p>
<blockquote>
<p>For the derivative operator $\nabla_a$ naturally associated with the metric, $\nabla_a g_{bc}=0$, we have $R_{abcd} = -R_{abdc}$.</p>
</blockquote>
<p>and</p>
<blockquote>
<p>The Bianchi identity holds: $\nabla_{[a}R_{bc]d}^{\quad \ \ e} = 0$</p>
</blockquote>
<p>Questions:</p>
<ol>
<li><p>What do the square brackets around "abc" mean?</p></li>
<li><p>Why does $R_{abc}^{\quad d}$ become $R_{abcd}$? What is the relation between the two?</p></li>
<li><p>What does having $R$ in the square brackets mean?</p></li>
</ol>
<p>Thank you.</p> | 1,393 | [
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<p>I was looking at the derivation of drift velocity of free electrons in a conductor in terms of relaxation time of electrons. </p>
<p>Here a metallic conductor XY of length $\ell$ is considered and having cross sectional area A. A potential difference $V$ is applied across the conductor XY. Due to this potential difference an electric field $E$ is produced. The magnitude of electric field strength is $E = V/\ell$. </p>
<p>I dont understand this part.
How is electric field strength potential difference divided by length. Where am I lagging in my concepts?</p> | 1,394 | [
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<p>What does the neutrino particles follow- Dirac or Majorana Statistics?</p> | 1,395 | [
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<p>A superposition of two probability wave creates standing wave. Well, that is convincing. Dose waves described by schrodinger's equation have other properties of wave? like reflection, refraction, polarization ect.</p> | 1,396 | [
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<p>I am trying to understand the gravitational sphere of influence (<a href="http://en.wikipedia.org/wiki/Sphere_of_influence_%28astrodynamics%29" rel="nofollow">SOI</a>), but all I get by searching is the formula that you can find on Wikipedia, that is</p>
<p>$$ r_{SOI} = a \left( \frac{m}{M} \right)^{2/5} $$</p>
<p>where </p>
<p>m: mass of orbiting (smaller) body</p>
<p>M: mass of central (larger) body</p>
<p>a: semi-major axis of smaller body</p>
<p>When inputing the Moon numbers in this formula, we get a SOI of 66,183 km for the Moon over the Earth. This is consistent with other sources on the web, for example the Apollo mission transcripts when they talk about entering the Moon SOI.</p>
<p>What I don't understand is that when I calculate the gravitational forces between different bodies using Newton's laws, an object placed at this distance between the Earth and the Moon still gets a bigger pull from the Earth. Say for example that we had an object with a mass of 100 kg, these are the gravitational pull (in Newtons) that it would receive from the Earth and the Moon at different distances :</p>
<p>Force from Earth on Earth's surface : 979.866 N</p>
<p>Force from Earth at 384400 km (Moon dist) : 0.27 N</p>
<p>Force from Moon at 66183 km from Moon : 0.112 N</p>
<p>Force from Earth at 318216 km (66183 km from Moon) : 0.394 N</p>
<p>Force from Moon at 38400 km from Moon : 0.333 N </p>
<p>Force from Earth at 346000 km (38400 km from Moon) : 0.333 N</p>
<p>As you can see, the pull from Earth and Moon cancel each other at around 38,000 km, not 66,000 km. This is somewhat counterintuitive to me, as I first thought that a spacecraft (for example) would get more pull from the Moon than from the Earth when it entered the Moon's gravitational sphere of influence. I suspect that it has to do with the fact that the Moon is in orbit around the Earth, i.e. it is in constant acceleration in the same direction as the Earth's pull, but I would like a clear explanation if somebody had one.</p> | 1,397 | [
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<p>As the resistance of a circuit goes down, the power increases because the current increases, assuming constant voltage. Why is this? I feel like resistance and current are inversely proportional, so lowering one should just raise the other, and power should stay relatively constant. What is it about power that makes it work like this?
Edit: Again, namely I find it confusing that decreasing resistance actually has a net increase in power used. I feel like while the current will increase, it should be countered by the lowered resistance...</p> | 1,398 | [
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<p>I was thinking how far can I hear sound coming from a concert. Today I was walking at night and I could hear sound from somewhere very far. I started following the sound but sound used to disappear momentarily and then reappear. I went atleast 2 miles but could not find where it was coming from. </p>
<p>Now I am really confused because sound is energy and because of energy losses, as I understand sound cannot go as far as 20 miles. The nearest concert was atleast 20 miles away. </p>
<p>I came back and read articles(I cannot proof the reliability) where sound had traveled 200 miles but it was mostly sounds from explosion or volcanoes. Maybe this makes sense because explosions or volcanoes have alot of energy but how can one explain sound being heard from a concert very far away(approx 20 miles). </p> | 1,399 | [
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0.005273390095680952,
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0.027732275426387787,
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0.004558575805276632,
0.05464126914739609,
0.05173129960894585,
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0.0013330664951354265,
0.007359616924077272,
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0.04782823100686073,
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0.014694202691316605,
-0.01060098223388195,
-0.012627697549760342
] |
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