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<p>Just a simple question. Why is it, that when a material becomes superconducting, and by that gets zero resistivity, the electrons don't hit impurities in the material?
For the material to have zero resistivity, that means that the electrons can just flow without any disturbance at all?</p>
<p>Is it because of the Cooper pair creation? In that case, why exactly?</p> | 1,684 | [
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<p>Why is the <a href="http://en.wikipedia.org/wiki/Scale_factor_%28cosmology%29" rel="nofollow">cosmological scale factor</a> (expansion rate of the universe) not simply the time $t$, i.e. the age of the universe?</p> | 1,685 | [
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<p>My question boils down to this. If the Universe was contracting the stars closer to the center would move faster to the middle than stars that were further away from the center. </p>
<p>That would also produce a red doppler shift. So why is it that the red shift is always linked to expansion? Human optimism? </p>
<p>Thanks,</p>
<p>Andrew</p> | 1,686 | [
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<p>I recently got into a lengthy debate about the exact nature of boundary layer separation. In common parlance, we have a tendency to talk about certain geometries as being too "sharp" for a viscous flow to remain attached to them. The flow can't "turn the corner" so to speak, and so it separates from the body. While I think this way of thinking can properly predict in which situations a flow might separate, I think it gets the underlying Physics completely wrong. From my understanding, what's happeneing is the adverse streamwise pressure gradient precludes the boundary layer from progressing downstream past a certain point, and the upstream flow subsequently has nowhere to go but up and off of the body. This is a very different causal relationship from the first explanation, where the flow lacks a sufficient streamwise-normal pressure gradient to overcome the centrifugal forces of a curved streamline. But which is correct?</p>
<p>Considering that normal shockwaves can produce extreme adverse pressure gradients (even along a streamline that is not curved), I figured that shock-induced flow separation might be a way to settle this matter. Any thoughts?</p> | 1,687 | [
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<p><a href="http://physics.stackexchange.com/questions/55686/gravitational-redshift-around-a-schwarzschild-black-hole">Another question</a> about the Schwarzschild solution of General Relativity:</p>
<p>In the derivation (shown below) of the Schwarzschild metric from the vacuum Einstein Equation, at the step marked "HERE," we have the freedom to rescale the time coordinate from $dt \rightarrow e^{-g(t)}dt$. However, later in the derivation, when we pick up an integration constant at the step marked "HERE2," we are unable to absorb that constant into the metric. Actually this constant becomes a fundamental part of the metric itself. Why can't we incorporate this constant into the differentials of the metric as well? </p>
<p>Here's the derivation: </p>
<p>$$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu} = 0$$</p>
<p>Assume the most general spherically symmetric metric solution to the above:</p>
<p>$$g_{\mu\nu} = \begin{pmatrix} -e^{2\alpha(r,t)} & 0 & 0 & 0\\ 0 & e^{2\beta(r,t)}&0&0\\0&0&r^2 & 0\\0 & 0 &0 & r^2\sin^2(\theta) \end{pmatrix}$$</p>
<p>Take the trace of both sides:</p>
<p>$$R + 2R = 0\\
\Rightarrow R = 0\\
R_{\mu\nu} - \frac{1}{2}(0)= 0\\
R_{\mu\nu}=0$$</p>
<p>We are free to say:</p>
<p>$$R_{01} = \frac{2}{r}\dot{\beta}=0\\
\Rightarrow \dot{\beta} = 0\\
\therefore \beta = \beta(r)
$$</p>
<p>Now, we can take a time derivative of $R_{22}$,</p>
<p>$$\dot{R_{22}} = \frac{d\left(e^{-2\beta}[r(\beta'-\alpha') - 1] + 1\right)}{dt} = 0 = e^{2\beta}r\dot{\alpha}'$$
(Because $\dot{\beta} = 0$, $r$ is just a spacetime coordinate (and is therefore independent of other spacetime coordinates), and $\dot{\beta}' = \partial_r\partial_t\beta = \partial_r(0)$.)
$$\dot{\alpha}' = 0\\
\therefore \alpha = f(r) + g(t)$$
because this is the only functional form for which $\partial_r\partial_t\alpha = \partial_t\partial_r\alpha = 0$. Now, the metric becomes:</p>
<p>$$g_{\mu\nu} = \begin{pmatrix} -e^{2f(r)}e^{2g(t)} & 0 & 0 & 0\\ 0 & e^{2\beta(r)}&0&0\\0&0&r^2 & 0\\0 & 0 &0 & r^2\sin^2(\theta) \end{pmatrix}$$</p>
<p><strong>HERE, at this point, we can rescale time: $dt \rightarrow e^{-g(t)}dt$, which makes the metric</strong>:</p>
<p>$$g_{\mu\nu} = \begin{pmatrix} -e^{2f(r)} & 0 & 0 & 0\\ 0 & e^{2\beta(r)}&0&0\\0&0&r^2 & 0\\0 & 0 &0 & r^2\sin^2(\theta) \end{pmatrix}$$</p>
<p>Setting $R_{11} = R_{00}$, we find:</p>
<p>$$e^{2(\beta-\alpha)}R_{00} + R_{11} = 0\\
\frac{2}{r}\alpha' + \frac{2}{r}\beta' = 0\\
\Rightarrow \alpha = -\beta + const.$$</p>
<p><strong>This constant is absorbed into $dr^2$.</strong></p>
<p><strong>HERE2,</strong> Now comes the crucial part, which I don't understand: looking back at $R_{22} = 0$, we have </p>
<p>$$(re^{2\alpha})' = 1\\
\int (re^{2\alpha})'= \int 1\\
re^{2\alpha} = r + \underbrace{\mu}_{\text{Constant of integration}}\\
e^{2\alpha} = 1 + \frac{\mu}{r}$$</p>
<p>So the metric finally ends up being:</p>
<p>$$ds^2 = -(1 + \frac{\mu}{r})dt^2 + (1 + \frac{\mu}{r})^{-1}dr^2 + r^2 d\Omega^2.$$</p>
<p><strong>Why can't $\frac{\mu}{r}$ be absorbed into $dt^2$ and $dr^2$ like $g(t)$ and the other integration constant in this derivation were?? I don't understand the distinction at all, any help would be greatly appreciated!!</strong></p> | 1,688 | [
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<p>Intuitively it's easy to accept that the usual variables like temperature, internal energy, etc. are 'macroscopic', but does there exist a formal definition of a macroscopic variable? </p>
<p>In other words, is there a clear way to separate the set of all observables (and functions of observables) on a system into ones we would describe as 'macroscopic' and ones we would not?</p>
<p>EDIT: Since apparently the answer is not completely straightforward, I'm interested in hearing any definitions which have appeared in literature, even if they are only conventions. I'm also interested in any necessary or sufficient conditions.</p> | 1,689 | [
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<p>I'm reading Arnold's "Mathematical Methods of Classical Mechanics" but I failed to find rigorous development for the allowed forms of Hamiltonian.</p>
<p>Space-time structure dictates the form of Hamiltonian. Indeed, we know how the free particle should move in inertial frame of references (straight line) so Hamiltonian should respect this.</p>
<p>I know how the form of the free particle Lagrangian can be derived from Galileo transform (see Landau's mechanics).</p>
<p>I'm looking for a text that presents a rigorous incorporation of space-time structure into Hamiltonian mechanics. I'm not interested in Lagrangian or Newtonian approach, only Hamiltonian. The level of the abstraction should correspond to the one in Arnolds' book (symplectic manifolds, etc).</p>
<p>Basically, I want to be able to answer the following question: "Given certain metrics, find the form of kinetic energy".</p> | 1,690 | [
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<p>I'm interested in knowing whether sigma models with an $n$-sheeted Riemann surface as the target space have been considered in the literature. To be explicit, these would have the action \begin{align*}
S=\frac{1}{2}\int d^2x\, \left(\partial_a R\partial^a R+R^2\partial_a \theta {\partial}^a\theta\right),
\end{align*}
where $R$ and $\theta$ represent radial and angular coordinates on the target space respectively. Also, $\theta\sim \theta+2\pi n$ for an $n$-sheeted Riemann surface.</p>
<p>Has anyone seen anything like this? One thing that I would be particularly happy to see is a computation of the partition function.</p> | 1,691 | [
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<ul>
<li>Is it possible to measure or calculate the total entropy of the Sun?</li>
<li>Assuming it changes over time, what are its current first and second derivatives w.r.t. time? </li>
<li>What is our prediction on its asymptotic behavior (barring possible collisions with other bodies)?</li>
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<p>The <a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis" rel="nofollow">Mathematical universe hypothesis</a>, mainly by Max Tegmark and <a href="http://en.wikipedia.org/wiki/A_New_Kind_of_Science" rel="nofollow">A new Kind of Science</a>, mainly by Stephen Wolfram both claim (as least as I understand it) that at its innermost core reality <em>is</em> mathematics.</p>
<p>Can this statement be made more precise, i.e. what is the exact relationship between these two hypotheses?</p> | 1,693 | [
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<p>When using two component notation people often prefer to refrain from using arrows in Feynman diagrams to denote charge flow as is done in four-component notation. Instead, if understand correctly, they use arrows to denote chirality. I'd like to know what is the prescription to draw out the diagrams. I have read <a href="http://arxiv.org/abs/0812.1594" rel="nofollow">here</a> (pg. 39) that </p>
<blockquote>
<p>arrows indicate the spinor index structure, with fields of undotted indices flowing into any vertex and field of dotted indices flowing out of any vertices</p>
</blockquote>
<p>(see the reference above for many examples).
However, trying this out on Majorana and Dirac mass terms, this doesn't seem to be correct. A Majorana mass term, $\psi ^\alpha \psi_\alpha +h.c.$, is thus composed only of undotted indices. With the reasoning above, it should have two arrows pointing into the vertex,</p>
<p><img src="http://i.stack.imgur.com/8B0z7.png" alt="enter image description here"></p>
<p>However, I'm pretty sure that this is a Dirac mass, and not a Majorana mass. What am I missing?</p> | 1,694 | [
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<p><img src="http://i.stack.imgur.com/6UfAY.jpg" alt="enter image description here">
Laminar flow is a streamlined steady flow with a uniform gradient of velocity across the diameter of the pipe. I am familiar with the elementary treatment of laminar flow, like basic velocity profile, shear stresses etc. But my question is:- </p>
<blockquote>
<p>The velocity of the fluid particles at the edges is zero while it is
maximum at the centre. This means, that in a specified interval of
time, the centre bulges out more than the progress of the
surrounding regions, while the fluid at the edges will remain
stationary. <strong>Then, with passage of time, the centre should continue to
bulge out relative to the rest of the flowing fluid, causing a very
steep spike in the middle of the cylindrical flow. Does this really happen?</strong>
It seems very counterintuitive and against everyday observations, if
the centre does bulge out creating a steep "pyramidal" (parabolic, to be rigourous) shape of the
leading surface. Also, <strong>after a definite bulge, should not gravity effects distort the shape?</strong></p>
</blockquote> | 1,695 | [
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<p>I'm currently studying field theory and I'm having some trouble with conserved charge given in field components. If we have a complex scalar action of a field $\phi=(\phi_1,\phi_2)^T$ that is</p>
<p>$$S[\phi]={\int}\left[ \partial^\mu\phi^\dagger\partial_\mu\phi-m^2\phi^\dagger\phi-\frac{1}{2}\lambda(\phi^\dagger\phi)^2\right]d^4x.$$</p>
<p>In a global $SU(2)$-symmetry when $\phi$ belongs to $SU(2)$ two dimensional representation $$\phi\rightarrow\phi'=g\phi,\quad g=e^{i\chi_iT^i},$$</p>
<p>where $T^i$ is the generator chosen as $T^i=\frac{1}{2}\tau^i$ and $\tau^i$ are Pauli matrices. Now I've gotten that the conserved charges are</p>
<p>$$Q_i=-i\int d^3x\:\phi^\dagger \overset{\leftrightarrow}{\partial^0}(T^i\phi).$$ </p>
<p>Now the next part says that written with the components of the field these should produce</p>
<p>$$Q_1=\mathrm{Im}\int d^3x\:\phi_2^\dagger \overset{\leftrightarrow}{\partial^0}\phi_1,\quad Q_2=\mathrm{Re}\int d^3x\:\phi_2^\dagger \overset{\leftrightarrow}{\partial^0}\phi_1, \quad Q_1=\frac{i}{2}\int d^3x\:(\phi_2^\dagger \overset{\leftrightarrow}{\partial^0}\phi_2-\phi_1^\dagger \overset{\leftrightarrow}{\partial^0}\phi_1)$$</p>
<p>Now I have no idea on how to get these. Why does the generator $T^i$ vanish for example?</p> | 1,696 | [
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<p>The string theory landscape seems to this outside observer to be an intermediate step in the intellectual progress toward a more robust theory that explains why our one universe has the particular properties that it has. is this the majority opinion or do most string theorists view the landscape as a plausibly being included in the final form of the theory?</p> | 1,697 | [
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<p>I'm working through some exam problems, and I came across this one - the solution of which baffles me considerably.</p>
<blockquote>
<p>A two-dimensional jet emerges from a narrow slit in a wall into fluid which is at rest. If the jet is thin, so that velocity $\vec u = (u, v)$ varies much more rapidly across the jet than along it, the fluid equation becomes:<br />
$u\frac{\partial u}{\partial x} +v\frac{\partial u}{\partial y} = \nu \frac{\partial^2 u}{\partial y^2}$<br />
where constant $\nu$ is the viscosity coefficient. The bounday conditions are that the velocity and its derivatives tend to zero as we leave the jet (that is as $|y|\rightarrow\infty$) and $\frac{\partial u}{\partial y}$ at $y = 0$, as the motion is symmetrical about the x-axis.</p>
</blockquote>
<p>The first question involves integrating across the jet to show that $\int u^2 dy$ is $x$ independent. So you end up with three integrals (subscripts denoting derivatives),</p>
<p>1) $\int u u_x dy$ <br/>
2) $\int v u_y dy$<br/>
3) $\nu\int u_{yy} dy$<br/></p>
<p>Somehow, for 1), you can write $\int u u_x dy = \frac{1}{2}\partial_x\int u^2 dy$ - is this an identity?</p>
<p>Secondly for 2) the solution states that: $\int v u_y dy = -\int v_y u dy$ from which you use the incompressibility condition to give $\int u_x u dy = \frac{1}{2}\partial_x\int u^2 dy$. Where does $vu_y=-v_yu$ come from, and again, is there an identity used in the final step?</p>
<p>And thirdly, this is a smaller issue but again, one that confused me a bit, the second question supposes that the streamfunction is self-similar and takes the form: $\psi = x^a f(\eta)$, $\eta = yx^b$. Which is fine, sub in for $u^2 = \psi_y^2$ and equate the power of the factor of $x$ that comes out to be zero. However, in the solution, it comes out as $2a = -b$ and I don't see how that works. </p>
<p>Suppose you take: $\eta = yx^b$, differentiate to get $\frac{d}{d\eta}\eta = \frac{dy}{d\eta}x^b$ so that $dy = d\eta x^{-b}$. When that's substituted into the integral, the factor that comes out is $x^{2a-b}$.</p>
<p>I have noticed some errors in the solutions, so just checking it's me, not them!</p>
<p>I'm sure this is a simple problem once you see the trick, thanks very much!</p> | 1,698 | [
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<p>In quantum mechanics, uncertainty principle states that we can only measure the quantity of spin in one axis but not others.</p>
<p>Then what about in superstring theory? As quantum mechanics is basically three-dimensional world, this does make sense, but superstring theory adopts more than three-dimensional space...</p> | 1,699 | [
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<p>If I had a mass of $100\:\rm{kg}$ accelerating due to gravity, using $F=ma$:</p>
<p>$F = 100\:\rm{kg} \times 9.8\:\rm{m/s^2}$</p>
<p>$F = 980 \:\rm N$...</p>
<p>If I increased the mass to 200kg, the force would be 1960 N:</p>
<p>$F = 200\:\rm{kg} \times 9.8\:\rm{m/s^2}$</p>
<p>$F = 1960 \:\rm{N}$</p>
<p>Now, finally getting to my question: Does this increase in force (which is supposed to be a push/pull) mean that the object would fall faster when it weighs more?</p> | 1,700 | [
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<p>The usual explanation of spontaneous radiation is that the energy eigenstates are perturbed by QED interaction, so that the eigenstates obtained from single-particle QM are no longer eigenstates of the full Hamiltonian, and in turn different (original) eigenstates can mix after some time-evolution. However I'm not quite convinced, why in the first place the states must be in single-particle Hamiltonian eigenstates instead of full Hamiltonian?</p>
<p>EDIT:I need to rephrase it a bit since the answers reflect that my statement has been misleading. I'm not questioning the fundamental principle of superposition, I'm more concerned about the phenomenological fact that spontaneous emissions always happen experimentally after you excite an atom(if I'm not mistaken). But isn't it possible to excite the atom to an QED eigenstate such that it stays there forever?</p> | 1,701 | [
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<p>I believe the answer to be yes, but I realize that sometimes physicists place additional constraints that might not be obvious. If superalgebras are clifford algebras, why make a literary distinction?</p> | 1,702 | [
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<p>Does wave-function collapse cause the entropy of the atom (ie. the sub-atomic particle system that makes up the atom) to increase? </p> | 1,703 | [
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<p>I have been searching for a straight forward answer to this question for ages now and it is driving me crazy.</p>
<p>Here is what I know:
If an object is moving at a constant speed the force of friction must equal the applied force (Please assume the applied force is horizontal) and for it to be accelerating or decelerating, the force of friction and the applied force must be uneven.</p>
<p>I know that FK = uK * FN</p>
<p>I have the applied force, the normal force and the coefficient of friction.</p>
<p>This is what I do not understand:
If in order to accelerate the applied force must be greater than the friction, how would that work? because surely the object would continue accelerating to infinity?
Or does the friction force change to level out and equal the applied force to change the objects state from accelerating to constant velocity?
If so, how do i work out the rate of the change in the friction force, surely it does not instantly change otherwise acceleration cannot occur.</p>
<p>Here is a sample situation that may help me understand it better if it is answered</p>
<p>Say I have a box that has a mass of 10 kg and I apply a force horizontally to the box of 50N and the coefficient of kinetic friction is 0.5.
How long does it take for the box to finish accelerating and reach a constant velocity? And what is the rate of friction force change?</p>
<p>Then say I suddenly increase the force I am applying to 60N what is the rate of change of the friction force and how long does it take to reach constant velocity?</p>
<p>To recap my questions are:
1. Does the object accelerate to infinity or does the friction force change to equal the applied force?
2. If the friction force does change, how do I work out the rate of change and how long it takes?
3. In the situation above, both times a new force is applied how do I work out change in friction force and how long it takes? Could you also show working so I can understand it properly.
4. If the above is way off and the friction does not change how do I work out the change of velocity over time using friction depending on a change of the applied force?</p>
<p>Sorry for the detailed question I just want to make sure I understand the answer properly :)</p>
<p>PS A simple solution is all I need as I am using this in a simple physics engine for a game.</p>
<p>Thanks, Arch</p> | 1,704 | [
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<p>In a specific date what law gives us perfect measurements and how will we measure if latitude is given?</p> | 28 | [
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<p>Say there were 2 objects with certain masses (e.g. $m_1$ and $m_2$). If they were close together gravity would attract the 2 objects. If they were a large distance apart the expansion of the universe (dark energy) would pull them apart. What is the distance apart that they must be (I assume that it is relative to their masses) in order for gravitational attraction and dark energy repulsion to cancel out, so that the objects remain motionless?</p> | 1,705 | [
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<p>I read a saying in wiki of asymptotically flat spacetime
<a href="http://en.wikipedia.org/wiki/Asymptotically_flat_spacetime" rel="nofollow">http://en.wikipedia.org/wiki/Asymptotically_flat_spacetime</a></p>
<p>"In general relativity, an asymptotically flat vacuum solution models the exterior gravitational field of an isolated massive object. Therefore, such a spacetime can be considered as an isolated system: a system in which exterior influences can be neglected."</p>
<p>So my question is :"Is the spacetime generated by isolated system always asymptotic flat? "
I remember that there exist vaccum solutions that are not asymptotically flat.
So can an isolated system generate a vaccum solution outside the system that is not asymptotically flat? Is it possible that although two objects are far enough away from each other, the gravitational effect cannot be neglected?</p>
<p>For example,
$$ds^2=-2xydt^2+2dtdz+dx^2+dy^2$$
the determinant is $-1$ for all $t,x,y,z$, the Riemann Tensor is constant but not zero in all spacetime. And the Ricci tensor is zero in all spacetime. So this is a vaccum solution even in global spacetime, while it is still not a asymptotically flat spacetime. So why we can say that an object far away from other object can be treated as an isolated system, i.e. can neglect the influence by others? </p> | 1,706 | [
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<p>I've read somewhere that one does not need to prove Lorentz invariance of the Maxwell
equations
$F_{\mu\nu,\sigma}+F_{\nu\sigma,\mu}+F_{\sigma\mu,\nu}=0$
because it is "manifestly Lorentz invariant" or "because they are tensor equations"? What is meant by that? I've read that this could mean that space and time are treated "on equal footing". How can this replace a mathematical proof?</p> | 1,707 | [
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<p>I know the wave nature of electrons was evoked to explain why atoms are stable but I thought waves could be put in the same state like photons yet electrons can not exist in the same state.</p> | 1,708 | [
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<p>My understanding about conservative force is a force that its work is independent of path such that we can construct another form of the work called potential to make our life easier.</p>
<p>For friction, if I start from microscopic point of view, it should be the macroscopic effect of the electric force or gravity which are both conservative force.</p>
<p>Why do we initially have description by conservative forces (electric/gravity force) but end up with a macroscopic description of nonconservative force(friction)?</p> | 1,709 | [
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<p>In <a href="http://www.if.ufrj.br/~mbr/warp/etc/cqg15_2523.pdf">this paper</a>, D. H. Coule argues that warp drive metrics, like the one proposed by Alcubierre, require the exotic matter to be laid beforehand on the travel path by conventional travel. At section 5 of this paper "Alteration of the light-cone structure" he basically goes for the same trick that Alcubierre did: write some metric up from his sleeve with the desired properties, and see what properties the energy-stress tensor needs to have in order to satisfy Einstein equations. The metric he writes describes a spacetime with a bigger effective speed of light in the $x$ axis. He explains that this demands a $T_{\mu \nu}$ that violates the dominant energy condition. </p>
<p>Now, if you head to the variable speed of light wikipedia page and head to the <a href="http://en.wikipedia.org/wiki/Variable_speed_of_light#The_varying_speed_of_light_cosmology">cosmology section</a>, none of the papers by Magueijo or Albrecht rely on a mechanism of this sort to obtain variable speed of light; They go instead by the arguably harder route of proposing alternative lagrangians or theories. I'm wondering if either there is something so blatantly wrong the Coule metric alteration that the VLS authors didn't dignify it with a rebuttal, or if they simply ignore this simple mechanism to stretch light cones?</p>
<p>We know that for explaining cosmic inflation, we need a scalar field that violates the dominant energy condition, but if this scalar had important rotational anisotropies, such a model (if correct) would imply that far-away parts of the primordial universe might have been causally connected after all (at least in some directions that would change from place to place) which might explain a lot of the long scale uniformity we are seeing? isn't this a better start for a simpler explanation that uses well-known gravitational theory?</p> | 1,710 | [
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<p>Suppose we have a man traveling in an open car (roof open) with speed $v$ towards right (man faces right). He throws a stone (mass $m$) towards right, in his frame-forward with speed $V$. </p>
<p>In the car's frame, the total energy imparted to it was $$E=E_f-E_i=\frac{1}{2}mV^2$$<br>
In the ground frame, the total energy given to it was $$E=E_f-E_i=\frac{1}{2}m((v+V)^2-v^2)=\frac{1}{2}m(V^2+2Vv)$$
What part am I getting wrong on? Is it okay to get such difference?</p> | 1,711 | [
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<p>In QED, one can relate the two-particle scattering amplitude to a static potential in the non-relativistic limit using the Born approximation. E.g. in Peskin and Schroeder pg. 125, the tree-level scattering amplitude for electron-electron scattering is computed, and in the non-relativistic limit one finds the Coulomb potential. If one allows for 1/c^2 effects in the non-relativistic expansion, one also finds spin-dependent interactions (e.g. spin-orbit, see Berestetskii, Lifshitz, Pitaevskii pg. 337).</p>
<p>Are there any alternative methods for calculating a two-particle non-relativistic potential? </p> | 1,712 | [
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<p>This is related to a <a href="http://physics.stackexchange.com/questions/57733/hamiltonian-of-polymer-chain">question about a simple model of a polymer chain</a> that I have asked yesterday. I have a Hamiltonian that is given as:</p>
<p>$H = \sum\limits_{i=1}^N \frac{p_{\alpha_i}^2}{2m} + \frac{1}{2}\sum\limits_{i=1}^{N-1} m \omega^2(\alpha_i - \alpha_{i+1})^2 $</p>
<p>where $\alpha_i$ are generalized coordinates and the $p_{\alpha_i}$ are the corresponding conjugate momenta. I want to find the equations of motion. From Hamilton's equations I get</p>
<p>$\frac{\partial H}{\partial p_{\alpha_i}} = \dot{\alpha_i} = \frac{p_{\alpha_i}}{m} \tag{1}$</p>
<p>$- \frac{\partial H}{\partial {\alpha_i}} = \dot{p_{\alpha_i}} = -m \omega^2 (\alpha_i - \alpha_{i+1} ) \tag{2}$</p>
<p>, for $i = 2,...,N-1$. Comparing this to my book, (1) is correct, but (2) is wrong. (2) should really be</p>
<p>$- \frac{\partial H}{\partial {\alpha_i}} = \dot{p_{\alpha_i}} = -m \omega^2 (2\alpha_i - \alpha_{i+1} - \alpha_{i-1}) \tag{$2_{correct}$}$</p>
<p>Clearly, I am doing something wrong. I suspect that I'm not chain-ruling correctly. But I also don't get, where the $\alpha_{i-1}$ is coming from. Can anybody clarify?</p> | 1,713 | [
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<p>In acoustic metamaterials we have simultaneously negative bulk modulus, $\beta$, and effective mass density, $\rho$.</p>
<p>I understand how one can obtain a -ve $\rho$ by constructing a solid-solid system with vastly different speeds of sound, as this can be considered as a mass-in-mass system connected by springs see <a href="http://syen.ualr.edu/glhuang/papers/Hhhuang2010.pdf" rel="nofollow">here</a>.</p>
<p>But how to get a negative bulk modulus eludes me, I am aware that in doubly negative acoustic metamaterials the negative bulk modulus is achieved by having a sphere of water containing a gas i.e. bubble-contained-water spheres. But I can't see why this would result in a negative composite bulk modulus.</p>
<p>I mean obviously water has an extremely high bulk modulus and is practically incompressible, whilst air is highly compressible so external pressure on the system would fail to compress the water however the pressure would be transmitted to the gas which would compress, thus creating an extremely low-pressure region in between the gas and water (the water wouldn't expand due to it's large bulk modulus right?) and thus the gas would then re-expand and possibly exert pressure on the water - I suppose if this pressure exceeded the external pressure then the sphere might expand resulting in an expansion of the system upon application of external pressure and thus a negative bulk modulus???</p>
<p>I believe the answer may lie in <a href="http://pre.aps.org/abstract/PRE/v70/i5/e055602" rel="nofollow">this paper</a> and I will attempt to get my University vpn to work to see if I can access it. </p>
<p>Any information about acoustic metamaterials would be greatly appreciated.</p>
<p>P.S: How is metamaterials not an existing tag?!</p> | 1,714 | [
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<p>I've been doing some research into the analysis used in particle physics when determining the significance of a finding (e.g. the recent Higgs candidate was announced as a boson in the 125-126 GeV/$c^{2}$ mass range with a $5\sigma$ significance).</p>
<p>I believe this confidence level is determined by estimating the Standard Model background cross-section they should observe if all known processes except for Higgs production occurred and then taking the ratio of the observed cross-section with that predicted. </p>
<p>I am interested in how they determine the background cross-section. I believe that they use a Monte-Carlo simulation normalized to fit with well-known processes such as $Z^{0}Z^{0}$-production, but how exactly does this work?</p>
<p>I am aware that the tool primarily used in High-energy physics for this kind of modelling is <a href="http://geant4.cern.ch">Geant</a>, and I would like to know more about how this works. I have looked through the source code, but it is very fragmented and thus quite hard to understand, especially since I am not 100% certain what it is that should be occuring in the code.</p> | 1,715 | [
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<p>Does someone know the historical reason behind the difference in physical units between nautical and terrestrial miles?</p> | 1,716 | [
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<p>I have a system of two spin 1/2 particles in a superposition of spin states in the z-direction given by:</p>
<p>$\psi = \frac{1}{2} |+ +\rangle + \frac{1}{2} |+ -\rangle + \frac{1}{\sqrt{2}} |- -\rangle$ </p>
<p>where $+$ designates spin up, $-$ designates spin down and the first particle's state is the first term in each ket and the second particles' state is the second term in each ket. If I measure the spin on the first particle and get a value of $-\hbar / 2$ (corresponding to a spin down state) is the new state of the particles simply</p>
<p>$\psi = | - - \rangle$</p>
<p>meaning that the first particle is now "set" to being spin down? And if I determine the spin on the first particle to be spin up, would the subsequent state be</p>
<p>$\psi = \frac{1}{\sqrt{2}} |+ +\rangle + \frac{1}{\sqrt{2}} |+ - \rangle$ ?</p>
<p>Basically, my question is once I make a measurement of a spin of a particle, does the wavefunction stay collapsed on the spin determined? And does having a second particle affect this in any way?</p> | 1,717 | [
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<p>In a <a href="http://radio.seti.org/episodes/Skeptic_Check_Science_Blunders" rel="nofollow">recent episode</a> of the <em>Big Picture Science</em> podcast, there was an interview with <a href="http://www.columbia.edu/cu/biology/faculty-data/stuart-firestein/faculty.html" rel="nofollow">Stuart Firestein</a> (chair of the Columbia University Biology Department) in which he discussed his book <a href="http://rads.stackoverflow.com/amzn/click/0199828075" rel="nofollow"><em>Ignorance: How It Drives Science</em></a>. In particular, he mentioned that for the most part he dislikes hypotheses---they tend to pigeonhole people's idea of what "good data" is, so that if when unexpected happens, it's often thrown out as "bad data".</p>
<p>Is this a problem in physics? If so, how much of a problem is it, and what, if anything, do physicists do to attempt to mitigate it? </p>
<p>My instinct tells me that this is a problem, albeit possibly not as much as in so-called "soft" fields such as psychology and sociology where eliminating variables and bias is much more difficult (at least I assume it's much more difficult).</p> | 1,718 | [
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<p>I have a question please about <a href="http://en.wikipedia.org/wiki/Renormalization" rel="nofollow">renormalization</a> in QFT. Why a renormalizable theory requires only a finite number of counter-terms?</p> | 1,719 | [
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<p>What is the force between two perpendicular wire carrying current, one to the north and one to the east?</p> | 1,720 | [
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<p>I'm currently working on a problem which is really giving me some issues.</p>
<p>The problem concerns the force required to expel water from a syringe. We have a 20 ml syringe (which is $2\times10^{-5}$ meters cubed) with a diameter of 1 cm, full of water. The needle of the syringe is 40 mm in length and has a diameter of 0.2 mm. All of the water must be expelled from the syringe in 20 s. How much force must be applied to the syringe head to achieve this?</p>
<p>Ordinarily this is fine, but we have to include the pressure loss as a result of the friction in the needle. I'm using the Darcy–Weisbach equation to determine this. I calculated the speed the fluid needs to flow at by dividing the flow rate by the cross-sectional area of the needle. I've used a Moody chart to get $f_D$ as 0.046, and I'm using $\rho = 998.21$. I'm guessing the pressure loss in the needle is therefore
$$0.046\times\frac{0.04}{0.002}\times\frac{998.21\times31.8^2}{2} = 4.64\,\mathrm{MPa}$$
Is that correct? In which case, how do I now get to the force from here?</p> | 1,721 | [
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<p>Consider Faraday's flux law for the EMF generated in a conductor loop:</p>
<p>$$ \varepsilon = - \frac{d \phi}{dt},$$</p>
<p>where $\varepsilon$ is the EMF, and $\phi$ is the magnetic flux through the loop.<p></p>
<p>There are two possible causes for the flux to variate over time: variations in the magnetic field ("transformer EMF") and variations in the area enclosed by the loop ("motional EMF").<br>
Feynman has noted that this is a unique case where a single rule is explained by two different phenomena:</p>
<blockquote>
<p>We know of no other place in physics where such a simple and accurate general principle requires for its real understanding an analysis in terms of <em>two different phenomena</em>. Usually such a beautiful generalization is found to stem from a single deep underlying principle. Nevertheless, in this case there does not appear to be any such profound implication.<br>
—Richard P. Feynman, <em>The Feynman Lectures on Physics</em> (Volume II, 17-2). </p>
</blockquote>
<p><strong>Is it really true that there is no way to view these two phenomena as one?</strong><br>
For example, it says in <a href="http://en.wikipedia.org/wiki/Electromagnetic_induction" rel="nofollow">this Wikipedia article</a> that this apparent dichotomy was part of what led Einstein to develop special relativity.<br>
<strong>Does special relativity give us a unifying principle to derive Faraday's law from?</strong></p> | 1,722 | [
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<p>I'm curious about the radiant intensity distribution of pulsars: what's the general dependence of intensity on angle, and what are typical angular beam widths? How much does the beam width vary between pulsars? (Presumably this is tied to magnetic field strength.)</p>
<p>Even something as simple as a very sketchy plot of intensity vs. angle would be great. It's easy enough to find plots of observed intensity vs. time for individual pulsars, but it takes a bit to get from those to the distribution at the source.</p> | 1,723 | [
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<p>Given enough time, where are the Voyager spacecrafts heading? (Assuming some alien civilization doesn't pick them up.)</p>
<p>Will they pass by any interesting stars on the way to the black hole at the center of our galaxy or will it perhaps leave the galaxy?</p>
<p>What are the highlights on their journey that we can reasonably predict?</p> | 1,724 | [
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<p>Various websites today are reporting with photos and videos of <a href="http://en.wikipedia.org/wiki/C/2011_W3_%28Lovejoy%29" rel="nofollow">Comet Lovejoy</a>. However, I can't seem to find a definition of which direction to look for it tomorrow morning. I'm in <a href="http://en.wikipedia.org/wiki/Christchurch" rel="nofollow">Christchurch</a>, <a href="http://en.wikipedia.org/wiki/New_Zealand" rel="nofollow">New Zealand</a> (roughly 43°S 173°E).</p> | 1,725 | [
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<p>If an effective field theory has a chiral anomaly it means that chiral symmetry isn't a symmetry of the underlying theory which has been cut off to make the EFT. My question is whether there's a good example where this can seen explicitly. The kind of thing I'm picturing would be an EFT at cutoff A which has no chiral symmetry, while the same EFT at cutoff B << A has chiral symmetry which is anomalous. </p> | 1,726 | [
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<p>From what I've learned, the more an object travels closer and closer to the speed of light, the more time will slow down for that object.. at least from an outside perspective.. </p>
<p>It was shown that atomic clocks run slower in high speed orbit than clocks on earth.. I assume that the rate of radioactive decay (for example) is also slowed down at high speeds (correct me at any time, please). </p>
<p>We are moving through space right now at <a href="http://www.brainstuffshow.com//blog/good-question-how-fast-are-you-moving-through-the-universe-right-now/" rel="nofollow">760 miles per second</a> (0.40771% the speed of light), which I can only assume is <em>our</em> current "cosmic clock", which also regulates how fast radioactive decay happens on earth (if we continue with that example). </p>
<p>When an astronaut is traveling at high velocity, his/her velocity is being added to the overall velocity of our galaxy moving through space, right?</p>
<p>So my question is this:</p>
<p>What will happen if an object were to stay completely <em>stationary</em> in space-time? Far away from any galaxy.. Will time go infinitely fast for that object? Will it instantly decay? </p>
<p>Since space is expanding, I realize you can't really stay "stationary".. but I mean: not having velocity of moving <em>through</em> space.</p>
<p>Thanks :)</p> | 1,727 | [
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<p>I'm trying to understand the derivation of the angular momentum commutator relations. How is</p>
<p>$$[zp_y, zp_x] ~=~ 0?$$ </p>
<p>How is</p>
<p>$$[yp_z, zp_x] ~=~ y[p_z, z]p_x?$$ </p> | 1,728 | [
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<p>My question is: <strong>is there a simple and truly general equation for the resistance between two electrical equipotential surfaces?</strong>. Obviously, if so, what is it, and if not, why? It would be very difficult to solve, granted, but I just want to see a calculus equation that is fully descriptive. I have two frameworks under which this could be entertained, I'll write those out and then explain the motivation.</p>
<p>To start with, we need propose that the volume separating the two surfaces has a volumetric resistivity, $\rho$ in units of $(\Omega m)$.</p>
<h1>Single Volume Framework</h1>
<p>We can limit the discussion to a defined volume, then the surfaces reside in that volume or on the surface of it. This volume may have a constant resistivity $\rho$ while everywhere outside the volume is completely electrically insulating.</p>
<h1>Infinite Volume Framework</h1>
<p>An alternative to the above approach that might make the task more or less difficult would be to replace a constant resistivity with a spatial dependence $\rho(\vec{r})$ and no longer require a boundary condition. In that case we only have 3 mathematical inputs to the problem, which is the resistivity defined for all $\vec{r}$ and a definition of the two surfaces, $S_1$ and $S_2$.</p>
<h1>Known Algebraic Analogs</h1>
<p>The <a href="http://en.wikipedia.org/wiki/Electrical_conductivity" rel="nofollow">basic algebraic formulation</a> that I find insufficient is:</p>
<p>$$R = \rho \frac{\ell}{A}$$</p>
<p>Where $l$ is the length of the restive material that is any shape which has translation symmetry over that length, and $A$ is the cross-sectional area. Obviously, this is a rather simple equation that won't apply to more complicated geometry. Even more <a href="http://iopscience.iop.org/0143-0807/30/4/L01" rel="nofollow">sophisticated academic sources</a> seem to give equations that fall short of what I'm asking. For example:</p>
<p>$$R = \rho \int_0^l \frac{1}{A(x)} dx$$</p>
<p>I think it's obvious that an equation such as this is built upon a myriad of assumptions. For a thought experiment, imagine that the area starts out as very small and then pans out to very large quickly. Well, accounting for the larger area in the above sense underestimates the resistance, because the charge has to diffuse out perpendicular to the average direction of flow as well as parallel to it.</p>
<p>I have some reasons to suspect this might actually be rather difficult. A big reason is that all the approaches I'm familiar with require the flow paths to be established beforehand, which can't be done for what I'm asking. So maybe this will result in two interconnected calculus equations.</p>
<h1>Motivation</h1>
<p>I had an interest in <a href="http://courseweb.stthomas.edu/apthomas/SquishyCircuits/" rel="nofollow">Squishy Circuits</a>, and it occurred to me that I can't quickly and simply write down the equation for resistance between two points. The unique thing about Squishy Circuits is that it calls for two types of dough, one that conducts and one that is mostly insulating. However, the recipes aren't perfect and because of that, the young children who play with these circuits regularly encounter the limits of conductor and insulator definitions. If you make your conductor dough too long and/or too thin, you will encounter dimming of the light you connect with it. Similarly, a thin insulator layer will lead to a lot of leakage current which also dims the light.</p> | 1,729 | [
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<p>Suppose I want to launch a rocket from earth to some point $O$ between the center of earth and the center of moon (on a straight line connecting their centers), where the gravitational force of the moon 'cancels out' the gravitational force of the earth (this point is located at $\approx 54 R_E$ from the center of earth where $R_E$ is the radius of earth). I want to know how much energy I should spend in order for the rocket to get there (neglecting the atmosphere and the rotation of the earth around its axis). So, I know that this is basically the difference between the potential energy at the start point and at the end point of the destination. However, $O$ is located not only in the gravitational field of the earth, but also in the gravitational field of the moon. And it seems that I cannot neglect the gravitational potential energy of the body at the moon's gravitational field. So my question is - how can I combine these two? How can I calculate the total GPE of the body in two (or even more) intersecting gravitational fields?</p> | 1,730 | [
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<p>Are there physical theories in use, which don't fit into the frameworks of either <em>Thermodynamics</em>, <em>Classical Mechanics</em> (including General Relativity and the notion of classical fields) or <em>Quantum Mechanics</em> (including Quantum Field Theory and friends)?</p> | 1,731 | [
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] |
<p>As the <a href="http://en.wikipedia.org/wiki/Earth_wobble" rel="nofollow">Earth wobbles</a> during rotation, does the higher gravity at the equator tend to pull the moon toward an equatorial orbit even as the earth does that thousands of years wobble cycle? It would seem to me that the higher gravity, due to the larger diameter at the equator, would keep the Moon's orbit close to the equator. Or does the orbit of the moon with respect to the earth stay relatively stable with respect to the solar plane around the sun?</p> | 1,732 | [
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<p>What is the moment of inertia of a pizza slice that has a radius r, an angle (radians) of theta, and a height of h about the center point perpendicular to the cheese plane?</p> | 1,733 | [
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<p>How do I prove that $$\int \dot{x}^2 dt\geq \int \langle \dot{x}\rangle^2 dt $$</p>
<p>(i.e a free particle not in any external potential field movies with uniform veloctiy)</p> | 1,734 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/21360/cosmic-radiation-cutoff-at-low-energies">Cosmic radiation cutoff at LOW energies?</a> </p>
</blockquote>
<p>The energy spectrum of the cosmic radiation (not CMB) is limited on both sides.</p>
<p>I know about the GZK-cutoff at high energies. Basically, the interaction probability for photons of energies above 10^20 eV becomes so high that all have interacted before they can reach us.</p>
<p>But why is there a limit at lower energies? Earth's magnetic field, atmosphere, and/or radiation belt? Perhaps someone can explain that to me.</p> | 29 | [
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<p>I've been solving a problem in quantum mechanics, and I was deriving the standard deviation of $P$, knowing that $\langle P\rangle=0$. Because $\Delta P=\sqrt{\langle P^2 \rangle - \langle P \rangle ^2} = \sqrt{\langle P^2 \rangle}$, I was trying to calculate the expectation value of the square of the momentum. The wave function was given by $\psi(x)=\sqrt{\alpha}e^{-\alpha|x|}$ where $\alpha>0$. </p>
<p>Here is what I've done. $$\langle P^2\rangle = \int_{-\infty}^{\infty}\psi^*(x) \left(-\hbar^2 \frac{d^2}{dx^2}\psi(x)\right)dx = -\hbar^2\alpha^2$$
Now, we have negative expectation value of the square of the momentum, which I think is wierd, and we have to take square root of that value. That's impossible. I couldn't find out what's wrong with my idea. Can somebody help me with this?</p> | 1,735 | [
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] |
<p>There are a bunch of stars orbiting the black hole in the center of <a href="http://en.wikipedia.org/wiki/Milky_Way" rel="nofollow">our galaxy</a>.</p>
<p>These stars move at huge speed. Why do we see this? Why do the black hole not impose any noticeable time dilation on these stars?</p> | 1,736 | [
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<p>Let's say I light a wall with two spotlights: One red and one green one. Where they overlap, I'll see a yellow area at the wall.</p>
<p>My question is, whether this is caused by an modification of the frequency/wavelength or simply by my eye combining the two incoming lights.</p>
<p>Light is "added", wavelength is modified:
<img src="http://i.stack.imgur.com/6DOCu.png" alt="enter image description here"></p>
<p>The eye combines two separate lights:
<img src="http://i.stack.imgur.com/bdPjA.png" alt="enter image description here"></p> | 1,737 | [
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<p><a href="http://www.linkedin.com/profile/view?id=14132627&authType=NAME_SEARCH&authToken=Q-WY&locale=en_US&srchid=120ba6d2-2971-4f13-8f93-22815076ab52-0&srchindex=1&srchtotal=10&goback=.fps_PBCK_Jay+Wacker_%2a1_%2a1_%2a1_%2a1_%2a1_%2a1_%2a2_%2a1_Y_%2a1_%2a1_%2a1_false_1_R_%2a1_%2a51_%2a1_%2a51_true_%2a2_%2a2_%2a2_%2a2_%2a2_%2a2_%2a2_%2a2_%2a2_%2a2_%2a2_%2a2_%2a2_%2a2_%2a2_%2a2_%2a2_%2a2_%2a2_%2a2_%2a2&pvs=ps&trk=pp_profile_name_link" rel="nofollow">Jay Wacker</a><sup>1</sup> (professor of physics at the <a href="http://en.wikipedia.org/wiki/SLAC_National_Accelerator_Laboratory" rel="nofollow">SLAC National Accelerator Laboratory</a>) <a href="http://www.quora.com/Life/The-Earth-is-4-55-billion-years-old-and-our-galaxy-is-at-least-13-billion-years-old-Why-didnt-the-Earth-and-people-evolve-earlier-than-they-did" rel="nofollow">stated</a>:</p>
<blockquote>
<p>The first stars (known as <a href="http://en.wikipedia.org/wiki/Metallicity#Population_III_stars" rel="nofollow">Pop III</a>) were made out of hydrogen and helium. They had no planets.</p>
</blockquote>
<p>Why couldn't they have had <a href="http://en.wikipedia.org/wiki/Gas_giant" rel="nofollow">gas planets</a>?</p>
<p><sub>[1] Requires login at <a href="http://en.wikipedia.org/wiki/LinkedIn" rel="nofollow">LinkedIn</a></sub></p> | 1,738 | [
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] |
<p>My son got an orion starblast 4.5 for Christmas. It comes with orion explorer II 17mm and 6mm eyepieces. We are looking at some additional accessories and wondering what you would recommend as "first accessories" to get the most out of the telescope. Our initial inclination is towards:</p>
<ol>
<li>A barlow lens. Probably the orion shorty 2x barlow. Is the shorty-plus twice as good? (It is twice the price.)</li>
<li>A 25mm eye piece. From what we've read, a reflector with these specs is best for wide field views.</li>
<li>A solar filter. We think it would be cool to look at the sun.</li>
</ol> | 1,739 | [
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<p>How can I determine whether the mass of an object is evenly distributed without doing any permanent damage? Suppose I got all the typical lab equipment. I guess I can calculate its center of mass and compare with experiment result or measure its moment of inertia among other things, but is there a way to be 99.9% sure?</p> | 1,740 | [
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<p>Let us only consider classical field theories in this discussion.</p>
<p>Noether's theorem states that for every global symmetry, there exists a conserved current and a conserved charge. The charge is the generator of the symmetry transformation. Concretely, if $\phi \to \phi + \varepsilon^a\delta_a \phi$ is a symmetry of the action, then there exists a set of conserved charges $Q_a$, such that
$$
\{Q_a, \phi\} = \delta_a \phi
$$</p>
<p>If the symmetry is a gauge symmetry, then no such conserved current or charge exists (although certain constraints can be obtained via Noether's second theorem). However, I have the following question</p>
<blockquote>
<p>Is there a quantity that, like $Q_a$ generates the gauge transformation?</p>
</blockquote>
<p>This quantity need not be conserved. To be precise, if an action is invariant under local transformations $\phi \to \phi + [\varepsilon^a(x) \delta_a] \phi$. Here, heuristically, the gauge transformation might look like $[\varepsilon^a \delta_a] \phi \sim d \varepsilon + \varepsilon \phi + \cdots$ where $d \varepsilon$ is heuristically some derivative on $\varepsilon$. Does there exist a quantity such that
$$
\{ Q_{\varepsilon}, \phi \} = [ \varepsilon^a \delta_a] \phi
$$
I know there exist such charges in some theories, and I have explicitly computed them as well. I was wondering if there is a general formalism to construct such quantities?</p>
<p>PS - I have some ideas about this, but nothing quite concrete. </p> | 1,741 | [
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<p>Consider the following question:</p>
<blockquote>
<p>In which of the following situations would a force be exerted on an object and no work be done on the object? </p>
<p>I. a centripetal force is exerted on a moving object</p>
<p>II. a force in the opposite direction as the object is moving</p>
<p>III. a force is exerted on an object that remains at rest</p>
<p>A.) I only </p>
<p>B.) I and II </p>
<p>C.) I and III </p>
<p>D.) II and III </p>
<p>E.) I, II, and III</p>
</blockquote>
<p>My response to this question is only II, but this is not a choice. I do not understand how I or III could have work. My logic is that in I, the force is centripetal and therefore is not parallel to the object's path, in any case it would be perpendicular because centripetal is towards the center. Work must be parallel. In III, if an object remains at rest, $d = 0$. And $W = fd$, so $W = 0$. </p>
<p>I can't find the flaw in my logic. Any help is appreciated!</p>
<p>Thanks!</p> | 1,742 | [
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<p>At the mean-field level, the dynamics of a polariton condensate can be described by a type of nonlinear Schrodinger equation (Gross-Pitaevskii-type), for a classical (complex-number) wavefunction $\psi_{LP}$. Its form in momentum space reads:</p>
<p>\begin{multline}
i \frac{d}{dt}\psi_{LP}(k) =\left[\epsilon(k)
-i\frac{\gamma(k)}{2}\right] \psi_{LP}(k)
+F_{p}(k)\,\, e^{-i\omega_{p}t} \\
+ \sum_{q_1,q_2} g_{k,q_1,q_2}\, \psi^{\star}_{LP}(q_1+q_2-k)
\, \psi_{LP}(q_1)\, \psi_{LP}(q_2).
\end{multline}</p>
<p>The function $\epsilon(k)$ is the dispersion of the particles (polaritons). The polaritons are a non-equilibrium system, due to their finite lifetime (damping rate $\gamma$). Therefore, they need continuous pumping with amplitude $F_p$ at energy $\omega_p$.
Finally, there exists a momentum-dependent nonlinear interaction $g_{k,q_1,q_2}$ that depends of the so-called Hopfield coefficients $X$ (simple functions of momentum) as:</p>
<p>\begin{equation}
g_{k,q_1,q_2}=g\, X^{\star}(k)\, X^{\star}(q_1+q_2-k)\, X(q_1)\, X(q_2)
\end{equation}</p>
<p>How can one transform the equation for $\psi$ to real-space?</p> | 1,743 | [
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<p>Most universities provide an experiment about the photoelectric effect to determine $h$ by measuring the stop voltage against the light frequency and calculating the slope $h/e$.</p>
<p>But mostly they also talk about the contact voltage, which just changes the offset, but not the slope.</p>
<p>I have several related questions regarding this contact voltage in combination with the photoelectric effect:</p>
<ul>
<li>Why do they talk about contact voltages at all if it's irrelevant for the slope?</li>
<li>Why are the photo cells build with two different materials at all? The same material would give no contact voltage and everything would be fine.</li>
<li>Books and Wikipedia tell me contact voltage is the voltage between two materials that are connected <em>directly</em>. But the photo cell's electrodes are not contacted directly, but normaly through copper cables and with several devices inbetween. Doesn't this change the contact voltage inpredictable?</li>
<li>Are there papers or books or something that explains contact electricity from ground up?</li>
</ul> | 1,744 | [
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<p>There are a lot of supersymmetric theories, and, sometimes,in the Lagrangian, there are interacting terms between bosonic and fermionic degrees of freedom, and sometimes not. Why ?</p>
<p>For instance, for the basic chiral superfield without superpotential, there is no interaction, while for 11-D supergravity, there is an interaction term between the (bilinear expression of) gravitinos and the strength 4-form.</p> | 1,745 | [
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<p>I have a very simple question with regard to numerical methods in physics.</p>
<p>I want to solve the eigenvalue problem for a particle moving in an arbitrary potential. Let's take 1D to be concrete. I.e. I want to find $(E,\psi(x))$ satisfying</p>
<p>\begin{align}
\left[-\frac{1}{2}\partial_x^2 + V(x) \right]\psi(x)=E \psi(x).
\end{align}</p>
<p>Now how do I do it exactly? Naively I would implement the following algorithm:</p>
<p>1) Pick some $E$. </p>
<p>2) I want to find $\psi(x)$ which is normalizable. So I could pick a large $L > 0 $, set $\psi(-L) = \epsilon > 0$ and $\psi'(-L) = \epsilon'>0$ and numerically integrate from there using the Schrodinger equation. </p>
<p>3) If I encounter a solution which is exponentially small far to the right of the origin, then I say the solution is normalizable (since it is decaying at $|x|\to\infty$), and I accept the pair $(E,\psi(x))$.</p>
<p>4) I increment $E \to E + dE$ and I repeat the process.</p>
<p>In doing so, I should get the spectrum around my starting value of $E$.</p>
<p>Does this algorithm actually work?? It also seems to me like a very uncontrolled way of doing it; I have no idea how accurate the spectrum is going to be. For example, would changing $L, \epsilon, \epsilon'$ make a difference?</p>
<p>The thing is, I know from Sturm-Liouville theory that the spectrum $E$ is going to be discrete (given $V(x)$ satisfying some nice properties). So the spectrum is going to be a set of measure 0 amongst the entire real line that $E$ lives in. This means that I'm almost surely (i.e. with probability 1) never going to get a solution that is normalizable, and whatever solution I try to numerically integrate from my starting point is always going to blow up having integrated far enough to the right.</p>
<p>So, what algorithm do people use to numerically obtain the spectrum and the eigenvalues? How do I also control accuracy of the spectrum generated?</p> | 1,746 | [
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<p>For example, imagine an object of a certain mass which is attached to a rope and is dangling in the air. The rope needs to have a certain tension to be able to hold the object up, that is, negate the effects of gravity pulling it down. I'm wondering, if you took the other end of the rope and started pulling it upwards, would the tension need to be higher to still keep the object from falling? For example if a helicopter is carrying an object attached to a rope under it, would the tension of the rope need to be different for when the helicopter is hovering in the air and when it's flying upwards?</p>
<p>I'm wondering this because it feels like that should be the case, especially when I think about how swinging the rope rapidly in circles might make the object detach and fly off, even if it would've stayed on had the rope not been in motion.</p> | 1,747 | [
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<p>What would be an experimental test of AdS/CFT correspondence? Or it's extensions?</p>
<p>I've heard that people are studying AdS/CMT (condensed matter) correspondence, but I don't know the details of it?</p>
<p>But in general, how would you test it?</p> | 1,748 | [
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<p>I am doing a simple project for a self-learn class with a friend to understand physics better over the summer.</p>
<p>We slide down a hill on a tarp on an ice-block and then half way down, we hit grass and we want to compare the friction between the grass and the tarp.</p>
<p>We calculated the angle of the hill as 9.7 degrees, the mass of myself and the ice-block is 63.5 kilograms. </p>
<p>How can I calculate the force of friction at this point since we do not have the coefficient of friction?</p> | 1,749 | [
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<p>In a forward-biased PN junction, the potential barrier decreases, allowing more majority carriers from one side to diffuse to the other side where they are minority carriers. After they cross the potential barrier, they form a diffusion current, the drift current of minority carriers is insignificant, then they recombine with majority carriers and form a drift current under the effect of the applied electric field.</p>
<p>Why do minority carriers form a diffusion current not a drift current after they cross the potential barrier? It is counter-intuitive that the main current is diffusion when there is an applied electric field.</p>
<p>This is according to all the microelectronics book I'm currently reading. There is one which says this can be proved but without providing anything. Can someone please provide a proof for this.</p> | 1,750 | [
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] |
<p>How was it discovered that the electric field of a negative charge points towards the charge itself? Is it true? </p>
<p><img src="http://i.stack.imgur.com/bh9uO.png" alt="field of a negative point charge"></p>
<p>(Courtesy of wikipedia)</p> | 1,751 | [
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<p>As we all know the speed of light is the limit at which energy/matter can travel through our universe. </p>
<p>My question being: is there a similar limit for acceleration? Is there a limit to how quickly we can speed something up and if so why is this the case?</p>
<p>Short question, not sure what else I can elaborate on! Thanks in advance!</p> | 1,752 | [
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<p>I am searching for not too old literature on the quantum description of unstable particles. I am referring to something beyond the ad-hoc S-matrix description based on the optical theorem common to textbooks such as those given by Peskin and Schröder or Weinberg etc. The book "Open Quantum Systems and Feynman Integrals" by Exner seems to go in this direction. But I find the formulation there very mathematical. Certainly it is possible to understand it but I am worried if I will get the connection to physics.</p> | 1,753 | [
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] |
<p>I am confused on few topics...</p>
<p>What is meant by "Frequency of Light"? Does the Photon(s) vibrate, that is known as its frequency? If the Photons vibrate, then they have a specific frequency, then What is meant by "Higher frequency light" as used in Photo-electric Effect? In which directions/axis do they vibrate to have a specific frequency. Why is it that nothing can go faster than the Speed of light. Does these two things have any relation to each other?</p>
<p>-Thanks.</p> | 1,754 | [
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<p>I have a simple question. I hope I don't get a stupid answer. Where does the magnetic field of a permanent magnet comes from AND why is it permanent (are we dealing with perpetual motion)?</p>
<p>This is what wiki is saying:
"The spin of the electrons in atoms is the main source of ferromagnetism, although there is also a contribution from the orbital angular momentum of the electron about the nucleus. When these tiny magnetic dipoles are aligned in the same direction, their individual magnetic fields add together to create a measurable macroscopic field."</p>
<p>What the wiki is saying is correct, but we have to think further or go one step further... So put your head out of the box:</p>
<p>THIS IS NOT THE ANSWER I WANT TO HEAR. Because then I can ask why is the electron spinning (or what is the origin of spin) and that for an infinit time (CRAZY)? This is of course a physics question......We are missing an important energy source, unless you know where this form of energy is coming from and why it is permanent.</p>
<p>Thank you very much!</p>
<p>Edit:</p>
<p>Wikipedia says: <strong>"Truly isolated systems cannot exist in nature..."</strong> Just be aware of this, before you are answering. (So before you are answering or commenting ==>THINK!)</p> | 1,755 | [
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<p>I know that this isn't the place for such basic questions, but I didn't find the answer to this anywhere else. It's pretty simple: some particle moves in straight line under constant acceleration from one point $x_0$ to another point $x_1$ during the time interval $\Delta t_1$. When the particle reaches the point $x_1$ it reverses it's movement and goes to another point $x_2$ during another time interval $\Delta t_2$. I want to determine $x_2$, however I don't understand how to do it. </p>
<p>My try was: Let $\Delta x_1 =x_1 -x_0$ be the first displacement and let $\Delta x_2 = x_2 - x_1$ be the second displacement. Then I can calculate two velocities:</p>
<p>$$v_1 = \frac{\Delta x_1}{\Delta t_1}$$</p>
<p>$$v_2 = \frac{\Delta x_2}{\Delta t_2}$$</p>
<p>My thought is then to find the acceleration as:</p>
<p>$$a = \frac{v_2-v_1}{\Delta t_2 + \Delta t_1}$$</p>
<p>But I'm not sure it'll work, since the movement reverses at $x_1$ and since I'm assuming the velocities constant on the intervals. </p>
<p>Can someone help me how to think with this problem, and how to solve it?</p> | 1,756 | [
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<p>When sitting in a gravity well, as we do on earth, does our effective mass become smaller than our rest mass due to having negative potential energy? Correspondingly, does a free falling mass (from infinity, radial path) have an effective mass equal to its rest mass, as here potential and kinetic energies are ballanced? The question is based on the concept of equivalence of energy and mass.</p> | 1,757 | [
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<p>I have a very simple mental picture that earthquake waves travel like shear (transverse) waves through the earth.</p>
<p>a. Does the speed of this wave give any valuable information about the mechanical properties of the geological medium?</p>
<p>b. Can this data be used to characterize mechanical properties of the earth?</p>
<p>c. Can the speed of this wave be estimated by recording the time taken for the
earthquake wave to reach sensors at different locations?</p> | 1,758 | [
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<p>I have been trying to understand "wave-particle duality" and other cases related to it. I am currently a college level student. I have few question which I am not getting answers clearly. </p>
<p>In double slit experiment, A particle behave like a wave, then how is "wave-particle duality" explained? I mean, If the particle behave like wave, then is it generating a wave or behaving as a wave? Is that wave going horizontally through slits(Double Slit Experiment) or vertically up and down or in which direction/axis the particle is vibrating to have a specific frequency? How does light behave as waves in it? and how does observer modify the experiment?</p>
<p>I may be thankful to you, If you clear my problems. I have read the theories on Wikipedia and other informational sites and tried to understand it.</p> | 1,759 | [
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<p>I'm writing up a lab report and have a question about the following formula</p>
<p>$$N = N_0e^{-\lambda t}$$</p>
<p>$N$ indicates the number of nuclei left after a time $t$ and $N_0$ indicates how much there was to begin with. In the experiment, we used a scintillation counter (consists of a crystal and a photomultiplier tube) to detect the gamma rays beginning emitted by a decaying metastable Ba-137 to Ba-137. After recording the "counts" $C$, we corrected for the background radiation counts $BG$ by calculating $C -BG$. Then we plotted $\ln(C-BG)$ vs $t$. We did this because from the above formula</p>
<p>$$C-BG = (C-BG)_0e^{-\lambda t}$$</p>
<p>Then,</p>
<p>$$\ln(C-BG) = \ln(C-BG)_0 - \lambda t$$</p>
<p>Thus a plot of $\ln(C-BG)$ vs $t$ yields a straight line with slope $-\lambda$. My question is, how does counts relate to $N$? I thought first formula above indicates the number of nuclei left. However, the detector is detecting gamma rays from decaying nuclei, i.e. counting how many nuclei are decay? </p>
<p>So why can I replace $N$ with counts?</p> | 1,760 | [
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<p>How does free electrons moving through a wire cause random vibrational motion of the positive ions?</p> | 1,761 | [
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<p>Can a laser powerful enough to cut(100w for example) be redirected using fiber optics ?
What issues need to be considered?</p>
<p>To be clear it should retain the ability to cut after exiting the fiber.</p> | 1,762 | [
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<p>Alice and Bob are moving in opposite direction around a circular ring of Radius $R$, which is at rest in an inertial frame. Both move with constant speed $V$ as measured in that frame. Each carries a clock, which they synchronize to zero time at a moment when they are at the same position on the ring. Bob predicts that when they next meet, Alice's clock will read less than his because of the time dilation arising because she is moving relative to him. Alice predicts that Bob's clock will read less with the same reason. They both can't be right. What's wrong with their arguments? What will the clocks really read?</p>
<p>I have try to answer it that their all wrong, Since the they are all moving at the same speed, and they will all cover the same distance ($\pi R$), so they will be at same place with the same time? but I am not sure about my reasoning,
Then the clock reading will be $\Delta t=\sqrt{1-\frac{V^2}{c^2}}\Delta t_B.$</p>
<p>Can any one give me a clear reasons on What's wrong with their arguments?</p> | 1,763 | [
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] |
<p>Temperature is disordered kinetic energy, with 0K being 0 Joules disordered kinetic energy, if I"m not mistaken. So, given required parameters(temperature, number of particles, mass of each particle, whatever is required) can you find the disordered kinetic energy of each particle?</p> | 1,764 | [
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<p>Are there any handy exercises about nuclear weapon design that are suitable for advanced undergrads in a nuclear physics or similar level physics course? I'm most curious about questions that actually appeared in textbooks or course homeworks.</p>
<p>It doesn't matter if it's about gun weapon design or implosion weapon design, but it should be suitably idealized so as to be a reasonable exercise for an individual homework assignment in some undergraduate course. </p> | 1,765 | [
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<p>I have a couple of conceptual questions that I have always been asking myself.</p>
<p>Suppose we have an electron and a proton at very large distance apart, with nothing in their way. They would feel each the other particle's field - however weak - and start accelerating towards each other.</p>
<p>Now:</p>
<p>1) Do they collide and bounce off? (conserving momentum) </p>
<p>2) Does the electron get through the proton, i.e. between its quarks?</p>
<p>3) Do both charges give off Brehmsstrahlung radiation while moving towards each other?</p>
<p>Different scenario:</p>
<p>Suppose I can control the two particles, and I bring them very close to each other (but they are not moving so quickly as before, so they have almost no momentum).
Then I let them go:</p>
<p>1) Would an atom be spontaneously formed? </p>
<p>2) If anything else happens: what kind of assumptions do we make before solving the TISE for an Hydrogen atom? Does the fact that the electron is bound enter in it?
This is to say: is quantum mechanics (thus solving the Schrödinger equation) the answer to all my questions here?</p> | 1,766 | [
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<p>If the universe is infinite, by virtue of chance it means that every possible configuration of matter must exist somewhere (according to <a href="http://www.youtube.com/watch?v=iBsywDWmwyA" rel="nofollow">this</a> documentary). </p>
<p>Therefore, if we accept that the universe is infinite and it's possible to travel through time and space near-instantaneously, by sheer chance there must be a version of me out there that can do so and that wants to contact me.</p>
<p>Since this has not happened yet, can I conclude that either near-instantaneous travel through time and space is impossible or that the universe isn't infinite?</p>
<p>On a broader scale: given the infinity of the universe, should certain occurences that occur everywhere at the same time not occur always throughout the universe? Or do such occurences simply not exist?</p> | 1,767 | [
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] |
<p>I am currently trying to learn some basic quantum mechanics and I am a bit confused. Wikipedia defines a <a href="http://en.wikipedia.org/wiki/Photon" rel="nofollow">photon</a> as a quantum of light, which it further explains as some kind of a <a href="https://en.wikipedia.org/wiki/Wave_packet" rel="nofollow">wave-packet</a>.</p>
<blockquote>
<p><em>What exactly is a quantum of light?</em></p>
</blockquote>
<p>More precisely, is a quantum of light meant to be just a certain number of wavelengths of light (something like "1 quantum = a single period of a sine wave" perhaps?), or is the concept completely unrelated to wavelengths? In other words, how much is a single quantum?</p> | 562 | [
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<p>Specifically: What are empirically well-understood examples of (integrable) Hamiltonian systems whose Hamiltonians include polynomial expressions, in the canonical coordinates $\{q^i,p_i\mid i=1,\ldots,n\}$, having degree greater than 2?</p>
<p>Below are follow up questions/replies in response to the comments/questions of Ron Maimon:</p>
<p>With respect to integrable motion in one dimension, what are physical examples of one-dimensional potentials containing polynomial expressions of degree greater than 2?</p>
<p>Beyond the integrable motion of a single particle in one dimension, what are empirically well-understood examples of many-body one-dimensional (integrable) Hamiltonian systems whose Hamiltonians include polynomial expressions, in the canonical coordinates $\{q^i,p_i\mid i=1,\ldots,n\}$, having degree greater than 2?</p>
<p>What are well-understood examples of single particle (or many-body) greater than one-dimensional (integrable) Hamiltonian systems whose Hamiltonians include polynomial expressions, in the canonical coordinates $\{q^i,p_i\mid i=1,\ldots,n\}$, having degree greater than 2?</p>
<p>As to the naturalness of these questions, the restriction is to the at least cubic polynomials in the Poisson algebra of classical polynomial observables in the $q^i$ and $p_i$ on phase space (presumably ${\mathbb R}^{2n}$ with $n>1$ in the single particle case). That said, could you expand on your observation concerning the naturalness of this restriction in the context of QFTs?</p> | 1,768 | [
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<p>I`ve just learned that electrically charged particles and magnetically charged monopoles in QED are S-dual to each other such that it depends on the value of the fine structure constant which of the two is the ligher (and more fundamental) one and which is heavier (and more complicated).</p>
<p>Whereas I think I understand reasonably well how and why the roles of the light and the heavy particle concerning their mass get interchanged as the coupling sonstant increases from $\alpha << 1$ to $\alpha >> 1$, I fail to grasp how the electric charge is turned into a magnetic charge in the course of this (hypothetical) process.</p>
<p>Should an electrically charged particle, such as an electron for example, not be fundamental different to some kind of a magnetically charged (hypothetical) monopole (I mean two different fundamental particles)?
Non of the different "pictures" I`ve seen so far to "visualize" S-duality explain what happens to the kind of charge ...</p> | 1,769 | [
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<p>As a thought experiment let us assume that we have isolated a magnetic domain. This domain is of finite size and we know its dimensions. Assuming that we can measure an infinitesimal field, will there be a certain region beyond which the field won't be applicable?</p>
<p>The instinctive answer to this question is no, but if you think about it we see the magnet's influence on the space around it as the result of equipotential regions then the contention is that only so many discrete equipotential regions are possible (the fact that something is not countable doesn't automatically mean it's infinite). Hence, that line of thought goes, there should be a limit theoretically and practically until which a field can exist.</p>
<p>Can you please clarify this sticking point for me? Am I pushing the analogies we use to understand fields too far? What conceptual mistake am I making over here?</p> | 1,770 | [
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<p>I have a bunch of magnets (one of those game-board thingies) given to me when I was a school-going lad over 20 years ago, and the magnets feel just as strong as it was the day it was given. </p>
<p>As a corollary to this question <a href="http://physics.stackexchange.com/questions/8036/do-magnets-lose-their-magnetism">Do magnets lose their magnetism?</a>, is there a way to determine how <em>long</em> a permanent magnet will remain a magnet? </p>
<p>Addendum:
Would two magnets remain a magnet for a shorter duration if they were glued in close proximity with like poles facing each other?</p> | 1,771 | [
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<p>If one covers up one eye, then he loses depth perception (two dimensional perspective). When we uncover that eye, we can now see depth (three dimensional perspective). My question is if we had four eyes, would we be able to see from a four dimensional perspective?</p> | 1,772 | [
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<p>When someone performs Young's Double Slit experiment, the person sees an interference pattern on the screen. What is the time taken to for the pattern to appear on the screen? Is it distance between slit and the screen divided by speed of light? Another way to put the question is when photons are converted to waves is wave propagation speed = speed of light ?</p> | 1,773 | [
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<p>This thought has completely changed my perspective towards matter. If the matter in a star can collapse to a point to form a Black hole, surely the true nature of matter should be able explain this behavior. I find this collapse easy to imagine if I visualize Matter as something which lies in space, but which <em>itself</em> is not space, or <em>space-like</em>... It does not occupies space... What I mean is this... Why should matter occupy space? <em>Space-fulness is the nature of space, why associate it with matter?</em> </p>
<p>I use this modified picture of matter to explain the collapse of a star to a point. Suppose matter in a star has <em>0 volume</em>, and it is the <em>space</em> in between matter that does the job of occupying the volume, then we can easily eliminate this <em>space</em> to explain a point sized infinitely dense black hole...!</p>
<p>A particle of matter with no volume is hardly like a particle I had imagined earlier. It is more surprising...</p>
<p><em>I think that matter lies in space, but does not occupies it.</em> </p>
<p>Is the general association of space-fulness with matter a misunderstanding? Is it correct to view our matter as a space-less entity? Or matter is actually woven to the fiber with space, such that it itself behaves a bit like space, by occupying it?</p>
<p>Does Matter REALLY occupies space? Am I am under a misconception?</p> | 1,774 | [
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<p>So I'm using the following definition for the Reflection coefficient :</p>
<p>$$\frac{\vec{j}_{reflected}}{\vec{j}_{incident}}$$</p>
<p>Hence, since :</p>
<p>$$\psi_{reflected}=Be^{-ikx}$$ and $$\psi^{*}_{reflected}=B^{*}e^{ikx}$$</p>
<p>We can perform the usual to obtain the incident probability density current as : $$\vec{j}_{reflected} =2ik\lvert{B}\rvert^{2}$$</p>
<p>Now for the incident we use :</p>
<p>$$\psi_{incident}=e^{ikx}$$ and $$\psi^{*}_{incident}=e^{-ikx}$$</p>
<p>Again we obtain : $$\vec{j}_{incident} =-2ik$$</p>
<p>This would imply a coefficient of $-\lvert{B}\rvert^{2}$ which is incorrect.</p>
<p>What have I done incorrectly ?</p> | 1,775 | [
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<p>What if I take a laser pen and direct it towards the middle of two wide slits? Will I get interference pattern or just two lines? Do I need to scatter photons before they hit the slits to observe interference pattern?</p> | 1,776 | [
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<p>I saw a claim in <a href="http://arxiv.org/abs/0902.2790" rel="nofollow">this paper</a> that holomorphic boundary CFT$_2$ primary operators correspond to massless states in the AdS$_3$ bulk. Specifically,</p>
<blockquote>
<p>As always, we simplify the situation by assuming the absence of holomorphic primary operators. (These would have a little group different from that of a massive particle in the bulk of AdS; therefore for small $\Lambda = −L^{-2}$ they can only correspond to massless states, which do not have a rest frame, or else to states which do not propagate into the bulk of AdS at all.)</p>
</blockquote>
<p>My question is: how did he arrive at this conclusion/where can I find an explanation? I can't figure it out, and nowhere near the claim does he give any relevant sources. It's certainly conceivable: holomorphic primary operators will include gauge fields, for example.</p> | 1,777 | [
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<p>More specifically, I want to understand why a <a href="http://en.wikipedia.org/wiki/Wave" rel="nofollow">wave</a> is a wave but a <a href="http://en.wikipedia.org/wiki/Wave_packet" rel="nofollow">wave packet</a> is not considered a wave (<a href="http://physics.stackexchange.com/questions/37891/what-is-the-mass-of-a-wave">as discussed in this question</a>).</p>
<p>I would think that if something have these characteristics: 1. Wavelength; 2. Frequency; 3. Period; 4. Amplitude; and 5. Wave velocity; then it must be a wave. Does a wave packet have these characteristics? Do you have a better set of rules to apply to test if something is a wave? </p> | 1,778 | [
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<p>I know that a capacitor with a dielectric can operate normally up till a certain voltage (AFAIK called breakdown voltage) which depends on the strength of the dielectric placed between the plates. After this voltage, the circuit becomes short and current flows between the plates and thus the capacitor breaks down. But i want to know what is exactly happening when we say a dielectric "breaks down" ? What I know about a dielectric is that due to the electric field (because of the plates of the capacitor) the molecules of the dielectric align themselves accordingly and set up an electric field in the opposite direction, thus decreasing the net electric field. So, please can anyone tell me what happens during breakdown?</p> | 1,779 | [
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<p>This may be anecdotal.
Playing in the kitchen I realized the frying pan comes with both a flat, and a concave bottom. So here's the question - </p>
<p>Given two pans made of brass, one has a concave base & the other a flat base, which of the two would use heat more efficiently? I would believe the latter because the flat base means any heat must necessarily travel through the cooking medium, and the recipe before reaching open air. </p> | 1,780 | [
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