Benji Remez
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 Dec 24 comment Finding interplanetary flight trajectory using calculus of variations? I edited the body of the original question to show my progress. Dec 22 comment Finding interplanetary flight trajectory using calculus of variations? Thanks! I'm having a look at that and trying to adapt it to the problem at hand. Dec 8 comment Partition function of bosons vs fermions I would just add a clarification that you consider states like $|12\rangle$ and $|21\rangle$ as indistinguishable (i.e., a proper boson\fermion state would be $|12\rangle \pm |21\rangle$) Dec 2 comment Is there any possibility in the future that domestic power consumption could be wholly solar powered? Well, yearly fluctuations and bad weather, I assume, would cancel out when averaged over an entire year. Unless the location of choice is close to either of the poles, adjusting for latitude isn't as crucial as coming up with a more reasonable figure for the actual collection efficiency. Nov 30 comment Is the momentum operator well-defined in the basis of standing waves? Alright, so here's a followup question. I agree that the mode counting operator produces equivalent information. But my problem is that these states do no distinguish between positive and negative momenta (they don't specify a sign). So, if I were to put wave packet in the box with some momentum, moving along, say, the positive $x$ axis, how would such a packet be decomposed in the basis of the eigenstates? And how would it differ if the initial momentum had been negative? Nov 30 comment Is the momentum operator well-defined in the basis of standing waves? I'm doing this indeed for numerical purposes. So technically speaking, since the matrix elements of $\hat{p}$ vanish when the difference between $m$ and $n$ increases, would taking a larger finite matrix result in the off-diagonal elements of $\hat{p}^2$ getting smaller? Nov 30 comment Is the momentum operator well-defined in the basis of standing waves? Shouldn't the momentum operator exist anyway, and any invariance of the Hamiltonian just imply that it (meaning its expected value) is conserved? Nov 28 comment How was the Oh-My-God particle observed? But how traceable are these jets, especially if a major portion of the particles produced do not reach any detector? (Also, what does PMT stand for?) Nov 23 comment Energy Question @dbaseman When the atoms decay, they don't vanish into thin air. Your expression for mass vanishes for t tending to infinity, which is wrong, since we still have a considerable mass of the ball as "radioactive waste". See my answer below. Oct 31 comment Cause of buoyant force? @leonardo - if you'd like a more rigorous proof, you can easily show that this is a consequence of the divergence theorem (Gauss's law). Oct 20 comment How does one calculate the volume of a nucleus and the volume of an atom (in this case hydrogen)? Do you mean, how do you find the volume from the given radii, or how these radii are obtained? The former is straightforward (see answer below), the latter is a bit more complicated. Oct 5 comment Acausality in solving time-domain inhomogeneous differential equations with Fourier transforms? So, if I get this right, for $t - t'>0$, I can only integrate the contour on the upper half of the plane, where the pole contributes a non-trivial residue, but for $t - t' < 0$ the exponent only vanishes in the lower half, so I must take the contour there - but no poles means no residue, so the expression is guaranteed to vanish, regardless of the choice of input? Neat. Very Neat. Oct 5 comment Acausality in solving time-domain inhomogeneous differential equations with Fourier transforms? Do you mean, show that the entire integral vanishes? Wouldn't it be equal to the residue at $i/RC$? Oct 5 comment How can such a high exponent arise in this physics equation? What I meant is that the quantity in question is my be an exponential function of the form $y(x) = e^x$. Both quantum tunneling and Boltzmann factors are exponential, and they both come into play in stellar fusion. Try looking up the Gamow Peak. Now, if you have some power law %f(r) = r^n%, notice that its logarithmic derivative %\frac{d(log(f(r))}{d(log(r))}$is equal to %n%, so this yields the exponent of the law. In principle, it may be applied to any general function, and its value would give the appropriate exponent should you wish to locally approximate that function as a power law. Oct 5 comment Acausality in solving time-domain inhomogeneous differential equations with Fourier transforms? So, in principle, could I construct a differential equation which would produce such an$H(\omega )$that violates causality? Oct 5 comment Acausality in solving time-domain inhomogeneous differential equations with Fourier transforms? I'll see if I can get that result this evening. Oct 4 comment Acausality in solving time-domain inhomogeneous differential equations with Fourier transforms? I agree, but does that solution not solve the original equation? Since the equation is causal, I expect the solution to maintain that property. I remember from school that when solving wave equations in classical EM, a delta function pops out during integration and enforces this causality (this raises an interesting question of what would happen when it is solved on a finite interval where the wavenumbers are discrete, but that's for another time) - but I can't see a way that such a causality-ex-machina would show up in any general problem. Oct 4 comment Acausality in solving time-domain inhomogeneous differential equations with Fourier transforms? Yes, but I can always expand this expression as$v_{o}(t)=\frac{1}{2\pi}\int\int d\omega dt'v_{i}(t')e^{j\omega (t-t')}$Integrating first over$\omega$, this clearly yields a delta function$\delta (t-t') $and causality is restored analytically for every input$v_i$. With the RC circuit, the extra$\omega$dependent terms in the integrand don't allow this triviality. And again, I'm concerned about this is a general issue, not necessarily in circuit analysis. Oct 4 comment Acausality in solving time-domain inhomogeneous differential equations with Fourier transforms? I agree that the problem may be solved like you have wrote, but what step goes wrong in the FT approach that seems to break causality? If the solution I outlined above it valid, what would happen if I try to evaluate it for some input function? I suspect that if I switch the order of integration, the first (homogeneous) solution term you wrote may pop out from the transform solution, but I've yet to check it out. Also, as I said in, I'm interested in this difficulty in a more general situation, and only used the RC circuit to illustrate the problem. Oct 4 comment How can such a high exponent arise in this physics equation? I would assume it's roughly the value of the power derivative$\frac{d ln(f(r))}{d ln(r)}\$, which gives the exponent of a the local approximate power law. It's probably an exponential law like @LubošMotl said. I remember once I did a calculation about stellar power production that yielded a power derivative of about 22, so why not 40?