Timeline for Can I swap quantum mechanical ground state for some classical trajectory distribution and have it sit still after the swap?
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| Sep 1, 2016 at 18:33 | comment | added | CR Drost | Hm. Peeking inside and ignoring $\lambda,\mu$ for a second gives $|x+i p\rangle=\exp(-(x^2+p^2)/2)\sum(x+ip)^n/\sqrt{n!}~|n\rangle$ where the $|n\rangle$ are SHO eigenfunctions; this seems to prove $\partial_x|x + i p\rangle =-x |x+ip\rangle + a^\dagger a/(x + i p) |x + i p\rangle,$ and the first $x$ can be of course replaced with $(a + a^\dagger)/2$ or whatever it is. Then this gets multiplied with $\partial H/\partial p,$ but if we did the same trick to $\hat H$ to get $H(x,p) = \langle x+ip|\hat H|x + i p\rangle$ we'd have a further set of terms. I'll definitely think about this more. | |
| Aug 31, 2016 at 22:09 | comment | added | Emilio Pisanty | Sure, it looks reasonable, but it's not at all obvious to me that that density will be stationary. You could say the same about the Husimi Q and the Sudarshan P representations (they're both baked from $|\psi⟩$ combined with $|\alpha⟩$), but if the same procedure works perfectly for all three then there's definitely something nontrivial going on there. | |
| Aug 31, 2016 at 22:05 | history | answered | CR Drost | CC BY-SA 3.0 |