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I'am having difficulties to understand the so-called classical limit in quantum mechanics. There is a popular method to transform the Schrödinger equation into two coupled equations that are the continuity equation of probability flow and the Hamilton-Jacobi equation with the quantum potential: The Ehrenfest’s theorem and the Quantum Hamilton-Jacobi equation

The limit $\hbar \rightarrow 0$ makes no sense in the Schrödinger picture or in the Heisenberg picture, but somehow it's supposed to make sense in the hydrodynamical picture. Thus, is it obvious that the quantum potential remains finite, as $\hbar \rightarrow 0$, to achieve the statistical version of the classical Hamilton-Jacobi equation?

Hulkster
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    More on classical limit of QM. – Qmechanic Jan 23 '19 at 18:13
  • @Qmechanic Thanks for the tip, but I have tried to search some answers already. Do you have any direct answers or references? – Hulkster Jan 23 '19 at 18:20
  • Your question appears to be if the "popular argument" is consistent with the Ehrenfest theorem. Please give a reference for this argument to improve the focus. – my2cts Jan 23 '19 at 20:28
  • Several of your questions are addressed in meticulous mathematical length in this required reading book. – Cosmas Zachos Jan 23 '19 at 22:31
  • Long and complex story. Bohmian mechanics is an obscure art... – Cosmas Zachos Jan 23 '19 at 23:21
  • @CosmasZachos I modified the question to concern only the consistency of the approach. Feel free to answer if you have something in mind. – Hulkster Jan 24 '19 at 12:05
  • Sorry, I am not a fan, personally, of the H-J bridge, even though it is crucial for the Path Integral formulation. I would, however, object to the judgement that the classical limit "makes no sense" in the Schroedinger picture (the von Neumann equation in the phase space formualtion, that is...). The do make quite a bit of sense. Hell, I have a short video on that in my Amazon author page..... – Cosmas Zachos Jan 24 '19 at 14:49
  • @CosmasZachos Thanks for answering! So there is a direct path from the Schrödinger equation to classical theory as $\hbar \rightarrow 0$ - without extra assumptions? – Hulkster Jan 24 '19 at 15:01
  • ...but here are some refs... i, ii, iii ... popular among chaos people. – Cosmas Zachos Jan 24 '19 at 15:10
  • Oh, I'm unclear as to what counts as "extra assumptions"... People grope for a picture and a plethora of such have emerged... Focusing of some pictures ignoring others is always risky.... – Cosmas Zachos Jan 24 '19 at 15:14
  • PS: A superior summary of your elliptical notes is in Sakurai-Napolitano QM, ISBN 978-0-8053-8291-4 , Ch 2.4; prob 2.29; Ch 6.5 (the Eikonal!) which have served as the baseline to your question for generations of students..... – Cosmas Zachos Jan 24 '19 at 15:33

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