# The classical limit of QM as a Hamilton-Jacobi equation?

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?

• More on classical limit of QM. – Qmechanic Jan 23 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 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 at 20:28
• Several of your questions are addressed in meticulous mathematical length in this required reading book. – Cosmas Zachos Jan 23 at 22:31
• Long and complex story. Bohmian mechanics is an obscure art... – Cosmas Zachos Jan 23 at 23:21