The quantum Hamilton Jacobi equation and the link between classical and quantum mechanics

TL;DR

This paper explores how classical mechanics emerges from quantum mechanics through the quantum Hamilton-Jacobi equation, highlighting the role of quantum fluctuations and coarse-graining in the classical limit.

Contribution

It provides a formal link between quantum and classical Hamilton-Jacobi functions, clarifying the transition in different regions and the role of quantum fluctuations.

Findings

Quantum quantities tend to classical ones in forbidden regions

Classical limit requires coarse-graining in allowed regions

Formal derivation of classical Hamilton-Jacobi scheme from quantum equations

Abstract

We study how the classical Hamilton's principal and characteristic functions are generated from the solutions of the quantum Hamilton-Jacobi equation. While in the classically forbidden regions these quantum quantities directly tend to the classical ones, this is not the case in the allowed regions. There, the limit is reached only if the quantum fluctuations are eliminated by means of coarse-graining averages. Analogously, the classical Hamilton-Jacobi scheme bringing to the motion's equations arises from a similar formal quantum procedure.