The two limits of the Schr\"odinger equation in the semi-classical approximation: discerned and non-discerned particles in classical mechanics

TL;DR

This paper investigates the semi-classical limit of quantum mechanics, revealing two distinct behaviors depending on initial conditions, which correspond to discerned and non-discerned particles in classical mechanics.

Contribution

It identifies two different limiting solutions of the Madelung equations as Planck's constant approaches zero, clarifying the transition from quantum to classical mechanics.

Findings

Quantum density converges to classical density in the semi-classical limit.

Different initial conditions lead to distinct classical behaviors.

The results distinguish between discerned and non-discerned particles in classical mechanics.

Abstract

We study, in the semi-classical approximation, the convergence of the quantum density and the quantum action, solutions to the Madelung equations, when the Planck constant h tends to 0. We find two different solutions which depend to the initial density . In the first case where the initial quantum density is a classical density rho_0(x), the quantum density and the quantum action converge to a classical action and a classical density which satisfy the statistical Hamilton-Jacobi equations. These are the equations of a set of classical particles whose initial positions are known only by the density rho_0(x). In the second case where initial density