标题： 薛定谔方程在半经典近似下的两个极限：经典力学中的可分辨粒子和不可分辨粒子 显示英文标题

标题： The two limits of the Schrödinger equation in the semi-classical approximation: discerned and non-discerned particles in classical mechanics

摘要： 我们在半经典近似下研究了当普朗克常数 \( h \) 趋于 0 时，满足马德隆方程的量子密度和量子作用量的收敛性。 我们发现了两种不同的解，它们取决于初始密度。 在第一种情况下，初始量子密度是一个经典密度 \( \rho_0(x) \)，量子密度和量子作用量收敛到一个经典作用量和一个经典密度，它们满足统计哈密顿-雅可比方程。 这些是这样一组经典粒子的方程，其初始位置仅由密度 \( \rho_0(x) \) 所知。 在第二种情况下，初始密度 显示英文摘要

摘要： 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