Classical echoes of quantum boundary conditions

TL;DR

This paper investigates how different quantum boundary conditions in a one-dimensional box influence the classical limit of the system, revealing that local conditions converge to a common classical boundary, while some non-local conditions retain quantum features.

Contribution

It classifies all selfadjoint boundary conditions for a quantum particle in a box and analyzes their classical limits through Wigner functions.

Findings

Local boundary conditions converge to the same classical boundary.

Non-local boundary conditions retain quantum characteristics in the classical limit.

Wigner functions effectively illustrate the classical transition.

Abstract

We consider a non-relativistic particle in a one-dimensional box with all possible quantum boundary conditions that make the kinetic-energy operator selfadjoint. We determine the Wigner functions of the corresponding eigenfunctions and analyze in detail their classical limit in the high-energy regime. We show that the quantum boundary conditions split into two classes: all local and regular boundary conditions collapse to the same classical boundary condition, while singular non-local boundary conditions slightly persist in the classical limit.