The reproduction of the dynamics of a quantum system by an ensemble of classical particles beyond de Broglie--Bohmian mechanics

TL;DR

This paper demonstrates that for any quantum system with a smooth potential, one can construct a classical ensemble whose distributions closely match quantum wave function distributions, extending beyond de Broglie--Bohmian mechanics.

Contribution

It introduces a classical ensemble model that reproduces quantum dynamics and distributions without relying on de Broglie--Bohmian trajectories.

Findings

Classical ensembles can approximate quantum distributions arbitrarily well.

The classical potential can differ from the original quantum potential on a measure zero set.

Classical trajectories need not match de Broglie--Bohmian trajectories.

Abstract

It is shown that for any given quantum system evolving unitarily with the Hamiltonian, $\hat{H} = \hat{\bf p}^2/(2m) + U({\bf q})$, [bold letters denote $D$-dimensional ($D \geqslant 3$) vectors] and with a sufficiently smooth potential $U({\bf q})$, there exits a classical ensemble with the Hamilton function, $\mathrsfs{H} ({\bf p}, {\bf q}) = {\bf p}^2/(2m) + U^{(\infty)} ({\bf q})$, where the potential $U^{(\infty)}({\bf q})$ coincides with $U({\bf q})$ for almost all ${\bf q}$ (i.e., $U^{(\infty)}$ can be different from $U$ only on a measure zero set), such that the square modulus of the wave function in the coordinate (momentum) representation approximately equals the coordinate (momentum) distribution of the classical ensemble within an arbitrary given accuracy. Furthermore, the trajectories of this classical ensemble, generally speaking, need not coincide with the trajectories obtained from de Broglie--Bohmian mechanics. Consequences of this result are discussed.