Exact classical limit of the quantum bouncer

TL;DR

This paper presents a systematic method to derive the classical probability density from quantum distributions for periodic systems, successfully applied to the quantum bouncer, revealing quantum corrections are negligible at macroscopic scales.

Contribution

The paper introduces a Fourier-based approach to connect quantum and classical probability densities, providing an exact classical limit for the quantum bouncer.

Findings

Classical probability density is exactly recovered from quantum distribution.

Quantum corrections are negligible (~10^{-10}) for realistic systems.

Fourier coefficients of quantum and classical densities converge at large quantum numbers.

Abstract

In this paper we develop a systematic approach to determine the classical limit of periodic quantum systems and applied it successfully to the problem of the quantum bouncer. It is well known that, for periodic systems, the classical probability density does not follow the quantum probability density. Instead, it follows the local average in the limit of large quantum numbers. Guided by this fact, and expressing both the classical and quantum probability densities as Fourier expansions, here we show that local averaging implies that the Fourier coefficients approach each other in the limit of large quantum numbers. The leading term in the quantum Fourier coefficient yields the exact classical limit, but subdominant terms also emerge, which we may interpret as quantum corrections at the macroscopic level. We apply this theory to the problem of a particle bouncing under the gravity field and show that the classical probability density is exactly recovered from the quantum distribution. We show that for realistic systems, the quantum corrections are strongly suppressed (by a factor of $\sim 10^{-10}$) with respect to the classical result.