A simple mathematical formulation of the correspondence principle

TL;DR

This paper introduces a straightforward mathematical method to derive classical probability densities from quantum systems using Bohr's correspondence principle, focusing on Fourier expansions and the classical limit.

Contribution

It presents a simple, generalizable Fourier-based procedure to connect quantum distributions with classical ones in the large quantum number limit.

Findings

Derived classical distributions from quantum ones analytically.

Identified correction terms as residual quantum effects.

Applied method to quantum harmonic oscillator as illustration.

Abstract

In this paper we suggest a simple mathematical procedure to derive the classical probability density of quantum systems via Bohr's correspondence principle. Using Fourier expansions for the classical and quantum distributions, we assume that the Fourier coefficients coincide for the case of large quantum numbers $n$. We illustrate the procedure by analyzing the classical limit for the quantum harmonic oscillator, although the method is quite general. We find, in an analytical fashion, the classical distribution arising from the quantum one as the zeroth order term in an expansion in powers of Planck's constant. We interpret the correction terms as residual quantum effects at the microscopic-macroscopic boundary.