Probabilistic Parallels in the Classical Limit of Quantum Mechanical Models

TL;DR

This paper explores how quantum probability densities for particles in simple systems approach classical results at large quantum numbers, highlighting the role of measurement resolution and angular momentum quantum numbers.

Contribution

It establishes a connection between quantum probability densities and classical distributions through coarse-graining and measurement resolution, especially in the context of angular momentum.

Findings

Quantum densities converge to classical results at large quantum numbers

Measurement resolution influences the quantum-to-classical transition

Angular momentum quantum numbers relate to measurement coarse-graining

Abstract

At large quantum numbers, the probability densities for particle-in-a-box or simple harmonic oscillator converge to the classical result upon coarse-graining the quantum mechanical probability densities by introducing a finite resolution in the measurement of the particle's position. This resolution in the position can be related to the resolution of the secondary total angular momentum quantum number ($m$) when interpreting the probabilistic outcomes of the Stern--Gerlach-type thought experiments for large values of the angular momentum quantum numbers ($j$).