Exact classical emergence from high-energy quantum superpositions

TL;DR

This paper rigorously demonstrates how high-energy quantum superpositions in an infinite square well asymptotically reproduce classical behavior, both in probability distribution and dynamics, as the number of states increases.

Contribution

It provides an exact analytical proof of the classical emergence from quantum superpositions in a well-defined high-energy limit, including static and dynamic aspects.

Findings

Probability density converges to a uniform classical distribution as states increase.

Expectation value of position matches classical trajectory asymptotically.

Quantum deviations are confined to boundary layers that vanish at macroscopic scales.

Abstract

We examine the correspondence principle for an equiprobable superposition of high-energy eigenstates of the infinite square well using a fully analytical Fourier-based approach. We derive a closed-form asymptotic expression for the interference terms $\rho_{\alpha}^{\text{a}}(x)$ by expanding them into a geometric series of quantum Fourier coefficients. We show these terms act as functional envelopes that do not vanish individually but become asymptotically equivalent in the large-$n$ limit. Furthermore, we prove the total probability density for a superposition of $2\Delta+1$ states converges exactly to the uniform classical distribution as $\Delta \to \infty$. Dynamically, the expectation value of position reproduces the classical triangular trajectory asymptotically. Residual quantum deviations remain confined to boundary layers whose relative width vanishes under macroscopic resolution. These results establish a rigorous asymptotic realization of the classical limit for isolated bound systems in both static and dynamical contexts.