Surprises in the suddenly-expanded infinite well

TL;DR

This paper investigates the surprising behavior of a quantum particle in an infinite well after sudden expansion, revealing constant-density plateaux and regular patterns at specific times through analytical methods.

Contribution

It provides a novel analysis of the time-evolution of a particle in an expanded infinite well, including analytical expressions and explanation of density pattern formations.

Findings

Plateaux of constant probability density appear at specific times.

Density patterns become independent of expansion size beyond a critical point.

Analytical expressions describe the organization of density patterns.

Abstract

I study the time-evolution of a particle prepared in the ground state of an infinite well after the latter is suddenly expanded. It turns out that the probability density $|\Psi(x, t)|^{2}$ shows up quite a surprising behaviour: for definite times, {\it plateaux} appear for which $|\Psi(x, t)|^{2}$ is constant on finite intervals for $x$. Elements of theoretical explanation are given by analyzing the singular component of the second derivative $\partial_{xx}\Psi(x, t)$. Analytical closed expressions are obtained for some specific times, which easily allow to show that, at these times, the density organizes itself into regular patterns provided the size of the box in large enough; more, above some critical time-dependent size, the density patterns are independent of the expansion parameter. It is seen how the density at these times simply results from a construction game with definite rules acting on the pieces of the initial density.