Semiclassical description of collapses and revivals of quantum wave packets in bounded domains

TL;DR

This paper provides a rigorous semiclassical analysis of quantum wave packet dynamics on a circle and in a box, detailing stages like collapse and revival, and compares classical mechanics models based on quantum limits.

Contribution

It introduces a detailed semiclassical framework for quantum wave packet evolution, including uniform distribution and revival phenomena, and evaluates classical mechanics models using quantum Husimi functions.

Findings

Wave packets are mostly uniformly distributed over time.

Functional classical mechanics is valid at larger time scales than Newtonian.

Quantum probability distributions tend to a limit distribution over time.

Abstract

We study a special kind of semiclassical limit of quantum dynamics on a circle and in a box (infinite potential well with hard walls) as the Planck constant tends to zero and time tends to infinity. The results give detailed information about all stages of evolution of quantum wave packets: semiclassical motion, collapses, revivals, as well as intermediate stages. In particular, we rigorously justify the fact that the spatial distribution of a wave packet is most of the time close to uniform. This fact was previously known only from numerical calculations. We apply the obtained results to a problem of classical mechanics: deciding whether recently suggested functional classical mechanics is preferable to traditional Newtonian one from the quantum-mechanical point of view. To do this, we study the semiclassical limit of the Husimi functions of quantum states. We show that functional mechanics remains valid at larger time scales than Newtonian one and, therefore, is preferable. Finally, we analyse the quantum dynamics in a box in case when the size of the box is known with a random error. We show that, in this case, the probability distribution of the position of a quantum particle is not almost periodic, but tends to a limit distribution as time indefinitely increases.