Semi-classical behaviour of Schr\"odinger's dynamics : revivals of wave packets on hyperbolic trajectory

TL;DR

This paper investigates the semi-classical dynamics of quantum wave packets along hyperbolic trajectories, revealing short-term classical behavior and long-term revival phenomena similar to elliptic cases.

Contribution

It extends the understanding of wave packet evolution to hyperbolic trajectories, demonstrating revival phenomena in this less-studied context.

Findings

Wave packets follow classical motion initially

Revivals occur at large times

Behavior parallels elliptic trajectory cases

Abstract

The aim of this paper is to study the semi-classical behaviour of Schr\"odinger's dynamics for an one-dimesional quantum Hamiltonian with a classical hyperbolic trajectory. As in the regular case (elliptic trajectory), we prove, that for an initial wave packets localized in energy, the dynamics follows the classical motion during short time. And, for large time, a new period for the dynamics appears : the wave packets revivals.