Wave packet propagation and the materialization of classical trajectories

TL;DR

This paper demonstrates how unbound quantum wave packets evolve into classical trajectories at large scales, providing a rigorous foundation for the classical assumptions used in particle detection and quantum reaction analysis.

Contribution

It introduces the asymptotic imaging theorem, linking quantum wave packet propagation to classical trajectories, and illustrates this transition with a simple free propagation example.

Findings

Wave packets become proportional to their initial momentum distribution at large times

Classical trajectories connect asymptotic coordinates and initial momenta

The theorem justifies classical particle assumptions in quantum detection

Abstract

Unbound wave packets propagating to macroscopic space and time coordinates become proportional to their (Fourier transform) momentum distribution at earlier times whereby the asymptotic coordinates and the initial momenta are connected everywhere by appropriate classical trajectories. This asymptotic imaging theorem is relevant to every quantum reaction involving macroscopic extraction and detection of fragments emerging from a microscopic volume. It justifies the usual assumption of classical particle motion used in the design of particle detectors. We illustrate this quantum to semiclassical transition with the example of the free propagation of a wave packet in one dimension, a standard problem treated in introductory lectures on quantum mechanics. We indicate generalizations appropriate for more advanced discussions.