The evolution of quantal uncertainties in at most quadratic potentials

TL;DR

This paper studies how momentum and position uncertainties evolve over time for wave packets in quadratic potentials, highlighting the roles of mixed uncertainty and chirp in understanding wave packet dynamics.

Contribution

It introduces a comprehensive analysis of uncertainty evolution in quadratic potentials, emphasizing the importance of mixed uncertainty and chirp for arbitrary wave packet shapes.

Findings

Mixed uncertainty and chirp are key to understanding uncertainty dynamics.

Wave packets exhibit spreading and narrowing behaviors.

Analysis applies to all initial conditions and potential cases.

Abstract

We investigate the time evolution of momentum and position uncertainties for wave packets of arbitrary shape in at most quadratic potentials. We consider all possible cases of potentials and initial conditions. Doing so we see that the mixed uncertainty and for Gaussian wave packets moreover the chirp are convenient and important tools to identify the characteristic features of uncertainty dynamics. Special attention is given to the spreading and narrowing of wave packets.