Evolution of the wave function's shape in a time-dependent harmonic potential

Université de Lyon, ENS de Lyon, CNRS, Laboratoire de Physique LPENSL - 69007 Lyon, France and Perimeter Institute for Theoretical Physics - Waterloo, Ontario N2L 2Y5, Canada

^(a) E-mail: etera.livine@ens-lyon.fr (corresponding author)

Received: 23 May 2023
Accepted: 7 August 2023

Abstract

An effective operational approach to quantum mechanics is to focus on the evolution of wave packets, for which the wave function can be seen in the semi-classical regime as representing a classical motion dressed with extra degrees of freedom describing the shape of the wave packet and its fluctuations. These quantum dressing are independent degrees of freedom, mathematically encoded in the higher moments of the wave function. We review how to extract the effective dynamics for Gaussian wave packets evolving according to the Schrödinger equation with time-dependent potential in a 1 + 1-dimensional spacetime, and derive the equations of motion for the quadratic uncertainty. We then show how to integrate the evolution of all the higher moments for a general wave function in a time-dependent harmonic potential.