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

TL;DR

This paper develops an effective approach to analyze the evolution of wave-packets in a time-dependent harmonic potential, focusing on the shape and fluctuations of the wave-function in quantum mechanics.

Contribution

It introduces a method to derive the dynamics of higher moments of wave-functions, extending the analysis beyond Gaussian wave-packets in time-dependent potentials.

Findings

Derived equations for quadratic uncertainty evolution

Presented a framework for higher moments in wave-function dynamics

Applied the method to a 1+1-dimensional quantum system

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 Schrodinger 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.