Geometry and Dynamics of Gaussian Wave Packets and their Wigner Transforms

TL;DR

This paper explores the symplectic geometric relationship between Gaussian wave packets and their Wigner transforms, revealing how this connection leads to improved dynamical accuracy through Hamiltonian corrections.

Contribution

It establishes a symplectic geometric framework linking Gaussian wave packet dynamics with Wigner functions, introducing Hamiltonian corrections that enhance approximation accuracy.

Findings

The momentum map relates covariance matrices to symplectic group actions.

Hamiltonian corrections improve the accuracy of wave packet dynamics.

Numerical results show enhanced approximation of observable expectation values.

Abstract

We find a relationship between the dynamics of the Gaussian wave packet and the dynamics of the corresponding Gaussian Wigner function from the Hamiltonian/symplectic point of view. The main result states that the momentum map corresponding to the natural action of the symplectic group on the Siegel upper half space yields the covariance matrix of the corresponding Gaussian Wigner function. This fact, combined with Kostant's coadjoint orbit covering theorem, establishes a symplectic/Poisson-geometric connection between the two dynamics. The Hamiltonian formulation naturally gives rise to corrections to the potential terms in the dynamics of both the wave packet and the Wigner function, thereby resulting in slightly different sets of equations from the conventional classical ones. We numerically investigate the effect of the correction term and demonstrate that it improves the accuracy of the dynamics as an approximation to the dynamics of expectation values of observables.