Symplectic Semiclassical Wave Packet Dynamics

TL;DR

This paper presents a symplectic-geometric framework for semiclassical Gaussian wave packet dynamics, revealing its Hamiltonian structure, applying symplectic reduction, and illustrating the theory with a harmonic oscillator example and numerical tunneling results.

Contribution

It introduces a symplectic-geometric approach to semiclassical wave packet dynamics, including reduction, phases, and practical corrections, advancing the understanding of semiclassical mechanics.

Findings

Gaussian wave packet dynamics is a Hamiltonian system.

Explicit calculation of dynamic and geometric phases.

Numerical demonstration of semiclassical tunneling.

Abstract

The paper gives a symplectic-geometric account of semiclassical Gaussian wave packet dynamics. We employ geometric techniques to "strip away" the symplectic structure behind the time-dependent Schr\"odinger equation and incorporate it into semiclassical wave packet dynamics. We show that the Gaussian wave packet dynamics is a Hamiltonian system with respect to the symplectic structure, apply the theory of symplectic reduction and reconstruction to the dynamics, and discuss dynamic and geometric phases in semiclassical mechanics. A simple harmonic oscillator example is worked out to illustrate the results: We show that the reduced semiclassical harmonic oscillator dynamics is completely integrable by finding the action--angle coordinates for the system, and calculate the associated dynamic and geometric phases explicitly. We also propose an asymptotic approximation of the potential term that provides a practical semiclassical correction term to the approximation by Heller. Numerical results for a simple one-dimensional example show that the semiclassical correction term realizes a semiclassical tunneling.