Hamiltonian Dynamics of Semiclassical Gaussian Wave Packets in Electromagnetic Potentials

TL;DR

This paper develops a Hamiltonian formulation for semiclassical Gaussian wave packet dynamics in electromagnetic fields, incorporating $ ext{O}( ext{ extonehalf} ext{ exttwosuperior})$ quantum corrections, and demonstrates improved accuracy over classical and previous models through numerical examples.

Contribution

It introduces a Hamiltonian-based semiclassical wave packet method with $ ext{O}( ext{ extonehalf} ext{ exttwosuperior})$ corrections for electromagnetic potentials, extending prior work to include vector potentials.

Findings

The equations recover Zhou's in linear vector and quadratic scalar potentials.

Numerical results show $ ext{O}( ext{ extonehalf} ext{ exttwosuperior})$ corrections improve short-time accuracy.

Solutions converge faster to quantum expectation values than classical or Zhou's equations.

Abstract

We extend our previous work on symplectic semiclassical Gaussian wave packet dynamics to incorporate electromagnetic interactions by including a vector potential. The main advantage of our formulation is that the equations of motion derived are naturally Hamiltonian. We obtain an asymptotic expansion of our equations in terms of $\hbar$ and show that our equations have $\mathcal{O}(\hbar)$ corrections to those presented by Zhou, whereas ours also recover the equations of Zhou in the case of a linear vector potential and quadratic scalar potential. One and two dimensional examples of a particle in a magnetic field are given and numerical solutions are presented and compared with the classical solutions and the expectation values of the corresponding observables as calculated by the Egorov or Initial Value Representation (IVR) method. We numerically demonstrate that the $\mathcal{O}(\hbar)$ correction terms improve the accuracy of the classical or Zhou's equations for short times in the sense that our solutions converge to the expectation values calculated using the Egorov/IVR method faster than the classical solutions or those of Zhou as $\hbar \to 0$.