The Anisotropic Gaussian Semi-Classical Schr\"{o}dinger Propagator

TL;DR

This paper develops an anisotropic Gaussian semi-classical Schrödinger propagator using Fourier integral operators, deriving algebraic relations and the Van Vleck formula to enhance wave packet analysis in quantum dynamics.

Contribution

It introduces a novel construction of the anisotropic Gaussian semi-classical propagator and derives key algebraic relations for variational matrices related to wave packet evolution.

Findings

Derived algebraic relations of variational matrices

Established invariances of the dynamics

Connected the propagator to the Van Vleck formula

Abstract

We present a construction of the Anisotropic Gaussian Semi-Classical Schr\"{o}dinger Propagator, emblematic of a class of Fourier Integral Operators of quadratic phase kernels related to the Schr\"{o}dinger equation. We deduce a set of algebraic relations of the variational matrices, solutions of the variational system pertaining to single Gaussian wave packet semi-classical time evolution, representing the symplectic and other invariances of the dynamics, which are subsequently used to derive the Van Vleck formula from the semi-classical propagator, as an argument for the practical importance of the later relations in the relevant wave packet calculus.