Hamilton-jacobi dynamics for the solution of time dependent quantum problems I. Formalism and wave packet propagation in one dimension

Abstract

Two methods for the numerical integration of the time-dependent Schrödinger equation with given initial conditions (initial wave packet) are presented. The first method (method A) is based on the Schrödinger representation of the quantum-dynamical system while the second one (method B) is based upon the intermediate representation. In both cases the quantum dynamical equation is transformed into a system of Hamilton-Jacobi type equations of motion as occurring in multi particle classical dynamics, i.e. standard molecular dynamics techniques can be applied for the integration. The dynamics of a minimum uncertainty Gaussian wave packet in a strongly anharmonic oscillator is taken as an example.