A phase space moment method for classical and quantal dynamics

Abstract

A hierarchy of translational moment equations for the position and momentum observables is considered. The resulting set of equations is truncated in a manner that includes only first and second order moments. This closure procedure is consistent with the conservation of energy and, for spherically symmetric potentials, with the conservation of angular momentum. The method is valid for both classical and quantal mechanics with the only difference being that for quantal systems, the dispersions in momentum and position must satisfy Heisenberg's uncertainty principle. With finite dispersion in the position, the average position and momentum do not satisfy Hamilton's equations of motion since the average acceleration is modulated by the variance–covariance in the position. For classical systems, the second order moments may be taken to zero in which case Hamilton's equations are obtained. For quantal systems, a comparision is made with Heller's Gaussian wave packet approach. The moment method is illustrated by applying it to the reflection of a phase space packet from a one dimensional repulsive inverse square potential. Applications to the calculation of collision cross sections are envisaged.