A new method to derive low-lying N-dimensional quantum wave functions by quadratures along a single trajectory

We present a new method to derive low lying N-dimensional quantum wave func tions by quadrature along a single trajectory. The N-dimensional Schroeding er equation is cast into a series of readily integrable first-order ordinar y differential equations. Our approach resembles the familiar W.K.B. approx imation in one dimension, but is designed to explore the classically forbid den region and has a much wider applicability than W.K.B. The current metho d also provides a perturbation series expansion and the Green functions of the wave equation in the N-dimension, all by quadratures along a single tra jectory. A number of examples are given for illustration, including a simpl e algorithm to evaluate the Stark effect in closed form to any finite order of the electric field. (C) 2001 Academic Press.