Progressively accurate WKB approximations through factorization and successive boost transformations

Abstract

This paper develops a procedure for obtaining progressively improving approximate solutions of the WKB (semiclassical ) model of the stationary Schroedinger equation through factorization and successive boost transformations of the resultant equivalent matrix equation. Each order of approximation provides an amplitude-modulated “plane” wave function specified by a renormalized momentum. A simple binomial expansion of the renormalized momentum allows exact evaluation of the phase accumulation integral for studying basic features of the dynamics in arbitrary potentials. For a linear potential, the probability density profile reveals the expected confinement of the particle within the allowed energy region.