Working with WKB waves far from the semiclassical limit

Abstract

WKB wave functions are expected to be accurate approximations of exact quantum mechanical wave functions mainly near the semiclassical limit of the quantum mechanical Schrödinger equation. The accuracy of WKB wave functions is, however, a local property of the Schrödinger equation, and the failure of the WKB approximation may be restricted to a small “quantal region” of coordinate space, even under conditions which are far from the semiclassical limit and close to the anticlassical or extreme quantum limit of the Schrödinger equation. In many physically important situations, exact or highly accurate approximate wave functions are available for the quantal region where the WKB approximation breaks down, and together with WKB wave functions in residual space provide highly accurate solutions of the full problem. WKB wave functions can thus be used to derive exact or highly accurate quantum mechanical results, even far from the semiclassical limit. We present a wide range of applications, including the derivation of properties of bound and continuum states near the threshold of a potential, which are important for understanding many results observed in experiments with cold atoms.