Reflection above potential steps

Abstract

Classically forbidden reflection by a potential step is analyzed by matching WKB waves to solutions of a Schr\odinger equation involving the tail of the potential on the up side of the step. Analytic expressions for the leading deviation of the reflection probability from unity and the phase of the reflection amplitude at the top of the step are derived for potentials decaying as an inverse power of the coordinate $V(x)\ensuremath{\propto}\ensuremath{-}{1/x}^{\ensuremath{\alpha}}, \ensuremath{\alpha}g2$.