Higher corrections to the WKB approximation

Abstract

A systematic method of successive approximations is described, of which the first step is the WKB approximation. The higher corrections consist of successive recombinations of the two WKB waves, rather than a perturbation expansion in their interaction. At the same time the local wave number (or refractive index) undergoes successive corrections. Using this method it is shown that the quantummechanical reflection by a potential barrier is a small quantity of order ħ ∞, provided that the barrier is smooth, and so low that there is no classical turning point.