Uniform JWKB amplitudes and phases for turning-point problems

Semiclassical wave functions written in the form A (E,R) sin[Φ (E,R)] can be used in conjunction with the method of stationary phase to obtain analytic expressions for T-matrix elements. Such expressions are functions of the amplitudes, phases, and their derivatives at the points of stationary phase. Generally, simple JWKB formulas for amplitude [As(E,R) =k (R)−1/2] and phase [Φs(E,R) =ℱRRtk (r) dr+1/4π] have been used. Unfortunately these become invalid for stationary-phase points that are near a classical turning point. This occurs at a crucial time, since T-matrix elements are often unusually large under these circumstances. To preserve the advantages of semiclassical stationary-phase methods, expressions for modified amplitudes, phases, and their derivatives are presented that remain uniformly valid both near to and far from the turning points in the three most commonly encountered problems: the single turning-point of a simple free state, the double set of a bound state, and the triple set associated with a centrigual barrier. These expressions are analytic functions of the simple JWKB amplitude [As(E,R)]. Explicit numerical criteria are given that one can employ to determine when the simple amplitude and phase are no longer sufficiently accurate, but instead should be used to calculate the uniform values.