Accuracy of analytic semiclassicalT-matrix elements

Abstract

The accuracy of two different semiclassical methods of evaluating T-matrix elements by stationary-phase techniques are reported. Their predictions are compared to quantal results for the far-red-wing emission spectrum of Rb collisionally broadened by Xe perturbers. One method is based on the direct use of uniform semiclassical wave functions in the overlap integral of the T-matrix elements. It is shown to give very accurate results (generally to within 3% of quantal values) for both classically allowed and classically forbidden transitions, whether they are near to or far from the classical singularity associated with the collisional turning points. The other semiclassical method is based on a cubic expansion of the phases of the semiclassical wave functions. If simple semiclassical amplitudes and phases are used in the resulting expression, individual T-matrix elements can have significant errors. However, the sum over angular momenta tends to compensate for the individual discrepancies so that the final differential intensity is generally within 10% of quantal values, except for narrow spikes due to the turning-point singularities. The use of uniform semiclassical amplitudes and phases removes these spikes and reduces the error in the total differential intensity given by the cubic expression to only a few percent.