Uniform semiclassical evaluation of Franck–Condon factors and inelastic atom–atom scattering amplitudes

Abstract

The problem of obtaining uniform semiclassical approximations for Franck–Condon factors and inelastic atom–atom scattering amplitudes is considered. Starting from Langer’s uniform Airy approximation for the wave function and an Airy function representation of the inelastic S matrix element, it is shown how both these problems can be expressed in terms of a multidimensional oscillatory integral, the exponent of which possesses two coalescing saddle points. Using a general formula for the uniform asymptotic evaluation of this multidimensional integral, simple derivations are presented for the uniform semiclassical approximation to the Franck–Condon factor and inelastic scattering amplitude. Previous derivations are shown to be special cases of the more general derivations given here. The use of catastrophe theory for choosing the simplest and most convenient canonical form for the transformation of the exponent in the uniform asymptotic evaluation of multidimensional integrals is discussed.