Uniform Semiclassical Approximations for Elastic Scattering and Eigenvalue Problems

Abstract

The uniform semiclassical wavefunction of Langer is used to obtain a modified quantum condition. The result [contained in Eqs. (1) and (2)] has the integral quantum numbers replaced by known, nonintegral constants and gives considerably better results for the example of the quartic oscillator. An approximation for the Franck-Condon factor is obtained which is qualitatively correct for all values of the parameters involved. The result, contained in Eqs. (4) and (5), reduces to the known semiclassical results in the appropriate regions and is uniformly valid in the transition regions. The method used by Carrier for evaluating integrals by the method of stationary phase in the case of more than one point of stationary phase is used to extend the semiclassical scattering amplitude of Ford and Wheeler to include the glory and rainbow regions uniformly. The result is contained in Eq. (6) of the text.