Semiclassical Treatment of Multiple Turning-Point Problems—Phase Shifts and Eigenvalues

Abstract

A simple, physically intuitive, expression has been obtained for the scattering phase shift $\mathrm{\delta \hspace{0.17em}(l,\hspace{0.17em}E)}$ in the case that the effective radial potential possesses a maximum. The result [Eq. (3)] is seen to modify the usual WKB phase shift [Eq. (5)] by replacing a step function by a particular “smooth step function.” On the basis of this result, resonances in the energy dependence of the total elastic cross section are discussed. From qualitative arguments it is seen that metastable states more than ${\text{∼0.6\hspace{0.17em}ℏω}}_B$ energy units above the top of the barrier are too short lived to have physical significance (${\omega }_B$ is the harmonic frequency related to the inverted barrier). Application is also made to the eigenvalues of a hindered rotor.