Phase shifts due to a mixture of long- and short-range potentials

Abstract

We show that the generalized or modified WKB method can be readily applied to a mixture of long- and short-range potentials. The phase shifts of such a mixed potential are calculated to order ${\ensuremath{\hbar}}^{2}$ by using the solutions of the long-range potential as the bases of the approximation and treating the differences between the long-range and mixed potential as a WKB perturbation. We demonstrate the method by a simple example using a Coulomb and inverse-square potential. Comparison of the phase shifts with the exact results shows excellent agreement. The phase shifts can then be separated into contributions due to the long-range and short-range potentials plus a contribution due to the mixture or coupling of the potentials. For small angular momenta, the contribution due to coupling is of the same order of magnitude as the Coulomb contribution, but rapidly decreases with higher angular momentum.