Asymptotic solution of an anisotropic scattering potential

Abstract

An example of a long-ranged anisotropic scattering potential is investigated in this paper. Using the known properties of the confluent hypergeometric function and by applying the Liouville-Green approximation, we were able to solve the Schrödinger equation asymptotically. From the resulting far-field form of the wave function, the differential scattering cross section is obtained. We have found that the latter is different compared with that of the Coulomb scattering in terms of incident energy. For our potential which behaves as r^−1/2, the resulting differential scattering cross section behaves as E⁻⁴, while for a Coulombic potential which varies as r⁻¹ we will get E⁻².