Analytic WKB energy expression for the linear plus Coulomb potential

Abstract

A direct evaluation of the lowest-order WKB integral is carried out for the linear plus Coulomb potential V(r)= mu r-( beta /r). The implicit relation for the bound-state energies defined by the WKB quantisation condition is expressed in terms of complete elliptic integrals. An approximate non-perturbative inversion of this relation provides an explicit analytic expression for the energy which reproduces known numerical results quite well. A criterion for the applicability of the approximation scheme is given.