Semiclassical Approach to Dipole Radial Integrals forNonhydrogenic Ions in Supersymmetric Quantum Mechanics

Abstract

Supersymmetric WKB (SWKB) wave functions diverging at the turningpoints and quantization relations up to the order ℏ6are derived directly from the supersymmetric partner Schrödingerequations. The resulting supersymmetric partner wave functions areused, without any Langer-type correction, for dipole radial integralcalculations for bound-bound transitions in the Coulomb approximation.To our knowledge, this is the first analytical calculation of matrixelements using SWKB wave functions. The bosonic SWKB partner wavefunction yields the WKB radial integral formula of Pankratov andMeyer-ter-Vehn which was derived by applying the Langer correction tothe centrifugal potential. The new dipole radial integral expressionobtained from the fermionic SWKB partner wave function overestimatesthe oscillator strengths with respect to accepted values. However, weverify that in the classical limit both supersymmetric partner SWKBresults converge to the same value.