The radial reduced Coulomb Green’s function

Abstract

The reduced Green’s function, g_nl(r,r′), of the radial hydrogenic Schrödinger equation is simplified for all values of n and l, l⩽n−1, to a closed form appropriate for analytic treatments of Rayleigh–Schrödinger perturbation theory. Integral moments of the form ∫₀^∞dr′ g_nl(r,r′) (r′)^k+2exp(−Zr′/n) are given. It is also shown how g_nl is connected to g_n,l±1 by the ladder operators of the factorization method. Recursion relations are derived between integrals that arise in perturbation theory. The above results are generalized to the case l⩾n, which occurs in the Green’s functions required for the Rayleigh–Schrödinger perturbation treatment even though it does not arise for the eigenfunctions. As an example of the use of the reduced Green’s functions, the first‐order wave function and second‐order energy corresponding to the spin‐orbit interaction is evaluated for any bound state.