A new recurrence relation for hydrogenic radial matrix elements

Abstract

A recurrence relation is presented which connects the general (n' not=n) hydrogenic radial matrix element of rbeta , where beta is a complex number to matrix elements involving exponents beta +2, beta -1, and beta -2 but the same sets of quantum numbers. This identity is found solely from the boundary conditions on the wavefunctions and the radial differential equation and should be of use in simplifying rk forms given elsewhere. For n'=n, the recurrence relation reveals a proportionality between the - beta -2, beta -1 matrix elements that includes the Pasternack-Sternheimer identity as a special case. Further results may be obtained by beta -differentiation, and new identities for matrix elements of rk log (r) are given. The n'=n matrix elements of rk are studied for k>or=-1; closed-product expressions are found for k-values near the sum and difference of the angular momenta, so that simple prescriptions can be given for the iterative calculation of n-dependent explicit forms. It is suggested that complex-variable theory be applied to a study of these results.