Evaluation techniques for quantum mechanical integrals involving orthogonal polynomials

Abstract

Quantum mechanical matrix elements often involve orthogonal polynomials, whose properties can be exploited for evaluation. Nullity conditions are considered for hydrogenic and harmonie oscillator radial integrals, and Gaunt's triangular condition for integrals over triple products of associated Legendre functions is generalized so that the superscripts form a triangle of even perimeter. Where the integral is nonzero, Gaussian quadratures are exact for polynomial integrands and converge under very general conditions. When the integrand is a product of three functions, one of which is a linear polynomial, the quadrature is equivalent to first developing summation-orthogonal expansions of the component functions and then integrating the product exactly.