Evaluation techniques for quantum mechanical integrals

The nullity of quantum-mechanical integrals can often be deduced by examining the orthogonal properties of the polynomial components in the integrand. Where the integral is nonzero, Gaussian quadratures are simple, direct and accurate, and often obviate the need for expanding the components in orthogonal polynomials [1]. On the other hand, the summation-orthogonal expansion coefficients associated with Gaussian quadratures can provide information on quadrature accuracy. The usual convergence test comparing results from two different formulas often yields false convergence signals, but can be strengthened by comparing these expansion coefficients also [2].