Numerical and perturbation theoretic methods for the solution of the Schrödinger equation
In the present work we present a numerical and perturbation theoretic approach to the solution of the one-particle Schrödinger equation. The numerical methods developed can be used to find energy eigenvalues for one-dimensional problems as well as for radial ones. Expectation values are determined by an approach based on eigenvalue calculations, without the explicit use of wave functions. Hypervirial and Hellmann-Feynman theorems are used to obtain perturbation series to high order for polynomial type radial perturbations of the hydrogen atom. One such perturbation leads to an apparently new phenomenon in Rayleigh-Schrödinger perturbation theory. Wynn's algorithm is used to get Padé approximants for the perturbation series. The series for the energy and for the quantities < rᴺ > are treated, and both types of series can be found using the hypervirial method. Several applications of the numerical techniques are given; it is emphasized that theoretical manipulations are needed to transform the problem to an appropriate numerical form. It is demonstrated that a slight modification in the numerical techniques developed permits treatment of quasi-bound states as well as bound states. It is also shown how to calculate a local quantity Ψ(O), using energy calculations, and how to reduce the problem of treating angular terms in the quadratic Zeeman effect problem to a radial integration problem.