Perburbation and non-perburbation numerical calculations to compute energy eigenvalues for the Schrödinger equation with various types of potential
The present work is concerned with methods of finding the energy eigenvalues of the one-particle Schrödinger equation for various model potentials in one, two, three and N-dimensional space. One major theme of this thesis is the study of diverent Rayleigh-Schrödinger perturbation series which are encountered in non-relativistic quantum mechanics and on the behaviour of the series coefficients E(n) in the energy expansion E(λ):E(O)+∑ E(n)λⁿ. Several perturbative techniques are used. Hypervirial and Hellmann-Feynman theorems with renormalised constants are used to obtain perturbation series for large numbers of potentials. Pade approximant methods are applied to various problems and also an inner product method with a renormalised constant is used to calculate energy eigenvalues with very high accuracy. The non-perturbative methods which are used to calculate energy eigenvalues include finite difference and power series methods. Expectation values are determined by an approach based on eigenvalue calculations, without the explicit use of wave functions. The first chapter provides a glance back into history and a preview of the problems and ideas to be investigated. Chapter two deals with one dimensional problems, including the calculation of the energy eigenvalues for quasi-bound states for some types of perturbation (λx²ⁿ⁺¹). Chapter three is concerned with two, three and N-dimensional problems. Chapter four deals with non-polynomial potentials in one and three dimensions. The final chapter is devoted to a variety of eigenvalue problems. Most of the energy eigenvalues are computed by more than one method with double precision accuracy, and the agreement between the results serves to illustrate the accuracy of the methods.