The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Methods

TLDR

The study focuses on determining eigenvalues and eigenfunctions of the Schrödinger equation for a non‑Coulomb central potential expressed as a function of radial distance. The authors employ a two‑point shooting technique that integrates the modified Schrödinger equation from both r = 0 and r = ∞, matching solutions at an intermediate radius to identify eigenvalues, while using variation equations, recurrence relations, and perturbation formulas to reduce numerical effort and facilitate efficient computation.

Abstract

The paper is concerned with the practical determination of the characteristic values and functions of the wave equation of Schrodinger for a non-Coulomb central field, for which the potential is given as a function of the distance r from the nucleus. The method used is to integrate a modification of the equation outwards from initial conditions corresponding to a solution finite at r = 0, and inwards from initial conditions corresponding to a solution zero at r = ∞, with a trial value of the parameter (the energy) whose characteristic values are to be determined; the values of this parameter for which the two solutions fit at some convenient intermediate radius are the characteristic values required, and the solutions which so fit are the characteristic functions (§§ 2, 10). Modifications of the wave equation suitable for numerical work in different parts of the range of r are given (§§ 2, 3, 5), also exact equations for the variation of a solution with a variation in the potential or of the trial value of the energy (§ 4); the use of these variation equations in preference to a complete new integration of the equation for every trial change of field or of the energy parameter avoids a great deal of numerical work. For the range of r where the deviation from a Coulomb field is inappreciable, recurrence relations between different solutions of the wave equations which are zero at r = ∞, and correspond to terms with different values of the effective and subsidiary quantum numbers, are given and can be used to avoid carrying out the integration in each particular case (§§ 6, 7). Formulae for the calculation of first order perturbations due to the relativity variation of mass and to the spinning electron are given (§ 8). The method used for integrating the equations numerically is outlined (§ 9).