Exact Solutions of the Schrödinger Equation

Abstract

In classical mechanics the problem of determining the forms of potential function which permit solution in terms of known functions received considerable attention. The present paper is a partial study of the same problem in quantum mechanics. A method is given for determining the forms of potential function which permit an exact solution of the one-dimensional Schr\odinger equation in terms of series whose coefficients are related by either two or three term recursion formulas. The more interesting expressions for the potential energy have been tabulated. A correspondence is found between these solutions and the solutions of the corresponding Hamilton-Jacobi equation. It is shown that whenever the Hamilton-Jacobi equation is soluble in terms of circular or exponential functions, the corresponding Schr\odinger equation is soluble in terms of a series whose coefficients are related by a two-term recursion formula. Whenever the Hamilton-Jacobi equation is soluble in terms of elliptic functions, the corresponding Schr\odinger equation is soluble in terms of a series whose coefficients are related by a three-term recursion formula. For the first case the quantized values of the energy are found by restricting the series to a polynomial and in the second by finding the roots of a continued fraction. A brief discussion of the technique of solution of continued fractions is given.