Exact Solutions of the Schrödinger Equation

TLDR: In this article, 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.

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.