Exact Solutions of the Schrödinger Equation
TL;DR: 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.
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TL;DR: In Eq.
Abstract: ' N Eq. (28), 0.11.87 should be replaced by +~ (0.1173). J ls In the third member of Eq. (32), 15/8 should be replaced by 15/32. On the top of page 8, 0.96 Nk/2S should be replaced by 0.96 (2NkS). In Eq. (97), a term —NzJS should be added to the right-hand sIde. In Eq. (120), 0.1187 should be replaced by 0.1173/(SI—S2) and 2JSIS2 should be replaced by 4JSIS2. Below Eq. (121), SIS2/S{SI—S2) should appear as twice this quantity, and (gIS&—g2S&)p should appear multiplied by a factor (q).
352 citations
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284 citations
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TL;DR: In this article, the authors give explicit point canonical transformations which map twelve types of shape invariant potentials (which are known to be exactly solvable) into two potential classes.
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203 citations
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139 citations