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Acta Physica Polonica A

Simplified Solving Procedure for the One-Dimensional Schrodinger Equation with Asymmetrical Boundary Conditions

H. Puszkarski

Published Sep 1, 1999• 0 citations
Publisher

Abstract

We propose a new simplified procedure for finding the analytical solutions of the stationary one-dimensional Schrödinger equation, with asymmetric boundary conditions imposed on the equation.The essence of the method consists in expressing the general solution explicitly in terms of the boundary parameters in a form which, by itself, satisfies one of the boundary conditions involved; then, the other boundary condition gives straightforwardly the characteristic equation.This method may turn out to be beneficial with regard to the recently growing interest in one-dimensional quantum systems.

DOI
10.12693/aphyspola.96.327

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A consistent approach to the description of a stationary motion of a quantum particle in an arbitrary one-dimensional potential has been developed. It has been proved that the wave function of an infinite motion can be expressed accurate to up two arbitrary constants with the use of one particular solution to the system of first-order linear differential equations. It has been shown that many well-known methods, such as the integral equation method, the transfer matrix method, the embedding method, and the method of combination of scattering parameters, are based on a general property of the solutions to the Schrödinger equation. Within the proposed approach, the relation between these methods becomes more transparent and their description can be well within a unified context.
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A new approach to the problem of description of the stationary motion of a quantum particle in the field of a one-dimensional arbitrary potential is developed. It is shown that the wave function of infinite motion, with the accuracy to two arbitrary constants, can be expressed by means of an arbitrary single solution for some set of linear differential equations of the first order. It is shown that one general property of the Schrödinger equation solutions lies in the basis of many known methods of the problem consideration, such as the method of integral equations, transfer-matrix method, imbedding method, and method of combination of scattering parameters. Within the framework of the proposed approach, the connection between the above-mentioned methods becomes more transparent.
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