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Physical review. B, Solid state

Solution to One-Dimensional Schrödinger Equation for an Arbitrary Potential; Application to Radial Equation in Three Dimensions

James L. Sigel

Published Nov 15, 1972• 1 citation
Publisher

Abstract

The work of Reading and Sigel on the solution of the one-dimensional Schr\"odinger equation for a particle moving in the presence of a collection of $\ensuremath{\delta}$-function potentials of arbitrary strength and position is generalized so that any arbitrary potential $V(x)$ can be handled. A close connection is shown to exist between the methods described in this note and those, involving Jost functions, which have been developed for handling the radial Schr\"odinger equation in three dimensions.

DOI
10.1103/physrevb.6.4084

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The work of Reading and Sigel on the solution of the one-dimensional Schr"odinger equation for a particle moving in the presence of a collection of \ensuremathδ\ensuremath{\delta}\ensuremathδ-function potentials of arbitrary strength and position is generalized so that any arbitrary potential V(x)V(x)V(x) can be handled. A close connection is shown to exist between the methods described in this note and those, involving Jost functions, which have been developed for handling the radial Schr"odinger equation in three dimensions.
A simple exact solution of a one-dimensional Schr\"odinger equation which describes the motion of a particle in a continuous potential V(x)V(x)V(x) and scattering from a finite set of $\ensuremath{\delta}$-function potentials of arbitrary positions and strengths is found.
The exact solution to the one-dimensional Schr\"odinger equation which describes the motion of a particle scattering off a finite set of $\ensuremath{\delta}$-function potentials of arbitrary strengths and positions is found. The behavior of the TTT matrix both on and off the energy shell is also precisely given. Localization of states is discussed.
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.
A practical method is developed for calculating numerically one-dimensional path integrals for an arbitrary potential V(x)V(x)V(x). For the oscillator potential V(x)=kx22V(x)=\frac{k{x}^{2}}{2}V(x)=2kx2​ the well-known analytic solution is obtained. To illustrate the numerical convergence of this method the path integral for the Ginzburg-Landau potential, $V(x)=\ensuremath{-}\frac{A{x}^{2}}{2}+\frac{B{x}^{4}}{4}$, is calculated over a range of the positive constants AAA and BBB and compared with numerical solutions of the Schr\"odinger equation.
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.