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American Journal of Physics

The remarkable capabilities of recursive relations

V. K. Ignatovich

Published Oct 1, 1989• 14 citations
Publisher

Abstract

The recursive relations method is presented. To illustrate its capability a new way is presented to solve the one-dimensional Schrödinger equation with a stepwise or any periodic potential (semi-infinite or finite). The method is shown to be also applicable to other physical problems, for instance, to molecular gas diffusion in linear and branched guides and to the scattering of a quantum mechanical particle on a spherically symmetric nut-shell potential.

DOI
10.1119/1.15838

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The recursive relations method is presented. To illustrate its capability a new way is presented to solve the one-dimensional Schrödinger equation with a stepwise or any periodic potential (semi-infinite or finite). The method is shown to be also applicable to other physical problems, for instance, to molecular gas diffusion in linear and branched guides and to the scattering of a quantum mechanical particle on a spherically symmetric nut-shell potential.
This chapter solves the eigen-problems for a few simple one-dimensional systems to acquire experience in solving the Schrodinger eigen-equation. One-dimensional problems are physically interesting not only because they are simple, but also because a number of more complicated higher-dimensional problems can be reduced to the solutions of equations in one dimension. For a symmetric potential, an eigenfunction of the Schrödinger equation can be always chosen to have a definite parity. The probability density of the plane wave solution is a constant, independent of the spatial coordinate of the particle. The eigensolution of a particle in an infinite square well consists of an infinite number of discrete energy levels. In classical mechanics, a particle cannot penetrate a potential barrier whose height is higher than the energy of the particle.
The transfer-matrix method was proposed in the 1940s to solve linear ordinary differential equations.It is broadly studied due to its efficiency and simplicity for solving one-dimensional physical problems.In this work, the method is applied to solve the timeindependent Schrödinger equation, in order to improve both the formalism and the computational treatment.Closed-form solutions are derived for quantum wells and periodic potentials of arbitrary profiles.Numerical results are given for symmetric finite-barrier quantum wells and sinusoidal potentials.
In the previous chapters, we have shown how to solve the Schrodinger equation for several systems as a particle in an infinite and finite potential well, scattering, harmonic oscillator, and hydrogen atom. For all these systems, we were able to solve Schrodinger's equation exactly. However, there are complicated potentials for which Schrodinger's equation cannot be solved exactly. For such situations, we resort to approximation methods for finding approximate solutions to the Schrodinger equation. In this chapter, we discuss several approximate methods for solving Schrodinger's equation.