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Mathematical Proceedings of the Cambridge Philosophical Society

The use of the Airy functions in a potential barrier problem

Bertha Swirles Jeffreys

Published Apr 1, 1956• 8 citations
Publisher

Abstract

ABSTRACT The exact solution for a trapezoidal potential barrier is given in terms of Airy functions and trigonometrical functions. Numerical values are found for the ratio of the amplitudes of the transmitted and reflected waves. These are compared with those given by the WKBJ approximation.
DOI
10.1017/s030500410003125x

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A simple yet exact method of solving the Schrodinger equation across an arbitrary one-dimensional piecewise-linear potential is described. It is based on the analytical solution of the Schrodinger equation across a linear potential, where the solution can be expressed as a linear combination of the Airy functions. Proper boundary conditions are imposed at the interface between adjacent linear intervals of the potential, and a transfer-matrix procedure is utilized in the derivation. The quantum mechanical transmission coefficient across a potential barrier and eigenenergies of a potential well can also be easily calculated using this method.