A “canonical functions” approach to the eigenvalues of a system of two coupled Schrödinger equation

Abstract

AbstractThe problem of the determination of eigenvalues for two coupled Schrödinger equations is considered. A new method to solve this problem is presented. This method replaces the use of the wave functions (with unknown initial values) by eight canonical functions αij and βij (i = 1,2; j = 1,2) having well‐defined initial values at an arbitrary “origin” r0. These functions are collected in four couples; each one is the solution of the given coupled equations. For a given E, an “eigenvalue functions” D(E) is defined by an analytical expression depending on αij (r) and βij (r) at r = 0 and r = ∞ only. The successive eigenvalues En of the given system are precisely the successive intersection of the graph D(E) with the E‐axis. The present method eliminates the conventional use of wave function initial values as well as the conventional problem of the prior guess of the limit points; it determines these points automatically. It eliminates also the use of trial values for E and the need of iterations for its correction. The numerical application of a standard example used by Friedman and co‐workers (1990) shows that the eigenvlues computed by the present method are highly accurate for low and high levels; the average relative discrepancy between computed and exact levels is about 3.4 × 10 −15 (this discrepancy never exceeds 1.6 × 10 −14), which is almost the precision of the computer. © 1994 John Wiley & Sons, Inc.