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

Abstract

The problem of the determination of eigenvalues for two coupled Schradinger 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 cyij and pi, ( i = 1,2; j = 1,2) having well-defined initial values at an arbitrary "origin" ro. These functions are collected in four couples; each one is the solution of the given coupled equations. For a given E, an "eigenvalue function" D(E) is defined by an analytical expression depending on ai,(r) and p i j ( r ) at r = 0 and r = m 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 eigenvalues 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 X which is almost the precision of the computer.

The Canonical Function Method and its applications in quantum physics

“…When it tends to a constant limit, we consider that the asymptotic region has been reached and the phaseshift is determined modulo 2π . Numerically, the phaseshift is actually calculated with the following steps (see Fakhreddine et al papers [13][14][15]):…”

Distorted waves with exact nonlocal exchange: A canonical function approach

“…Essentially, we need only to develop a single algorithm for the solution of a system of two coupled equations of the form (7) or ( 9) and then apply it to the different cases represented by the intial conditions (8) or (10).…”

“…In section 3 we describe the use of the the Canonical Fuction approach to solve the resulting coupled differential equations. It is interesting to note that the present treatment of exchange can ultimately be combined with the Canonical Function approach to the solution of coupled equations without exchange [7,8,12] so as to get a full solution of the Close Coupling equations including all potentials and non-local exchange terms (diagonal and off-diagonal).…”