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The system of two coupled Schrödinger differential equations (CE) arising in the close-coupling description of electron-atom scattering is considered. A “canonical functions” approach is used. This approach replaces the integration of CE for the eigenvector with unknown initial values by the integration of CE for “canonical” functions having well-defined initial values at an arbitrary origin r0(0 < r0 < ∞). The terms of the reactance matrix are then deduced from these canonical functions. The results are independent from the choice of r0. It is shown that many conventional difficulties in the numerical application are avoided, mainly that of the choice of the starting boundary variable rs∼0 which is automatically determined. The results of the present method are compared to those of other methods and show excellent agreement with previous accurate results

a b s t r a c tThe Canonical Function Method (CFM) is a powerful method that solves the radial Schrödinger equation for the eigenvalues directly, without having to evaluate the eigenfunctions. It is applied to various quantum mechanical problems in Atomic and Molecular physics in the presence of regular or singular potentials. It has also been developed to handle single and multiple channel scattering problems, where phaseshift is required for the evaluation of the scattering cross-section. Its controllable accuracy makes it a valuable tool for the evaluation of vibrational levels of cold molecules, a sensitive test of the Bohr correspondence principle and a powerful method for tackling local and non-local spin dependent problems.

The determination of the phase-shift 6,(E) (related to a central potential V ( r ) , a total energy E, and an angular momentum p) is considered. The "canonical functions" approach already used for the eigenvalue problem is adapted to that of 6. The conventional approach computes the radial wave functiony,(E; r) starting at r, -0 (with convenient initial values) and stepping on toward a large value r = R -m, where y , is matched to its asymptotic value y,,(R) -a sin(kRp 7r/2 + 6,) and 6 is deduced. The present approach starts at any "origin" ro, replaces the use of the wave functiony by that of the "canonical functions" a and p (well defined for given V , E, and p) and defines two functions q(r) and Q(r) in terms of a and p. When r 4 0, q(r) approaches a constant limit giving Q(ro), and thus the starting problem is avoided. Using this value Q(ro), the function Q(r) is generated for r > ro. The function Q(r) reaches a constant limit when r + m; this limit is precisely tan 8; thus, the "final" matching problem is avoided. The present method is applied to the Lennard-Jones potential function for low and high E and for low and high p . The comparison of the results of the present method with those of confirmed numerical methods show that the present method is competitive.

ABSTRACT:The eigenvalue problem for a system of N coupled one-dimensional Schrodinger equations, arising in bound state in quantum mechanics, is considered. Ä canonical approach for the calculation of the energy eigenvalues of this system is presented. This method replaces the use of the wave functions by 2 N canonical functions Ž . having well-defined initial values at an arbitrary point r . An eigenvalue function D E 0 is associated with the system, where the energy E is considered as a variable. It is shown that the energy eigenvalues are the zeros of this function. This method is new in the sense that it reduces the eigenvalues search to that of finding the zeros of the function Ž . D E , which is defined and constructed from the canonical functions independently from the wave function. A numerical application to a model problem proposed by Freidman w x et al. Freidman, R. S.; Jamieson, M. J Comput Phys Commun 1989, 55, 137 is presented. It is shown that the eigenvalues computed by the present method are highly accurate for low and high levels; the average relative discrepancy between computed and exact one is about 1.5 = 10 y11 .

The eigenvalue problem for a system of N coupled one‐dimensional Schrödinger equations, arising in bound state in quantum mechanics, is considered. A canonical approach for the calculation of the energy eigenvalues of this system is presented. This method replaces the use of the wave functions by 2N canonical functions having well‐defined initial values at an arbitrary point r0. An eigenvalue function D(E) is associated with the system, where the energy E is considered as a variable. It is shown that the energy eigenvalues are the zeros of this function. This method is new in the sense that it reduces the eigenvalues search to that of finding the zeros of the function D(E), which is defined and constructed from the canonical functions independently from the wave function. A numerical application to a model problem proposed by Freidman et al. [Freidman, R. S.; Jamieson, M. J Comput Phys Commun 1989, 55, 137] is presented. It is shown that the eigenvalues computed by the present method are highly accurate for low and high levels; the average relative discrepancy between computed and exact one is about 1.5×10−11. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 325–332, 1999

The canonical function method (CFM) is a powerful means for solving the Radial Schrdinger Equation. The mathematical difficulty of the RSE lies in the fact it is a singular boundary value problem. The CFM turns it into a regular initial value problem and allows the full determination of the spectrum of the Schrdinger operator without calculating the eigenfunctions.Following the parametrisation suggested by Klapisch and Green, Sellin and Zachor we develop a CFM to optimise the potential parameters in order to reproduce the experimental Quantum Defect results for various Rydberg series of He, Ne and Ar as evaluated from Moore's data.

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.