Coupled radial Schrödinger equations written as Dirac-type equations: application to an amplitude-phase approach

Authors: Thylwe, K.-E., McCabe, P.
Document Type: Article
Pubstate: Published
Journal: Journal of Physics A: Mathematical and Theoretical
Volume: 45   135302
Year: 2012


The classical amplitude-phase method due to Milne, Wilson, Young and Wheeler in the 1930s is known to be a powerful computational tool for determining phase shifts and energy eigenvalues in cases where a sufficiently slowly varying amplitude function can be found. The key for the efficient computations is that the original single-state radial Schro ?dinger equation is transformed to a nonlinear equation, the Milne equation. Such an equation has solutions that may or may not oscillate, depending on boundary conditions, which then requires a robust recipe for locating the (optimal) ‘almost constant’ solutions for its use in the method. For scattering problems the solutions of the amplitude equations always approach constants as the radial distance r tends to infinity, and there is no problem locating the ‘optimal’ amplitude functions from a low-order semiclassical approximation. In the present work, the amplitude- phase approach is generalized to two coupled Schro ?dinger equations similar to an earlier generalization to radial Dirac equations. The original scalar amplitude then becomes a vector quantity, and the original Milne equation is generalized accordingly. Numerical applications to resonant electron–atom scattering are illustrated.