Solving Schrödinger’s equation by B-spline collocation

B-splines and collocation techniques have been applied to the solution of Schrödinger’s equation in quantum mechanics since the early 1970s, but one aspect that is noticeably missing from this literature is the use of Gaussian points (i.e., the zeros of Legendre polynomials) as the collocation points, which can significantly reduce approximation errors. Authors in the past have used equally spaced or nonlinearly distributed collocation points (noticing that the latter can increase approximation accuracy) but, strangely, have continued to avoid Gaussian collocation points so there are no published papers employing this approach. Using the methodology and computer routines provided by Carl de Boor’s book A Practical Guide to Splines as a `numerical laboratory’, the present dissertation examines how the use of Gaussian collocation points can interact with other features such as box size, mesh size and the order of polynomial approximants to affect the accuracy of approximations to Schrödinger’s bound state wave functions for the electron in the hydrogen atom. In particular, we explore whether or not, and under what circumstances, B-spline collocation at Gaussian points can produce more accurate approximations to Schrödinger’s wave functions than equally spaced and nonlinearly distributed collocation points. We also apply B-spline collocation at Gaussian points to a Schrödinger equation with cubic nonlinearity which has been used extensively in the past to study nonlinear phenomena. Our computer experiments show that in the case of the hydrogen atom, collocation at Gaussian points can be a highly successful approach, consistently superior to equally spaced collocation points and often superior to nonlinearly distributed collocation points. However, we do encounter some situations, typically when the mesh is quite coarse relative to the box size for the hydrogen atom, and also in the cubic Schrödinger equation case, in which nonlinearly distributed collocation points perform significantly better than Gaussian collocation points.

Full dissertation can be found on ArXiv.org.