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Hydrogen Atom in a Finite Linear Space

✍ Scribed by Rafael G. Campos; L.O. Pimentel

Publisher: Elsevier Science
Year: 2000
Tongue: English
Weight: 170 KB
Volume: 160
Category: Article
ISSN: 0021-9991
DOI: 10.1006/jcph.2000.6446

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✦ Synopsis

Galerkin-collocation-type technique for solving numerically differential boundary value problems was developed several years ago. Such a method is based on a certain finite-dimensional matrix representation of the derivative d/dx obtained through Lagrange's interpolation. Recently, an extension to separable multivariate problems has been given; in this context, the authors have found a matrix representation of the quantum angular momentum, yielding the precise eigenvalues and finite-dimensional vectors that coincide exactly with the spherical harmonics evaluated at a certain set of points. The aim of this paper is to give additional properties of such a matrix representation and to show how these findings can be applied to obtain binding energies and eigenfunctions for the hydrogen atom. We consider three cases: the Coulomb potential, the fine-structure splitting, and the hydrogen atom in a uniform magnetic field. Since this last case is a nonseparable problem in the coordinates, the method requires a modification that is introduced in this paper.

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