Hydrogen Atom in a Finite Linear Space

A 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 Lagrangeʹs 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.