An operator solution for the hydrogen atom using the phase as an additional variable

Abstract

We discuss an operator solution for the bound states of the non-relativistic hydrogen atom. The method adds the phase of a state and its associated operator to the set of variables of the system. The augmented set of operators is found to form a closed set of commutation relations thus comprising an operator Lie algebra. From these relations, the energy spectrum and bounded radial eigenfunctions are calculated. Our approach is analogous to the one employed to compute the angular momentum spectrum and eigenfunctions but with operators satisfying an su(1,1) Lie algebra instead of su(2). This method, with the same operator algebra and minor modifications, may be used to solve the Dirac relativistic hydrogen atom. (c) 2007 American Association of Physics Teachers.