The Dirac H-atom using the phase as an extra variable

Abstract

In this paper, an algebraic solution for the bound states of the relativistic hydrogen atom is presented. The method discussed here adds an operator associated with the phase of the energy eigenstates to the set of variables of the problem. In terms of this set, appropriate ladder operators are constructed in order to express the full solution of the Dirac hydrogen equation. These ladder operators are used to form the Lie algebra of a su(1, 1) group, in the same way as that applied to angular momentum algebra and the ladder operators L±. The elements of the vector space associated with the representation of this algebra are related to a generalization of the Laguerre polynomials of the non integer index, also known as Sonine polynomials. In addition, we find that the eigenvalues of the operator constructed with the sum of the square of the three su(1, 1) generators gives precisely the relativistic energy spectrum of the hydrogen atom, including its angular momentum dependency.