An Algebraic Approach to Bound States of Simple One-Electron Systems

A very general realization of the so(2, 1) algebra, which easily follows from the basic commutation relations that are satisfied by any pair of mutually conjugate generalized coordinates and momenta, is constructed. Using special cases of this general realization, and closely following the well known derivation of the eigenvalue spectra of the angular-momentum operators, based on the so(3) algebra, we derive the energy spectrum for the N-dimensional isotropic harmonic oscillator, and for both the nonrelativistic and the relativistic N-dimensional hydrogen atom. Special attention is given to a simple derivation of the form of these Hamiltonians in terms of the so(2, 1) algebra generators. In particular, the usually exploited tilting transformation is avoided, and the whole derivation is presented in an extremely simple and straightforward way. The present approach stresses the similarity and mutual relationship between the systems studied and, in addition to introducing some novel techniques and providing considerable insight into the overall structure of these problems, also has a definite pedagogical value.