Variational energy spectra of relativistic Hamiltonians

Abstract

In ordinary and relativistic quantum mechanics the energy spectra of most of the Hamiltonians cannot be obtained exactly. Approximate methods have to be used among which the variational one is particuarly popular. The purpose of this paper is to show that when the matrices appearing in a Dirac equation with interaction, are replaced by a direct product of matrices associated with ordinary and what we call sign spins, then a standard complete set of non-relativistic wavefunctions can be used to carry out the variational calculations. To illustrate the power of our method we analyse first the variational energies of ordinary and Dirac relativistic oscillator Hamiltonians, and then indicate the procedure for the general one-particle case. The extensions to higher spins, or to a larger number of particles, are briefly mentioned in the conclusion.