The Dirac oscillator

Abstract

Dirac's free particle equation originated in an attempt to express linearly the relativistic quadratic relation between energy and momentum. The authors introduce a Dirac equation which, besides the momentum, is also linear in the coordinates. They call it the Dirac oscillator because in the nonrelativistic limit it becomes a harmonic oscillator with a very strong spin-orbit coupling term. The eigenstates and eigenvalues of the Dirac oscillator can be obtained in an elementary fashion, with the degeneracy of the latter being quite different from that of the ordinary oscillator. They briefly mention the symmetry Lie algebra responsible for this degeneracy and the generalisation of the problem to many-particle systems.