Harmonic oscillators in relativistic quantum mechanics

For the Dirac equation in one space dimension, we examine the possibility of harmonic oscillator (HO) potentials. A HO potential is such that it has only bound states and the energy levels are equally spaced. Regarding their Lorentz transformation properties, there are three types of potentials for the Dirac equation: vector, scalar, and pseudoscalar. HO potentials are possible for the scalar and pseudoscalar types, but not for the vector type. We show how HO potentials can be constructed by means of the “inverse scattering method.” We also examine the behavior of a wave packet in the HO potentials so constructed. The behavior is, in most of the cases, very similar to that of Schrödinger’s coherent wave packet of the nonrelativistic harmonic oscillator.