The One-dimensional Thermal Properties for the Relativistic Harmonic Oscillators

The relativistic harmonic oscillator is one of the most important quantum system, as it is one of the very few that can be solved exactly. The Dirac relativistic oscillator (DO) interaction is an important potential both for theory and application. It was for the first time studied by Ito et al [1]. They considered a Dirac equation in which the momentum p is replaced by p − imβω r, with r being the position vector, m the mass of particle, and ω the frequency of the oscillator. The interest in the problem was revived by Moshinsky and Szczepaniak [2], who gave it the name of Dirac oscillator (DO) because, in the non-relativistic limit, it becomes a harmonic oscillator with a very strong spin-orbit coupling term. Physically, it can be shown that the (DO) interaction is a physical system, which can be interpreted as the interaction of the anomalous magnetic moment with a linear electric field [3, 4]. The electromagnetic potential associated with the DO has been found by Benitez et al[5]. The Dirac oscillator has attracted a lot of interest both because it provides one of the examples of the Dirac’s equation exact solvability and because of its numerous physical applications [6, 7, 8, 9].