In the present article we have found the complete energy spectrum and the corresponding eigenfunctions of the Dirac oscillator in two spatial dimensions. We show that the energy spectrum depends on the spin of the Dirac particle.
Exact Solution of a Spin-21 Particle for a Linear Potential
The problem of a spin-21 particle moving in a linear potential field in two-dimensions is searched to obtain for nonzero energy eigenvalues and the corresponding normalized eigenfunctions. The zero-mode (E=0) eigenfunctions are also studied and it is seen that they are normalizable. The variation of the non-zero eigenfunctions and also eigenvalues are according to the position and potential parameter γ, respectively, given in the text.
We extend the notion of Dirac oscillator in two dimensions, to construct a set of potentials. These potentials become exactly and quasi-exactly solvable potentials of nonrelativistic quantum mechanics when they are transformed into a Schrödinger-like equation. For the exactly solvable potentials, eigenvalues are calculated and eigenfunctions are given by confluent hypergeometric functions. It is shown that, our formulation also leads to the study of those potentials in the framework of the supersymmetric quantum mechanics.
The energy and eigenstate spectrum of a charged particle in the electric field of a 2D anisotropic oscillator and in a uniform magnetic field is considered. The exact analytic solution to the problem is obtained for an arbitrary magnetic field strength. The characteristic features of variation of the energy spectrum depending on the magnetic field strength are analyzed. The results of this study are of interest for the quantum-mechanical theory of magnetism and can be used to simulate the magnetic properties of atoms and molecules.