The Statistical Properties of the q-Deformed Dirac Oscillator in One and Two Dimensions

We study the behavior of the eigenvalues of the one and two dimensions of q-deformed Dirac oscillator. The eigensolutions have been obtained by using a method based on the q-deformed creation and annihilation operators in both dimensions. For a two-dimensional case, we have used the complex formalism which reduced the problem to a problem of one-dimensional case. The influence of the q-numbers on the eigenvalues has been well analyzed. Also, the connection between the q-oscillator and a quantum optics is well established. Finally, for very small deformation eta, we (i) showed the existence of well-known q-deformed version of Zitterbewegung in relativistic quantum dynamics and (ii) calculated the partition function and all thermal quantities such as the free energy, total energy, entropy, and specific heat. The extension to the case of Graphene has been discussed only in the case of a pure phase (q = e(i eta)).