The Dirac-Moshinsky oscillator: theory and applications

C. Quesne

Emerson Sadurní, R. Bijker, Octavio Castaños +2 more

This work summarizes the most important developments in the construction and application of the Dirac‐Moshinsky oscillator (DMO). The literature on the subject is voluminous, mostly because of the avenues that exact solvability opens towards our understanding of relativistic quantum mechanics. Here we make an effort to present the subject in chronological order and also in increasing degree of complexity. We start our discussion with the seminal paper by Moshinsky and Szczepaniak and the immediate implications stemming from it. Then we analyze the extensions of this model to many particles. The one‐particle DMO is revisited in the light of the Jaynes‐Cummings model in quantum optics and exactly solvable extensions are presented. Applications and implementations in hexagonal lattices are given, with a particular emphasis in the emulation of graphene in electromagnetic billiards.

Toshihiro Iwai, Boris I Zhilinskií

J. A. Franco-Villafañe, Emerson Sadurní, Sonja Barkhofen +3 more

We present the first experimental microwave realization of the one-dimensional Dirac-Moshinsky oscillator, a paradigm in exactly solvable relativistic systems. The experiment relies on a relation of the Dirac-Moshinsky oscillator to a corresponding tight-binding system. This tight-binding system is implemented as a microwave system by a chain of coupled dielectric disks, where the coupling is evanescent and can be adjusted appropriately. The resonances of the finite microwave system yields the spectrum of the one-dimensional Dirac-Moshinsky oscillator with and without mass term. The flexibility of the experimental set-up allows the implementation of other one-dimensional Dirac type equations.

D. Ojeda-Guillén, R. D. Mota, M. Salazar–Ramírez +1 more

We extend the (1 + 1)-dimensional Dirac–Moshinsky oscillator by changing the standard derivative by the Dunkl derivative. We demonstrate in a general way that for the Dirac–Dunkl oscillator be parity invariant, one of the spinor component must be even, and the other spinor component must be odd, and vice versa. We decouple the differential equations for each of the spinor component and introduce an appropriate su(1, 1) algebraic realization for the cases when one of these functions is even and the other function is odd. The eigenfunctions and the energy spectrum are obtained by using the su(1, 1) irreducible representation theory. Finally, by setting the Dunkl parameter to vanish, we show that our results reduce to those of the standard Dirac-Moshinsky oscillator.