Three-Dimensional Dirac Oscillator with Minimal Length: Novel Phenomena for Quantized Energy

Abstract

We study quantum features of the Dirac oscillator under the condition that the position and the momentum operators obey generalized commutationrelations that lead to the appearance of minimal length with the order of the Planck length,∆xmin=ℏ3β+β′, whereβandβ′are two positive small parameters. Wave functions of the system and the corresponding energy spectrum are derived rigorously. The presence of the minimal length accompanies a quadratic dependence of the energy spectrum on quantum numbern, implying the property of hard confinement of the system. It is shown that the infinite degeneracy of energy levels appearing in the usual Dirac oscillator is vanished by the presence of the minimal length so long asβ≠0. Not only in the nonrelativistic limit but also in the limit of the standard case(β=β′=0), our results reduce to well known usual ones.

Exact Solutions of the (2+1)-Dimensional Dirac Oscillator under a Magnetic Field in the Presence of a Minimal Length in the Non-commutative Phase Space

“…In this context, many papers were published where a different quantum system in space with the Heisenberg algebra was studied. They are the Abelian Higgs model [19], the thermostatics with minimal length [20], the one-dimensional hydrogen atom [21], the casimir effect in minimal length theories [22], the effect of minimal lengths on electron magnetism [23], the DO in one and three dimensions [24][25][26][27][28], the noncommutative (NC) (2+1)-dimensional DO and quantum phase transition [10], the solutions of a two-dimensional Dirac equation in the presence of an external magnetic field [29], the NC phase space Schrödinger equation [30], and the Schrödinger equation with harmonic potential in the presence of a magnetic field [31].…”

The exact solutions of a (2 + 1)-dimensional Dirac oscillator under a magnetic field in the presence of a minimal length

“…In this context, many papers were published where a different quantum system in space with Heisenberg algebra was studied. They are: the Abelian Higgs model [15], the thermostatics with minimal length [16], the one-dimensional Hydrogen atom [17], the casimir effect in minimal length theories [18], the effect of minimal lengths on electron magnetism [19], the Dirac oscillator in one and three dimensions [20][21][22][23][24], the solutions of a two-dimensional Dirac equation in presence of an external magnetic field [25], the noncommutative phase space Schrödinger equation [26], Schrödinger equation with Harmonic potential in the presence of a Magnetic Field [27].…”

Novel characteristics of energy spectrum for 3D Dirac oscillator analyzed via Lorentz covariant deformed algebra

“…Its asymptotic behavior will be estimated in both non-relativistic and non-deformed cases. An interesting discrepancy between our energy spectrum and the one obtained by Kempf deformed algebra 34 will be addressed.…”

“…(1) does not affect the commutation relations and only modify the squeezing factor of the momentum space measure. In fact, the inner product is now defined by Now, let us introduce the (3 + 1) dimensional Lorentz-covariant algebra 33 34 . In this case we have to make the following substitution in Eqs.…”