Massive fermions interacting via a harmonic oscillator in the presence of a minimal length uncertainty relation

Abstract

We derive the relativistic energy spectrum for the modified Dirac equation by adding a harmonic oscillator potential where the coordinates and momenta are assumed to obey the commutation relation $\left[\hat{x},\hat{p}\right]=i\hslash \left(1+\eta p^2\right)$. In the nonrelativistic limit, our results are in agreement with the ones obtained {previously}. Furthermore, the extension to the construction of creation and annihilation operators for the harmonic oscillators with minimal length uncertainty relation is presented. Finally, we show that the commutation relation of the $su(1,1)\sim so(2,1)$ algebra is satisfied by the {operators $\widehat{{\mathcal{L}}_{\pm }}$ and $\widehat{{\mathcal{L}}_z}$}.

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