1 Introduction

A long time ago in the past, in order to regularize the divergences in quantum field theories, Snyder [1, 2] proposed a model of noncommutative spacetime by means of the projective geometry approach to the de Sitter momentum space. The Snyder model is invariant under the action of the Lorentz group, and is based on the modification of the Heisenberg algebra of quantum mechanics by a fundamental minimal length scale. The existence of fundamental minimal length scale is predicted by various research fields as string theory [3], black hole physics [4], and quantum gravity [5]. For example, in the case of string theory, it’s assumed that a particle described as a string does not interact at distances smaller than its size.

The deformed quantum algebra introduced by Snyder is based on following modified commutation relation:

$$\begin{aligned} \left[ X_{\mu },P_{\nu }\right] =i\hslash \left( \eta _{\mu \nu }+\beta P_{\mu }P_{\nu }\right) ; \quad \left[ P_{\mu },P_{\nu }\right] =0; \quad \left[ X_{\mu },X_{\nu }\right] =i\hslash \beta J_{\mu \nu }, \end{aligned}$$
(1)

where \(J_{\mu \nu }=X_{\mu }P_{\nu }-X_{\nu }P_{\mu }\) are the generators of the Lorentz symmetry and \(\beta \) is a coupling constant to be of the order of the Planck length. Due to presence of \(\beta \), the Snyder model can be interpreted as an example of a doubly (deformed) special relativity (DSR) [6, 7], namely, a theory where there exist two observers-independent scales, velocity which is identified by the speed of light, and the length which is identified by the Planck length.

There exists in some papers another deformation of the standard commutation relation [8,9,10]. Recently by using Beltrami coordinate, Mignemi [10] showed that on the (anti-) de Sitter background, the Heisenberg uncertainty principle must be modified by introducing a small correction, this modification was called extended uncertainty principle (EUP). On (anti-) de Sitter background, the extended commutation relations introduced by Mignemi are given by

$$\begin{aligned} \left[ X_{\mu },P_{\nu }\right] =i\hslash \left( \eta _{\mu \nu }+\alpha X_{\mu }X_{\nu }\right) , \end{aligned}$$
(2)

and also

$$\begin{aligned} \left[ X_{\mu },X_{\nu }\right] =0;\qquad \qquad \qquad \left[ P_{\mu },P_{\nu }\right] \ne 0, \end{aligned}$$
(3)

where \(\alpha \prec 0\) for de Sitter spacetime, and \(\alpha \succ 0\) for anti-de Sitter spacetime, which characterized by the presence of a nonzero minimum uncertainty in momentum (MUM).

On the other hand, an extensive effort has been made to extend the study of Snyder model in flat spaces to a de Sitter curved spacetime, by introducing a third invariant parameter \(\alpha \) [11,12,13], this model was named triply special relativity (TSR) or Snyder (anti-)de Sitter (SdS) model, it’s characterized by the modification of standard commutation relation between the position operators \(X_{\mu }\) and the momentum operators \(P_{\nu }\)

$$\begin{aligned} \left[ X_{\mu },P_{\nu }\right] =i\hslash \left( \eta _{\mu \nu }+\beta P_{\mu }P_{\nu }+\alpha X_{\mu }X_{\nu }+...\right) , \end{aligned}$$
(4)

and the appearance of both minimal position and momentum uncertainties.

Recently, various studies about the effects of the deformed canonical commutation relations have been done in the literature, among them: the relativistic and non-relativistic harmonic oscillator [14,15,16,17,18], the hydrogen atom in one [19] and three dimensions [20], the Cusp potential [21], the Casimir effect has been investigated in [22] and Scattering states of Woods-Saxon interaction by [23].

The Dirac oscillator (DO) has attracted particular interest in the last years. It was introduced, for the first time, in [24, 25] by the simple replacement of the momentum operator \(\overrightarrow{p}\) \(\rightarrow \overrightarrow{p}-im\omega \gamma ^{0}\overrightarrow{r}\). Moshinsky was the first who named it “DO” because, in the non-relativistic limit, it reduces to a quantum simple harmonic oscillator with a strong spin-orbit coupling term plus a constant term. On the other hand, it has been shown in [26] that the Hamiltonian of DO describes the interaction of a neutral particle with a linearly electric field via its anomalous magnetic dipole moment.

In this work, we consider the three and two-dimensionals Dirac oscillator in deformed space obeying the algebras (2) and (3) which gives rise to the appearance of the minimal uncertainty in momentum. Our manuscript is organized as follows: in Sect. 2, we give a brief introduction of the extended uncertainty principle. In Sect. 3, using the position space representation, we solve exactly the (\(1+3\)) Dirac equation with an oscillator-like interaction in the framework of EUP, the energy eigenvalue equation are obtained and the corresponding wave function are calculated in terms of orthogonal Jacobi polynomials. In Sect. 4, we consider the case of massless Dirac–Weyl electron moves with an effective Fermi velocity in graphene and subjected to the action of a uniform magnetic field, the energy eigenvalues and their corresponding wave functions are examined. By using the experimental results of the relativistic Landau levels in graphene we find upper bounds on the value of the EUP parameter. Finally, in Sect. 5, we present the conclusion.

2 Quantum Mechanics with Extended Heisenberg Relation

In more than one dimension, the modified commutation relations between the operators of position \(X_{i}\) and momentum \(P_{j}\) in AdS space read [27]

$$\begin{aligned} \left[ X_{i},P_{j}\right] =i\hslash \left( \delta _{ij}+\alpha X_{i}X_{j}\right) ;\quad i=j=1;2;...;D. \end{aligned}$$
(5)

This deformed commutation relations (5) add some problems when one tries to study quantum mechanical problems. To our knowledge, only a few works have been studied in connection with the EUP, among them we cite [27,28,29,30,31,32,33,34,35] and the higher order generalized uncertainty principle (HGUP) case [36]. The modified commutator (5) implies a modification of the uncertainty relations. They are given by

$$\begin{aligned} \left( \triangle X_{i}\right) \left( \triangle P_{i}\right) \succeq \frac{ \hslash }{2}\left( 1+\alpha \left( \triangle X_{i}\right) ^{2}\right) , \end{aligned}$$
(6)

this modification implies a nonzero minimal uncertainty in momentum (MUM). The minimization of (6) with respect to \(\left( \triangle X\right) \) gives

$$\begin{aligned} \left( \triangle P_{k}\right) _{\min }=\frac{\hslash \sqrt{\alpha }}{2},\quad \forall \quad k. \end{aligned}$$
(7)

The operators of position and momentum satisfying equation (5) can be represented by

$$\begin{aligned} X_{i}&=x_{i}, \end{aligned}$$
(8)
$$\begin{aligned} P_{i}&=\frac{\hslash }{i}f\left( x\right) \frac{\partial }{\partial x_{i}} =\left( \delta _{ij}+\alpha x_{i}x_{j}\right) p_{j}, \end{aligned}$$
(9)

where the operators \(x_{i}\) and \(p_{j}\) satisfy the canonical commutation relation \(\left[ x_{i},p_{j}\right] =i\hslash \delta _{ij}\). Using the symmetricity condition of the operators of position and momentum, the modified scalar product can be written as

$$\begin{aligned} \left\langle \phi \right. \left| \psi \right\rangle =\int \frac{d^{D} \vec {r}}{\left( 1+\alpha r^{2}\right) ^{\frac{1+D}{2}}}\phi ^{\times }\left( \vec {r}\right) \psi \left( \vec {r}\right) ;\quad \text { where }r=\sum _{i=1}^{D}x_{i}^{2}\text {.} \end{aligned}$$
(10)

3 (\(1+3\))d Dirac Oscillator

In this section, we are interested in solving the (\(1+3\))-dimensional Dirac oscillator, in position space with deformed commutation relations. In this case, the stationary equation describing the Dirac oscillator in (\(1+3\))-dimension is given by [25]:

$$\begin{aligned} \left[ \gamma ^{0}E+\vec {\gamma }\left( \vec {P}-im\omega \gamma ^{0}\vec {r} \right) -m\right] \psi =0, \end{aligned}$$
(11)

where m is the rest mass, and \(\omega \) is the classical frequency of the oscillator, the \(4\times 4\) Dirac matrices \(\gamma ^{\mu }=\left( \gamma ^{0}, \vec {\gamma }\right) \) are given by given by

$$\begin{aligned} \gamma ^{0}=\left( \begin{array}{cc} 1 &{} 0 \\ 0 &{} -1 \end{array} \right) ;\vec {\gamma }=\left( \begin{array}{cc} 0 &{} \vec {\sigma } \\ -\vec {\sigma } &{} 0 \end{array} \right) , \end{aligned}$$
(12)

where \(\vec {\sigma }\) are \(2\times 2\) Pauli matrices, and \(\psi \) is 4-component spinor, which can be written in the form

$$\begin{aligned} \psi ^{T}=\left( \begin{array}{cc} \Phi&\Xi \end{array} \right) , \end{aligned}$$
(13)

where \(\Phi \) and \(\Xi \) are the large and small components respectively of the Dirac wavefunction, satisfy the following equations

$$\begin{aligned} \left( E-m\right) \Phi +\vec {\sigma }.\left( \vec {P}+im\omega \vec {r}\right) \Xi&=0, \end{aligned}$$
(14)
$$\begin{aligned} \left( E+m\right) \Xi +\vec {\sigma }.\left( \vec {P}-im\omega \vec {r}\right) \Phi&=0. \end{aligned}$$
(15)

This system gives the following differential equation for the component \(\Phi \)

$$\begin{aligned} \left[ \left( E^{2}-m^{2}\right) -\vec {\sigma }.\left( \vec {P}+im\omega \vec {r }\right) \vec {\sigma }.\left( \vec {P}-im\omega \vec {r}\right) \right] \Phi =0, \end{aligned}$$
(16)

Applying the definition of the position and momentum operators reported in Sect. 2, and using the well-known relations:

$$\begin{aligned} \left( \vec {\sigma }.\vec {A}\right) \left( \vec {\sigma }.\vec {B}\right) =\vec {A }.\vec {B}+i\vec {\sigma }.\left( \vec {A}\wedge \vec {B}\right) , \end{aligned}$$
(17)

we obtain the following differential equation

$$\begin{aligned} \left( m^{2}-E^{2}\right) \Phi =\left\{ \left[ \left( 1+\alpha r^{2}\right) \frac{\partial }{\partial r}\right] ^{2}+\frac{2}{r}\left( 1+\alpha r^{2}\right) \frac{\partial }{\partial r}-\frac{\hat{L}^{2}}{r^{2}} -m^{2}\omega ^{2}r^{2}+m\omega \left( 3+\alpha r^{2}\right) +2m\omega \vec { \sigma }.\vec {L}\right\} \Phi , \end{aligned}$$
(18)

where \(\vec {L}\) is the orbital angular momentum, and \(\vec {\sigma }=2\vec {S}\) is the spin-1/2 operator. It is necessary to note that, Eq. (18) describing a Klein-Gordon equation with harmonic oscillator interaction plus a spin-orbit coupling term. It is clear that the right-hand side of (18) commutes with the total angular momentum of the Dirac oscillator \(\vec {J}= \vec {L}+\vec {S}.\) Thus, it’s appropriate to split the energy eigenfunction \( \Phi \) into a radial part and an angular part as:

$$\begin{aligned} \Phi =\mathcal {F}_{n,\ell ,j}\left( r\right) \mathcal {Y}_{\kappa }^{\mu }, \end{aligned}$$
(19)

where \(\mathcal {Y}_{\kappa }^{\mu }\) are the eigenfunction of the spin-angular part. Now we consider the action of \(\vec {\sigma }.\vec {L}\) on the spin-angular function [37, 38]

$$\begin{aligned} \vec {\sigma }.\vec {L}\mathcal {Y}_{\kappa }^{\mu }=\kappa \mathcal {Y}_{\kappa }^{\mu }=\left[ s\left( 2j+1\right) -1\right] \mathcal {Y}_{\kappa }^{\mu }; \quad \kappa =\left( -1,1,-2,2,...\right) . \end{aligned}$$
(20)

This allows us to rewrite Eq. (18) as

$$\begin{aligned} \left[ \left[ \left( 1+\alpha r^{2}\right) \frac{\partial }{\partial r} \right] ^{2}+\frac{2}{r}\left( 1+\alpha r^{2}\right) \frac{\partial }{ \partial r}-\frac{\ell \left( \ell +1\right) }{r^{2}}-m^{2}\omega ^{2}r^{2}+m\omega \alpha r^{2}+\lambda _{n,j}\right] \mathcal {F}_{n,\ell ,j}\left( r\right) =0, \end{aligned}$$
(21)

where

$$\begin{aligned} \lambda _{n,j}=E^{2}-m^{2}+2m\omega \left[ s\left( 2j+1\right) -1\right] +3m\omega . \end{aligned}$$
(22)

To solve this equation, we begin by making the following change of variable

$$\begin{aligned} \sqrt{\alpha }\rho =\tan ^{-1}\sqrt{\alpha }r, \end{aligned}$$
(23)

which maps the interval \(r\in \left] 0,\infty \right[ \) to \(\rho \in \left] 0,\frac{\pi }{2\sqrt{\alpha }}\right[ \) and brings Eq. (21) to the following form

$$\begin{aligned} \left[ \frac{\partial ^{2}}{\partial \rho ^{2}}+\frac{2\sqrt{\alpha }}{\tan \left( \sqrt{\alpha }\rho \right) }\frac{\partial }{\partial \rho }-\frac{ \alpha \ell \left( \ell +1\right) }{\tan ^{2}\left( \sqrt{\alpha }\rho \right) }-m\omega \left( \frac{m\omega }{\alpha }-1\right) \tan ^{2}\left( \sqrt{\alpha }\rho \right) +\lambda _{n,j}\right] \mathcal {F}_{n,\ell ,j}\left( \rho \right) =0. \end{aligned}$$
(24)

To eliminate the first derivative, we make the subtitution

$$\begin{aligned} \mathcal {F}_{n,\ell ,j}\left( \rho \right) =e^{-\sqrt{\alpha }\int ^{\rho }\tan \left( \sqrt{\alpha }\rho ^{\prime }\right) d\rho ^{\prime }}g_{n,\ell ,j}\left( \rho \right) , \end{aligned}$$
(25)

after some manipulation, we obtain the following equation for \(g_{n,\ell , j}\left( \rho \right) \)

$$\begin{aligned} \left[ \frac{\partial ^{2}}{\partial \rho ^{2}}-\frac{\alpha \ell \left( \ell +1\right) }{\tan ^{2}\left( \sqrt{\alpha }\rho \right) }-m\omega \left( \frac{m\omega }{\alpha }-1\right) \tan ^{2}\left( \sqrt{\alpha }\rho \right) +\lambda _{n,j}+\alpha \right] g_{n,\ell ,j}\left( \rho \right) =0. \end{aligned}$$
(26)

Making now the following change of function

$$\begin{aligned} g_{n,\ell ,j}\left( \rho \right) =\sin ^{\ell +1}\left( \sqrt{\alpha }\rho \right) \cos ^{\vartheta }\left( \sqrt{\alpha }\rho \right) \Psi _{n,\ell ,j}\left( \rho \right) , \end{aligned}$$
(27)

where \(\vartheta \) is a constant to be determined letter. By means of the substitution given in Eq. (26), the differential equation for \(\Psi _{n,\ell ,j}\) can be reduced to the following form:

$$\begin{aligned} \left[ \begin{array}{c} \frac{\partial ^{2}}{\partial \rho ^{2}}+2\sqrt{\alpha }\left( \frac{\left( \ell +1\right) }{\tan \left( \sqrt{\alpha }\rho \right) }-\vartheta \tan \left( \sqrt{\alpha }\rho \right) \right) \frac{\partial }{\partial \rho } -\alpha \vartheta \left( 2\ell +3\right) \\ +\alpha \left[ \vartheta \left( \vartheta -1\right) -\frac{m\omega }{\alpha } \left( \frac{m\omega }{\alpha }-1\right) \right] \tan ^{2}\left( \sqrt{ \alpha }\rho \right) +\lambda _{n,j}-\ell \alpha \end{array} \right] \Psi _{n,\ell ,j}\left( \rho \right) =0. \end{aligned}$$
(28)

Here, we select \(\vartheta \) to eliminate the term \(\tan ^{2}\left( \sqrt{ \alpha }\rho \right) \) by demanding

$$\begin{aligned} \vartheta \left( \vartheta -1\right) -\frac{m\omega }{\alpha }\left( \frac{ m\omega }{\alpha }-1\right) =0, \end{aligned}$$
(29)

then it leads to the following expression of \(\vartheta \)

$$\begin{aligned} \vartheta _{+}=\frac{m\omega }{\alpha },\vartheta _{-}=1-\frac{m\omega }{ \alpha }. \end{aligned}$$
(30)

Among these two solutions, the physically acceptable one is only \(\vartheta _{+},\) the second solution leads to a non physically acceptable wave function. Then Eq. (28) simplifies to

$$\begin{aligned} \left[ \frac{\partial ^{2}}{\partial \rho ^{2}}+2\sqrt{\alpha }\left( \frac{ \ell +1}{\tan \left( \sqrt{\alpha }\rho \right) }-\frac{m\omega }{\alpha } \tan \left( \sqrt{\alpha }\rho \right) \right) \frac{\partial }{\partial \rho }-\left( 2\ell +3\right) m\omega +\lambda _{n,j}-\alpha \ell \right] \Psi _{n,\ell ,j}=0. \end{aligned}$$
(31)

At this stage, we introduce another change of variable defined by

$$\begin{aligned} \zeta =2\sin ^{2}\left( \sqrt{\alpha }\rho \right) -1. \end{aligned}$$
(32)

The range of the new variable is \(-1\preceq \zeta \preceq 1\). Then Eq. (31) reduces to

$$\begin{aligned} \left[ \left( 1-\zeta ^{2}\right) \frac{\partial ^{2}}{\partial \zeta ^{2}} +\left( \ell -\frac{m\omega }{\alpha }+1-\left( \ell +\frac{m\omega }{\alpha }+2\right) \zeta \right) \frac{\partial }{\partial \zeta }+\frac{\lambda _{n,j}-\alpha \ell -m\omega -2\left( \ell +1\right) m\omega }{4\alpha } \right] \Psi _{n,\ell ,j}=0. \end{aligned}$$
(33)

We require a polynomial solution to Eq. (33) to guarantee regularity of the function \(\Psi _{n,\ell ,j}\) at \(\zeta =\pm 1\). This is obtained by imposing the following constraint

$$\begin{aligned} \frac{\lambda _{n,\ell }-\alpha \ell -m\omega -2\left( \ell +1\right) m\omega }{4\alpha }=n\left( n+\eta +\tau +1\right) , \end{aligned}$$
(34)

where n is non-negative integer, and the parameters \(\eta \), \(\tau \) are defined by

$$\begin{aligned} \eta =\ell +\frac{1}{2};\quad \tau =\frac{m\omega }{\alpha }-\frac{1 }{2}. \end{aligned}$$
(35)

Then Eq. (33) takes the form

$$\begin{aligned} \left\{ \left( 1-\zeta ^{2}\right) \frac{\partial ^{2}}{\partial \zeta ^{2}} +\left[ \eta -\tau -\left( \eta +\tau +2\right) \zeta \right] \frac{\partial }{\partial \zeta }+n\left( n+\eta +\tau +1\right) \right\} \Psi _{n,\ell ,j}=0, \end{aligned}$$
(36)

whose solution is given in terms of Jacobi polynomials as

$$\begin{aligned} \Psi _{n,j,\ell }\left( \zeta \right) =P_{n}^{\left( \frac{m\omega }{\alpha } -\frac{1}{2},\ell +\frac{1}{2}\right) }\left( \zeta \right) . \end{aligned}$$
(37)

Using the old variable r, the large component of the Dirac wavefunction \( \Phi \) is then given by :

$$\begin{aligned} \Phi _{n,j,\ell ,\mu ,s}=Const\times \frac{r^{\ell +1}}{\left( 1+\alpha r^{2}\right) ^{\frac{m\omega }{2\alpha }+\frac{2+\ell }{2}}}P_{n}^{\left( \frac{m\omega }{\alpha }-\frac{1}{2},\ell +\frac{1}{2}\right) }\left( \frac{ \alpha r^{2}-1}{1+\alpha r^{2}}\right) \mathcal {Y}_{\kappa }^{\mu }. \end{aligned}$$
(38)

Using the properties of the Jacobi polynomials, and with the aid of the following relations [39]

$$\begin{aligned} \sigma _{i}\frac{\partial }{\partial x_{i}}=\frac{\partial }{\partial r}+ \frac{\vec {\sigma }.\vec {L}+2}{r}=\frac{\left( \vec {\sigma }.\vec {r}\right) }{r }\left( \frac{\partial }{\partial r}-\frac{\vec {\sigma }.\vec {L}}{r}\right) , \end{aligned}$$
(39)

and

$$\begin{aligned} \frac{\left( \vec {\sigma }.\vec {r}\right) }{r}\mathcal {Y}_{\kappa }^{\mu }=- \mathcal {Y}_{-\kappa }^{\mu }. \end{aligned}$$
(40)

The total wave function of Dirac oscillator is

$$\begin{aligned} \psi _{n,j,\ell ,\mu ,s}= & {} \mathcal {C}\left( \begin{array}{c} \mathcal {Y}_{\kappa }^{\mu } \\ \frac{-i}{\left( E+m\right) }\left[ \left( 1+\alpha r^{2}\right) \frac{ \partial }{\partial r}-\frac{s\left( 2j+1\right) -1}{r}+m\omega r\right] \mathcal {Y}_{-\kappa }^{\mu } \end{array} \right) \frac{r^{\ell +1}}{\left( 1+\alpha r^{2}\right) ^{\frac{m\omega }{ 2\alpha }+\frac{2+\ell }{2}}}\times \nonumber \\&P_{n}^{\left( \frac{m\omega }{\alpha }-\frac{1}{ 2},\ell +\frac{1}{2}\right) } \left( \frac{\alpha r^{2}-1}{1+\alpha r^{2}} \right) , \end{aligned}$$
(41)

where \(\mathcal {C}\) is the normalization constant, can be calculated through the modified normalization condition

$$\begin{aligned} \int \frac{d\vec {r}}{\left( 1+\alpha r^{2}\right) ^{2}}\bar{\psi }\gamma ^{0}\psi =1. \end{aligned}$$
(42)

In order to obtain the energy spectrum of Dirac oscillator, we use the expressions of n,  \(\eta ,\tau \) and \(\lambda _{n,\ell }\) given in Eqs. (34), (35) and (22). Astraightforward calculation leads to

$$\begin{aligned} E_{N,j}=\pm \sqrt{m^{2}+2m\omega \left( N-j\right) +m\omega +\alpha \left( N-j\right) \left( N+j+\frac{1}{2}\right) +\frac{\alpha }{2}\left( N+3j-\frac{ 1}{2}\right) }, \end{aligned}$$
(43)

for \(\ell =j-\frac{1}{2},\) and

$$\begin{aligned} E_{N,j}=\pm \sqrt{m^{2}+2m\omega \left( N+j\right) +3m\omega +\alpha \left( N-j\right) \left( N+j+\frac{3}{2}\right) -\frac{\alpha }{2}\left( N-j+\frac{1 }{2}\right) }, \end{aligned}$$
(44)

for \(\ell =j+\frac{1}{2}\). The energy spectrum has been written in terms of a quantum number \(N=2n+\ell \), that is, commonly introduced in ordinary (\(1+3\))d Dirac oscillator. Notice that the energy levels depend on the square of the quantum number N. This effect is due to the modification of the Heisenberg algebra. In Fig. 1, we plot the energy level as a function of quantum number n for various values of \( \alpha ,\) we have chosen \(\alpha =0,0.1,0.5\) and \(\ell =0\), \(s=1/2.\) As a result, we remark that for a fixed value of n, the energy E increases monotonically with the increase of the EUP parameter. The effect of the EUP parameter on the energy levels is observable, where \(\alpha =0\) corresponding to the case of the normal quantum mechanics. Expanding the expression of the energy levels to first order in \(\alpha \), we obtain

$$\begin{aligned} E_{N,j}=\pm \left\{ \begin{array}{c} \sqrt{m^{2}+2m\omega \left( N-j\right) +m\omega }\left( 1+\left( \Delta P\right) _{\min }\frac{4\left( N-j\right) \left( N+j+\frac{1}{2}\right) +2\left( N+3j-\frac{1}{2}\right) }{m^{2}+2m\omega \left( N-j\right) +m\omega }\right) ;\text { for }\ell =j-\frac{1}{2} \\ \sqrt{m^{2}+2m\omega \left( N+j\right) +3m\omega }\left( 1+\left( \Delta P\right) _{\min }\frac{4\left( N-j\right) \left( N+j+\frac{3}{2}\right) -2\left( N-j+\frac{1}{2}\right) }{m^{2}+2m\omega \left( N+j\right) +3m\omega }\right) ;\text { for }\ell =j+\frac{1}{2} \end{array} \right. . \end{aligned}$$
(45)

The first term is the energy spectrum of the ordinary 3d Dirac oscillator, while the second term is the corrections brought about by the existence of nonzero minimal uncertainty in momentum, and when we study the limit \(\alpha \rightarrow 0,\) we obtain

$$\begin{aligned} E_{n}^{\alpha =0}=\pm \left\{ \begin{array}{c} \sqrt{m^{2}+2m\omega \left( N-j\right) +m\omega };\text { for }\ell =j-\frac{1 }{2} \\ \sqrt{m^{2}+2m\omega \left( N+j\right) +3m\omega };\text { for }\ell =j+\frac{1 }{2} \end{array} \right. , \end{aligned}$$
(46)

which is the same result in ordinary case [25].

Fig. 1
figure 1

Energy spectrum as a function of n for the different values of the parameter \(\alpha \)

Full size image

4 (\(1+2\))d Massless Dirac Equation

The electron in quantum theory of graphene is massless fermion that move with a velocity \(V_{F}=(1.12\pm 0.02)\times 10^{6}\,\mathrm{m}\,\mathrm{s}^{-1}\) called Fermi velocity verify the relativistic massless Dirac equation. The discovered of graphene give us the opportunity of testing various effects of QED, such as “Klein paradox” because this effect is unobservable in particle physics [40]. In this section, we are interested in solving the (\(1+2\))-dimensional massless Dirac equation in the presence of external constant magnetic field \(\vec {A}=\frac{B}{2}\left( -y,x,0\right) \). In this case, the Dirac equation read

$$\begin{aligned} H\psi =E\psi , \end{aligned}$$
(47)

where the eigenstate \(\psi ^{T}=\left( \begin{array}{cc} \psi ^{K}&\psi ^{K^{\prime }} \end{array} \right) \) in the graphene case describes the electron states around the Dirac points K and \(K^{\prime }\), and \(\psi ^{K},\) \(\psi ^{K^{\prime }}\) are 2-dimensional eigenfunctions,

$$\begin{aligned} \psi ^{K}=\left( {\begin{array}{c}\psi ^{A}\\ \psi ^{B}\end{array}}\right) ;\psi ^{K^{\prime }}=\left( {\begin{array}{c}\psi ^{A^{\prime }}\\ \psi ^{B^{\prime }}\end{array}}\right) . \end{aligned}$$
(48)

In the presence of external constant magnetic field, the Hamiltonian can write as

$$\begin{aligned} H=\left( \begin{array}{cc} H_{K} &{} 0 \\ 0 &{} H_{K^{\prime }} \end{array} \right) , \end{aligned}$$
(49)

where

$$\begin{aligned} H_{K}=V_{F}\left( \begin{array}{cc} 0 &{} P_{x}-iP_{y}+\frac{eB}{2c}\left( y+ix\right) \\ P_{x}+iP_{y}+\frac{eB}{2c}\left( y-ix\right) &{} 0 \end{array} \right) , \end{aligned}$$
(50)

and

$$\begin{aligned} H_{K^{\prime }}=V_{F}\left( \begin{array}{cc} 0 &{} P_{x}+iP_{y}+\frac{eB}{2c}\left( y-ix\right) \\ P_{x}-iP_{y}+\frac{eB}{2c}\left( y+ix\right) &{} 0 \end{array} \right) . \end{aligned}$$
(51)

To obtain the energy eigenvalue of Eq. (50) at the Dirac point K, one has to solve the following eigenvalue problem

$$\begin{aligned} \left[ P_{x}-iP_{y}+\frac{eB}{2c}\left( y+ix\right) \right] \psi ^{B}&= \frac{E}{V_{F}}\psi ^{A}, \end{aligned}$$
(52)
$$\begin{aligned} \left[ P_{x}+iP_{y}+\frac{eB}{2c}\left( y-ix\right) \right] \psi ^{A}&= \frac{E}{V_{F}}\psi ^{B}. \end{aligned}$$
(53)

Eliminating \(\psi ^{B}\), we obtain

$$\begin{aligned} \left\{ \begin{array}{c} P_{x}^{2}+P_{y}^{2}+i\left[ P_{x},P_{y}\right] +\frac{eB}{2c}\left( P_{x}y+yP_{x}-P_{y}x-xP_{y}\right) +\frac{ieB}{2c}\left[ y,P_{y}\right] + \frac{ieB}{2c}\left[ x,P_{x}\right] \\ +\frac{e^{2}B^{2}}{4c^{2}}\left( y^{2}+x^{2}\right) -\frac{E^{2}}{V_{F}^{2}} \end{array} \right\} \psi ^{A}=0, \end{aligned}$$
(54)

Now, in order to solve the last equation in the polar coordinates, we apply the definition of the position and momentum operators reported in section (2), and we use the following definition

$$\begin{aligned} P_{x}&=-i\hbar \left( \left( 1+\alpha r^{2}\right) \cos \theta \frac{ \partial }{\partial r}-\frac{\sin \theta }{r}\frac{\partial }{\partial \theta }\right) , \end{aligned}$$
(55)
$$\begin{aligned} P_{y}&=-i\hbar \left( \left( 1+\alpha r^{2}\right) \sin \theta \frac{ \partial }{\partial r}+\frac{\cos \theta }{r}\frac{\partial }{\partial \theta }\right) . \end{aligned}$$
(56)

With the aid of these expressions, it is not difficult to verify that the \( \psi ^{A}\) components satisfy the following differential equation

$$\begin{aligned} \left\{ \left( \left( 1+\alpha r^{2}\right) \frac{\partial }{\partial r} \right) ^{2}+\frac{1}{r}\left( 1+\alpha r^{2}\right) \frac{\partial }{ \partial r}-\frac{\mu ^{2}}{r^{2}}+\frac{eB}{\hbar c}-\frac{eB}{2\hbar c} \left( \frac{eB}{2\hbar c}-\alpha \right) r^{2}+\mu \left( \frac{eB}{\hslash c}+\alpha \right) +\frac{E^{2}}{\hbar ^{2}V_{F}^{2}}\right\} \phi _{\mu }^{A}=0, \end{aligned}$$
(57)

where \(\psi ^{A}=\frac{e^{i\mu \theta }}{\sqrt{2\pi }}\phi _{\mu }^{A},\) and \(\mu =0;\pm 1;2;...\) is the angular momentum quantum number. With the aid of the new variable \(\rho \) defined by:

$$\begin{aligned} \sqrt{\alpha }\rho =\tan ^{-1}\left( \sqrt{\alpha }r\right) . \end{aligned}$$
(58)

Then, the Eq. (57) becomes

$$\begin{aligned} \left\{ \begin{array}{c} \frac{\partial ^{2}}{\partial \rho ^{2}}+\frac{\sqrt{\alpha }}{\tan \left( \sqrt{\alpha }\rho \right) }\frac{\partial }{\partial \rho }-\frac{\alpha \mu ^{2}}{\tan ^{2}\left( \sqrt{\alpha }\rho \right) }-\frac{1}{2\ell _{B}^{2}}\left( \frac{1}{2\ell _{B}^{2}\alpha }-1\right) \tan ^{2}\left( \sqrt{\alpha }\rho \right) \\ +\mu \left( \frac{1}{\ell _{B}^{2}}+\alpha \right) +\frac{E^{2}}{\hbar ^{2}V_{F}^{2}}+\frac{1}{\ell _{B}^{2}} \end{array} \right\} \phi _{\mu }^{A}=0, \end{aligned}$$
(59)

where \(\ell _{B}=\sqrt{\frac{\hslash c}{eB}}\) is the fundamental length scale in the presence of a magnetic field. We use the following ansatz:

$$\begin{aligned} \phi _{\mu }^{A}=\sin ^{\mu }\left( \sqrt{\alpha }\rho \right) \cos ^{\frac{1 }{2\alpha \ell _{B}^{2}}}\left( \sqrt{\alpha }\rho \right) \chi _{\mu }^{A}, \end{aligned}$$
(60)

then Eq. (59) simplifies to

$$\begin{aligned} \left\{ \frac{\partial ^{2}}{\partial \rho ^{2}}+\left( \frac{\left( 2\mu +1\right) \sqrt{\alpha }}{\tan \left( \sqrt{\alpha }\rho \right) }-\frac{1}{ \alpha \ell _{B}^{2}}\sqrt{\alpha }\tan \left( \sqrt{\alpha }\rho \right) \right) \frac{\partial }{\partial \rho }-\frac{1}{\ell _{B}^{2}}+\frac{E^{2} }{\hbar ^{2}V_{F}^{2}}+\frac{1}{\ell _{B}^{2}}\right\} \chi _{\mu }^{A}=0. \end{aligned}$$
(61)

At this stage, we introduce another change of variable defined by

$$\begin{aligned} \xi =\sin ^{2}\left( \sqrt{\alpha }\rho \right) . \end{aligned}$$
(62)

Then Eq. (61) reduced to the Hypergeometric equation form

$$\begin{aligned} \left\{ \left( 1-\xi \right) \xi \frac{\partial ^{2}}{\partial \xi ^{2}} +\left( \left( \mu +1\right) -\left( \mu +\frac{1}{2\alpha \ell _{B}^{2}}+ \frac{3}{2}\right) \xi \right) \frac{\partial }{\partial \xi }-\frac{1}{ 4\alpha \ell _{B}^{2}}+\frac{E^{2}}{4\alpha \hbar ^{2}V_{F}^{2}}+\frac{1}{ 4\alpha \ell _{B}^{2}}\right\} \chi _{\mu }^{A}=0. \end{aligned}$$
(63)

The general solution of Eq. (63) is given in terms of the hypergeometric function

$$\begin{aligned} \chi _{\mu }^{A}=const\times F\left( a;b;c,\xi \right) , \end{aligned}$$
(64)

where the parameters a,  b, and c are given by

$$\begin{aligned} \left\{ \begin{array}{c} a=\frac{1}{4\alpha \ell _{B}^{2}}+\frac{1}{4}+\frac{\mu }{2}+\frac{1}{2} \sqrt{\mu \left( \mu +1\right) +\frac{1}{4}+\frac{\mu }{\alpha \ell _{B}^{2}} +\frac{E^{2}}{\alpha \hbar ^{2}V_{F}^{2}}+\frac{1}{2\alpha \ell _{B}^{2}} \left( \frac{1}{2\alpha \ell _{B}^{2}}+1\right) } \\ b=\frac{1}{4\alpha \ell _{B}^{2}}+\frac{1}{4}+\frac{\mu }{2}-\frac{1}{2} \sqrt{\mu \left( \mu +1\right) +\frac{1}{4}+\frac{\mu }{\alpha \ell _{B}^{2}} +\frac{E^{2}}{\alpha \hbar ^{2}V_{F}^{2}}+\frac{1}{2\alpha \ell _{B}^{2}} \left( \frac{1}{2\alpha \ell _{B}^{2}}+1\right) } \\ c=\mu +1 \end{array} \right. . \end{aligned}$$
(65)

The hypergeometric function \(F\left( a;b;c,\xi \right) \) reduces to a polynomial of degree n in \(\xi \). This is known to occur when a or b equals a negative integer, and then

$$\begin{aligned} E_{n,\mu }^{\alpha }=\pm \frac{\hbar V_{F}}{\ell _{B}}\sqrt{2n+ \frac{\hslash c}{eB}\alpha \left( 2n+4n\mu +4n^{2}\right) }. \end{aligned}$$
(66)

It’s clear that the energy at the Dirac point K depends explicitly on the noncommutative parameter associated with the momenta, and the zero-energy level for this system is \(E_{0,\mu }^{\alpha }=0\). in the limit \(\alpha \rightarrow 0\), we obtain:

$$\begin{aligned} E_{n}=\pm \frac{\hbar V_{F}}{\ell _{B}}\sqrt{2n}, \end{aligned}$$
(67)

which coincides exactly with the result of the [41].

Indeed, using the experimental results of the relativistic Landau levels in graphene [42], we can be obtained an upper bound on the EUP parameter \( \alpha .\)In the absence of EUP, for a magnetic field of strength \(B=18T\) , the energy for the Landau levels \(n=1\) is \(E=(172\pm 3)meV.\) In the presence of EUP, by putting \(n=1\), \(\mu =0\) in the energy level Eq.(66) and considering that the uncertainty in the energy is 6meV, we get

$$\begin{aligned} \Delta E=E_{1,0}^{\alpha }-E_{1}^{\alpha =0}=E_{1}^{\alpha =0}\left[ \sqrt{ 1+3\frac{\hslash c}{eB}\alpha }-1\right] \prec 6\, \mathrm{m eV} . \end{aligned}$$
(68)

Therefore, the upper bound of the EUP parameter is

$$\begin{aligned} \sqrt{\alpha }\le 0.2\, 52\times 10^{6}m^{-1}. \end{aligned}$$
(69)

Thus, the upper bound of the MUM is found as

$$\begin{aligned} \left( \triangle P\right) _{\min }=\frac{\hslash \sqrt{\alpha }}{2}\le 1.\, 328\,8\times 10^{-27} \,\mathrm{J sm}^{-1}. \end{aligned}$$
(70)

This assumption was also considered in order to obtain the upper bound of generalized uncertainty principle parameter [43]. It is worthwhile to mention that our result (69) as expected coincides with the result calculated in [34].

5 Conclusion

In this contribution, we have investigated the three and two-dimensionals Dirac oscillator in the presence of minimal uncertainty in momentum. For the 3-dimensionals Dirac oscillator, according to the symmetry of the system, we used the adequate radial representation, the problem has been converted to the case of the Klein Gordon oscillator with an orbit spin coupling term . The energy eigenvalues and their corresponding eigenfunctions are analytically obtained and are given in terms of Jacobi polynomials. A numerical study is presented and the energy of 3-dimensionals Dirac oscillator is represented for values of the energy parameter.

For the second problem, which is of great importance, may have applications in phenomenology, we have examined the massless Dirac–Weyl particle moves with an effective Fermi velocity in graphene and subjected to the action of a uniform magnetic field. The wave functions are determined and expressed according to the hypergeometric function. The corresponding exact energy spectrum is extracted, contains an additional corrections depends on the deformation parameter and its deviation grows quickly with \(n^{2}\), which it is a sign of the confinement phenomenon and increases monotonically with the increase of the EUP parameter. It is remarkable to note that this system is associated a zero point energy ( as the vacuum energy in standard model, in the gauge fields and in the electroweak Higgs field) and the limit case is obtained and agrees with those in the literature. Finally, in order to see the effect of the deformation on the physical systems and to compare them with the experimental results of the relativistic Landau levels in graphene, we have determined a satisfactory value of the upper bound on the EUP.