The bound–state solutions of the modified Dirac equation (m.d.e.) for the modified anharmonic oscillator (m.a.o.) are presented exactly for arbitrary spin–orbit quantum number $k\left(\tilde{k}\right)$  by means Bopp’s shift method instead to solving (m.d.e.) with star product, in the framework of noncommutativity three dimensional real space (NC: 3D–RS). The exact corrections for $n^{th}$ excited states are found straightforwardly for interactions in one–electron atoms by applying the standard perturbation theory. Furthermore, the obtained corrections of energies are depended on two infinitesimal parameters $\left({\Theta }_{ij},{\chi }_{ij}\right)\equiv {\epsilon }_{ij}^k\left({\Theta }_k,{\chi }_k\right)$  which induced by position–position noncommutativity, in addition to the non–relativistic quantum mechanics $\left(n,j,l=l\pm 1/2,m\right)$  and $\left(n,j=\tilde{l}\pm \tilde{s},\tilde{m}\right)$  under spin–symmetry and p–spin symmetry in (NC: 3D–RS), respectively. In limit of parameters $\left({\Theta }_k,{\chi }_k\right)\to \left(0,0\right)$ , the energy equation is consistent with the results of ordinary relativistic quantum mechanics.

Keywords: anharmonic oscillator, noncommutative space, star product, bopp’s shift method, dirac equation

MOA: Modified Anharmonic Oscillator; (NC: 3D–RS): Noncommutativity Three Dimensional Real Space; MDE: Modified Dirac Equation; (NCCRs): NC Canonical Commutations Relations

One of the interesting problems of the relativistic quantum mechanics is to find exact solutions to the Klein–Gordon (to the treatment of a zero–spin particle) and Dirac (spin ½ particles and anti–particles) equations for certain potentials of the physical interest, in recent years, considerable efforts have been done to obtain the analytical solution of central and non–central physics problems for different areas of atoms, nuclei, and hadrons, numerous papers of the physicist have discussed in details all the necessary information for the quantum system and in particularly the bound states solutions.^1–21 Some of these potentials are known to play important roles in many fields, one of such potential is the anharmonic oscillator has been a subject of many studies, it is a central potential of nuclear shell model, etc.^20,21 The ordinary quantum structures obey the standard Weyl–Heisenberg algebra in both Schrödinger and Heisenberg (the operators are depended on time) pictures, respectively, as (Throughout this paper the natural unit are employed):

$\begin{array}{l}\left[x_i,p_j\right]=\left[x_i\left(t\right),p_j\left(t\right)\right]=i{\delta }_{ij}\\ \left[x_i,x_j\right]=\left[p_i,p_j\right]=\left[x_i\left(t\right),x_j\left(t\right)\right]=\left[p_i\left(t\right),p_j\left(t\right)\right]=0\end{array}$   (1)

where the two operators $\left(x_i\left(t\right),p_i\left(t\right)\right)$ in Heisenberg picture are related to the corresponding operators $\left(x_i,p_i\right)$  in Schrödinger picture from the following projections relations:

$\begin{array}{l}{x}_i\left(t\right)=\exp (i\widehat{H}\left(t-t_0\right))x_i\exp (-i\widehat{H}\left(t-t_0\right))\\ p_i\left(t\right)=\exp (i\widehat{H}\left(t-t_0\right))p_i\exp (-i\widehat{H}\left(t-t_0\right))\end{array}$   (2)

here $\widehat{H}$  denote to the ordinary quantum Hamiltonian operator. In addition, for spin ½ particles described by the Dirac equation, experiment tells us that must satisfy Fermi Dirac statistics obey the restriction of Pauli, which imply to gives the only non–null equal–time anti–commutator for field operators as follows:

$\left\{{\Psi }_{\alpha }\left(t,\text{r}\right),{\overline{\Psi }}_{\beta }\left(t,\text{r'}\right)\right\}=i{\left({\gamma }^0\right)}_{\alpha \beta }{\delta }^{\text{3}}\left(\text{r-r'}\right)$   (3)

 with ${\stackrel{-}{\Psi }}_{\beta }(t,r^1)={\Psi }^{+}{}_{\beta }(t,r^1){\gamma }^0$ . Very recently, many authors have worked on solving these equations with physical potential in the new structure of quantum mechanics, known by NC quantum mechanics, which known firstly H Snyder.²² to obtaining profound and new applications for different areas of matter sciences in the microscopic and nano scales.^23–68 It is important to noticing that, the new quantum structure of NC space based on the following NC canonical commutations relations (NCCRs) in both Schrödinger and Heisenberg pictures, respectively, as follows.^23–60

$\begin{array}{l}\left[{\widehat{x}}_i\stackrel{\ast }{,}{\widehat{p}}_j\right]=\left[{\widehat{x}}_i\left(t\right)\stackrel{\ast }{,}{\widehat{p}}_{{}_j}\left(t\right)\right]=i{\delta }_{ij},\left[{\widehat{x}}_i\stackrel{\ast }{,}{\widehat{x}}_j\right]=\left[{\widehat{x}}_i\left(t\right)\stackrel{\ast }{,}{\widehat{x}}_j\left(t\right)\right]=i{\theta }_{ij}\\ \left[{\widehat{p}}_i\stackrel{\ast }{,}{\widehat{p}}_j\right]=\left[{\widehat{p}}_i\left(t\right)\stackrel{\ast }{,}{\widehat{p}}_j\left(t\right)\right]=0\end{array}$   (4)

Where the two new operators $\left({\widehat{x}}_i\left(t\right),{\widehat{p}}_i\left(t\right)\right)$ in Heisenberg picture are related to the corresponding new operators $\left({\widehat{x}}_i,{\widehat{p}}_i\right)$  in Schrödinger picture from the new projections relations:

$\begin{array}{l}{\widehat{x}}_i\left(t\right)=\exp (i{\widehat{H}}_{nc}\left(t-t_0\right))*{\widehat{x}}_i*\exp (-i{\widehat{H}}_{{}_{nc}}\left(t-t_0\right))\\ {\widehat{p}}_i\left(t\right)=\exp (i{\widehat{H}}_{nc}\left(t-t_0\right))*{\widehat{p}}_i*\exp (-i{\widehat{H}}_{nc}\left(t-t_0\right))\end{array}$   (5)

with ${\widehat{H}}_{nc}$ being the Hamiltonian operator of the quantum system described on (NC: 3D–RS) symmetries. The very small parameters ${\theta }^{\mu \nu }$  (compared to the energy) are elements of anti symmetric real matrix of dimension $\frac{{\left(\text{length}\right)}^2}{\hslash }$  and $(\ast )$  denote to the new star product (the Moyal–Weyl product), which is generalized between two arbitrary functions $f\left(x\right)\to \widehat{f}\left(\widehat{x}\right)$ and $g\left(x\right)\to \widehat{g}\left(\widehat{x}\right)$ to $\widehat{f}\left(\widehat{x}\right)\widehat{g}\left(\widehat{x}\right)\equiv \left(f\ast g\right)\left(x\right)$  instead of the usual product $\left(fg\right)\left(x\right)$  in ordinary three dimensional spaces.^23–68

$\widehat{f}\left(\widehat{x}\right)\widehat{g}\left(\widehat{x}\right)\equiv \left(f\ast g\right)\left(x\right)\equiv \exp (\frac{i}{2}{\theta }^{\mu \nu }{\partial }_{\mu }^x{\partial }_{\nu }^x\left(fg\right)\left(x,p\right)\equiv {(fg-\frac{i}{2}{\theta }^{\mu \nu }{\partial }_{\mu }^{x}f{\partial }_{\nu }^{x}g|}_{\left(x^{\mu }=x^{\nu }{}^{\nu }\right)}+O\left({\theta }^2\right)$   (6)

where $\widehat{f}\left(\widehat{x}\right)$  and $\widehat{g}\left(\widehat{x}\right)$  are the new function in (NC: 3D–RS), the following term ($-\frac{i}{2}{\theta }^{mn}{\partial }_{\mu }^{x}f\left(x\right){\partial }_{\nu }^{x}g\left(x\right)$ ) is induced by (space–space) noncommutativity properties and $O\left({\theta }^2\right)$  stands for the second and higher order terms of $\theta$ , a Bopp’s shift method can be used, instead of solving any quantum systems by using directly star product procedure.^23–55

$\left[{\widehat{x}}_i,{\widehat{x}}_j\right]=\left[{\widehat{x}}_i\left(t\right),{\widehat{x}}_j\left(t\right)\right]=i{\theta }_{ij}\begin{array}{cc}\text{and}& \left[{\widehat{p}}_i,{\widehat{p}}_j\right]=\left[{\widehat{p}}_i\left(t\right),{\widehat{p}}_j\left(t\right)\right]=0\end{array}$   (7)

The three–generalized coordinates $\left(\widehat{x}={\widehat{x}}_1,\widehat{y}={\widehat{x}}_2,\widehat{z}={\widehat{x}}_3\right)$  in the NC space are depended with corresponding three–usual generalized positions $\left(x,y,z\right)$  and momentum coordinates $\left(p_x,p_y,p_z\right)$  by the following relations, as follows.^{25,28,29,32–34,37–47}

$\begin{array}{l}\widehat{x}=x-\frac{{\theta }_{12}}{2}{p}_y-\frac{{\theta }_{13}}{2}{p}_z,\widehat{y}=y-\frac{{\theta }_{21}}{2}{p}_x-\frac{{\theta }_{23}}{2}{p}_z\\ \widehat{z}=z-\frac{{\theta }_{31}}{2}{p}_x-\frac{{\theta }_{32}}{2}{p}_y\end{array}$   (8)

 The non–vanish–commutators in (NC–3D: RS) can be determined as follows:

$\begin{array}{l}\left[\widehat{x},{\widehat{p}}_x\right]=\left[\widehat{y},{\widehat{p}}_y\right]=\left[\widehat{z},{\widehat{p}}_z\right]=i,\\ \left[\widehat{x},\widehat{y}\right]=i{\theta }_{12},\left[\widehat{x},\widehat{z}\right]=i{\theta }_{13},\left[\widehat{y},\widehat{z}\right]=i{\theta }_{23}\end{array}$   (9)

which allow us to getting the operator ${\widehat{r}}^2$  on NC three dimensional spaces as follows.^{25,28,29,32,33,34,37–48}

${\widehat{r}}^2=r^2-\overrightarrow{L}\overrightarrow{\Theta }$   (10)

Where the coupling $L\Theta$  is given by $\left({\Theta }_{ij}={\theta }_{ij}/2\right)$ :

$L\Theta \equiv L_x{\Theta }_{12}+L_y{\Theta }_{23}+L_z{\Theta }_{13}$   (11)

with $L_x=y{p}_z-z{p}_y$ , $L_y={\text{zp}}_{\text{x}}{\text{-xp}}_{\text{z}}$ and $L_z=x{p}_y-y{p}_x$ . Furthermore, the new equal–time anti–commutator for fermionic field operators’ noncommutative spaces can be expressed in the following postulate relations:

$\begin{array}{l}\left\{{\widehat{\Psi }}_{\alpha }\left(t,\text{r}\right)\stackrel{*}{{\displaystyle ,}}{\widehat{\overline{\Psi }}}_{\beta }\left(t,\text{r'}\right)\right\}=i{\left({\gamma }^0\right)}_{\alpha \beta }{\delta }^{\text{3}}\left(\text{r-r'}\right)\\ \left\{{\widehat{\Psi }}_{\alpha }\left(t,\text{r}\right)\stackrel{*}{{\displaystyle ,}}{\widehat{\Psi }}_{\alpha }\left(t,\text{r'}\right)\right\}=\left\{{\widehat{\overline{\Psi }}}_{\alpha }\left(t,\text{r}\right)\stackrel{*}{{\displaystyle ,}}{\widehat{\overline{\Psi }}}_{\beta }\left(t,\text{r'}\right)\right\}=i{\theta }_{\alpha \beta }{\delta }^{\text{3}}\left(\text{r-r'}\right)\end{array}$   (12)

Here T is the time–ordered product. The purpose of the present work is to extend and present the solution of the Dirac equation with spin–1/2 particle moving in (m.a.o.) potential of the new form:

$V_{ao}\left(\widehat{r}\right)=\frac{1}{2}M{\omega }^2{r}^2+\frac{\alpha }{2M{r}^2}+\{\begin{array}{l}\left(\frac{\alpha }{2M{r}^4}-\frac{1}{2}M{\omega }^2\right)\overrightarrow{L}\overrightarrow{\Theta }\begin{array}{cc}\text{ for}& \text{the spin symmetric case }\end{array}\\ \left(\frac{\alpha }{2M{r}^4}-\frac{1}{2}M{\omega }^2\right)\tilde{\overrightarrow{L}}\overrightarrow{\Theta }\begin{array}{cc}\text{ for}& \text{the p-spin symmetric case }\end{array}\end{array}$   (13)

In (NC: 3D–RS) using the generalization Bopp’s shift method to discover the new symmetries and a possibility to obtain another applications to this potential in different fields. This work based essentially on our previously works.^23–48 The outline of our recently article is as follows: In next section, we briefly review the Dirac equation with anharmonic oscillator on based to.^18–21 In section three, we give a description of the Bopp’s shift method for the (m.d.e.) with (m.a.o).Then in section four, we apply standard perturbation theory to establish exact modifications at first order of infinitesimal parameters $\left(\Theta ,\chi \right)$  for the perturbed Dirac equation in (NC–3D: RS) for spin–orbital (pseudo–spin orbital) and the relativistic magnetic spectrum for (m.a.o.). In the fifth section, we resume the global spectrum and corresponding NC Hamiltonian for (m.a.o.). Finally, some important concluding remarks are drawn from the present study in last section.

Review the Dirac equation for anharmonic oscillator in ordinary quantum

We start this section by considering a relativistic particle in spherically symmetric for the potential $V(r,\theta )$  which known by anharmonic oscillator, given by in the main reference.²¹

$V\left(r,\theta \right)=\frac{1}{2}M{\omega }^2{r}^2+\frac{\alpha }{2M{r}^2}+\frac{\eta }{2M{r}^2\sin \left(\theta \right)}\stackrel{\eta \to 0}{\to }V\left(r\right)=\frac{1}{2}M{\omega }^2{r}^2+\frac{\alpha }{2M{r}^2}$   (14)

where M, $\omega$ , (α and η)denote the rest mass, frequency of particle and dimensionless parameters. The Dirac equation describing a fermionic particle (spin–1/2 particle) with scalar $S(r,\theta )$  and vector $V(r,\theta )$  potentials is given by.^18–21

$\left(\alpha \text{P}+\beta (M+S(r,\theta ))\right)\Psi \left(r,\theta ,\varphi \right)=\left(E-V\left(r,\theta \right)\right)\Psi \left(r,\theta ,\varphi \right)$   (15)

here M are E the fermions’ mass and the relativistic energy while ( ${\alpha }_i=\left(\begin{array}{cc}0& {\sigma }_i\\ {\sigma }_i& 0\end{array}\right)$ ,$\beta =\left(\begin{array}{cc}{I}_{2\times 2}& 0\\ 0& I_{2\times 2}\end{array}\right)$ ) are the usual Dirac matrices, the spinor $\Psi (r,\theta ,\phi )$  can be expressed as.²¹

${\Psi }_{nk}\left(r,\theta ,\varphi \right)=\left(\begin{array}{c}{f}_{nk}\left(\overrightarrow{r}\right)\\ g_{nk}\left(\overrightarrow{r}\right)\end{array}\right)=\frac{1}{r}\left(\begin{array}{l}{F}_{nk}\left(r\right)Y_{jm}^l\left(\theta ,\varphi \right)\\ i{G}_{n\tilde{k}}\left(r\right)Y_{jm}^{\tilde{l}}\left(\theta ,\varphi \right)\end{array}\right)$   (16)

where ${\sigma }_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ ,${\sigma }_2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)$ and ${\sigma }_3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$  and are $2\times 2$  three Pauli matrices while $k\left(\tilde{k}\right)$  is related to the total angular momentum quantum numbers for spin symmetry $l$  and p–spin symmetry $\tilde{l}$  as.^18–21

$k=\{\begin{array}{l}-\left(l+1\right)\text{ if -}\left(\text{j}+\text{1/2}\right)\text{,}\left({\text{s}}_{\text{1/2}},p_{3/2},etc\right)\text{, }j=l+\frac{1}{2},\text{ aligned spin }\left(\text{k}\langle \text{0}\right)\\ +l\text{ if }j=l+\frac{1}{2},\left({\text{p}}_{\text{1/2}},d_{3/2},etc\right)\text{, }j=l-\frac{1}{2},\text{ unaligned spin }\left(\text{k}\rangle \text{0}\right)\end{array}$   (17)

and

$\tilde{k}=\{\begin{array}{l}-\tilde{l}\text{ if -}\left(\text{j}+\text{1/2}\right)\text{,}\left({\text{s}}_{\text{1/2}},p_{3/2},etc\right)\text{, }j=\tilde{l}-\frac{1}{2},\text{ aligned spin }\left(\text{k}\langle \text{0}\right)\\ +\left(\tilde{l}+1\right)\text{ if }j=\tilde{l}+\frac{1}{2},\left({\text{p}}_{\text{1/2}},d_{3/2},etc\right)\text{, }j=\tilde{l}+\frac{1}{2},\text{ unaligned spin }\left(\text{k}\rangle \text{0}\right)\end{array}$   (18)

The radial functions ($F_{nk}\left(r\right)$ , $G_{nk}\left(r\right)$ ) are obtained by solving the following differential equations.^18–21

$\left[\frac{d^2}{d{r}^2}-\frac{k(k+1)}{r^2}-\left(M+E_{nk}-\Delta \left(r\right)\left(M-E_{nk}+\Sigma \left(r\right)\right)+\frac{\frac{d\Delta \left(r\right)}{dr}\left(\frac{d}{dr}+\frac{k}{r}\right)}{M-E_{nk}+\Sigma \left(r\right)}\right)\right]F_{nk}\left(r\right)=0$   (19)

and

$\left[\frac{d^2}{d{r}^2}-\frac{k(k-1)}{r^2}\left(M+E_{nk}-\Delta \left(r\right)\left(M-E_{nk}+\Sigma \left(r\right)\right)+\frac{\frac{d\Sigma \left(r\right)}{dr}\left(\frac{d}{dr}+\frac{k}{r}\right)}{M+E_{nk}-\Delta \left(r\right)}\right)\right]G_{n\tilde{k}}\left(r\right)=0$   (20)

The exact spin symmetry corresponding $\frac{d\Delta \left(r\right)}{dr}=0$ , thus the radial function $F_{nk}\left(r\right)$  satisfying the following like Schrödinger equation.²¹

$\left[\frac{d^2}{d{r}^2}-\frac{k(k+1)}{r^2}-M^2-E^2\left(M-E\right)\left(\frac{1}{2}M{\omega }^2{r}^2+\frac{\alpha }{2M{r}^2}\right)\right]F_{nk}\left(r\right)=0$   (21)

The relativistic energy $E_{n,k}$  and radial upper wave $F_{nk}\left(r\right)$  are given by.²¹

$\frac{M^2-E_{n,k}^2}{\sqrt{M\left(M+E_{n,k}\right)}}+4n+2L+3=0$   (22)

and

$F_{n,k}\left(r\right)=C_n\exp \left(-\frac{\sqrt{M\left(M+E_{n,k}\right)}}{2}{r}^2\right)r^{L+1}{L}_n^{L+1/2}\left(\sqrt{M\left(M+E_{n,k}\right)}{r}^2\right)$   (23)

where $L_n^{L+1/2}\left(\sqrt{M\left(M+E_{n,k}\right)}{r}^2\right)$  stands for the associated Laguerre functions. For, the exact pseudospin symmetry which corresponds $\frac{d\sum \left(r\right)}{dr}=0$ , the relativistic energy $E_{n,k}$  and radial lower wave $G_{nk}\left(r\right)$  are given by.²¹

$\frac{M^2-E_{n,k}^2}{\sqrt{M\left(M-E_{n,k}\right)}}+4n+2\tilde{L}+3=0$   (24)

and

$G_{n,k}\left(r\right)=C_n\exp \left(-\frac{\sqrt{M\left(M-E_{n,k}\right)}}{2}{r}^2\right)r^{\tilde{L}+1}{L}_n^{\tilde{L}+1/2}\left(\sqrt{M\left(M-E_{n,k}\right)}{r}^2\right)$   (25)

Formalism of bopp’s shift method

In this section I first highlight in brief the basics of the concepts of the quantum noncommutative quantum mechanics in the framework of relativistic Dirac equation for modified an harmonic oscillator $V_{ao}\left(\widehat{r}\right)$  on based to our works.^{25,28,29,32,–48}

1. Ordinary Dirac Hamiltonian operator $\widehat{H}\left(p_i,x_i\right)$ replace by NC Dirac Hamiltonian operator ${\widehat{H}}_{{}_{nc-ao}}\left({\widehat{p}}_i,{\widehat{x}}_i\right)$
2. Ordinary spinor $\Psi \left(\overrightarrow{r}\right)$ replace by new spinor $\stackrel{⌢}{\Psi }\left(\overleftrightarrow{\stackrel{⌢}{r}}\right)$ ,
3. Ordinary relativistic energy ${\text{E}}_{\text{nk}}$ replaces by new relativistic energy $E_{nc-ao}$ and ordinary product replace by new star product.

Thus, the Dirac equation in ordinary quantum mechanics will change into the Dirac equation in extended quantum mechanics for the (m.a.o.) as follows:

${\widehat{H}}_{{}_{nc-ao}}\left({\widehat{p}}_i,{\widehat{x}}_i\right)\ast \stackrel{⌢}{\Psi }\left(\overleftrightarrow{\stackrel{⌢}{r}}\right)=E_{nc-ao}\stackrel{⌢}{\Psi }\left(\overleftrightarrow{\stackrel{⌢}{r}}\right)$ (26)

The Bopp’s shift method permutes to reduce the above NC equation to simplest form with usual product and translations applied to in space and phase operators:

$H_{nc-ao}\left({\widehat{p}}_i,{\widehat{x}}_i\right)\psi \left(\overrightarrow{r}\right)=E_{nc-ao}\psi \left(\overrightarrow{r}\right)$   (27)

Where the new Hamiltonian operator $H_{nc-ao}\left({\widehat{p}}_i,{\widehat{x}}_i\right)$  can be expressed in three general varieties: both NC space and NC phase (NC–3D: RSP), only NC space (NC–3D: RS) and only NC phase (NC: 3D–RP) as, respectively:

$H_{nc-ao}\left({\widehat{p}}_i,{\widehat{x}}_i\right)\equiv H\left({\widehat{p}}_i=p_i-\frac{1}{2}{\overline{\theta }}_{ij}{x}_j;{\widehat{x}}_i=x_i-\frac{1}{2}{\theta }_{ij}{p}_j\right)\begin{array}{cc}\text{ for}& \left(\text{NC-3D: RSP}\right)\end{array}$ (28)

$H_{nc-ao}\left({\widehat{p}}_i,{\widehat{x}}_i\right)\equiv H\left({\widehat{p}}_i=p_i;{\widehat{x}}_i=x_i-\frac{1}{2}{\theta }_{ij}{p}_j\right)\text{ for }\left(\text{NC-3D: RS}\right)$ (29)

$H_{nc-ao}\left({\widehat{p}}_i,{\widehat{x}}_i\right)\equiv H\left({\widehat{p}}_i=p_i-\frac{1}{2}{\overline{\theta }}_{ij};x_j,{\widehat{x}}_i=x_i\right)\begin{array}{cc}\text{for}& \left(\text{NC-3D: RP}\right)\end{array}$   (30)

In recently work, we are interest with the second variety which present by eq. (29) and by the means of the auxiliary two variables ${\widehat{x}}_i=x_i-\frac{1}{2}{\theta }_{ij}{p}_j$  and ${\widehat{p}}_i=p_i$ , the new modified Hamiltonian $H_{nc-ao}\left({\widehat{p}}_i,{\widehat{x}}_i\right)$  may be written as follows

$H_{nc-ao}\left({\widehat{p}}_i,{\widehat{x}}_i\right)=\alpha \widehat{p}+\beta (M+S(\widehat{r}))\text{+}{V}_{ao}\left(\widehat{r}\right)$   (31)

where the modified anharmonic oscillator $V_{ao}\left(\widehat{r}\right)$  is given by:

$V_{ao}\left(\widehat{r}\right)=\frac{1}{2}M{\omega }^2{\widehat{r}}^2+\frac{\alpha }{2M{\widehat{r}}^2}$ (32)

The Dirac equation in the presence of above interaction $V_{ao}\left(\widehat{r}\right)$  can be rewritten according Bopp shift method as follows:

$\left(\alpha \text{P}+\beta (M+S(\widehat{r}))\right)\Psi \left(r,\theta ,\varphi \right)=\left(E_{nc-ao}-V_{ao}\left(\widehat{r}\right)\right)\Psi \left(r,\theta ,\varphi \right)$   (33)

The radial functions ($F_{nk}\left(r\right)$ ,$G_{nk}\left(r\right)$ ) are obtained by solving two equations:

$\left[\frac{d}{dr}+\frac{k}{r}\right]F_{nk}\left(r\right)=\left[M+E_{nc-kb}-\Delta \left(\widehat{r}\right)\right]G_{nk}\left(r\right)$   (34)

$\left[\frac{d}{dr}+\frac{k}{r}\right]G_{nk}\left(r\right)=\left[M-E_{nc-kb}+\Sigma \left(\widehat{r}\right)\right]G_{nk}\left(r\right)$   (35)

with $\Delta \left(\widehat{r}\right)=V\left(\widehat{r}\right)-S\left(\widehat{r}\right)$ and $\Sigma \left(\widehat{r}\right)=V\left(\widehat{r}\right)+S\left(\widehat{r}\right)$ , eliminating $F_{nk}\left(r\right)$ and $G_{nk}\left(r\right)$ from Eqs. (34) and (35), we can obtain the following two Schrödinger–like differential equations in (NC–3D: RS) symmetries as follows:

$\left[\frac{d^2}{d{r}^2}-\frac{k(k+1)}{r^2}-\left(M+E_{nc-ao}-\Delta \left(\widehat{r}\right)\right)\left(M-E_{nc-ao}+\Sigma \left(\widehat{r}\right)\right)\right]F_{nk}\left(r\right)=0$   (36)

and

$\left[\frac{d^2}{d{r}^2}-\frac{k(k-1)}{r^2}-\left(M+E_{nc-ao}-\Delta \left(\widehat{r}\right)\right)\left(M-E_{nc-ao}+\Sigma \left(\widehat{r}\right)\right)\right]G_{nk}\left(r\right)=0$ (37)

After straightforward calculations one can obtains the following two terms: $\frac{1}{2}M{\omega }^2{\widehat{r}}^2$  and $\frac{\alpha }{2M{\widehat{r}}^2}$  in (NC–3D: RS) as follows:

$\begin{array}{l}\frac{1}{2}M{\omega }^2{\widehat{r}}^2=\frac{1}{2}M{\omega }^2{r}^2-\frac{1}{2}M{\omega }^2\overrightarrow{L}\overrightarrow{\Theta }\\ \frac{\alpha }{2M{\widehat{r}}^2}=\frac{\alpha }{2M{r}^2}+\frac{\alpha \overrightarrow{L}\overrightarrow{\Theta }}{2M{r}^4}\end{array}$   (38)

Which allow us to writing the (m.a.o.) potential $V_{ao}\left(\widehat{r}\right)$ in (NC–3D: RS) as follows:

$\begin{array}{l}{V}_{ao}\left(\widehat{r}\right)=\frac{1}{2}M{\omega }^2{r}^2+\frac{\alpha }{2M{r}^2}\\ +\{\begin{array}{l}{\widehat{V}}_{1p-ao}\left(r,\Theta ,M,\omega \right)=\left(\frac{\alpha }{2M{r}^4}-\frac{1}{2}M{\omega }^2\right)\overrightarrow{L}\overrightarrow{\Theta }\begin{array}{cc}\text{ for}& \text{the spin symmetric case }\end{array}\\ {\widehat{V}}_{2p-ao}\left(r,\Theta ,M,\omega \right)=\left(\frac{\alpha }{2M{r}^4}-\frac{1}{2}M{\omega }^2\right)\tilde{\overrightarrow{L}}\overrightarrow{\Theta }\begin{array}{cc}\text{ for}& \text{the p-spin symmetric case }\end{array}\end{array}\end{array}$   (39)

It’s clearly that, the first 2–terms represent the ordinary anharmonic oscillator while the rest two parts ${\widehat{V}}_{1p-ao}\left(r,\Theta ,M,\omega \right)$  and ${\widehat{V}}_{2p-ao}\left(r,\Theta ,M,\omega \right)$  are produced by the deformation of space, this allows writing the (m.a.o.) in the NC case as an equation similarly to the usual Dirac equation of the commutative type with a non local potential. Furthermore, using the unit step function (also known as the Heaviside step function or simply the theta function) we can rewrite the modified anharmonic oscillator to the following form:

$V_{ao}\left(\widehat{r}\right)=\frac{1}{2}M{\omega }^2{r}^2+\frac{\alpha }{2M{r}^2}+\theta \left(E_{nc-ao}\right){\widehat{V}}_{1p-ao}\left(r,\Theta ,M,\omega \right)+\theta \left(-E_{nc-ao}\right){\widehat{V}}_{2p-ao}\left(r,\Theta ,M,\omega \right)$ (40)

Where

$\theta \left(x\right)=\{\begin{array}{l}1\begin{array}{cc}\text{ for}& \text{x}\rangle \text{0 }\end{array}\\ 0\begin{array}{cc}\text{ for}& \text{x}\langle \text{0}\end{array}\end{array}$   (41)

We generalized the constraint for the pseudospin (p–spin) symmetry $\Delta \left(r\right)=V\left(r\right)$  and $\Sigma \left(r\right)=C_{ps}=\text{constants}$ which presented in refs.^18–21 into the new form $\Delta \left(\widehat{r}\right)=V\left(\widehat{r}\right)$  and $\Sigma \left(\widehat{r}\right)={\widehat{C}}_{ps}=\text{constants}$  in (NC–3D: RS) and inserting the potential $V_{ao}\left(\widehat{r}\right)$  in eq. (39) into the two Schrödinger–like differential equations (36) and (37), one obtains:

$\left[\begin{array}{l}\frac{d^2}{d{r}^2}-\frac{k(k+1)}{r^2}-\left(M+E_{nc-kb}\right)\left(M-E_{nc-kb}+C_{ps}\right)\\ -\left(a{r}^2+br-\frac{c}{r}\right)\left(M-E_{nc-kb}+C_{ps}\right)\\ -\left(\left(\frac{c}{2r^3}-\frac{b}{2r}-a\right)\right)\left(M-E_{nc-kb}+{\widehat{C}}_{ps}\right)\overrightarrow{L}\overrightarrow{\Theta }\end{array}\right]F_{nk}\left(r\right)=0$   (42)

$\left[\begin{array}{l}\frac{d^2}{d{r}^2}-\frac{k(k-1)}{r^2}-\left(M+E_{nc-kb}\right)\left(M-E_{nc-kb}+C_{ps}\right)\\ -\left(a{r}^2+br-\frac{c}{r}\right)\left(M-E_{nc-kb}+C_{ps}\right)\\ -\left(\left(\frac{c}{2r^3}-\frac{b}{2r}-a\right)\right)\overrightarrow{L}\overrightarrow{\Theta }\left(M-E_{nc-kb}+{\widehat{C}}_{ps}\right)\end{array}\right]G_{nk}\left(r\right)=0$  (43)

It’s clearly that, the additive two parts ${\widehat{V}}_{1p-ao}\left(r,\Theta ,M,\omega \right)$  and ${\widehat{V}}_{2p-ao}\left(r,\Theta ,M,\omega \right)$  are proportional with infinitesimal parameter $\Theta$ , thus we can considered as a perturbations terms.

The exact relativistic spin–orbital hamiltonian for (m.a.o.) in (nc: 3d– rs) symmetries for one–electron atoms:

 Again, the two perturbative terms ${\widehat{V}}_{1p-ao}\left(r,\Theta ,M,\omega \right)$  and ${\widehat{V}}_{2p-ao}\left(r,\Theta ,M,\omega \right)$  can be rewritten to the equivalent physical form for (m.a.o.) potential as follows:

$\{\begin{array}{l}{\widehat{V}}_{1p-ao}\left(r,\Theta ,M,\omega \right)=\Theta \left(\frac{\alpha }{2M{r}^4}-\frac{1}{2}M{\omega }^2\right)\overrightarrow{L}\overrightarrow{S}\begin{array}{cc}\text{ for}& \text{the spin symmetric case }\end{array}\\ {\widehat{V}}_{2p-ao}\left(r,\Theta ,M,\omega \right)=\Theta \left(\frac{\alpha }{2M{r}^4}-\frac{1}{2}M{\omega }^2\right)\tilde{\overrightarrow{L}}\overrightarrow{\tilde{S}}\begin{array}{cc}\text{ for}& \text{the p-spin symmetric case }\end{array}\end{array}$   (44)

Furthermore, the above perturbative terms ${\widehat{V}}_{1p-ao}\left(r,\Theta ,M,\omega \right)$  and ${\widehat{V}}_{2p-ao}\left(r,\Theta ,M,\omega \right)$  can be rewritten to the following new equivalent form for (m.a.o.) potential: 

$\{\begin{array}{l}{\widehat{V}}_{1p-ao}\left(r,\Theta ,M,\omega \right)=\frac{1}{2}\Theta \left(\frac{\alpha }{2M{r}^4}-\frac{1}{2}M{\omega }^2\right)\left({\overleftrightarrow{J}}^2-{\overleftrightarrow{L}}^2-{\overleftrightarrow{S}}^2\right)\\ \begin{array}{cc}\text{ for}& \text{the spin symmetric case }\end{array}\\ {\widehat{V}}_{2p-ao}\left(r,\Theta ,M,\omega \right)=\frac{1}{2}\Theta \left(\frac{\alpha }{2M{r}^4}-\frac{1}{2}M{\omega }^2\right)\left({\overleftrightarrow{J}}^2-{\overleftrightarrow{L}}^2-{\overleftrightarrow{\tilde{S}}}^2\right)\\ \begin{array}{cc}\text{ for}& \text{the p-spin symmetric case }\end{array}\end{array}$   (45)

To the best of our knowledge, we just replace the two spin–orbital coupling $\overleftrightarrow{S}\overleftrightarrow{L}$  and $\tilde{\overrightarrow{L}}\overrightarrow{\tilde{S}}$  by the expression $\frac{1}{2}\left({\overleftrightarrow{J}}^2-{\overleftrightarrow{L}}^2-{\overleftrightarrow{S}}^2\right)$  and $\frac{1}{2}\left({\overleftrightarrow{J}}^2-{\overleftrightarrow{L}}^2-{\overleftrightarrow{\tilde{S}}}^2\right)$ , in relativistic quantum mechanics.The set ($H_{nc-kb}\left({\widehat{p}}_i,{\widehat{x}}_i\right)$ ,${\text{J}}^{\text{2}}$ ,${\text{L}}^2$ ,S ^˜2 and $J_z)$  forms a complete of conserved physics quantities and the spin–orbit quantum number $k$  ($\tilde{k}$ ) is related to the quantum numbers for spin symmetry $l$  and p–spin symmetry $\tilde{l}$  as follows.^18–21

$k=\{\begin{array}{l}{k}_1\equiv -\left(l+1\right)\text{ if -}\left(\text{j}+\text{1/2}\right)\text{,}\left({\text{s}}_{\text{1/2}},p_{3/2},etc\right)\text{, }j=l+\frac{1}{2},\text{ aligned spin }\left(\text{k}\langle \text{0}\right)\\ k_2\equiv +l\text{ if }\left(j=l+\frac{1}{2}\right),\left({\text{p}}_{\text{1/2}},d_{3/2},etc\right)\text{, }j=l-\frac{1}{2},\text{ unaligned spin }\left(\text{k}\rangle \text{0}\right)\end{array}$   (46)

and

$\tilde{k}=\{\begin{array}{l}{\tilde{k}}_1\equiv -\tilde{l}\text{ if -}\left(\text{j}+\text{1/2}\right)\text{,}\left({\text{s}}_{\text{1/2}},p_{3/2},etc\right)\text{, }j=\tilde{l}-\frac{1}{2},\text{ aligned spin }\left(\text{k}\langle \text{0}\right)\\ {\tilde{k}}_2\equiv +\left(\tilde{l}+1\right)\text{ if }\left(j=\tilde{l}+\frac{1}{2}\right),\left({\text{p}}_{\text{1/2}},d_{3/2},etc\right)\text{, }j=\tilde{l}+\frac{1}{2},\text{ unaligned spin }\left(\text{k}\rangle \text{0}\right)\end{array}$   (47)

With $\tilde{k}\left(\tilde{k}-1\right)=\tilde{l}\left(\tilde{l}+1\right)$  and $k\left(k-1\right)=l\left(l+1\right)$ , which allows us to form two diagonal $\left(3\times 3\right)$  matrixes ${\widehat{H}}_{so-ao}\left(k_1,k_2\right)$  and ${\widehat{\tilde{H}}}_{so-ao}\left({\tilde{k}}_1,{\tilde{k}}_2\right)$ , for (m.a.o.), respectively, in (NC: 3D–RS) as:

$\begin{array}{l}{\left({\widehat{H}}_{so-ao}\right)}_{11}\left(k_1\right)=k_1\Theta \left(\frac{\alpha }{2M{r}^4}-\frac{1}{2}M{\omega }^2\right)\\ \text{if -}\left(\text{j}+\text{1/2}\right)\text{,}\left({\text{s}}_{\text{1/2}},p_{3/2},etc\right)\text{, }j=l+\frac{1}{2},\text{ aligned spin }\left(\text{k}\langle \text{0}\right)\\ {\left({\widehat{H}}_{so-ao}\right)}_{22}\left(k_2\right)=k_2\Theta \left(\frac{\alpha }{2M{r}^4}-\frac{1}{2}M{\omega }^2\right)\\ \text{if }\left(j=l+\frac{1}{2}\right),\left({\text{p}}_{\text{1/2}},d_{3/2},etc\right)\text{, }j=l-\frac{1}{2},\text{ unaligned spin }\left(\text{k}\rangle \text{0}\right)\\ {\left({\widehat{H}}_{so-ao}\right)}_{33}=0\end{array}$   (48)

and

$\begin{array}{l}{\left({\widehat{H}}_{so-ao}\right)}_{11}\left({\tilde{k}}_1\right)={\tilde{k}}_1\Theta \left(\frac{\alpha }{2M{r}^4}-\frac{1}{2}M{\omega }^2\right)\\ \text{if -}\left(\text{j}+\text{1/2}\right)\text{,}\left({\text{s}}_{\text{1/2}},p_{3/2},etc\right)\text{, }j=\tilde{l}-\frac{1}{2},\text{ aligned spin }\left(\text{k}\langle \text{0}\right)\\ {\left({\widehat{H}}_{so-ao}\right)}_{22}\left({\tilde{k}}_2\right)={\tilde{k}}_2\Theta \left(\frac{\alpha }{2M{r}^4}-\frac{1}{2}M{\omega }^2\right)\\ \text{if }\left(j=\tilde{l}+\frac{1}{2}\right),\left({\text{p}}_{\text{1/2}},d_{3/2},etc\right)\text{, }j=\tilde{l}+\frac{1}{2},\text{ unaligned spin }\left(\text{k}\rangle \text{0}\right)\\ {\left({\widehat{H}}_{so-ao}\right)}_{33}=0\end{array}$  (49)

The exact relativistic spin–orbital spectrum for (m.a.o.) potential symmetries for n^thexcited states for one–electron atoms in (NC: 3D– RSP) symmetries:

 In this sub section, we are going to study the modifications to the energy levels $E_{nc-per:u}\left(\Theta ,k_1,E_{nk},M,\omega \right)$  and $E_{nc-per:d}\left(\Theta ,k_2,E_{nk},M,\omega \right)$  for ($\text{-}\left(\text{j}+\text{1/2}\right)\text{,}\left({\text{s}}_{\text{1/2}},p_{3/2},etc\right)\text{, }j=l+\frac{1}{2},\text{ aligned spin }\left(\text{k}\langle \text{0}\right)$  and spin–up) and ( $\left(j=l+\frac{1}{2}\right),\left({\text{p}}_{\text{1/2}},d_{3/2},etc\right)\text{, }j=l-\frac{1}{2},\text{ unaligned spin }\left(\text{k}\rangle \text{0}\right)$  and spin down), respectively, at first order of infinitesimal parameter $\Theta$ , for ground state, obtained by applying the standard perturbation theory, using Eqs. (22), (44)and (45) as:

$\begin{array}{l}{\displaystyle \int {\stackrel{+}{{\displaystyle \Psi }}}_{nk}\left(r,\theta ,\varphi \right)\left[\begin{array}{l}\theta \left(E_{nc-ao}\right){\widehat{V}}_{1p-ao}\left(r,\Theta ,E_{nk},M,\omega \right)\\ +\theta \left(-E_{nc-ao}\right){\widehat{V}}_{2p-ao}\left(r,\Theta ,E_{nk},M,\omega \right)\end{array}\right]{\Psi }_{nk}\left(r,\theta ,\varphi \right)r^{2}drd\Omega }=\\ =\theta \left(E_{nc-ao}\right){\displaystyle \int F^{*}{}_{nk}\left(r\right){\widehat{V}}_{1p-kb}\left(r,\Theta ,E_{nk},M,\omega \right)F_{nk}\left(r\right)dr}\\ -\theta \left(E_{nc-ao}\right){\displaystyle \int G_{n\tilde{k}}{}^{*}\left(r\right){\widehat{V}}_{2p-ao}\left(r,\Theta ,E_{nk},M,\omega \right)G_{n\tilde{k}}\left(r\right)dr}\end{array}$   (50)

The first parts represent the modifications to the energy levels for the spin symmetric cases $E_{nc-per:u}\left(\Theta ,k_1,E_{nk},M,\omega \right)$  and $E_{nc-per:d}\left(\Theta ,k_2,E_{nk},M,\omega \right)$  while the second part represent the modifications to the energy levels ($E_{nc-per:d}\left(\Theta ,{\tilde{k}}_1,E_{0k},E_{nk},M,\omega \right)$ ,$E_{nc-per:u}\left(\Theta ,{\tilde{k}}_2,E_{0k},E_{nk},M,\omega \right)$ ) for the spin spin–symmetry, then we have explicitly:

$E_{nc-per:u}\left(\Theta ,k_1,E_{nk},M,\omega \right)\equiv \theta \left(E_{nc-kb}\right)k_1\Theta {\displaystyle \int {\stackrel{*}{F}}_{nk}\left(r\right)\left(\frac{\alpha }{2M{r}^4}-\frac{1}{2}M{\omega }^2\right)F_{nk}\left(r\right)dr}$   (51)

$E_{nc-per:u}\left(\Theta ,k_2,E_{nk},M,\omega \right)\equiv \theta \left(E_{nc-kb}\right)k_2\Theta {\displaystyle \int {\stackrel{*}{F}}_{nk}\left(r\right)\left(\frac{\alpha }{2M{r}^4}-\frac{1}{2}M{\omega }^2\right)F_{nk}\left(r\right)dr}$   (52)

Inserting the radial function $F_{nk}(r)$  given by Eq. (23) into the above two Eqs. (51) and (52) to obtain:

$\begin{array}{l}{E}_{nc-per:u}\left(\Theta ,k_2,E_{nk},M,\omega \right)\equiv \theta \left(E_{nc-ao}\right)\Theta k_1{\left|C_n\right|}^2\\ {\displaystyle \underset{0}{\overset{+\infty }{\int }}\exp \left(-\sqrt{M\left(M+E_{n,k}\right)}{r}^2\right)r^{2L+2}{\left[L_n^{L+1/2}\left(\sqrt{M\left(M+E_{n,k}\right)}{r}^2\right)\right]}^2\left(\frac{\alpha }{2M{r}^4}-\frac{1}{2}M{\omega }^2\right)dr}\end{array}$   (53)