Energy level splitting of a 2D hydrogen atom with Rashba coupling in non-commutative space
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S Aghababaei,∗, G RezaeiDepartment of Physics, College of Sciences, Yasouj University, Yasouj, 75918-74934, Iran
First author contact: Author to whom any correspondence should be addressed. Received:2020-06-4Revised:2020-07-16Accepted:2020-07-29Online:2020-11-23
Abstract We explore the non-commutative (NC) effects on the energy spectrum of a two-dimensional hydrogen atom. We consider a confined particle in a central potential and study the modified energy states of the hydrogen atom in both coordinates and momenta of non-commutativity spaces. By considering the Rashba interaction, we observe that the degeneracy of states can also be removed due to the spin of the particle in the presence of NC space. We obtain the upper bounds for both coordinates and momenta versions of NC parameters by the splitting of the energy levels in the hydrogen atom with Rashba coupling. Finally, we find a connection between the NC parameters and Lorentz violation parameters with the Rashba interaction. Keywords：quantum mechanics;non-commutative space;spin-orbit interaction
PDF (325KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article S Aghababaei, G Rezaei. Energy level splitting of a 2D hydrogen atom with Rashba coupling in non-commutative space. Communications in Theoretical Physics, 2020, 72(12): 125101- doi:10.1088/1572-9494/abb7cc
1. Introduction
The hydrogen atom has played an important role in fundamental physics such as the Bohr Model, Sommerfeld's relativistic hydrogen atom, the development of QED to explain the Lamb shift, the anomalous magnetic moment of electrons, the detection of hydrogen Bose-Einstein condensation, current experiments on anti-hydrogen, and its role in the proton size puzzle [1-3]. The hydrogen atom is also a model which is considered for several number of exotic atoms including positronium, muonium and muonic hydrogen [4]. The two-dimensional (2D) hydrogen atom, where an electron is confined to move in a plane around an attractive Coulomb potential, is a theoretical structure that is not of interest within a model but also has a considerable interest in physical realizations [5]. It can describe indeed the effect of a charged impurity in 2D systems such as quantum wells and surface states in condensed matter physics. Theoretical studies on the 2D quantum structures of hydrogen atoms have been carried out during recent years [6].
The non-commutative (NC) theory, which was introduced by Heisenberg and then formalized by Snyder in 1947 [7], was suggested due to the physical results from string theory and quantum gravity [8]. In recent decades, many researchers have shown a great deal of interest in solving Schrödinger, Klein-Gordon and Dirac equations for various potentials in the NC theory with both coordinate and momentum spaces to obtain profound interpretations at a microscopic scale. Techniques from NC space have played a major role in low-dimensional physics such as the quantum Hall effect [9] and the gravitational quantum well [10]. There have been many phenomenological consequences of the NC theory in different physical problems in both quantum mechanics [11] and quantum field theory [12] frameworks.
On the other hand, the spin structure of half-spin particles recently has become important in condensed matter physics. The arising spin-orbit (SO) interaction from the structural and bulk inversion asymmetries are characterized in several low-dimensional systems [13]. The corresponding Rashba coupling of the SO interaction, which originates from structural inversion asymmetry, gives rise to the splitting energy spectrum depending on the material characteristics which range from a few to hundreds of meV [14]. Several studies have already been devoted to the effects of Rashba coupling and show that splitting levels with the same orbital angular momenta but different spins could change the physical properties of low-dimensional systems [15]. Since the bound states around charged impurities in 2D systems with structural inversion asymmetry can be described with 2D hydrogen atoms in the presence of the Rashba interaction, we consider in this paper an electron confined within the Coulomb potential of a hydrogen atom in 2D and investigate the deformation of the discrete spectrum in the presence of NC space for both coordinates and momenta spaces where the Rashba coupling effects are also considered.
The paper is organized as follows: in section 2, we briefly introduce the energy spectrum of a 2D hydrogen atom and present the modified energy states with the Rashba term in usual (commutative) space. In section 3, we consider the non-commutativity of the coordinates and momenta in the quantum mechanics framework and examine the deformation of energy states in the 2D hydrogen atom in both approaches. Moreover, we investigate both NC and Rashba coupling effects on the energy spectrum of 2D hydrogen. We find the corresponding bounds on NC parameters from the splitting of energy levels in the 2D hydrogen atom with the Rashba interaction. Finally, we mention the Lorentz symmetry breaking effects in the 2D hydrogen atom and express some concluding results in section 4.
2. Energy spectrum of 2D hydrogen atom
The hydrogen atom is generally a system composed of an electron with effective mass of μ in the positively charged nucleus central potential (Ze) as$\begin{eqnarray}V(r)=-\displaystyle \frac{{{Ze}}^{2}}{r},\end{eqnarray}$which is located at the origin of the coordinate system. The three-dimensional (3D) hydrogen atom has played an important role in the formulation and development of quantum mechanics [1-4]. However, if the motion of the electron around the nucleus is constrained in a plane by certain boundary conditions, then such a system is called the 2D hydrogen atom. This system is not an exact sense for all fields including the electromagnetic field, photon emission, angular momentum, and spin which are not confined to a plane. Nevertheless, the analytic solutions of a 2D hydrogen atom is of interest to the study of anisotropic effects in solids and its quantum mechanical properties are investigated in [16]. Within the framework of the non-relativistic quantum mechanics, the Schrödinger equation of a 2D hydrogen atom in the polar coordinates applying ℏ=c=1, in the natural unit, is given by$\begin{eqnarray}\left(\displaystyle \frac{{\overrightarrow{p}}^{2}}{2\mu }+V(r)\right){\psi }_{{nm}}^{(0)}(r,\varphi )={E}_{n}^{(0)}{\psi }_{{nm}}^{(0)}(r,\varphi ),\end{eqnarray}$where$\begin{eqnarray}\begin{array}{rcl}{\psi }_{{nm}}^{(0)}(r,\varphi ) & = & {N}_{{nm}}{\left({\beta }_{n}r\right)}^{| m| }{{\rm{e}}}^{-{\beta }_{n}\tfrac{r}{2}}{\rm{F}}(-n+| m| \\ & & +1,2| m| +1;{\beta }_{n}r){{\rm{\Phi }}}_{m}(\varphi ),\end{array}\end{eqnarray}$$\begin{eqnarray}{E}_{n}^{(0)}=-\displaystyle \frac{{Z}^{2}{e}^{2}}{2{\left(n-\tfrac{1}{2}\right)}^{2}{a}_{0}},\qquad n=1,2,3,\ldots \end{eqnarray}$with introducing$\begin{eqnarray}{\beta }_{n}=\displaystyle \frac{2Z\mu {e}^{2}}{n-\tfrac{1}{2}}.\end{eqnarray}$Nnm is a normalization constant, F is the confluent hypergeometric function, and ${{\rm{\Phi }}}_{m}(\varphi )=\tfrac{1}{\sqrt{2\pi }}{{\rm{e}}}^{{\rm{i}}m\varphi }$. The Bohr radius, ${a}_{0}=\tfrac{1}{\mu {e}^{2}}$, n and $| m| =0,1,2,\ldots $ are the principle and magnetic quantum numbers, respectively. It should be mentioned that equation (4) denotes the discrete part of the hydrogen spectrum in 2D and $r=\sqrt{{x}^{2}+{y}^{2}}$. As a single hydrogen atom must be treated as a 3D system, one can examine to recover the corresponding formulas for the 3D hydrogen atom by replacing $n-\tfrac{1}{2}$ with n in equation (4) [16].
2.1. Energy spectrum of 2D hydrogen atom in the presence of Rashba coupling
The 2D problem for an electron interacting with Rashba potential arising from structural inversion asymmetry in the direction perpendicular to the 2D plane is numerically solved [14]. It is shown that the strong Rashba interaction removes the initial degeneracy of the 2D hydrogen atom. In this section we obtain this feature in the explicit form with the Baker-Hausdorff formula. To this end, one can consider the Hamiltonian of a 2D hydrogen atom in the presence of the Rashba potential in the form of$\begin{eqnarray}{H}_{R}=\displaystyle \frac{{\vec{p}}^{2}}{2\mu }+V(r)+{\lambda }_{R}({p}_{y}{\sigma }_{x}-{p}_{x}{\sigma }_{y}),\end{eqnarray}$where λR denotes the strength of Rashba coupling, and σx and σy are Pauli matrices. It is evident that the given Rashba Hamiltonian is not diagonal in this representation and not useful for the calculation of eigenvalues. Hence, we apply the Baker-Hausdroff lemma to find the diagonalized Hamiltonian in the spin space. The respect Hamiltonian is given by$\begin{eqnarray}{H}_{R}^{D}={U}^{\dagger }{H}_{R}U,\end{eqnarray}$where the transformed operator is $U={{\rm{e}}}^{-{\rm{i}}\beta \hat{{\rm{O}}}}$ and the diagonalized Hamiltonian would be written at the leading order as follows:$\begin{eqnarray}{H}_{R}^{D}={H}_{R}+{\rm{i}}\beta [\hat{O},{H}_{R}]-\displaystyle \frac{{\beta }^{2}}{2}[\hat{O},[\hat{O},{H}_{R}]]+{ \mathcal O }({\beta }^{3}).\end{eqnarray}$By introducing the operator$\begin{eqnarray}\hat{O}=x{\sigma }_{y}-y{\sigma }_{x},\end{eqnarray}$and considering the β parameter as$\begin{eqnarray}\beta =\mu {\lambda }_{R},\end{eqnarray}$one can write down the Hamiltonian of the system as follows:$\begin{eqnarray}{H}_{R}={H}_{0}-\mu {\lambda }_{R}^{2}-\mu {\lambda }_{R}^{2}{L}_{z}{\sigma }_{z},\end{eqnarray}$where ${H}_{0}=\tfrac{{\vec{p}}^{2}}{2\mu }+V(r)$. Therefore, the corresponding energy eigenvalues of the system would be given by$\begin{eqnarray}{E}_{{nm}}={E}_{n}^{(0)}-\mu {\lambda }_{R}^{2}\mp m\mu {\lambda }_{R}^{2},\end{eqnarray}$where ${E}_{n}^{(0)}$ denotes the energy spectrum of the 2D hydrogen atom in the absence of the Rashba term (equation (4)) and m is the magnetic quantum number. A glance at equation (12) reveals that the energy spectrum of our system is comparable with the results reported in [17] where the energy spectrum of 2D hydrogen with Rashba coupling is evaluated. In addition, if we set ${\lambda }_{R}\approx {10}^{-4}$, in the natural unit, we then find the splitting energy in an order of meV which is compatible with other low-dimensional systems. By introducing the Rashba energy ${E}_{R}=\tfrac{\mu {\lambda }_{R}^{2}}{2}$ and $\eta =2\mu {Z}^{2}{e}^{4}$, removing the energy levels degeneracy is shown in figure 1, where there is the lack of spatial inversion symmetry induced by Rashba coupling.
Figure 1.
New window|Download| PPT slide Figure 1.Splitting energy levels (n, m) for a 2D hydrogen atom with Rashba coupling ${\lambda }_{R}\ne 0$. Since $[{H}_{R},{ \mathcal P }]\ne 0$ where ${ \mathcal P }\equiv {\gamma }^{0}$ as the spatial inversion operator, the inversion asymmetry from Rashba coupling is shown for $m\ne 0$ where the spin of the particle removes the degeneracy of the energy spectrum.
3. Energy spectrum of 2D hydrogen atom in the presence of NC space
In this section, we use NC theories to examine the non-relativistic spectrum energy of the 2D hydrogen atom. To analyze the NC effects, we construct a NC version of quantum mechanics by replacing the ordinary product between operators with its NC counterpart based on the Weyl-Moyal product [18]. The new star product is defined by a power series expansion at the leading order of$\begin{eqnarray}A\,\star \,B=A\cdot B+{\rm{i}}\displaystyle \frac{1}{2}{\theta }^{\mu \nu }(x){\partial }_{\mu }A\cdot {\partial }_{\nu }B+{ \mathcal O }({\theta }^{2}),\end{eqnarray}$where ${\theta }^{\mu \nu }$ is a constant antisymmetric tensor. This product leads to non-zero commutators of position and momentum operators as follows:$\begin{eqnarray}\begin{array}{rcl}\left[{\hat{x}}^{i},{\hat{x}}^{j}\right] & = & {\rm{i}}{\theta }^{{ij}},\\ \left[{\hat{p}}^{i},{\hat{p}}^{j}\right] & = & {\rm{i}}{\bar{\theta }}^{{ij}},\\ \left[{x}^{i},{p}^{j}\right] & = & {\rm{i}}{\delta }^{{ij}},\end{array}\end{eqnarray}$which θij, ${\bar{\theta }}^{{ij}}$, and δij are elements of a real constant matrix. Notice that the dimensions of θij and ${\bar{\theta }}^{{ij}}$ are respectively ${(\mathrm{Energy})}^{-2}$ and (Energy)2. The hat symbol denotes the NC version of coordinates and their conjugate momentum operators. Therefore, the Heisenberg algebra is deformed in (14) by θ and $\bar{\theta \,}$ parameters which are well known as NC parameters. These matrices can be divided into two parts; the time-like components ${\theta }^{0i}({\bar{\theta }}^{0i})$ and the space-like components ${\theta }^{{ij}}({\bar{\theta }}^{{ij}})$ where i, j, k=x, y, z. We consider only the space-like components due to avoiding the unitary problem in this theory [19]. The NC operators can be represented by the Bopp's shift [20] as follows:$\begin{eqnarray}\begin{array}{rcl}{\hat{x}}_{i} & = & {x}_{i}-\displaystyle \frac{1}{2}\displaystyle \sum _{j}{\theta }_{{ij}}{p}_{j},\\ {\hat{p}}_{i} & = & {p}_{i}+\displaystyle \frac{1}{2}\displaystyle \sum _{j}{\bar{\theta }}_{{ij}}{x}_{j},\end{array}\end{eqnarray}$where coordinates xi and momenta pi satisfy the ordinary commutation relations. It is interesting to mention that the non-vanishing right-hand side of equation (14) manifestly causes breaking of rotational and time-reversal symmetries, which leads to the Lorentz symmetry violation. As any particle Lorentz transformation ${\theta }_{{ij}}(\,{\bar{\theta }}_{{ij}}\,)$ is mannered as a fixed vector like $\vec{\theta }=({\theta }_{{yz}},{\theta }_{{zx}},{\theta }_{{xy}})$ the NC parameters obviously violate particle Lorentz symmetry [21]. To investigate the effects of NC space, we consider the non-commutativity of coordinates and momenta which are well known as space-space and momentum-momentum non-commutativity, respectively. NC coordinates The 2D Schrödinger equation with a potential of hydrogen atoms can be written in the NC space of coordinates as follows:$\begin{eqnarray}\hat{H}=\displaystyle \frac{{\vec{p}}^{2}}{2\mu }-\displaystyle \frac{{{Ze}}^{2}}{\hat{r}},\end{eqnarray}$where by using the relations in (15), one can obtain$\begin{eqnarray}\displaystyle \frac{1}{\hat{r}}=\displaystyle \frac{1}{r}-\displaystyle \frac{1}{2{r}^{3}}\vec{\theta }\cdot \vec{L}+{ \mathcal O }({\theta }^{2}).\end{eqnarray}$By substituting (17) into (16), the full Hamiltonian would be decreased at the leading order as$\begin{eqnarray}\hat{H}={H}_{0}+\displaystyle \frac{{{Ze}}^{2}}{2{r}^{3}}\vec{\theta }\cdot \vec{L}+{ \mathcal O }({\theta }^{2}).\end{eqnarray}$As the NC parameter is so small, the energy spectrum of a 2D hydrogen atom by using the perturbation theory can be written as$\begin{eqnarray}{\hat{E}}_{{nm}}={E}_{n}^{(0)}+{\rm{\Delta }}{E}_{m},\end{eqnarray}$where$\begin{eqnarray}{\rm{\Delta }}{E}_{m}=\langle {\psi }_{{nm}}^{(0)}| \displaystyle \frac{{{Ze}}^{2}}{2{r}^{3}}\vec{\theta }\cdot \vec{L}| {\psi }_{{nm}}^{(0)}\rangle .\end{eqnarray}$One can express the NC correction by considering the $\vec{\theta }$ vector in the z-direction (restrict to the case θyz=θzx=0, θxy=θ) as follows:$\begin{eqnarray}{\rm{\Delta }}{E}_{m}=\displaystyle \frac{{{mZe}}^{2}\theta }{2}\langle {r}^{-3}{\rangle }_{{nm}},\end{eqnarray}$where m is the magnetic quantum number and the exception value of $\langle {r}^{-3}{\rangle }_{{nm}}$ can be calculated from [16]. For instance, $\langle {r}^{-3}{\rangle }_{21}$ would be found as$\begin{eqnarray}\langle {r}^{-3}{\rangle }_{21}=\displaystyle \frac{1}{6}{\left(\displaystyle \frac{4Z}{3{a}_{0}}\right)}^{3},\end{eqnarray}$where the corresponding splitting of levels would be in the presence of NC space as follows:$\begin{eqnarray}\delta {\hat{E}}_{21}={\hat{E}}_{21}-{\hat{E}}_{2-1}=\displaystyle \frac{Z\theta }{\mu {a}_{0}}\displaystyle \frac{1}{6}{\left(\displaystyle \frac{4Z}{3{a}_{0}}\right)}^{3}.\end{eqnarray}$Thus, the NC coordinates can deform the energy spectrum of a 2D hydrogen atom related to the magnetic quantum number at the leading order. This subject is discussed for ordinary hydrogen atoms in [1, 22]. Equation (23) shows that there is an enhancement on splitting levels of a 2D hydrogen atom where the NC effects would be enhanced. NC momenta The 2D hydrogen atom in the NC momenta can be expressed by$\begin{eqnarray}\hat{H}=\displaystyle \frac{{\vec{\hat{p}}}^{2}}{2\mu }+V(r),\end{eqnarray}$where$\begin{eqnarray}{\vec{\hat{p}}}^{2}={\vec{p}}^{2}-\vec{\bar{\theta \,}}\cdot \vec{L}+{ \mathcal O }({\bar{\theta }}^{2}).\end{eqnarray}$Therefore, the corresponding NC Hamiltonian becomes$\begin{eqnarray}\hat{H}={H}_{0}-\displaystyle \frac{\vec{\bar{\theta \,}}\cdot \vec{L}}{2\mu }+{ \mathcal O }({\bar{\theta }}^{2}),\end{eqnarray}$where, by considering the small NC parameter in the z-direction, we obtain the correction energy spectrum as$\begin{eqnarray}{\hat{E}}_{{nm}}={E}_{n}^{(0)}+{\rm{\Delta }}{E}_{m},\end{eqnarray}$where$\begin{eqnarray}{\rm{\Delta }}{E}_{m}=-\left\langle {\psi }_{{nm}}^{(0)}\left|\displaystyle \frac{\bar{\theta \,}{L}_{z}}{2\mu }\right|{\psi }_{{nm}}^{(0)}\right\rangle =-m\displaystyle \frac{\bar{\theta \,}}{2\mu }.\end{eqnarray}$So that, the deformation energy spectrum in the non-commutativity of the momenta version depends on the magnetic quantum number as an order of $\left|\tfrac{\,\bar{\theta \,}}{2\mu }\right|$. The splitting energy of state $\delta {\hat{E}}_{21}$ would be an order of $\left|\tfrac{\,\bar{\theta \,}}{\mu }\right|$, which is not depend on the Bohr radius in comparison to equation (23).
3.1. 2D hydrogen atom in the presence of Rashba coupling and NC space
NC theory could deform the spin structure of a charged particle at least in the linear approximation [23]. In this section, we study directly the SO effects which could be realized in the NC geometry due to the spin structure. We show manifestly this feature by considering the explicit term of Rashba coupling in the NC space for hydrogen-like systems. To this end, we write down the Hamiltonian of a hydrogen atom with the Rashba term for both coordinates and momenta of non-commutativity approaches, respectively. NC coordinates The corresponding Hamiltonian in this approach is given by$\begin{eqnarray}{\hat{H}}_{R}=\displaystyle \frac{{\vec{p}}^{2}}{2\mu }+\hat{V}(\hat{r})+{\lambda }_{R}({p}_{y}{\sigma }_{x}-{p}_{x}{\sigma }_{y}),\end{eqnarray}$where by applying equation (14), the 2D Hamiltonian (${\hat{H}}_{R}$) would be expressed by$\begin{eqnarray}{\hat{H}}_{R}={H}_{0}+\displaystyle \frac{{{Ze}}^{2}\theta }{2{r}^{3}}{L}_{z}+{\lambda }_{R}({p}_{y}{\sigma }_{x}-{p}_{x}{\sigma }_{y}),\end{eqnarray}$where ${L}_{z}={{xp}}_{y}-{{yp}}_{x}$. The last term in equation (30) depends on the spin which makes the Hamiltonian non-diagonal so that we use the Baker-Hausdroff lemma to find the diagonalized Hamiltonian in spin space, up to the second-order of λR. It is required that the energy scale corresponding to the SO interaction must be much smaller than the spin-independent Hamiltonian. The transformed Hamiltonian is given by (7), by introducing the operator$\begin{eqnarray}\hat{O}=({p}_{x}-{Ay}){\sigma }_{x}+({p}_{y}+{Ax}){\sigma }_{y},\end{eqnarray}$by considering$\begin{eqnarray}A\equiv -\displaystyle \frac{2}{\theta },\end{eqnarray}$$\begin{eqnarray}\beta \equiv -\mu {\lambda }_{R}\theta +{ \mathcal O }({\theta }^{2}).\end{eqnarray}$Therefore, the full diagonalized Hamiltonian with a Rashba interaction in NC space of coordinates can be obtained as follows:$\begin{eqnarray}{\hat{H}}_{R}^{D}=\displaystyle \frac{{\vec{p}}^{2}}{2{M}^{* }}+V(r)+\displaystyle \frac{{{Ze}}^{2}\theta }{2{r}^{3}}{L}_{z}-\mu {\lambda }_{R}^{2}-\mu {\lambda }_{R}^{2}{L}_{z}{\sigma }_{z},\end{eqnarray}$where the corrected kinetic energy term is$\begin{eqnarray}\displaystyle \frac{1}{2{M}^{* }}=\displaystyle \frac{1}{2\mu }+\mu {\lambda }_{R}^{2}\theta {\sigma }_{z}.\end{eqnarray}$Here the 1 denotes the identity matrix and higher order contributions ${ \mathcal O }({\lambda }_{R}^{2})$ are negligible compares to the NC parameter. One can find the Rashba Hamiltonian of a 2D hydrogen atom in usual space (11) by considering $\theta \longrightarrow 0$ in (34) and (35). Therefore the corresponding 2D eigenvalue equation could be written as follows:$\begin{eqnarray}{H}_{R}^{D}{\psi }_{{nm}}^{(0)}(r,\varphi ){\chi }_{s}^{\pm }={E}_{{nms}}^{R}{\psi }_{{nm}}^{(0)}(r,\varphi ){\chi }_{s}^{\pm },\end{eqnarray}$where ${\chi }_{s}^{\pm }$ are the eigenvectors related to the z-component of the spin operator. Thus, one can write the corresponding eigenvalues of the Hamiltonian in terms of$\begin{eqnarray}\begin{array}{rcl}{E}_{{nms}}^{R} & = & {E}_{n}^{(0)}(\mu \longrightarrow {M}^{* })\\ & & +m\displaystyle \frac{{{Ze}}^{2}\theta }{2}\langle {r}^{-3}{\rangle }_{{nm}}-\mu {\lambda }_{R}^{2}\mp m\mu {\lambda }_{R}^{2}.\end{array}\end{eqnarray}$It shows that energy level splitting of the 2D hydrogen atom, by considering Rashba coupling and the NC parameter, provides a new quantum numbers set {n, m, s} where s is the spin quantum number. Thus θ and λR respectively break the rotation and inversion symmetries of the hydrogen atom and remove the degeneracy of energy levels, see figure 2. In this manner, considering the effective mass of electrons in the order of ∼MeV for the level (n=2, m=1) in equation (37), one can find the upper bound on the NC parameter related to the coordinates version of order $\theta \lesssim {(10\mathrm{MeV})}^{-2}$, while the splitting energy levels in the 2D hydrogen atom with the Rashba interaction are in the order of ${\rm{\Delta }}{E}^{R}\sim (0.1-0.01)$ meV. It should be noted that the latter bound is 102 times smaller than one obtained using the Lamb shift effect in the usual hydrogen atom [24]. This result is in agreement with the order of ${\left(\tfrac{{\lambda }_{R}}{\alpha }\right)}^{2}$, where α denotes the fine structure constant in the usual hydrogen atom.
Figure 2.
New window|Download| PPT slide Figure 2.Splitting energy corresponding to the levels (2, 1) of the 2D hydrogen atom with Rashba coupling ${\lambda }_{R}\sim {10}^{-4}$, in the NC space. The black and red lines denote removing the degeneracy level for coordinates with $\theta \sim {(10\mathrm{MeV})}^{-2}$ and momenta non-commutativity $\bar{\theta \,}\,\sim {({10}^{-2}\mathrm{MeV})}^{2}$, respectively.
NC momenta In this case the Hamiltonian of the system is given by$\begin{eqnarray}{\hat{H}}_{R}=\displaystyle \frac{{\vec{\hat{p}}}^{2}}{2\mu }+V(r)+{\lambda }_{R}({\hat{p}}_{y}{\sigma }_{x}-{\hat{p}}_{x}{\sigma }_{y}),\end{eqnarray}$where the expansion of NC momenta with usual momenta (equation (15)) leads to$\begin{eqnarray}\begin{array}{rcl}{\hat{H}}_{R} & = & {H}_{0}-\displaystyle \frac{\,\bar{\theta \,}}{2\mu }{L}_{z}+{\lambda }_{R}({p}_{y}{\sigma }_{x}-{p}_{x}{\sigma }_{y})\\ & & +\displaystyle \frac{1}{2}\bar{\theta \,}{\lambda }_{R}(y{\sigma }_{y}+x{\sigma }_{x})+{ \mathcal O }({\bar{\theta }}^{2}).\end{array}\end{eqnarray}$In a similar manner one can diagonalize the above Hamiltonian by considering$\begin{eqnarray}\hat{O}=({p}_{x}-{Ay}){\sigma }_{x}+({p}_{y}+{Ax}){\sigma }_{y},\end{eqnarray}$with$\begin{eqnarray}A\equiv \displaystyle \frac{4\mu }{\bar{\theta \,}}\displaystyle \frac{{{Ze}}^{2}}{{r}^{3}}-\bar{\theta \,},\end{eqnarray}$$\begin{eqnarray}\beta \equiv \displaystyle \frac{\bar{\theta \,}{r}^{3}{\lambda }_{R}}{2{{Ze}}^{2}}.\end{eqnarray}$Therefore, the momentum version of the Hamiltonian is obtained as follows$\begin{eqnarray}\begin{array}{rcl}{\hat{H}}_{R}^{D} & = & \displaystyle \frac{{\vec{p}}^{2}}{2{M}^{* }}+V(r)-\displaystyle \frac{\,\bar{\theta \,}}{2\mu }{L}_{z}+\mu \bar{\theta \,}{\lambda }_{R}^{2}{r}^{2}{\sigma }_{z}\\ & & -\mu {\lambda }_{R}^{2}-\mu {\lambda }_{R}^{2}{L}_{z}{\sigma }_{z}+{ \mathcal O }({\bar{\theta }}^{2}),\end{array}\end{eqnarray}$where$\begin{eqnarray}\displaystyle \frac{1}{2{M}^{* }}=\displaystyle \frac{1}{2\mu }-\displaystyle \frac{\bar{\theta \,}{\lambda }_{R}^{2}{r}^{3}}{2{{Ze}}^{2}}{\sigma }_{z}.\end{eqnarray}$Finally, the spin up and spin down Hamiltonian in the NC momenta approach become$\begin{eqnarray}\begin{array}{rcl}{\hat{H}}_{R}^{\uparrow } & = & \displaystyle \frac{{\vec{p}}^{2}}{2{M}_{\uparrow }^{* }}+V(r)-\displaystyle \frac{\bar{\theta \,}}{2\mu }{L}_{z}+\mu \bar{\theta \,}{\lambda }_{R}^{2}{r}^{2}-\mu {\lambda }_{R}^{2}-\mu {\lambda }_{R}^{2}{L}_{z},\\ {\hat{H}}_{R}^{\downarrow } & = & \displaystyle \frac{{\vec{p}}^{2}}{2{M}_{\downarrow }^{* }}+V(r)-\displaystyle \frac{\bar{\theta \,}}{2\mu }{L}_{z}-\mu \bar{\theta \,}{\lambda }_{R}^{2}{r}^{2}-\mu {\lambda }_{R}^{2}+\mu {\lambda }_{R}^{2}{L}_{z}.\end{array}\end{eqnarray}$One can easily find the effect of Rashba coupling in usual space in equation (11) and the momentum non-commutativity effect in equation (26) by applying $\bar{\theta \,}\,\longrightarrow 0$ and ${\lambda }_{R}\longrightarrow 0$, respectively. The splitting energy state, for instance, the state n=2 and m=1 in the momentum version by considering$\begin{eqnarray}\begin{array}{rcl}\langle {r}^{2}{\rangle }_{21} & = & \displaystyle \frac{20}{{\beta }_{2}^{2}},\\ \langle {r}^{3}{\rangle }_{21} & = & \displaystyle \frac{120}{{\beta }_{2}^{3}},\end{array}\end{eqnarray}$are shown in figure 2. Therefore, one can limit the NC parameter of momenta version of order $\bar{\theta \,}\,\lesssim {({10}^{-2}\mathrm{MeV})}^{2}$ by using equations (45) and (46) in which the theoretical precision of removing the degeneracy of the energy spectrum 2D hydrogen atom in the presence of the Rashba SO interaction is considered as an order of meV. The corresponding momentum scale from the latter bound is 1010 times greater than the former bounds obtained in [25]. As it was observed, both NC coordinates and momenta forms can remove the degeneracy energy spectrum of the 2D hydrogen atom, which are proportional to the NC parameters. We also can see from equation (35) and equation (44) that the Rashba coupling and NC parameters can modify the kinetic energy of the system. These modified terms show manifestly the breaking of Lorentz symmetry. With respect to the non-relativistic Lorentz violation Hamiltonian which is presented in [26], we can find that the modified kinetic energy in the NC theory with Rashba coupling have similar terms in the Lorentz violation theory. Thus, we should mention that the Lorenz violation effect induced by Rashba coupling [27] in space-space non-commutativity depends on the mass of the particle according to the LV parameters $\propto {\mu }^{3}{\lambda }_{R}^{2}\theta $ while in the momentum-momentum version it is also related to the Bohr radius, it means that the LV parameters are $\propto {\mu }^{3}{\lambda }_{R}^{2}\bar{\theta \,}{\,a}_{0}^{4}$. From these relations, one can understand the effects of Lorentz violation with Rashba coupling from the enhancement of NC space.
4. Conclusion
In this paper, we have studied the energy spectrum of the 2D hydrogen atom in the presence of Rashba coupling. We have shown that the spin of an electron has a considerable effect on the energy states of the 2D hydrogen atom. We also have investigated some aspects of non-commutative theory in the framework of quantum mechanics. We have shown that the energy spectrum of such potential is modified by considering the non-commutative coordinates and momenta approaches. Deforming energy levels are found for coordinates and momenta non-commutativity versions in equations (20) and (27) for $m\ne 0$, where the non-commutative parameter has no effect on the ground state of the system. Another contribution to splitting energy levels has been studied with the Rashba coupling term in the hydrogen potential in 2D. After diagonalization of the corresponding Hamiltonian, by the Baker-Hausdroff lemma, we have shown the energy levels' degeneracy would be removed for spin up and down of the particle in both non-commutative approaches. It means that there are double splitting levels in the 2D hydrogen atom spectrum due to Rashba coupling and the non-commutative parameter by introducing quantum numbers {n, m, s}. Therefore, we have shown that the symmetries of the hydrogen atom can be removed in the corresponding degeneracy by considering non-commutative and Rashba coupling. We have estimated the upper bounds on non-commutative parameters of orders $\theta \lesssim {(10\mathrm{MeV})}^{-2}$ and $\bar{\theta \,}\,\lesssim {({10}^{-2}\mathrm{MeV})}^{2}$ for coordinates and momenta versions, receptively. Furthermore, we have expressed a possible deviation of the Lorentz symmetry which occurs at high energy in a non-relativistic 2D hydrogen model. It is clear that the correction energy spectrum of a 2D hydrogen atom in both approaches is related to the Lorentz violation effects. It comes from considering both Rashba coupling and non-commutative space influences. Finally, we have obtained an expression for the Lorentz violation with non-commutative parameters and Rashba coupling.
Acknowledgments
S Aghababaei would like to thank the National Elites Foundation (INEF), Iran for financial support under research project No. 15/6072.
,∗, G RezaeiDepartment of Physics, College of Sciences, 
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