Exact Noncommutative Two-Dimensional Hydrogen Atom
Beatriz Wang, Emanuel Brenag, Ronni Amorim, Vinicius Rispoli, Sergio Ulhoa
aa r X i v : . [ h e p - t h ] J a n Exact Noncommutative Two-DimensionalHydrogen Atom
B.C. Wang a , E.C. Brenag b R.G.G. Amorim a,c,d , V.C. Rispoli a , S.C. Ulhoa c,da Universidade de Brasília, Faculdade Gama,72444-240, Brasília, DF, Brazil b Universidade de Brasília, Faculdade de Tecnologia,70910-900, Brasília, DF, Brazil c International Centre for Condensed Matter Physics , Instituto de Física, Universidade de Brasília,70910-900, Brasília, DF, Brazil d Canadian Quantum Research Center,204-3002 32 Ave Vernon, BC V1T 2L7 CanadaJanuary 26, 2021
Abstract
In this work, we present an exact analysis of two-dimensional non-commutative hydrogen atom. In this study, it is used the Levi-Civitatransformation to perform the solution of the noncommutative Schrödingerequation for Coulomb potential. As an important result, we determinethe energy levels for the considered system. Using the result obtainedand experimental data, a bound on the noncommutativity parameterwas obtained.
The concept of noncommutativity in physical theories was formally intro-duced by Snyder in 1947 [1, 2, 3]. In a seminal paper, Snyder stated thatspatial coordinates would not commute with each other at small distances.1n this sense, a new paradigm was proposed in which the space-time shouldbe understood as a collection of tiny cells of minimum size, where thereis no such idea of a point. So far, once the minimum size is reached, inthe realm of some high energy phenomenon, the position should be givenby the noncommutative coordinate operators. As a direct consequence, itwould be impossible to precisely measure a particle position. Over the lastyears, the interest of the scientific community on noncommutative geometryhas increased due to works on non-abelian theories [4], gravitation [5, 6, 7],standard model [8, 9, 10] and quantum Hall effect [11]. More recently, thediscovery that the dynamics of an open string can be associated with non-commutative spaces has contributed to the last revival of noncommutativetheories [12].From the mathematical point of view, the simplest algebra obeyed by thecoordinates operators b x µ is [ b x µ , b x ν ] = i Θ µν , where Θ µν is a skew-symmetric constant tensor called noncomutativity pa-rameter. It is worth to point out that the mean values of the position oper-ators do correspond to the actual position observed, thus it is said that suchoperators are Hermitian ones. It is well known in quantum mechanics that anoncommutative relation between two operators lead to a specific uncertainrelation, hence the above expression yields ∆ b x µ ∆ b x ν ≥ | Θ µν | , in which implies a set of uncertainty relations between position coordinatesanalogous to the Heisenberg uncertainty principle. Following the ideas intro-duced by Snyder, it is possible to associate the minimum size with a distanceof the p | Θ µν | order of magnitude. Thus the noncommutative effects turn outto be relevant at such scales. Usually the noncommutativity is introducedby means of the Moyal product defined as [2] f ( x ) ⋆ g ( x ) ≡ exp (cid:18) i µν ∂∂x µ ∂∂y ν (cid:19) f ( x ) g ( y ) | y → x , (1)with a constant Θ µν . Then, the usual product is replaced by the Moyal prod-uct in the classical Lagrangian density. In similar perspective, the noncom-mutative quantum mechanics is introduced by imposing further commutationrelations between the position coordinates themselves.2n this perspective, we aim to apply the ideas about noncommutativityof space in the hydrogen atom. The hydrogen atom is an electrically neu-tral atom with a positively charged proton and a negatively charged electronbounded to the nucleus. This system plays a significant role in quantummechanics and field theory. There are many good reasons to address thehydrogen atom [14]. As an example, the hydrogen atom with high preci-sion measurements in atomic transitions is one of the best laboratories totest quantum electrodynamics theory [15]. Other applications of the hydro-gen atom appear in many occasions, such as to examine the constancy offine structure constant over a cosmological time scale [16]. In this paper,we analyze the two-dimensional noncommutative hydrogen atom. A two-dimensional hydrogen atom can be defined as a system in which the motionof the electron around the proton is constrained to be planar. Then, in thiswork, we consider that this plane is noncommutative. As a practical example,a semiconductor quantum-well under illumination is a quasi-two-dimensionalsystem, in which photoexcited electrons and holes are essentially confined toa plane [17, 18]. There are many works that consider the hydrogen atom ina noncommutative context, but they present disagreement in results. Ourmethod presents an approach using the Levi-Civita mapping, which allowsan exact treatment.This paper is organized as follows. In section 2, we present the mathe-matical framework of the two-dimensional hydrogen atom. The Levi-Civitamapping is presented in section 3. In section 4, we obtain the solution andspectrum for the noncommutative hydrogen atom. Finally, in the section 5,we present our concluding remarks. The Hamiltonian that define two-dimensional hydrogen atom is given by H = 12 m ( p x + p y ) − k ( x + y ) / , (2)where p x and p y stands for the momentum of the electron in directions x and y , respectively; x and y represents the electron coordinates and theconstant k = e πǫ o , where e is the elemental electrical charge and ǫ o is the3acuum electrical permissivity. To quantize the Hamiltonian given in Eq.(2),as usual, the momentum operators are given by b p x = − i ¯ h ∂∂x and b p y = − i ¯ h ∂∂y ,where ¯ h = h/ π and h is the Planck constant.In the noncommutative perspective, we define the following positions op-erators b x = x + i Θ2 ∂∂y , (3) b y = y − i Θ2 ∂∂x , (4)in which Θ is the noncommutativity parameter in Cartesian coordinates. Wenotice that [ b x, b y ] = i Θ , (5)as expected.However, the treatment of the Hamiltonian given in Eq.(2) is difficultbecause of the operators in the denominator of the potential energy term.For this reason, in the next section we present a transformation that put thesystem in a more suitable way. The Levi-Civita (also known as Bohlin) transformation is a parabolic co-ordinates mapping that is capable to convert the planar Coulomb problemin a two-dimensional harmonic oscillator [19, 20, 21, 22]. It is a R → R surjection defined by x = u − v ,y = 2 uv. (6)Given Eqs.(6) it is immediate to conclude that ∂∂u = 2 (cid:18) u ∂∂x + v ∂∂y (cid:19) ,∂∂v = 2 (cid:18) − v ∂∂x + u ∂∂y (cid:19) , (7)4nd, by inversion, ∂∂x = 12( u + v ) (cid:18) u ∂∂u − v ∂∂v (cid:19) ,∂∂y = 12( u + v ) (cid:18) v ∂∂u + u ∂∂v (cid:19) . (8)As a direct consequence of Eq.(8), the momentum operators can be rewrittenin this new coordinate system as p x = p u u − p v v u + v ) p y = p u v + p v u u + v ) . (9)It should be noted that the Levi-Civita mapping is a canonical transformation[21].Applying Eqs.(6) and Eqs.(9) in Eq.(2), we obtain the following trans-formed Hamiltonian H = 12 m (cid:20) p u + p v u + v ) (cid:21) − k ( u + v ) . (10)Finally, the hypersurface defined by H = E is given by m ( p u + p v ) − E ( u + v ) = 4 k. (11)Equation (11) is the main result of this section and is the one to be usedfrom now on. Applying the following set of operators in Eq.(11) b u = u + iθ ∂∂v , b v = v − iθ ∂∂u , b p u = − i ¯ h ∂∂u , b p v = − i ¯ h ∂∂v ,
5e obtain the modified Schrödinger equation as − (cid:18) ¯ h m − Eθ (cid:19) (cid:18) ∂ ψ∂u + ∂ ψ∂v (cid:19) − E (cid:20) ( u + v ) ψ − iθ (cid:18) v ∂ψ∂u − u ∂ψ∂v (cid:19)(cid:21) = 4 kψ, (12)where ψ ≡ ψ ( u, v ) , θ is the noncommutativity parameter in parabolic coor-dinates, [ b u, b v ] = iθ , [ b u, b p u ] = i ¯ h and [ b v, b p v ] = i ¯ h . Here is crucial to note thatthe order of θ is θ ∼ √ Θ , due to Eqs.(6).The solution of Eq.(12) can be obtained from the following change ofvariables z = u + v and then Eq.(12) can be rewritten as z d ψ ( z ) dz + dψ ( z ) dz + 1 β ( Ez + k ) ψ ( z ) = 0 , (13)with β = (cid:16) ¯ h m − Eθ (cid:17) . Defining ψ ( z ) = e − λz φ ( z ) , where λ = q − Eβ , Eq.(13)can be written as z d φ ( z ) dz + (1 − λz ) dφ ( z ) dz + (cid:18) kβ − λ (cid:19) φ ( z ) = 0 . (14)Performing the change of variable w = 2 λz , we finally obtain w d φ ( w ) dw + (1 − w ) dφ ( w ) dw + 12 λ (cid:18) kβ − λ (cid:19) φ ( w ) = 0 . (15)Note that Eq.(15) has the following form wφ ′′ + (1 − w ) φ ′ + lφ = 0 , which is the Laguerre differential equation. If l is an integer l = 0 , , , , . . . the solution of Laguerre’s equation is given by Laguerre polynomials L l ( x ) .In this sense, we obtain the solution ψ ( u, v ) = e − λ ( u + v ) L l (2 λ ( u + v )) . (16)The energy levels can be determined from l = 12 (cid:18) kλβ − (cid:19) , (17)6ith l = 0 , , , , . . . . Using the condition given in Eq.(17), the spectrumcan be calculated as E − ¯ h mθ E − k n θ = 0 , (18)where n = 2 l +1 . Solving Eq.(18) we obtain the spectrum of noncommutativetwo-dimensional hydrogen atom E n = ¯ h mθ − ¯ h mθ r k m θ ¯ h n . (19)Considering k m θ ¯ h n ≪ and using the binomial series (1 + x ) j = 1 + jx + j ( j − x / . . . we calculate the following approximation E n ≈ − me π ǫ o ¯ h n + e m θ π ǫ o ¯ h n . (20)Notice that in the limit θ → , we obtain the same result of the usual two-dimensional hydrogen atom given in the literature [15, 16]. Notice also thatthe first order term in θ do not contribute to energy of this system.Then, the noncommutative correction, ∆ E NC , for the energy is given by ∆ E NC ≈ e m θ π ǫ o ¯ h n . (21)The result given in Eq.(21) can be used to estimate the bound on the non-commutativity parameter θ . The experimental value for S → S frequencytransition in the hydrogen atom is ν S → S = (2922743278671 . ± . kHz [23]. The uncertainty in this experimental value ∆ ν = 0 . kHz can fix thelower bound on the parameter θ . In this sense, the theoretical value for theerror in transition S → S , denoted by ∆ E → , is given by ∆ E → ≈ m e θ π ǫ o ¯ h (cid:20) (cid:21) , Using the fact that in the bidimensional case the energy is four times biggerthan tridimensional case, then ∆ E → = hν S → S , where h is Planck constant.So, we have θ . (cid:20) π ǫ o ¯ h ∆ νm e (cid:21) / . θ . . · − m . In this case, thebound of the noncommutativity parameter Θ is Θ . . · − m . Usingthe definition of the length scale factor, Γ NC = p | Θ | , that is the length scalewhere the noncommutative effects of the space will be relevant, we found forthe considered case Γ NC . . · − m , which is about one thousand timessmaller than the proton radius r p ≈ , · − m . Using the Levi-Civita mapping, we treated the non-trivial problem of thenoncommutative hydrogen atom. As results, we obtained the solution ofSchrödinger equation for this system and calculate the energy levels. Us-ing the spectrum obtained and the experimental data, we estimated thenoncommutativity parameter Θ , which has the order of magnitude about − m , and the noncommutative effects will be relevant to lengths smallerthan − m . These results are in agreement with the literature [23, 24]. Acknowledgements
This work was partially supported by CNPq of Brazil.
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