The relativistic Aharonov-Bohm-Coulomb system with position-dependent mass
R. R. S. Oliveira, A. A. Araujo Filho, R. V. maluf, C. A. S. Almeida
aa r X i v : . [ qu a n t - ph ] D ec The relativistic Aharonov-Bohm-Coulomb system withposition-dependent mass
R. R. S Oliveira, ∗ A. A. Ara´ujo Filho, † R. V. Maluf, ‡ and C. A. S. Almeida § Universidade Federal do Cear´a (UFC), Departamento de F´ısica,Campus do Pici, Fortaleza - CE, C.P. 6030, 60455-760 - Brazil. (Dated: December 20, 2018)
Abstract
In this work, we study the Aharonov-Bohm-Coulomb (ABC) system for a relativistic Diracparticle with position-dependent mass (PDM). To solve our system, we use the lef t - handed and right - handed projection operators. Next, we explicitly obtain the eigenfunctions and the energyspectrum of the particle. We verify that these eigenfunctions are written in terms of the generalizedLaguerre polynomials and the energy spectrum depends on the parameters Z , Φ AB and κ . Wenotice that the parameter κ has the function of increasing the values of the energy levels of thesystem. In addition, the usual ABC system is recovered when one considers the limit of constantmass ( κ → Z = Φ AB = 0),the particle with PDM still has a discrete energy spectrum. ∗ Electronic address: rubensrso@fisica.ufc.br † Electronic address: dilto@fisica.ufc.br ‡ Electronic address: r.v.maluf@fisica.ufc.br § Electronic address: carlos@fisica.ufc.br . INTRODUCTION The in literature, there a significant number of works that investigate the dynamics ofparticles with constant mass interacting with the vector potential of the Aharonov-Bohm(AB) effect [1–4] and the 2D Coulomb potential [5–8]. A system described by the combina-tion of these two potentials is so-called Aharonov-Bohm-Coulomb (ABC) system [9–12]. Inparticular, the ABC system for spin-1/2 relativistic particles is studied in connection withthe Feynman path integrals [14, 15], scattering [16], magnetic monopole [17, 18], sponta-neous creation of fermions pairs [19], Coulomb impurity [20], etc. Recently, the ABC systemwas applied in a graphene ring [21] and studied together with the Dirac oscillator [22].Physical systems with effective mass, in special, with position-dependent mass (PDM)are particular interest in theoretical and experimental physics. For instance, using theSchr¨odinger equation (SE) with PDM, we can investigate the electronic properties of semi-conductors [23], quantum well and quantum dots [24], He clusters [25], quantum liquids[26], etc. However, the relativistic extension this formalism it has the advantage of themto eliminate the problem of the ordering ambiguity between the mass and the momentumoperator in the SE [27, 28]. In particular, using the Dirac equation (DE) with PDM, wecan study problems involving scattering [29], Coulomb field [30, 31], spin and pseudo-spinsymmetry [32], solid state physics [33], supersymmetry [34], PT-symmetry [35–37], infinitesquare well [38], generalized uncertainty principle [39], etc.In this work, we investigate the relativistic quantum dynamics of an electrically chargedDirac particle with PDM in an ABC system in the (2+1)-dimensional Minkowski spacetime.To solve exact our problem, we use the lef t - handed and right - handed projection operators.Yet, we assume that the PDM be relevant for distances of the order of magnitude of theCompton wavelength λ , otherwise, we obtain the rest mass m of the particle in the limit ρ → ∞ or λ →
0, being ρ the radial coordinate.This paper is organized as follows. In Section II, we introduce the DE in polar coor-dinates for an electrically charged particle with PDM in an ABC system. Next, we applythe lef t - handed and right - handed projection operators in the DE and we obtain a secondorder differential equation. In Section III, we determine the eigenfunctions and the energyspectrum for the bound-states of the particle. In Section IV, we present our conclusions.2 I. THE DIRAC EQUATION WITH POSITION-DEPENDENT MASS IN ANAHARONOV-BOHM-COULOMB SYSTEM
The (2 + 1)-dimensional DE that governs the dynamics of an electrically charged particlewith PDM in the presence of an external electromagnetic field A µ reads as follows (Gaussiansystem in natural units ~ = c = 1) [40][ γ µ Π µ − m ( r )]Ψ( t, r ) = 0 , ( µ = 0 , , , (1)where γ µ = ( γ , γ ) are the gamma matrices, Π µ = p µ − qA µ is the kinetic momentumoperator, being p µ = i∂ µ the momentum operator, q < t, r ) is the two-component Dirac spinor.Now, we introduce the following lef t - handed and right - handed projection operators [41] P L = 12 ( I × − γ ) , P R = 12 ( I × + γ ) , (2)which satisfy the properties P L = P L , P R = P R , { P L , P R } = 0 , P L + P R = I × and P R γ µ = γ µ P L , where γ = γ = σ . Then, applying P L in Eq. (1) and defining the lef t - handed and right - handed spinors as Ψ L ( t, r ) = P L Ψ( t, r ) and Ψ R ( t, r ) = P R Ψ( t, r ), we getΨ L ( t, r ) = 1 m ( r ) γ µ Π µ Ψ R ( t, r ) . (3)The above relation allows us to write the original Dirac spinor in the formΨ( t, r ) = 1 m ( r ) [ γ µ Π µ + m ( r )]Ψ R ( t, r ) , (4)where we used Ψ( t, r ) = Ψ L ( t, r ) + Ψ R ( t, r ).Substituting the spinor (4) into Eq. (1), we obtain[ γ µ Π µ − m ( r )][ γ µ Π µ + m ( r )]Ψ R ( t, r ) = 0 . (5)Adopting now the polar coordinates system ( t, ρ, θ ) where the metric tensor is given by g µν =diag(1 , − , − ρ ) [42], being ρ = p x + y > ≤ θ ≤ π the azimutal coordinate, Eq. (5) becomes A − A + Ψ R ( t, ρ, θ ) = 0 , (6)where the operators A ∓ are defined in the form A ∓ = (cid:20) γ (cid:18) i ∂∂t − qA (cid:19) + iγ ρ ∂∂ρ + γ θ (cid:18) iρ ∂∂θ + qA θ (cid:19) ∓ m ( ρ ) (cid:21) , (7)3ith γ = β , γ ρ = γ · ˆ e ρ = γ cos θ + γ sin θ and γ θ = γ · ˆ e θ = − γ sin θ + γ cos θ .Here, we are explicitly assuming that the radial component of the vector potential is null( A ρ = 0). Also, through a similarity transformation given by unitary operator U ( θ ) = e − iθσ ,we can reduce the matrices γ ρ and γ θ to the matrices γ and γ as follows [42] U − ( θ ) γ ρ U ( θ ) = γ , U − ( θ ) γ θ U ( θ ) = γ . (8)Since we are working in a (2 + 1)-dimensional Minkowski spacetime, it is convenientdefine the Dirac matrices γ = ( γ , γ ) = ( − γ , − γ ) and γ in terms of the Pauli matrices,i.e., γ = σ σ , γ = σ σ and γ = σ [40, 42]. Therefore, using this information and therelations (8), we rewrite Eq. (6) in the form B − B + ψ R ( t, ρ, θ ) = 0 , (9)where B ∓ = (cid:20) σ (cid:18) i ∂∂t − qA (cid:19) + σ ∂∂ρ + iσ (cid:18) iρ ∂∂θ + qA θ + σ ρ (cid:19) ∓ m ( ρ ) (cid:21) , (10) ψ R ( t, ρ, θ ) = U − ( θ )Ψ R ( t, ρ, θ ) . (11)Let us now consider the configurations of the vector potential of the AB effect and of the2D Coulomb potential. Explicitly, these configurations are given in the form [9–12, 21] A = A θ ˆ e θ = Φ2 πρ ˆ e θ , ( ρ > a ) , (12) V = qA = − Ze ρ , ( q = − e ) , (13)where Φ = πa B = const is the magnetic flux in the region intern of a solenoid of radius a electrically charged with a total charge Ze >
0, being Z the atomic number.With respect to the configuration of the variable mass, we consider the following PDM m ( ρ ) = m + κρ , (14)where m is the rest mass of the particle and κ > κ can be defined inthe form κ = m µλ , where λ is the Compton wavelength of the particle and µ is a realscale parameter with length inverse dimension. However, as the PDM (14) diverges at the4rigin, the parameter ν can be interpreted as a renormalization scale in quantum field theory(QFT) to eliminate the ultraviolet divergences that appear in high energy physics [30].Therefore, using the configurations (12), (13) and (14), we transform Eq. (9) as follows (cid:20) ∂ ∂ρ + 1 ρ ∂∂ρ − ρ − Γ ( Γ − ρ + ∆ ρ + Σ (cid:21) ψ R ( t, ρ, θ ) = 0 , (15)where we define the following operators Γ ≡ "(cid:18) L z + e Φ2 π (cid:19) − Z e + κ , Γ ≡ (cid:0) iL z σ σ + Ze σ σ − κσ + ie Φ AB σ σ (cid:1) , (16) ∆ ≡ (cid:18) iZe ∂∂t − m κ (cid:19) , Σ ≡ (cid:18) − ∂ ∂t − m (cid:19) , L z = − i ∂∂θ . (17)Writing the two-component Dirac spinor in the form [42] ψ R ( t, ρ, θ ) = e i ( m l θ − Et ) √ π φ + ( ρ ) φ − ( ρ ) , ( m l = ± / , ± / , . . . ) , (18)Eq. (15) becomes compacted in the following differential equation (cid:20) d dρ + 1 ρ ddρ − γ s ρ + (2 ZαE − m κ ) ρ + E − m (cid:21) φ s ( ρ ) = 0 , ( s = ± , (19)where γ s ≡ p ( m l + Φ AB ) − Z α + κ − s , (20)being φ s ( ρ ) real radial functions, E is the relativistic total energy of the particle, m l is theorbital magnetic quantum number, Φ AB = ΦΦ > = πe the magnetic flux quantum and α = e ∼ = is the Sommerfeld fine structure constant. III. BOUND-STATE SOLUTIONS AND ENERGY SPECTRUM
In order to solve Eq. (20), we will introduce now a new dimensionless variable given by z = 2 ηρ , where η = p m − E , being m > E . Thereby, Eq. (20) becomes (cid:20) d dz + 1 z ddz − γ s z + z z − (cid:21) φ s ( z ) = 0 , (21)where z ≡ ( ZαE − m κ ) η . (22)5nalyzing the asymptotic behavior of Eq. (21) for z → z → ∞ , we obtain φ s ( z ) = C s z | γ s | e − z R s ( z ) , (23)where C s are normalization constants and R s ( z ) are unknown functions to be determined.In this way, substituting (23) into Eq. (21), we have z d R s ( z ) dz + (2 | γ s | + 1 − z ) dR s ( z ) dz − (cid:18) | γ s | + 12 − z (cid:19) R s ( z ) = 0 . (24)It is not difficult to note that Eq. (24) has the form of a generalized Laguerre equation,whose solution are the generalized Laguerre polynomials R s ( z ) = L | γ s | n ( z ) [43]. Besides that,to φ s ( z ) be a normalizable solution, we must impose that the parameter | γ s | + − z to beequal to a non-positive integer number − n ( n = 0 , , , . . . ). Therefore, using this conditionand the relation (22), we obtain the following energy spectrum for the Dirac particle withPDM in an ABC system E + n,m l = Zαm κ [( n s + γ ) + ( Zα ) ] + m s ( Zακ ) [( n s + γ ) + ( Zα ) ] + ( n s + γ ) − κ [( n s + γ ) + ( Zα ) ] , (25)where n s = n + − s is a quantum number and γ ≡ p ( m l + Φ AB ) − Z α + κ >
0. Wesee that the energy spectrum (25) explicitly depends on the parameters Z and Φ AB thatcharacterize the ABC system and of the parameter κ that characterizes the PDM. Due tothe presence of the term s = ±
1, we see that the upper component of the Dirac spinorhas energy eigenvalues slightly larger than the lower component. In addition, the negativesignal in (25) was excluded because for an positively charged solenoid (
Zα >
E < z = n + | γ s | + > κ → κ = 0 or κ = 0,the energy spectrum of the ABC system still it has degeneracy finite. Also interestingto note that even in the absence of the ABC system ( Z = Φ AB = 0), the Dirac particlewith PDM still has a discrete energy spectrum. In this sense, we can interpret the termof the PDM that varies spatially as a type of scalar coupling in the DE, where we havethe Lorentz-scalar potential V s ( ρ ) = κρ , and whose energy spectrum is given as follows: E n,m l = ± m q − κ ( n s + p m l + κ ) − . 6ow, let us concentrate on the form of the original Dirac spinor. Substituting the variable z = 2 ηρ in the radial function (23), the spinor (18) becomes ψ R ( t, ρ, θ ) = e i ( m l θ − Et ) ¯ C + ρ | γ + | e − ηρ L | γ + | n (2 ηρ )¯ C − ρ | γ − | e − ηρ L | γ − | n (2 ηρ ) , (26)where ¯ C s ≡ C s (2 η ) | γ s | √ π . (27)Now, substituting the spinor (11) in the spinor (4) and using the relations (9), we obtainΨ = 1 m ( ρ ) U (cid:20) σ (cid:18) i ∂∂t + Ze ρ (cid:19) + m ( ρ ) + σ (cid:18) ∂∂ρ + 12 ρ (cid:19) + iσ (cid:18) iρ ∂∂θ − Φ AB (cid:19)(cid:21) ψ R . (28)Therefore, substituting the spinor (26) in (28), the original Dirac spinor is written in theform Ψ n,m l ( t, ρ, θ ) = e i [( m l − ) θ − Et ] [ F + ( ρ ) + iG − ( ρ )] e i [( m l + ) θ − Et ] [ F − ( ρ ) − iG + ( ρ )] , (29)where F s ( ρ ) = ¯ C s m ( ρ ) e − ηρ ρ | γ s | L | γ s | n (2 ηρ ) (cid:18) E + m + ( κ − sZe ) ρ (cid:19) , (30) G s ( ρ ) = ¯ C s m ( ρ ) e − ηρ ρ | γ s | (cid:20) L | γ s | n (2 ηρ ) (cid:18) η + ( sm l − − | γ s | + s Φ AB ) ρ (cid:19) + L | γ s | +1 n − (2 ηρ ) (cid:21) . (31) IV. CONCLUSION
In this paper, we study the (2+1)-dimensional DE for an electrically charged particle withPDM in an ABC system. Next, we applied the lef t - handed and right - handed projectionoperators in the DE and we obtain a second order differential equation. After analyzingthe asymptotic behavior this differential equation for z → z → ∞ , we obtain ageneralized Laguerre equation. As results, we observed that the energy spectrum of theparticle explicitly depends on the parameters Z and Φ AB that characterize the ABC systemand of the parameter κ that characterizes the PDM. We verify that the parameter κ hasthe finality of increasing the values of the energy levels of the system. We observed alsothat in the limit of the constant mass ( κ → Z = Φ AB = 0), the Dirac particle with PDM still has a discrete energy spectrum. In viewof this, we can interpret the PDM as a type of scalar coupling in the DE.7 cknowledgments The authors would like to thank the Funda¸c˜ao Cearense de apoio ao DesenvolvimentoCient´ıfico e Tecnol´ogico (FUNCAP) the Coordena¸c˜ao de Aperfei¸coamento de Pessoal deN´ıvel Superior (CAPES), and the Conselho Nacional de Desenvolvimento Cient´ıfico e Tec-nol´ogico (CNPq) for financial support. [1] Y. Aharonov, D. Bohm, Phys. Rev. , 485 (1959).[2] A. Tonomura, N. Osakabe, T. Matsuda, T. Kawasaki, J. Endo, S. Yano, H. Yamada, Phys.Rev. Lett. , 792 (1986).[3] P. Recher, B. Trauzettel, A. Rycerz, Y. M. Blanter, C. W. J. Beenakker, A. F. Morpurgo,Phys. Rev. B , 235404 (2007).[4] C. R. Hagen, Phys. Rev. Lett. , 2347 (1990).[5] A. C. Neto, F. Guinea, N. M. Peres, K. S. Novoselov, A. K. Geim, Rev. Mod. Phys. , 109(2009); V. M. Pereira, V. N. Kotov, A. C. Neto, Phys. Rev. B , 085101 (2008).[6] V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, A. C. Neto, Rev. Mod. Phys. , 1067(2012).[7] U. W. Rathe, C. H. Keitel, M. Protopapas, P. L. Knight, J. Phys. B , L531 (1997).[8] S. H. Dong, M. Lozada-Cassou, Phys. Lett. A , 168-172 (2004); S. H. Dong, Phys. Scr. , 89 (2003); S. H. Dong, G. H. Sun, Phys. Scr. , 161 (2004).[9] H. T. T. Nguyen, P. A. Meleshenko, A. V. Dolgikh, A. F. Klinskikh, Eur. Phys. J. D , 361(2011).[10] G. E. Dr˘ag˘anascu, C. Campigotto, M. Kibler, Phys. Lett. A , 339 (1992).[11] H. D. Doebner, E. Papp, Phys. Lett. A , 423 (1990).[12] C. R. Hagen, Phys. Rev. D , 5935 (1993); C. R. Hagen, D. K. Park, Ann. Phys. , 45(1996); C. R. Hagen, Phys. Rev. A , 036101 (2008).[13] E. Jung, M. R. Hwang, C. S. Park, D. Park, J. Phys. A: Math. Theor. , 055301 (2012).[14] D. H. Lin, J. Math. Phys. , 1246 (1999).[15] J. Bornales, C. C. Bernido, M. V. Carpio-Bernido, Phys. Lett. A , 447 (1999).[16] Q. G. Lin, J. Phys. A: Math. Gen. , 5049 (2000).
17] V. M. Villalba, J. Math. Phys. , 3332 (1995).[18] L. X. Hai, L. I. Komarov, T. S. Romanova, J. Phys. A: Math. Gen. , 6461 (1992).[19] V. R. Khalilov, Theor. Math. Phys. , 210 (2009); V. R. Khalilov, Eur. Phys. J. C , 2548(2013); V. R. Khalilov, I. V. Mamsurov, Mod. Phys. Lett. A , 1650032 (2016).[20] Y. Nishida, Phys. Rev. B , 085430 (2016).[21] E. Jung, M. R. Hwang, C. S. Park, D. Park, J. Phys A: Math. Theor. , 055301 (2012).[22] R. R. S. Oliveira, R. V. Maluf, C. A. S. Almeida, Ann. Phys. , 1 (2018).[23] O. Krebs, P. Voisin, Phys. Rev. Lett. , 1829 (1996).[24] P. Harrison, Quantum Wells, Wires and Dots (John Wiley and Sons, 2000); L. Serra and E.Lipparini, Europhys. Lett. , 667 (1997)[25] M. Barranco, M. Pi, S. M. Gatica, E. S. Hernandez, J. Navarro, Phys. Rev. B , 8997 (1997)[26] F. A. Saavedra, J. Boronat, A. Polls, A. Fabrocini, Phys. Rev. B , 4248 (1994).[27] A. R. Plastino, A. Rigo, M. Casas, F. Garcias, A. Plastino, Phys. Rev. A , 4318 (1999).[28] A. de Souza Dutra, C. A. S. Almeida, Phys. Lett. A , 25 (2000).[29] A. Alhaidari, H. Bahlouli, A. Al-Hasan, M. Abdelmonem, Phys. Rev. A , 062711 (2007).[30] A. Alhaidari, Phys. Lett. A , 72 (2004).[31] I. O. Vakarchuk, J. Phys. A: Math. Gen. , 4727 (2005).[32] S. M. Ikhdair, R. Sever, Appl. Math. Comput. , 545 (2010).[33] R. Renan, M. Pacheco, C. A. S. Almeida, J. Phys. A , L509 (2000).[34] C. L. Ho, P. Roy, Ann. Phys. , 161 (2004).[35] C. S. Jia, A. S. Dutra, Ann. Phys. , 566 (2008).[36] L. B. Castro, Phys. Lett. A , 2510 (2011).[37] O. Mustafa, S. H. Mazharimousavi, Int. J. Theor. Phys. , 1112 (2008).[38] P. Alberto, C. Fiolhais, V. M. S. Gil, Eur. J. Phys. , 19 (1996); P. Alberto, S. Das, E. C.Vagenas, Phys. Lett. A , 1436-1440 (2011).[39] P. Pedram, Phys. Lett. B , 295-298 (2011).[40] W. Greiner, Relativistic Quantum Mechanics: Wave Equations , Springer, Berlin, (1997).[41] P. R. Auvil, L. M. Brown, Am. J. Phys. , 679 (1978).[42] V. M. Villalba, A. R. Maggiolo, Eur. Phys. J. B , 31 (2001).[43] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions , Dover Publications Inc.,New York, (1965)., Dover Publications Inc.,New York, (1965).