Algebraic Approach to Exact Solution of the (2 + 1...
7
Research Article Algebraic Approach to Exact Solution of the (2 + 1)-Dimensional Dirac Oscillator in the Noncommutative Phase Space H. Panahi and A. Savadi Department of Physics, University of Guilan, Rasht 41635-1914, Iran Correspondence should be addressed to H. Panahi; [email protected] Received 28 April 2017; Revised 6 August 2017; Accepted 11 September 2017; Published 17 October 2017 Academic Editor: Elias C. Vagenas Copyright © 2017 H. Panahi and A. Savadi. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e publication of this article was funded by SCOAP 3 . We study the (2 + 1)-dimensional Dirac oscillator in the noncommutative phase space and the energy eigenvalues and the corresponding wave functions of the system are obtained through the sl(2) algebraization. It is shown that the results are in good agreement with those obtained previously via a different method. 1. Introduction Study on Dirac oscillator as an important potential has attracted a lot of attention and has found many physical applications in various branches of physics [1–6]. e Dirac oscillator was introduced for the first time by Itˆ o et al. [7], in which the momentum → in Dirac equation is replaced by → − 0 → , where → is the position vector and 0 , , and are the mass of particle, the frequency of the oscillator, and the usual Dirac matrices, respectively. Similar system was studied by Moshinsky and Szczepaniak [8], who gave it the name of Dirac oscillator; due to the nonrelativistic limit, it becomes a simple harmonic oscillator with strong spin-orbit coupling term. In quantum optics and for (2 + 1)-dimension space, it is seen that the Dirac oscillator system can be mapped into Anti-Jaynes-Cummings model [9–11] which describes the atomic transitions in a two-level system. In [5, 12], it has been shown that the Dirac oscillator interaction can be interpreted as the interaction of the anomalous magnetic moment with a linear electric field. Also the electromagnetic potential associated with the Dirac oscillator interaction has been found by Benitez et al. in [13]. On the other hand in recent decades, the noncommuta- tivity has become an extremely active area of research such as in string theories, quantum field theories, and quantum mechanics [14–24]. For quantum systems in noncommutative space, it is seen that assuming the noncommutativity may be a result of quantum gravity effects, also it is a fruitful theoretical laboratory where we can get some insight on the consequences of noncommutativity in field theory by using standard calculation techniques of quantum mechanics. e study on Dirac equation in noncommutative phase space has also attracted much attention in recent years [25–31]. For example, study on the relativistic Landau levels of Dirac equation in (2 + 1)-dimensional noncommutative phase space has been shown in [28] and one can see an exact mapping of this relativistic model to the AJC model. Also in [25], it is seen that, for Landau problem in two-dimensional noncommutative phase space, the equation of motion of a harmonic oscillator is similar to the equation of motion of a particle in a constant magnetic field and in the lowest Landau level. e energy gap of Dirac oscillator in a non- commutative phase space changes by noncommutative effect [25]. A noncommutative description of graphene, which consists of a Dirac equation for massless Dirac fermions plus noncommutative corrections, has been studied in [27] and it has been shown that the momentum noncommutativity affects the energy levels of grapheme. Since the appearance of quantum mechanics, consider- able efforts have been devoted to obtaining exact solutions of the relativistic and nonrelativistic wave equations, using different methods and techniques [32–36]. Now in this paper, we study the Dirac oscillator in noncommutative phase space Hindawi Advances in High Energy Physics Volume 2017, Article ID 1723567, 6 pages https://doi.org/10.1155/2017/1723567
Transcript of Algebraic Approach to Exact Solution of the (2 + 1...
Research ArticleAlgebraic Approach to Exact Solution of the (2 + 1)-DimensionalDirac Oscillator in the Noncommutative Phase Space
H Panahi and A Savadi
Department of Physics University of Guilan Rasht 41635-1914 Iran
Correspondence should be addressed to H Panahi t-panahiguilanacir
Received 28 April 2017 Revised 6 August 2017 Accepted 11 September 2017 Published 17 October 2017
Academic Editor Elias C Vagenas
Copyright copy 2017 H Panahi and A Savadi This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited The publication of this article was funded by SCOAP3
We study the (2 + 1)-dimensional Dirac oscillator in the noncommutative phase space and the energy eigenvalues and thecorresponding wave functions of the system are obtained through the sl(2) algebraization It is shown that the results are in goodagreement with those obtained previously via a different method
1 Introduction
Study on Dirac oscillator as an important potential hasattracted a lot of attention and has found many physicalapplications in various branches of physics [1ndash6] The Diracoscillator was introduced for the first time by Ito et al [7]in which the momentum 997888rarr119901 in Dirac equation is replaced by997888rarr119901 minus1198941198980120596120573997888rarr119903 where997888rarr119903 is the position vector and1198980120596 and 120573are themass of particle the frequency of the oscillator and theusual Diracmatrices respectively Similar systemwas studiedby Moshinsky and Szczepaniak [8] who gave it the name ofDirac oscillator due to the nonrelativistic limit it becomesa simple harmonic oscillator with strong spin-orbit couplingterm In quantum optics and for (2 + 1)-dimension spaceit is seen that the Dirac oscillator system can be mappedinto Anti-Jaynes-Cummings model [9ndash11] which describesthe atomic transitions in a two-level system In [5 12] ithas been shown that the Dirac oscillator interaction can beinterpreted as the interaction of the anomalous magneticmoment with a linear electric field Also the electromagneticpotential associated with the Dirac oscillator interaction hasbeen found by Benitez et al in [13]
On the other hand in recent decades the noncommuta-tivity has become an extremely active area of research suchas in string theories quantum field theories and quantummechanics [14ndash24] For quantum systems in noncommutative
space it is seen that assuming the noncommutativity maybe a result of quantum gravity effects also it is a fruitfultheoretical laboratory where we can get some insight on theconsequences of noncommutativity in field theory by usingstandard calculation techniques of quantum mechanics Thestudy on Dirac equation in noncommutative phase spacehas also attracted much attention in recent years [25ndash31]For example study on the relativistic Landau levels of Diracequation in (2 + 1)-dimensional noncommutative phase spacehas been shown in [28] and one can see an exact mappingof this relativistic model to the AJC model Also in [25]it is seen that for Landau problem in two-dimensionalnoncommutative phase space the equation of motion of aharmonic oscillator is similar to the equation of motion ofa particle in a constant magnetic field and in the lowestLandau level The energy gap of Dirac oscillator in a non-commutative phase space changes by noncommutative effect[25] A noncommutative description of graphene whichconsists of a Dirac equation for massless Dirac fermions plusnoncommutative corrections has been studied in [27] andit has been shown that the momentum noncommutativityaffects the energy levels of grapheme
Since the appearance of quantum mechanics consider-able efforts have been devoted to obtaining exact solutionsof the relativistic and nonrelativistic wave equations usingdifferent methods and techniques [32ndash36] Now in this paperwe study the Dirac oscillator in noncommutative phase space
HindawiAdvances in High Energy PhysicsVolume 2017 Article ID 1723567 6 pageshttpsdoiorg10115520171723567
2 Advances in High Energy Physics
within the framework of representation theory of the sl(2) Liealgebra
A quantum system is exactly solvable (ES) if all theeigenvalues and the corresponding eigenfunctions can becalculated in an exact analytical manner In contrast aquantum system is quasi-exactly solvable (QES) if only a finitenumber of eigenvalues and eigenfunctions can be determinedexactly [37ndash40] In the case of ES models the Hamiltonianof the system can be diagonalized completely due to the factthat algebraic symmetry is complete but in the case of QESmodels the Hamiltonian is only block diagonalized Such afinite block can always be diagonalized which yields a finitepart of the spectrum algebraically
This paper is organized as follows In Section 2 based on[31] we briefly review the noncommutative phase space on (2+ 1)-dimensional Dirac oscillator Section 3 is devoted to 119904119897(2)algebraization of Dirac oscillator in the noncommutativephase space We obtain the energy eigenvalues and the cor-responding wave functions through the sl(2) representationand show that our results are in good agreement with theresults of [31] In Section 4 we present the conclusion
2 Review on (2 + 1)-DimensionalDirac Oscillator in the NoncommutativePhase Space
According to [31] the noncommutative phase space is char-acterized by the fact that their coordinate operators satisfy theequation
[119909(NC)119894 119909(NC)119895 ] = 119894120579119894119895[119901(NC)119894 119901(NC)119895 ] = 119894120579119894119895[119909(NC)119894 119901(NC)119895 ] = 119894ℏ120575119894119895(1)
where 120579119894119895 and 120579119894119895 are an antisymmetric tensor of space dimen-sion By using Bopprsquos shift method the two-dimensionalnoncommutative phase space can be considered as [29]
119909(NC) = 119909 minus 1205792ℏ119901119910119910(NC) = 119910 + 1205792ℏ119901119909119901(NC)119909 = 119901119909 + 1205792ℏ119910119901(NC)119910 = 119901119910 minus 1205792ℏ119909
(2)
TheDirac oscillator is also described by the following Hamil-tonian [8]
= 119888 sdot (997888rarr119901 minus 1198941198980120596120573997888rarr119903 ) + 12057311989801198882 (3)
where in two-dimensional noncommutative space it istransformed as119888120572119909 (119901(NC)119909 minus 1198941198980120596120573119909(NC))+ 119888120572119910 (119901(NC)119910 minus 1198941198980120596120573119910(NC)) + 12057311989801198882 120595119863= 119864NC120595119863
(4)
By considering the Pauli matrices as the representation ofDirac matrices in (2 + 1)-dimension that is
120572119909 = 120590119909 = (0 11 0) 120572119910 = 120590119910 = (0 minus119894119894 0 ) 120573 = (1 00 minus1)
(5)
and 120595119863 = (1205951 1205952)119879 (4) becomes
(11989801198882 119888119901minus119888119901+ minus11989801198882)(12059511205952) = 119864NC (12059511205952) (6)
where 119901minus 119901+ are as follows119901minus = 119901(NC)119909 minus 119894119901(NC)119910 + 1198941198980120596 (119909(NC) minus 119894119910(NC))= 1205881 (119901119909 minus 119894119901119910) + 11989411989801205961205882 (119909 minus 119894119910) 119901+ = 119901(NC)119909 + 119894119901(NC)119910 minus 1198941198980120596 (119909(NC) + 119894119910(NC))= 1205881 (119901119909 + 119894119901119910) minus 11989411989801205961205882 (119909 + 119894119910) (7)
with 1205881 = 1 + 11989801205962ℏ 1205791205882 = 1 + 12057921198980120596ℏ (8)
After some calculations it is easy to see that (6) gives1198882119901minus119901+ minus (119864NC2 minus 119898201198884) 1205951 = 0 (9)1198882119901+119901minus minus (119864NC2 minus 119898201198884) 1205952 = 0 (10)
Similar to [31 41] and by defining 119901119909 = 119901 cos 120579 119901119910 = 119901 sin 120579and 1199012 = 1199012119909 + 1199012119910 it is seen that the (7) transform into
119901minus = 119890minus119894120579 1205881119901 minus 120582( 120597120597119901 minus 119894119901 120597120597120579) 119901+ = 119890119894120579 1205881119901 + 120582( 120597120597119901 + 119894119901 120597120597120579) (11)
Advances in High Energy Physics 3
where
120582 = (1 + 12057921198980120596ℏ)1198980120596ℏ (12)
Substituting (11) into (9) and by considering 1205951(119901 120579) =119891(119901)119890119894119898120579 we have(1198892119891 (119901)1198891199012 + 1119901 119889119891 (119901)119889119901 minus 11989821199012 119891 (119901))+ (1205812 minus 11989621199012) 119891 (119901) = 0 (13)
where 1205812 = 21205821205881 (119898 + 1) + 1205761205822 1198962 = 120588211205822 (14)
with
120576 = 119864NC2 minus 1198982011988841198882 (15)
Here it is necessary to point out that in the literature there aremany different methods for solving quantum models exactly[32ndash36] Boumali and Hassanabadi [31] solved this problemanalytically and obtained the exact solutions in terms ofthe hypergeometric functions Within the present studywe intend to illustrate the exact solvability of the problemthrough the sl(2) algebraization [39] To do a comprehensiveanalysis we need to obtain the more general solution ofthe problem This is the main advantage of the presentmethod In fact by using the theory of representation spacewe construct the general matrix equation of the problemand calculate the closed-form expressions for energies andeigenfunctions Accordingly we can quickly obtain exactsolutions of any arbitrary state n without any cumbersomenumerical procedure or a complicated analytical one indetermining the solutions of the higher states It seems thatthe method is computationally much simpler than otheranalytical methods
Hence in the next section we solve (13) by Lie algebraicapproach related to representation theory of sl(2) Lie groupSimilar procedure can be also done on (10) for the 1205952 spinorwave function
3 Lie Algebraic Solution of the DiracOscillator in the NoncommutativePhase Space
In this section for solving (13) by the sl(2) Lie algebrarepresentation we first use similarity transformation as119891(119901) = 119901minus12119877(119901) such that the first-order derivative of thedifferential equation (13) can be eliminated and it is reducedto a Schrodinger-type operator as
( 11988921198891199012 + 14 minus 11989821199012 minus 1205812 minus 11989621199012)119877 (119901) = 0 (16)
Extracting the asymptotic behavior of thewavefunction at theorigin and infinity using the transformation119877 (119901) = 1199011205741198901205721199012119865 (119901) (17)
where 120572 lt 0 (to be physically meaningful) and 120574 areparameters to be determined later by the exact solvabilityconditions (see (26)) (16) is converted to1198892119865 (119901)1198891199012 + (2120574119901 + 4120572119901) 119889119865 (119901)119889119901 + ((41205722 minus 1198962) 1199012
+ 120574 (120574 minus 1) + 14 minus 11989821199012 + 4120572120574 + 2120572 + 1205812)119865 (119901)= 0
(18)
Using the change of variable 119903 = 1198961199012 we get11990311988921198651198891199032 + (2120572119896 119903 + 120574 + 12) 119889119865119889119903 + [(12057221198962 minus 14) 119903
+ 120574 (120574 minus 1) + 14 minus 11989824119903 + 120572120574119896 + 1205722119896 + 12058124119896]119865 = 0 (19)
According to [38] in one dimension the only Lie algebra ofthe first-order differential operators which possesses finite-dimensional representations is the sl(2) Lie algebra Hencewe consider the following realization of the operators
119869+119899 = 1199032 119889119889119903 minus 1198991199031198690119899 = 119903 119889119889119903 minus 1198992 119869minus119899 = 119889119889119903 (20)
which satisfy the sl(2) commutation relations[119869+119899 119869minus119899 ] = minus21198690119899 [119869plusmn119899 1198690119899] = ∓119869plusmn119899 (21)
and preserve the (119899 + 1)-dimensional linear space of polyno-mials with finite order119875119899+1 = ⟨1 119903 1199032 119903119899⟩ (22)
The most general second-order differential equation whichpreserves the space119875119899+1 can be also represented as a quadraticcombination of the sl(2) generators as119867 = sum
119886119887=0plusmn
119862119886119887119869119886119899119869119887119899 + sum119886=0plusmn
119862119886119869119886119899 + 119862 (23)
where 119862119886119887 119862119886 and 119862 are real parameters Substituting (20)into (23) yields the following differential form
119867 = 1198754 (119903) 11988921198891199032 + 1198753 (119903) 119889119889119903 + 1198752 (119903) (24)
4 Advances in High Energy Physics
where 119875119894(119903) are polynomials of degree 1198941198754 (119903) = 119862++1199034 + 119862+01199033 + 119862+minus1199032 + 1198620minus119903 + 119862minusminus1198753 (119903) = 119862++ (2 minus 2119899) 1199033 + (119862+ + 119862+0 (1 minus 31198992 )) 1199032+ (1198620 minus 119899119862+minus) 119903 + (119862minus minus 11989921198620minus)
1198752 (119903) = 119862++119899 (119899 minus 1) 1199032 + (11989922 119862+0 minus 119899119862+) 119903+ (119862 minus 11989921198620)
(25)
Comparing (24) with (19) gives119862++ = 119862+0 = 119862+minus = 119862minusminus = 119862+ = 0120572 = minus1198962 120574 = 119898 + 12 1198620minus = 11198620 = minus1
119862minus = 1198992 + 119898 + 1119862 = 12058124119896 minus 12 (119898 + 119899 + 1)
(26)
From (23) and (26) the Lie algebraic form of theHamiltonianis written as
119867 = 1198690119899119869minus119899 minus 1198690119899 + (1198992 + 119898 + 1) 119869minus119899+ (12058124119896 minus 12 (119898 + 119899 + 1)) (27)
which implies that this operator is ES due to the absenceof the terms of positive grading Therefore the operator 119867preserves the finite-dimensional representation space of thealgebra sl(2) as
119865 (119903) = 119899sum119895=0
119886119895119903119895 119899 = 0 1 2 (28)
Hence using (27) and (28) and doing some calculations thefollowing matrix equation can be obtained
((((((((((((((
12058124119896 minus 12 (119898 + 1) (119898 + 1) 0 sdot sdot sdot 00 12058124119896 minus 12 (119898 + 1) minus 1 2 + 2 (119898 + 1) sdot sdot sdot 00 0 d d
0 12058124119896 minus 12 (119898 + 1) minus (119899 minus 1) 119899 (119899 minus 1) + 119899 (119898 + 1)0 0 sdot sdot sdot 0 12058124119896 minus 12 (119898 + 1) minus 119899
))))))))))))))
((((
11988601198861119886119899minus1119886119899))))
= 0
(29)
where the necessary condition for the closed form of theenergy eigenvalues is as follows12058124119896 minus 12 (119898 + 1) minus 119899 = 0 119899 = 0 1 2 (30)
By substituting the values of 1205812 and 119896 into (30) we have119864NC119899
= plusmn11989801198882radic1 + 4119899 (1 + 11989801205962ℏ 120579)(1 + 12057921198980120596ℏ) 120596ℏ11989801198882 (31)
which is the same as the energy relation in [31]
Also from (29) the expansion coefficients 119886119898 satisfy thefollowing two-term recursion relation
(119895 (119895 + 1) + (119895 + 1) (119898 + 1)) 119886119895+1+ (12058124119896 minus 12 (119898 + 1) minus 119895) 119886119895 = 0 (32)
with the boundary conditions 119886minus1 = 0 and 119886119899+1 = 0Before we proceed further it would be well to note that
(31) may or may not possess real eigenvalues depending onthe choice of noncommutative parameters 120579 and 120579 to bepositive or negative values Of course it is not our aim to
Advances in High Energy Physics 5
study here but one can see a similar study in [42] wherethe Dirac oscillator in 2 + 1-dimensional noncommutativespace has been studied for the cases of positive and negativenoncommutative parameter Hence it seems that the resultscan be used to study the connection with non-Hermitianquantum mechanics Also the possible case of imaginaryenergy gives useful clue regarding the lack of bound-states inthe spectroscopy of relativistic fermions
Therefore for a given 119899 the energy eigenvalue is obtainedfrom (31) and the corresponding unnormalized wave func-tion can be obtained from (6) as
120595119863 equiv 120595119899119898 (119901 120579) = (1205951 (119901 120579)1205952 (119901 120579))
= ( 1119888119901+119864 + 11989801198882)1205951 (119901 120579) (33)
where
1205951 (119901 120579) = 119901119898119890minus(1198962)1199012119890119894119898120579 119899sum119895=0
119886119895119903119895 (34)
with 119903 = 1198961199012 Now for clarifying that the wave functionobtained from this method is in good agreement with theresults of [31] we calculate it for 119899 = 2 According to (28) and(32) the three-dimensional invariant subspace is obtained as
119865 (119903) = 1198860 + 1198861119903 + 11988621199032= 1198860 [1 + (minus12058124119896 + (12) (119898 + 1))119898 + 1 (1198961199012) + (minus12058124119896 + (12) (119898 + 1)) (minus12058124119896 + (12) (119898 + 1) + 1)(119898 + 1) (2 + 2 (119898 + 1)) (1198961199012)2] (35)
where by using the definition of the confluent hypergeometricfunction for 119899 = 2 we have119865 (119903) = 1198860 11198651 (minus119899119898 + 1 1198961199012) (36)
The above equation is exactly identical to equation (32) in [31]and so for other values of 119899 and after some calculations onecan show that 119865(119903) can be written in terms of the confluenthypergeometric function
4 Conclusions
Using the Lie algebraic approach we have solved the Diracoscillator in (2 + 1) dimension in the framework of non-commutative phase space We have obtained the energyeigenvalues and the corresponding wave functions throughthe sl(2) Lie algebra representation We have also shown thatour results are in good agreement with the results in [31] It isseen that the Lie algebraic approach is a powerful method forreproducing the exact analytical results
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
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[2] N Ferkous and A Bounames ldquoEnergy spectrum of a 2DDirac oscillator in the presence of the AharonovmdashBohm effectrdquoPhysics Letters A vol 325 no 1 pp 21ndash29 2004
[3] V M Villalba ldquoExact solution of the two-dimensional Diracoscillatorrdquo Physical Review A vol 49 no 1 586 pages 1994
[4] A Bermudez M A Martin-Delgado and E Solano ldquoExactmapping of the 2 + 1Dirac oscillator onto the Jaynes-Cummingsmodel Ion-trap experimental proposalrdquo Physical Review A vol76 no 4 Article ID 041801 2007
[5] M Moreno and A Zentella ldquoCovariance CPT and the Foldy-Wouthuysen transformation for the Dirac oscillatorrdquo Journal ofPhysics A Mathematical and General vol 22 no 17 L821 pages1989
[6] R de Lima Rodrigues ldquoOn the Dirac oscillatorrdquo Physics LettersA vol 372 no 15 pp 2587ndash2591 2008
[7] D Ito K Mori and E Carriere ldquoAn example of dynamicalsystems with linear trajectoryrdquo Il Nuovo Cimento A vol 51 no4 pp 1119ndash1121 1967
[8] M Moshinsky and A Szczepaniak ldquoThe Dirac oscillatorrdquoJournal of Physics A Mathematical and General vol 22 no 17L817 pages 1989
[9] E T Jaynes and FW Cummings ldquoComparison of quantum andsemiclassical radiation theories with application to the beammaserrdquo Proceedings of the IEEE vol 51 no 1 pp 89ndash109 1963
[10] J Larso ldquoDynamics of the JaynesndashCummings and Rabi modelsoldwine in newbottlesrdquoPhysica Scripta vol 76 no 2 146 pages2007
[11] L Allen and J H Eberly Optical Resonance and Two LevelAtoms Dover Publications Mineola New York USA 1987
[12] R P Martinez-y-Romero and A L Salas-Brito ldquoConformalinvariance in a Dirac oscillatorrdquo Journal of MathematicalPhysics vol 33 no 5 1831 pages 1992
[13] J Benitez R P Martnez y Romero H N Nuez-Yepez and AL Salas-Brito ldquoSolution and hidden supersymmetry of a Diracoscillatorrdquo Physical Review Letters vol 64 1643 no 14 1990
[14] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 no 09article 032 1999
[15] A Connes M R Douglas and A Schwarz ldquoNoncommutativegeometry and Matrix theoryrdquo Journal of High Energy Physicsvol 1998 003 no 02 1998
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[16] M R Douglas and N A Nekrasov ldquoNoncommutative fieldtheoryrdquo Reviews of Modern Physics vol 73 no 4 977 pages2001
[17] C S Chu and P M Ho ldquoNon-commutative open string and D-branerdquo Nuclear Physics B vol 550 no 1-2 pp 151ndash168 1999
[18] C-S Chu and P-M Ho ldquoConstrained quantization of openstring in background B field and non-commutative D-branerdquoNuclear Physics B vol 568 no 1-2 pp 447ndash456 2000
[19] F Ardalan H Arfaei and M M Sheikh-Jabbari ldquoDiracquantization of open strings and noncommutativity in branesrdquoNuclear Physics B vol 576 no 1-3 pp 578ndash596 2000
[20] J Jing and Z-W Long ldquoOpen string in the constant B-fieldbackgroundrdquo Physical Review D vol 72 no 12 126002 pages2005
[21] I Hinchliffe N Kersting and Y L Ma ldquoReview of thephenomenology of noncommutative geometryrdquo InternationalJournal of Modern Physics A Particles and Fields GravitationCosmology vol 19 no 2 179 pages 2004
[22] SMinwalla M V Raamsdonk andN Seiberg ldquoNoncommuta-tive perturbative dynamicsrdquo Journal of High Energy Physics vol2000 020 no 02 2000
[23] M V Raamsdonk and N Seiberg ldquoComments on noncommu-tative perturbative dynamicsrdquo Journal of High Energy Physicsvol 2000 035 no 03 2000
[24] R Gopakumar S Minwalla and A Strominger ldquoNoncommu-tative solitonsrdquo Journal of High Energy Physics vol 2000 020no 05 2000
[25] S Cai T Jing G Guo and R Zhang ldquoDirac oscillator in non-commutative phase spacerdquo International Journal of TheoreticalPhysics vol 49 no 8 pp 1699ndash1705 2010
[26] O Bertolami and R Queiroz ldquoPhase-space noncommutativityand the Dirac equationrdquo Physics Letters A vol 375 no 46 pp4116ndash4119 2011
[27] Z-Y Luo Q Wang X Li and J Jing ldquoDirac oscillatorin noncommutative phase space and (anti)-Jaynes-Cummingsmodelsrdquo International Journal of Theoretical Physics vol 51 no7 pp 2143ndash2151 2012
[28] B P Mandal and S K Rai ldquoNoncommutative Dirac oscillatorin an external magnetic fieldrdquo Physics Letters A vol 376 no 36pp 2467ndash2470 2012
[29] C Bastos O Bertolami N C Dias and J N Prata ldquoNoncom-mutative graphenerdquo International Journal of Modern Physics Avol 28 no 16 Article ID 1350064 13 pages 2013
[30] O Panella and P Roy ldquoQuantum phase transitions in thenoncommutative Dirac oscillatorrdquo Physical Review A vol 90no 4 Article ID 042111 2014
[31] A Boumali and H Hassanabadi ldquoExact solutions of the (2 +1)-dimensional dirac oscillator under a magnetic field in thepresence of a minimal length in the non-commutative phasespacerdquo Zeitschrift fur Naturforschung A - A Journal of PhysicalSciences vol 70 no 8 619 pages 2015
[32] H Ciftci R L Hall and N Saad ldquoAsymptotic iteration methodfor eigenvalue problemsrdquo Journal of Physics A Mathematicaland General vol 36 no 47 11807 pages 2003
[33] S H Dong D Morales and J Garcia-Ravelo ldquoExact quanti-zation rule and its applications to physical potentialsrdquo Interna-tional Journal of Modern Physics E vol 16 no 1 p 189 2007
[34] G FWei and S H Dong ldquoAlgebraic approach to energy spectraof the Scarf type and generalized PoschlndashTeller potentialsrdquoCanadian Journal of Physics vol 89 no 12 pp 1225ndash1231 2011
[35] H Hassanabadi S Zarrinkamar and A A Rajabi ldquoExact solu-tions of D-dimensional Schrodinger equation for an energy-dependent potential by NUmethodrdquo Communications inTheo-retical Physics vol 55 no 4 pp 541ndash544 2011
[36] H Hassanabadi A A Rajabi and S Zarrinkamar ldquoCornelland kratzer potentials within the semirelativistic treatmentrdquoModern Physics Letters A vol 27 no 10 Article ID 12500572012
[37] A G Ushveridze ldquoLie-algebraic approach to the problem ofquasi-exact solubility in quantum mechanicsrdquo Modern PhysicsLetters A Particles and Fields Gravitation Cosmology andNuclear Physics vol 5 no 23 pp 1891ndash1899 1990
[38] A V Turbiner ldquoQuasi-exactly-solvable problems and sl(2)algebrardquo Communications in Mathematical Physics vol 118 no3 pp 467ndash474 1988
[39] A V Turbiner ldquoLie-algebras and linear operators with invariantsubspacesrdquo Contemporary Mathematics vol 160 p 263 1994
[40] D Gomez-Ullate N Kamran and R Milson ldquoQuasi-exactsolvability beyond the sl(2) algebraizationrdquo Physics of AtomicNuclei vol 70 no 3 pp 520ndash528 2007
[41] L Menculini O Panella and P Roy ldquoExact solutions of the (2+ 1) dimensional Dirac equation in a constant magnetic field inthe presence of aminimal lengthrdquo Physical ReviewD vol 87 no6 Article ID 065017 2013
[42] Y-Q Luo Y Cui Z-W Long and J Jing ldquo2 + 1 Dimensionalnoncommutative dirac oscillator and (anti)-jaynes-cummingsmodelsrdquo International Journal of Theoretical Physics vol 50 no10 pp 2992ndash3000 2011
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
2 Advances in High Energy Physics
within the framework of representation theory of the sl(2) Liealgebra
A quantum system is exactly solvable (ES) if all theeigenvalues and the corresponding eigenfunctions can becalculated in an exact analytical manner In contrast aquantum system is quasi-exactly solvable (QES) if only a finitenumber of eigenvalues and eigenfunctions can be determinedexactly [37ndash40] In the case of ES models the Hamiltonianof the system can be diagonalized completely due to the factthat algebraic symmetry is complete but in the case of QESmodels the Hamiltonian is only block diagonalized Such afinite block can always be diagonalized which yields a finitepart of the spectrum algebraically
This paper is organized as follows In Section 2 based on[31] we briefly review the noncommutative phase space on (2+ 1)-dimensional Dirac oscillator Section 3 is devoted to 119904119897(2)algebraization of Dirac oscillator in the noncommutativephase space We obtain the energy eigenvalues and the cor-responding wave functions through the sl(2) representationand show that our results are in good agreement with theresults of [31] In Section 4 we present the conclusion
2 Review on (2 + 1)-DimensionalDirac Oscillator in the NoncommutativePhase Space
According to [31] the noncommutative phase space is char-acterized by the fact that their coordinate operators satisfy theequation
[119909(NC)119894 119909(NC)119895 ] = 119894120579119894119895[119901(NC)119894 119901(NC)119895 ] = 119894120579119894119895[119909(NC)119894 119901(NC)119895 ] = 119894ℏ120575119894119895(1)
where 120579119894119895 and 120579119894119895 are an antisymmetric tensor of space dimen-sion By using Bopprsquos shift method the two-dimensionalnoncommutative phase space can be considered as [29]
119909(NC) = 119909 minus 1205792ℏ119901119910119910(NC) = 119910 + 1205792ℏ119901119909119901(NC)119909 = 119901119909 + 1205792ℏ119910119901(NC)119910 = 119901119910 minus 1205792ℏ119909
(2)
TheDirac oscillator is also described by the following Hamil-tonian [8]
= 119888 sdot (997888rarr119901 minus 1198941198980120596120573997888rarr119903 ) + 12057311989801198882 (3)
where in two-dimensional noncommutative space it istransformed as119888120572119909 (119901(NC)119909 minus 1198941198980120596120573119909(NC))+ 119888120572119910 (119901(NC)119910 minus 1198941198980120596120573119910(NC)) + 12057311989801198882 120595119863= 119864NC120595119863
(4)
By considering the Pauli matrices as the representation ofDirac matrices in (2 + 1)-dimension that is
120572119909 = 120590119909 = (0 11 0) 120572119910 = 120590119910 = (0 minus119894119894 0 ) 120573 = (1 00 minus1)
(5)
and 120595119863 = (1205951 1205952)119879 (4) becomes
(11989801198882 119888119901minus119888119901+ minus11989801198882)(12059511205952) = 119864NC (12059511205952) (6)
where 119901minus 119901+ are as follows119901minus = 119901(NC)119909 minus 119894119901(NC)119910 + 1198941198980120596 (119909(NC) minus 119894119910(NC))= 1205881 (119901119909 minus 119894119901119910) + 11989411989801205961205882 (119909 minus 119894119910) 119901+ = 119901(NC)119909 + 119894119901(NC)119910 minus 1198941198980120596 (119909(NC) + 119894119910(NC))= 1205881 (119901119909 + 119894119901119910) minus 11989411989801205961205882 (119909 + 119894119910) (7)
with 1205881 = 1 + 11989801205962ℏ 1205791205882 = 1 + 12057921198980120596ℏ (8)
After some calculations it is easy to see that (6) gives1198882119901minus119901+ minus (119864NC2 minus 119898201198884) 1205951 = 0 (9)1198882119901+119901minus minus (119864NC2 minus 119898201198884) 1205952 = 0 (10)
Similar to [31 41] and by defining 119901119909 = 119901 cos 120579 119901119910 = 119901 sin 120579and 1199012 = 1199012119909 + 1199012119910 it is seen that the (7) transform into
119901minus = 119890minus119894120579 1205881119901 minus 120582( 120597120597119901 minus 119894119901 120597120597120579) 119901+ = 119890119894120579 1205881119901 + 120582( 120597120597119901 + 119894119901 120597120597120579) (11)
Advances in High Energy Physics 3
where
120582 = (1 + 12057921198980120596ℏ)1198980120596ℏ (12)
Substituting (11) into (9) and by considering 1205951(119901 120579) =119891(119901)119890119894119898120579 we have(1198892119891 (119901)1198891199012 + 1119901 119889119891 (119901)119889119901 minus 11989821199012 119891 (119901))+ (1205812 minus 11989621199012) 119891 (119901) = 0 (13)
where 1205812 = 21205821205881 (119898 + 1) + 1205761205822 1198962 = 120588211205822 (14)
with
120576 = 119864NC2 minus 1198982011988841198882 (15)
Here it is necessary to point out that in the literature there aremany different methods for solving quantum models exactly[32ndash36] Boumali and Hassanabadi [31] solved this problemanalytically and obtained the exact solutions in terms ofthe hypergeometric functions Within the present studywe intend to illustrate the exact solvability of the problemthrough the sl(2) algebraization [39] To do a comprehensiveanalysis we need to obtain the more general solution ofthe problem This is the main advantage of the presentmethod In fact by using the theory of representation spacewe construct the general matrix equation of the problemand calculate the closed-form expressions for energies andeigenfunctions Accordingly we can quickly obtain exactsolutions of any arbitrary state n without any cumbersomenumerical procedure or a complicated analytical one indetermining the solutions of the higher states It seems thatthe method is computationally much simpler than otheranalytical methods
Hence in the next section we solve (13) by Lie algebraicapproach related to representation theory of sl(2) Lie groupSimilar procedure can be also done on (10) for the 1205952 spinorwave function
3 Lie Algebraic Solution of the DiracOscillator in the NoncommutativePhase Space
In this section for solving (13) by the sl(2) Lie algebrarepresentation we first use similarity transformation as119891(119901) = 119901minus12119877(119901) such that the first-order derivative of thedifferential equation (13) can be eliminated and it is reducedto a Schrodinger-type operator as
( 11988921198891199012 + 14 minus 11989821199012 minus 1205812 minus 11989621199012)119877 (119901) = 0 (16)
Extracting the asymptotic behavior of thewavefunction at theorigin and infinity using the transformation119877 (119901) = 1199011205741198901205721199012119865 (119901) (17)
where 120572 lt 0 (to be physically meaningful) and 120574 areparameters to be determined later by the exact solvabilityconditions (see (26)) (16) is converted to1198892119865 (119901)1198891199012 + (2120574119901 + 4120572119901) 119889119865 (119901)119889119901 + ((41205722 minus 1198962) 1199012
+ 120574 (120574 minus 1) + 14 minus 11989821199012 + 4120572120574 + 2120572 + 1205812)119865 (119901)= 0
(18)
Using the change of variable 119903 = 1198961199012 we get11990311988921198651198891199032 + (2120572119896 119903 + 120574 + 12) 119889119865119889119903 + [(12057221198962 minus 14) 119903
+ 120574 (120574 minus 1) + 14 minus 11989824119903 + 120572120574119896 + 1205722119896 + 12058124119896]119865 = 0 (19)
According to [38] in one dimension the only Lie algebra ofthe first-order differential operators which possesses finite-dimensional representations is the sl(2) Lie algebra Hencewe consider the following realization of the operators
119869+119899 = 1199032 119889119889119903 minus 1198991199031198690119899 = 119903 119889119889119903 minus 1198992 119869minus119899 = 119889119889119903 (20)
which satisfy the sl(2) commutation relations[119869+119899 119869minus119899 ] = minus21198690119899 [119869plusmn119899 1198690119899] = ∓119869plusmn119899 (21)
and preserve the (119899 + 1)-dimensional linear space of polyno-mials with finite order119875119899+1 = ⟨1 119903 1199032 119903119899⟩ (22)
The most general second-order differential equation whichpreserves the space119875119899+1 can be also represented as a quadraticcombination of the sl(2) generators as119867 = sum
119886119887=0plusmn
119862119886119887119869119886119899119869119887119899 + sum119886=0plusmn
119862119886119869119886119899 + 119862 (23)
where 119862119886119887 119862119886 and 119862 are real parameters Substituting (20)into (23) yields the following differential form
119867 = 1198754 (119903) 11988921198891199032 + 1198753 (119903) 119889119889119903 + 1198752 (119903) (24)
4 Advances in High Energy Physics
where 119875119894(119903) are polynomials of degree 1198941198754 (119903) = 119862++1199034 + 119862+01199033 + 119862+minus1199032 + 1198620minus119903 + 119862minusminus1198753 (119903) = 119862++ (2 minus 2119899) 1199033 + (119862+ + 119862+0 (1 minus 31198992 )) 1199032+ (1198620 minus 119899119862+minus) 119903 + (119862minus minus 11989921198620minus)
1198752 (119903) = 119862++119899 (119899 minus 1) 1199032 + (11989922 119862+0 minus 119899119862+) 119903+ (119862 minus 11989921198620)
(25)
Comparing (24) with (19) gives119862++ = 119862+0 = 119862+minus = 119862minusminus = 119862+ = 0120572 = minus1198962 120574 = 119898 + 12 1198620minus = 11198620 = minus1
119862minus = 1198992 + 119898 + 1119862 = 12058124119896 minus 12 (119898 + 119899 + 1)
(26)
From (23) and (26) the Lie algebraic form of theHamiltonianis written as
119867 = 1198690119899119869minus119899 minus 1198690119899 + (1198992 + 119898 + 1) 119869minus119899+ (12058124119896 minus 12 (119898 + 119899 + 1)) (27)
which implies that this operator is ES due to the absenceof the terms of positive grading Therefore the operator 119867preserves the finite-dimensional representation space of thealgebra sl(2) as
119865 (119903) = 119899sum119895=0
119886119895119903119895 119899 = 0 1 2 (28)
Hence using (27) and (28) and doing some calculations thefollowing matrix equation can be obtained
((((((((((((((
12058124119896 minus 12 (119898 + 1) (119898 + 1) 0 sdot sdot sdot 00 12058124119896 minus 12 (119898 + 1) minus 1 2 + 2 (119898 + 1) sdot sdot sdot 00 0 d d
0 12058124119896 minus 12 (119898 + 1) minus (119899 minus 1) 119899 (119899 minus 1) + 119899 (119898 + 1)0 0 sdot sdot sdot 0 12058124119896 minus 12 (119898 + 1) minus 119899
))))))))))))))
((((
11988601198861119886119899minus1119886119899))))
= 0
(29)
where the necessary condition for the closed form of theenergy eigenvalues is as follows12058124119896 minus 12 (119898 + 1) minus 119899 = 0 119899 = 0 1 2 (30)
By substituting the values of 1205812 and 119896 into (30) we have119864NC119899
= plusmn11989801198882radic1 + 4119899 (1 + 11989801205962ℏ 120579)(1 + 12057921198980120596ℏ) 120596ℏ11989801198882 (31)
which is the same as the energy relation in [31]
Also from (29) the expansion coefficients 119886119898 satisfy thefollowing two-term recursion relation
(119895 (119895 + 1) + (119895 + 1) (119898 + 1)) 119886119895+1+ (12058124119896 minus 12 (119898 + 1) minus 119895) 119886119895 = 0 (32)
with the boundary conditions 119886minus1 = 0 and 119886119899+1 = 0Before we proceed further it would be well to note that
(31) may or may not possess real eigenvalues depending onthe choice of noncommutative parameters 120579 and 120579 to bepositive or negative values Of course it is not our aim to
Advances in High Energy Physics 5
study here but one can see a similar study in [42] wherethe Dirac oscillator in 2 + 1-dimensional noncommutativespace has been studied for the cases of positive and negativenoncommutative parameter Hence it seems that the resultscan be used to study the connection with non-Hermitianquantum mechanics Also the possible case of imaginaryenergy gives useful clue regarding the lack of bound-states inthe spectroscopy of relativistic fermions
Therefore for a given 119899 the energy eigenvalue is obtainedfrom (31) and the corresponding unnormalized wave func-tion can be obtained from (6) as
120595119863 equiv 120595119899119898 (119901 120579) = (1205951 (119901 120579)1205952 (119901 120579))
= ( 1119888119901+119864 + 11989801198882)1205951 (119901 120579) (33)
where
1205951 (119901 120579) = 119901119898119890minus(1198962)1199012119890119894119898120579 119899sum119895=0
119886119895119903119895 (34)
with 119903 = 1198961199012 Now for clarifying that the wave functionobtained from this method is in good agreement with theresults of [31] we calculate it for 119899 = 2 According to (28) and(32) the three-dimensional invariant subspace is obtained as
119865 (119903) = 1198860 + 1198861119903 + 11988621199032= 1198860 [1 + (minus12058124119896 + (12) (119898 + 1))119898 + 1 (1198961199012) + (minus12058124119896 + (12) (119898 + 1)) (minus12058124119896 + (12) (119898 + 1) + 1)(119898 + 1) (2 + 2 (119898 + 1)) (1198961199012)2] (35)
where by using the definition of the confluent hypergeometricfunction for 119899 = 2 we have119865 (119903) = 1198860 11198651 (minus119899119898 + 1 1198961199012) (36)
The above equation is exactly identical to equation (32) in [31]and so for other values of 119899 and after some calculations onecan show that 119865(119903) can be written in terms of the confluenthypergeometric function
4 Conclusions
Using the Lie algebraic approach we have solved the Diracoscillator in (2 + 1) dimension in the framework of non-commutative phase space We have obtained the energyeigenvalues and the corresponding wave functions throughthe sl(2) Lie algebra representation We have also shown thatour results are in good agreement with the results in [31] It isseen that the Lie algebraic approach is a powerful method forreproducing the exact analytical results
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] A Bermudez M A Martin-Delgado and A Luis ldquoChiralityquantum phase transition in the Dirac oscillatorrdquo PhysicalReview A vol 77 no 6 Article ID 063815 2008
[2] N Ferkous and A Bounames ldquoEnergy spectrum of a 2DDirac oscillator in the presence of the AharonovmdashBohm effectrdquoPhysics Letters A vol 325 no 1 pp 21ndash29 2004
[3] V M Villalba ldquoExact solution of the two-dimensional Diracoscillatorrdquo Physical Review A vol 49 no 1 586 pages 1994
[4] A Bermudez M A Martin-Delgado and E Solano ldquoExactmapping of the 2 + 1Dirac oscillator onto the Jaynes-Cummingsmodel Ion-trap experimental proposalrdquo Physical Review A vol76 no 4 Article ID 041801 2007
[5] M Moreno and A Zentella ldquoCovariance CPT and the Foldy-Wouthuysen transformation for the Dirac oscillatorrdquo Journal ofPhysics A Mathematical and General vol 22 no 17 L821 pages1989
[6] R de Lima Rodrigues ldquoOn the Dirac oscillatorrdquo Physics LettersA vol 372 no 15 pp 2587ndash2591 2008
[7] D Ito K Mori and E Carriere ldquoAn example of dynamicalsystems with linear trajectoryrdquo Il Nuovo Cimento A vol 51 no4 pp 1119ndash1121 1967
[8] M Moshinsky and A Szczepaniak ldquoThe Dirac oscillatorrdquoJournal of Physics A Mathematical and General vol 22 no 17L817 pages 1989
[9] E T Jaynes and FW Cummings ldquoComparison of quantum andsemiclassical radiation theories with application to the beammaserrdquo Proceedings of the IEEE vol 51 no 1 pp 89ndash109 1963
[10] J Larso ldquoDynamics of the JaynesndashCummings and Rabi modelsoldwine in newbottlesrdquoPhysica Scripta vol 76 no 2 146 pages2007
[11] L Allen and J H Eberly Optical Resonance and Two LevelAtoms Dover Publications Mineola New York USA 1987
[12] R P Martinez-y-Romero and A L Salas-Brito ldquoConformalinvariance in a Dirac oscillatorrdquo Journal of MathematicalPhysics vol 33 no 5 1831 pages 1992
[13] J Benitez R P Martnez y Romero H N Nuez-Yepez and AL Salas-Brito ldquoSolution and hidden supersymmetry of a Diracoscillatorrdquo Physical Review Letters vol 64 1643 no 14 1990
[14] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 no 09article 032 1999
[15] A Connes M R Douglas and A Schwarz ldquoNoncommutativegeometry and Matrix theoryrdquo Journal of High Energy Physicsvol 1998 003 no 02 1998
6 Advances in High Energy Physics
[16] M R Douglas and N A Nekrasov ldquoNoncommutative fieldtheoryrdquo Reviews of Modern Physics vol 73 no 4 977 pages2001
[17] C S Chu and P M Ho ldquoNon-commutative open string and D-branerdquo Nuclear Physics B vol 550 no 1-2 pp 151ndash168 1999
[18] C-S Chu and P-M Ho ldquoConstrained quantization of openstring in background B field and non-commutative D-branerdquoNuclear Physics B vol 568 no 1-2 pp 447ndash456 2000
[19] F Ardalan H Arfaei and M M Sheikh-Jabbari ldquoDiracquantization of open strings and noncommutativity in branesrdquoNuclear Physics B vol 576 no 1-3 pp 578ndash596 2000
[20] J Jing and Z-W Long ldquoOpen string in the constant B-fieldbackgroundrdquo Physical Review D vol 72 no 12 126002 pages2005
[21] I Hinchliffe N Kersting and Y L Ma ldquoReview of thephenomenology of noncommutative geometryrdquo InternationalJournal of Modern Physics A Particles and Fields GravitationCosmology vol 19 no 2 179 pages 2004
[22] SMinwalla M V Raamsdonk andN Seiberg ldquoNoncommuta-tive perturbative dynamicsrdquo Journal of High Energy Physics vol2000 020 no 02 2000
[23] M V Raamsdonk and N Seiberg ldquoComments on noncommu-tative perturbative dynamicsrdquo Journal of High Energy Physicsvol 2000 035 no 03 2000
[24] R Gopakumar S Minwalla and A Strominger ldquoNoncommu-tative solitonsrdquo Journal of High Energy Physics vol 2000 020no 05 2000
[25] S Cai T Jing G Guo and R Zhang ldquoDirac oscillator in non-commutative phase spacerdquo International Journal of TheoreticalPhysics vol 49 no 8 pp 1699ndash1705 2010
[26] O Bertolami and R Queiroz ldquoPhase-space noncommutativityand the Dirac equationrdquo Physics Letters A vol 375 no 46 pp4116ndash4119 2011
[27] Z-Y Luo Q Wang X Li and J Jing ldquoDirac oscillatorin noncommutative phase space and (anti)-Jaynes-Cummingsmodelsrdquo International Journal of Theoretical Physics vol 51 no7 pp 2143ndash2151 2012
[28] B P Mandal and S K Rai ldquoNoncommutative Dirac oscillatorin an external magnetic fieldrdquo Physics Letters A vol 376 no 36pp 2467ndash2470 2012
[29] C Bastos O Bertolami N C Dias and J N Prata ldquoNoncom-mutative graphenerdquo International Journal of Modern Physics Avol 28 no 16 Article ID 1350064 13 pages 2013
[30] O Panella and P Roy ldquoQuantum phase transitions in thenoncommutative Dirac oscillatorrdquo Physical Review A vol 90no 4 Article ID 042111 2014
[31] A Boumali and H Hassanabadi ldquoExact solutions of the (2 +1)-dimensional dirac oscillator under a magnetic field in thepresence of a minimal length in the non-commutative phasespacerdquo Zeitschrift fur Naturforschung A - A Journal of PhysicalSciences vol 70 no 8 619 pages 2015
[32] H Ciftci R L Hall and N Saad ldquoAsymptotic iteration methodfor eigenvalue problemsrdquo Journal of Physics A Mathematicaland General vol 36 no 47 11807 pages 2003
[33] S H Dong D Morales and J Garcia-Ravelo ldquoExact quanti-zation rule and its applications to physical potentialsrdquo Interna-tional Journal of Modern Physics E vol 16 no 1 p 189 2007
[34] G FWei and S H Dong ldquoAlgebraic approach to energy spectraof the Scarf type and generalized PoschlndashTeller potentialsrdquoCanadian Journal of Physics vol 89 no 12 pp 1225ndash1231 2011
[35] H Hassanabadi S Zarrinkamar and A A Rajabi ldquoExact solu-tions of D-dimensional Schrodinger equation for an energy-dependent potential by NUmethodrdquo Communications inTheo-retical Physics vol 55 no 4 pp 541ndash544 2011
[36] H Hassanabadi A A Rajabi and S Zarrinkamar ldquoCornelland kratzer potentials within the semirelativistic treatmentrdquoModern Physics Letters A vol 27 no 10 Article ID 12500572012
[37] A G Ushveridze ldquoLie-algebraic approach to the problem ofquasi-exact solubility in quantum mechanicsrdquo Modern PhysicsLetters A Particles and Fields Gravitation Cosmology andNuclear Physics vol 5 no 23 pp 1891ndash1899 1990
[38] A V Turbiner ldquoQuasi-exactly-solvable problems and sl(2)algebrardquo Communications in Mathematical Physics vol 118 no3 pp 467ndash474 1988
[39] A V Turbiner ldquoLie-algebras and linear operators with invariantsubspacesrdquo Contemporary Mathematics vol 160 p 263 1994
[40] D Gomez-Ullate N Kamran and R Milson ldquoQuasi-exactsolvability beyond the sl(2) algebraizationrdquo Physics of AtomicNuclei vol 70 no 3 pp 520ndash528 2007
[41] L Menculini O Panella and P Roy ldquoExact solutions of the (2+ 1) dimensional Dirac equation in a constant magnetic field inthe presence of aminimal lengthrdquo Physical ReviewD vol 87 no6 Article ID 065017 2013
[42] Y-Q Luo Y Cui Z-W Long and J Jing ldquo2 + 1 Dimensionalnoncommutative dirac oscillator and (anti)-jaynes-cummingsmodelsrdquo International Journal of Theoretical Physics vol 50 no10 pp 2992ndash3000 2011
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Advances in High Energy Physics 3
where
120582 = (1 + 12057921198980120596ℏ)1198980120596ℏ (12)
Substituting (11) into (9) and by considering 1205951(119901 120579) =119891(119901)119890119894119898120579 we have(1198892119891 (119901)1198891199012 + 1119901 119889119891 (119901)119889119901 minus 11989821199012 119891 (119901))+ (1205812 minus 11989621199012) 119891 (119901) = 0 (13)
where 1205812 = 21205821205881 (119898 + 1) + 1205761205822 1198962 = 120588211205822 (14)
with
120576 = 119864NC2 minus 1198982011988841198882 (15)
Here it is necessary to point out that in the literature there aremany different methods for solving quantum models exactly[32ndash36] Boumali and Hassanabadi [31] solved this problemanalytically and obtained the exact solutions in terms ofthe hypergeometric functions Within the present studywe intend to illustrate the exact solvability of the problemthrough the sl(2) algebraization [39] To do a comprehensiveanalysis we need to obtain the more general solution ofthe problem This is the main advantage of the presentmethod In fact by using the theory of representation spacewe construct the general matrix equation of the problemand calculate the closed-form expressions for energies andeigenfunctions Accordingly we can quickly obtain exactsolutions of any arbitrary state n without any cumbersomenumerical procedure or a complicated analytical one indetermining the solutions of the higher states It seems thatthe method is computationally much simpler than otheranalytical methods
Hence in the next section we solve (13) by Lie algebraicapproach related to representation theory of sl(2) Lie groupSimilar procedure can be also done on (10) for the 1205952 spinorwave function
3 Lie Algebraic Solution of the DiracOscillator in the NoncommutativePhase Space
In this section for solving (13) by the sl(2) Lie algebrarepresentation we first use similarity transformation as119891(119901) = 119901minus12119877(119901) such that the first-order derivative of thedifferential equation (13) can be eliminated and it is reducedto a Schrodinger-type operator as
( 11988921198891199012 + 14 minus 11989821199012 minus 1205812 minus 11989621199012)119877 (119901) = 0 (16)
Extracting the asymptotic behavior of thewavefunction at theorigin and infinity using the transformation119877 (119901) = 1199011205741198901205721199012119865 (119901) (17)
where 120572 lt 0 (to be physically meaningful) and 120574 areparameters to be determined later by the exact solvabilityconditions (see (26)) (16) is converted to1198892119865 (119901)1198891199012 + (2120574119901 + 4120572119901) 119889119865 (119901)119889119901 + ((41205722 minus 1198962) 1199012
+ 120574 (120574 minus 1) + 14 minus 11989821199012 + 4120572120574 + 2120572 + 1205812)119865 (119901)= 0
(18)
Using the change of variable 119903 = 1198961199012 we get11990311988921198651198891199032 + (2120572119896 119903 + 120574 + 12) 119889119865119889119903 + [(12057221198962 minus 14) 119903
+ 120574 (120574 minus 1) + 14 minus 11989824119903 + 120572120574119896 + 1205722119896 + 12058124119896]119865 = 0 (19)
According to [38] in one dimension the only Lie algebra ofthe first-order differential operators which possesses finite-dimensional representations is the sl(2) Lie algebra Hencewe consider the following realization of the operators
119869+119899 = 1199032 119889119889119903 minus 1198991199031198690119899 = 119903 119889119889119903 minus 1198992 119869minus119899 = 119889119889119903 (20)
which satisfy the sl(2) commutation relations[119869+119899 119869minus119899 ] = minus21198690119899 [119869plusmn119899 1198690119899] = ∓119869plusmn119899 (21)
and preserve the (119899 + 1)-dimensional linear space of polyno-mials with finite order119875119899+1 = ⟨1 119903 1199032 119903119899⟩ (22)
The most general second-order differential equation whichpreserves the space119875119899+1 can be also represented as a quadraticcombination of the sl(2) generators as119867 = sum
119886119887=0plusmn
119862119886119887119869119886119899119869119887119899 + sum119886=0plusmn
119862119886119869119886119899 + 119862 (23)
where 119862119886119887 119862119886 and 119862 are real parameters Substituting (20)into (23) yields the following differential form
119867 = 1198754 (119903) 11988921198891199032 + 1198753 (119903) 119889119889119903 + 1198752 (119903) (24)
4 Advances in High Energy Physics
where 119875119894(119903) are polynomials of degree 1198941198754 (119903) = 119862++1199034 + 119862+01199033 + 119862+minus1199032 + 1198620minus119903 + 119862minusminus1198753 (119903) = 119862++ (2 minus 2119899) 1199033 + (119862+ + 119862+0 (1 minus 31198992 )) 1199032+ (1198620 minus 119899119862+minus) 119903 + (119862minus minus 11989921198620minus)
1198752 (119903) = 119862++119899 (119899 minus 1) 1199032 + (11989922 119862+0 minus 119899119862+) 119903+ (119862 minus 11989921198620)
(25)
Comparing (24) with (19) gives119862++ = 119862+0 = 119862+minus = 119862minusminus = 119862+ = 0120572 = minus1198962 120574 = 119898 + 12 1198620minus = 11198620 = minus1
119862minus = 1198992 + 119898 + 1119862 = 12058124119896 minus 12 (119898 + 119899 + 1)
(26)
From (23) and (26) the Lie algebraic form of theHamiltonianis written as
119867 = 1198690119899119869minus119899 minus 1198690119899 + (1198992 + 119898 + 1) 119869minus119899+ (12058124119896 minus 12 (119898 + 119899 + 1)) (27)
which implies that this operator is ES due to the absenceof the terms of positive grading Therefore the operator 119867preserves the finite-dimensional representation space of thealgebra sl(2) as
119865 (119903) = 119899sum119895=0
119886119895119903119895 119899 = 0 1 2 (28)
Hence using (27) and (28) and doing some calculations thefollowing matrix equation can be obtained
((((((((((((((
12058124119896 minus 12 (119898 + 1) (119898 + 1) 0 sdot sdot sdot 00 12058124119896 minus 12 (119898 + 1) minus 1 2 + 2 (119898 + 1) sdot sdot sdot 00 0 d d
0 12058124119896 minus 12 (119898 + 1) minus (119899 minus 1) 119899 (119899 minus 1) + 119899 (119898 + 1)0 0 sdot sdot sdot 0 12058124119896 minus 12 (119898 + 1) minus 119899
))))))))))))))
((((
11988601198861119886119899minus1119886119899))))
= 0
(29)
where the necessary condition for the closed form of theenergy eigenvalues is as follows12058124119896 minus 12 (119898 + 1) minus 119899 = 0 119899 = 0 1 2 (30)
By substituting the values of 1205812 and 119896 into (30) we have119864NC119899
= plusmn11989801198882radic1 + 4119899 (1 + 11989801205962ℏ 120579)(1 + 12057921198980120596ℏ) 120596ℏ11989801198882 (31)
which is the same as the energy relation in [31]
Also from (29) the expansion coefficients 119886119898 satisfy thefollowing two-term recursion relation
(119895 (119895 + 1) + (119895 + 1) (119898 + 1)) 119886119895+1+ (12058124119896 minus 12 (119898 + 1) minus 119895) 119886119895 = 0 (32)
with the boundary conditions 119886minus1 = 0 and 119886119899+1 = 0Before we proceed further it would be well to note that
(31) may or may not possess real eigenvalues depending onthe choice of noncommutative parameters 120579 and 120579 to bepositive or negative values Of course it is not our aim to
Advances in High Energy Physics 5
study here but one can see a similar study in [42] wherethe Dirac oscillator in 2 + 1-dimensional noncommutativespace has been studied for the cases of positive and negativenoncommutative parameter Hence it seems that the resultscan be used to study the connection with non-Hermitianquantum mechanics Also the possible case of imaginaryenergy gives useful clue regarding the lack of bound-states inthe spectroscopy of relativistic fermions
Therefore for a given 119899 the energy eigenvalue is obtainedfrom (31) and the corresponding unnormalized wave func-tion can be obtained from (6) as
120595119863 equiv 120595119899119898 (119901 120579) = (1205951 (119901 120579)1205952 (119901 120579))
= ( 1119888119901+119864 + 11989801198882)1205951 (119901 120579) (33)
where
1205951 (119901 120579) = 119901119898119890minus(1198962)1199012119890119894119898120579 119899sum119895=0
119886119895119903119895 (34)
with 119903 = 1198961199012 Now for clarifying that the wave functionobtained from this method is in good agreement with theresults of [31] we calculate it for 119899 = 2 According to (28) and(32) the three-dimensional invariant subspace is obtained as
119865 (119903) = 1198860 + 1198861119903 + 11988621199032= 1198860 [1 + (minus12058124119896 + (12) (119898 + 1))119898 + 1 (1198961199012) + (minus12058124119896 + (12) (119898 + 1)) (minus12058124119896 + (12) (119898 + 1) + 1)(119898 + 1) (2 + 2 (119898 + 1)) (1198961199012)2] (35)
where by using the definition of the confluent hypergeometricfunction for 119899 = 2 we have119865 (119903) = 1198860 11198651 (minus119899119898 + 1 1198961199012) (36)
The above equation is exactly identical to equation (32) in [31]and so for other values of 119899 and after some calculations onecan show that 119865(119903) can be written in terms of the confluenthypergeometric function
4 Conclusions
Using the Lie algebraic approach we have solved the Diracoscillator in (2 + 1) dimension in the framework of non-commutative phase space We have obtained the energyeigenvalues and the corresponding wave functions throughthe sl(2) Lie algebra representation We have also shown thatour results are in good agreement with the results in [31] It isseen that the Lie algebraic approach is a powerful method forreproducing the exact analytical results
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] A Bermudez M A Martin-Delgado and A Luis ldquoChiralityquantum phase transition in the Dirac oscillatorrdquo PhysicalReview A vol 77 no 6 Article ID 063815 2008
[2] N Ferkous and A Bounames ldquoEnergy spectrum of a 2DDirac oscillator in the presence of the AharonovmdashBohm effectrdquoPhysics Letters A vol 325 no 1 pp 21ndash29 2004
[3] V M Villalba ldquoExact solution of the two-dimensional Diracoscillatorrdquo Physical Review A vol 49 no 1 586 pages 1994
[4] A Bermudez M A Martin-Delgado and E Solano ldquoExactmapping of the 2 + 1Dirac oscillator onto the Jaynes-Cummingsmodel Ion-trap experimental proposalrdquo Physical Review A vol76 no 4 Article ID 041801 2007
[5] M Moreno and A Zentella ldquoCovariance CPT and the Foldy-Wouthuysen transformation for the Dirac oscillatorrdquo Journal ofPhysics A Mathematical and General vol 22 no 17 L821 pages1989
[6] R de Lima Rodrigues ldquoOn the Dirac oscillatorrdquo Physics LettersA vol 372 no 15 pp 2587ndash2591 2008
[7] D Ito K Mori and E Carriere ldquoAn example of dynamicalsystems with linear trajectoryrdquo Il Nuovo Cimento A vol 51 no4 pp 1119ndash1121 1967
[8] M Moshinsky and A Szczepaniak ldquoThe Dirac oscillatorrdquoJournal of Physics A Mathematical and General vol 22 no 17L817 pages 1989
[9] E T Jaynes and FW Cummings ldquoComparison of quantum andsemiclassical radiation theories with application to the beammaserrdquo Proceedings of the IEEE vol 51 no 1 pp 89ndash109 1963
[10] J Larso ldquoDynamics of the JaynesndashCummings and Rabi modelsoldwine in newbottlesrdquoPhysica Scripta vol 76 no 2 146 pages2007
[11] L Allen and J H Eberly Optical Resonance and Two LevelAtoms Dover Publications Mineola New York USA 1987
[12] R P Martinez-y-Romero and A L Salas-Brito ldquoConformalinvariance in a Dirac oscillatorrdquo Journal of MathematicalPhysics vol 33 no 5 1831 pages 1992
[13] J Benitez R P Martnez y Romero H N Nuez-Yepez and AL Salas-Brito ldquoSolution and hidden supersymmetry of a Diracoscillatorrdquo Physical Review Letters vol 64 1643 no 14 1990
[14] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 no 09article 032 1999
[15] A Connes M R Douglas and A Schwarz ldquoNoncommutativegeometry and Matrix theoryrdquo Journal of High Energy Physicsvol 1998 003 no 02 1998
6 Advances in High Energy Physics
[16] M R Douglas and N A Nekrasov ldquoNoncommutative fieldtheoryrdquo Reviews of Modern Physics vol 73 no 4 977 pages2001
[17] C S Chu and P M Ho ldquoNon-commutative open string and D-branerdquo Nuclear Physics B vol 550 no 1-2 pp 151ndash168 1999
[18] C-S Chu and P-M Ho ldquoConstrained quantization of openstring in background B field and non-commutative D-branerdquoNuclear Physics B vol 568 no 1-2 pp 447ndash456 2000
[19] F Ardalan H Arfaei and M M Sheikh-Jabbari ldquoDiracquantization of open strings and noncommutativity in branesrdquoNuclear Physics B vol 576 no 1-3 pp 578ndash596 2000
[20] J Jing and Z-W Long ldquoOpen string in the constant B-fieldbackgroundrdquo Physical Review D vol 72 no 12 126002 pages2005
[21] I Hinchliffe N Kersting and Y L Ma ldquoReview of thephenomenology of noncommutative geometryrdquo InternationalJournal of Modern Physics A Particles and Fields GravitationCosmology vol 19 no 2 179 pages 2004
[22] SMinwalla M V Raamsdonk andN Seiberg ldquoNoncommuta-tive perturbative dynamicsrdquo Journal of High Energy Physics vol2000 020 no 02 2000
[23] M V Raamsdonk and N Seiberg ldquoComments on noncommu-tative perturbative dynamicsrdquo Journal of High Energy Physicsvol 2000 035 no 03 2000
[24] R Gopakumar S Minwalla and A Strominger ldquoNoncommu-tative solitonsrdquo Journal of High Energy Physics vol 2000 020no 05 2000
[25] S Cai T Jing G Guo and R Zhang ldquoDirac oscillator in non-commutative phase spacerdquo International Journal of TheoreticalPhysics vol 49 no 8 pp 1699ndash1705 2010
[26] O Bertolami and R Queiroz ldquoPhase-space noncommutativityand the Dirac equationrdquo Physics Letters A vol 375 no 46 pp4116ndash4119 2011
[27] Z-Y Luo Q Wang X Li and J Jing ldquoDirac oscillatorin noncommutative phase space and (anti)-Jaynes-Cummingsmodelsrdquo International Journal of Theoretical Physics vol 51 no7 pp 2143ndash2151 2012
[28] B P Mandal and S K Rai ldquoNoncommutative Dirac oscillatorin an external magnetic fieldrdquo Physics Letters A vol 376 no 36pp 2467ndash2470 2012
[29] C Bastos O Bertolami N C Dias and J N Prata ldquoNoncom-mutative graphenerdquo International Journal of Modern Physics Avol 28 no 16 Article ID 1350064 13 pages 2013
[30] O Panella and P Roy ldquoQuantum phase transitions in thenoncommutative Dirac oscillatorrdquo Physical Review A vol 90no 4 Article ID 042111 2014
[31] A Boumali and H Hassanabadi ldquoExact solutions of the (2 +1)-dimensional dirac oscillator under a magnetic field in thepresence of a minimal length in the non-commutative phasespacerdquo Zeitschrift fur Naturforschung A - A Journal of PhysicalSciences vol 70 no 8 619 pages 2015
[32] H Ciftci R L Hall and N Saad ldquoAsymptotic iteration methodfor eigenvalue problemsrdquo Journal of Physics A Mathematicaland General vol 36 no 47 11807 pages 2003
[33] S H Dong D Morales and J Garcia-Ravelo ldquoExact quanti-zation rule and its applications to physical potentialsrdquo Interna-tional Journal of Modern Physics E vol 16 no 1 p 189 2007
[34] G FWei and S H Dong ldquoAlgebraic approach to energy spectraof the Scarf type and generalized PoschlndashTeller potentialsrdquoCanadian Journal of Physics vol 89 no 12 pp 1225ndash1231 2011
[35] H Hassanabadi S Zarrinkamar and A A Rajabi ldquoExact solu-tions of D-dimensional Schrodinger equation for an energy-dependent potential by NUmethodrdquo Communications inTheo-retical Physics vol 55 no 4 pp 541ndash544 2011
[36] H Hassanabadi A A Rajabi and S Zarrinkamar ldquoCornelland kratzer potentials within the semirelativistic treatmentrdquoModern Physics Letters A vol 27 no 10 Article ID 12500572012
[37] A G Ushveridze ldquoLie-algebraic approach to the problem ofquasi-exact solubility in quantum mechanicsrdquo Modern PhysicsLetters A Particles and Fields Gravitation Cosmology andNuclear Physics vol 5 no 23 pp 1891ndash1899 1990
[38] A V Turbiner ldquoQuasi-exactly-solvable problems and sl(2)algebrardquo Communications in Mathematical Physics vol 118 no3 pp 467ndash474 1988
[39] A V Turbiner ldquoLie-algebras and linear operators with invariantsubspacesrdquo Contemporary Mathematics vol 160 p 263 1994
[40] D Gomez-Ullate N Kamran and R Milson ldquoQuasi-exactsolvability beyond the sl(2) algebraizationrdquo Physics of AtomicNuclei vol 70 no 3 pp 520ndash528 2007
[41] L Menculini O Panella and P Roy ldquoExact solutions of the (2+ 1) dimensional Dirac equation in a constant magnetic field inthe presence of aminimal lengthrdquo Physical ReviewD vol 87 no6 Article ID 065017 2013
[42] Y-Q Luo Y Cui Z-W Long and J Jing ldquo2 + 1 Dimensionalnoncommutative dirac oscillator and (anti)-jaynes-cummingsmodelsrdquo International Journal of Theoretical Physics vol 50 no10 pp 2992ndash3000 2011
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
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AstrophysicsJournal of
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Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
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AerodynamicsJournal of
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PhotonicsJournal of
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Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
4 Advances in High Energy Physics
where 119875119894(119903) are polynomials of degree 1198941198754 (119903) = 119862++1199034 + 119862+01199033 + 119862+minus1199032 + 1198620minus119903 + 119862minusminus1198753 (119903) = 119862++ (2 minus 2119899) 1199033 + (119862+ + 119862+0 (1 minus 31198992 )) 1199032+ (1198620 minus 119899119862+minus) 119903 + (119862minus minus 11989921198620minus)
1198752 (119903) = 119862++119899 (119899 minus 1) 1199032 + (11989922 119862+0 minus 119899119862+) 119903+ (119862 minus 11989921198620)
(25)
Comparing (24) with (19) gives119862++ = 119862+0 = 119862+minus = 119862minusminus = 119862+ = 0120572 = minus1198962 120574 = 119898 + 12 1198620minus = 11198620 = minus1
119862minus = 1198992 + 119898 + 1119862 = 12058124119896 minus 12 (119898 + 119899 + 1)
(26)
From (23) and (26) the Lie algebraic form of theHamiltonianis written as
119867 = 1198690119899119869minus119899 minus 1198690119899 + (1198992 + 119898 + 1) 119869minus119899+ (12058124119896 minus 12 (119898 + 119899 + 1)) (27)
which implies that this operator is ES due to the absenceof the terms of positive grading Therefore the operator 119867preserves the finite-dimensional representation space of thealgebra sl(2) as
119865 (119903) = 119899sum119895=0
119886119895119903119895 119899 = 0 1 2 (28)
Hence using (27) and (28) and doing some calculations thefollowing matrix equation can be obtained
((((((((((((((
12058124119896 minus 12 (119898 + 1) (119898 + 1) 0 sdot sdot sdot 00 12058124119896 minus 12 (119898 + 1) minus 1 2 + 2 (119898 + 1) sdot sdot sdot 00 0 d d
0 12058124119896 minus 12 (119898 + 1) minus (119899 minus 1) 119899 (119899 minus 1) + 119899 (119898 + 1)0 0 sdot sdot sdot 0 12058124119896 minus 12 (119898 + 1) minus 119899
))))))))))))))
((((
11988601198861119886119899minus1119886119899))))
= 0
(29)
where the necessary condition for the closed form of theenergy eigenvalues is as follows12058124119896 minus 12 (119898 + 1) minus 119899 = 0 119899 = 0 1 2 (30)
By substituting the values of 1205812 and 119896 into (30) we have119864NC119899
= plusmn11989801198882radic1 + 4119899 (1 + 11989801205962ℏ 120579)(1 + 12057921198980120596ℏ) 120596ℏ11989801198882 (31)
which is the same as the energy relation in [31]
Also from (29) the expansion coefficients 119886119898 satisfy thefollowing two-term recursion relation
(119895 (119895 + 1) + (119895 + 1) (119898 + 1)) 119886119895+1+ (12058124119896 minus 12 (119898 + 1) minus 119895) 119886119895 = 0 (32)
with the boundary conditions 119886minus1 = 0 and 119886119899+1 = 0Before we proceed further it would be well to note that
(31) may or may not possess real eigenvalues depending onthe choice of noncommutative parameters 120579 and 120579 to bepositive or negative values Of course it is not our aim to
Advances in High Energy Physics 5
study here but one can see a similar study in [42] wherethe Dirac oscillator in 2 + 1-dimensional noncommutativespace has been studied for the cases of positive and negativenoncommutative parameter Hence it seems that the resultscan be used to study the connection with non-Hermitianquantum mechanics Also the possible case of imaginaryenergy gives useful clue regarding the lack of bound-states inthe spectroscopy of relativistic fermions
Therefore for a given 119899 the energy eigenvalue is obtainedfrom (31) and the corresponding unnormalized wave func-tion can be obtained from (6) as
120595119863 equiv 120595119899119898 (119901 120579) = (1205951 (119901 120579)1205952 (119901 120579))
= ( 1119888119901+119864 + 11989801198882)1205951 (119901 120579) (33)
where
1205951 (119901 120579) = 119901119898119890minus(1198962)1199012119890119894119898120579 119899sum119895=0
119886119895119903119895 (34)
with 119903 = 1198961199012 Now for clarifying that the wave functionobtained from this method is in good agreement with theresults of [31] we calculate it for 119899 = 2 According to (28) and(32) the three-dimensional invariant subspace is obtained as
119865 (119903) = 1198860 + 1198861119903 + 11988621199032= 1198860 [1 + (minus12058124119896 + (12) (119898 + 1))119898 + 1 (1198961199012) + (minus12058124119896 + (12) (119898 + 1)) (minus12058124119896 + (12) (119898 + 1) + 1)(119898 + 1) (2 + 2 (119898 + 1)) (1198961199012)2] (35)
where by using the definition of the confluent hypergeometricfunction for 119899 = 2 we have119865 (119903) = 1198860 11198651 (minus119899119898 + 1 1198961199012) (36)
The above equation is exactly identical to equation (32) in [31]and so for other values of 119899 and after some calculations onecan show that 119865(119903) can be written in terms of the confluenthypergeometric function
4 Conclusions
Using the Lie algebraic approach we have solved the Diracoscillator in (2 + 1) dimension in the framework of non-commutative phase space We have obtained the energyeigenvalues and the corresponding wave functions throughthe sl(2) Lie algebra representation We have also shown thatour results are in good agreement with the results in [31] It isseen that the Lie algebraic approach is a powerful method forreproducing the exact analytical results
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] A Bermudez M A Martin-Delgado and A Luis ldquoChiralityquantum phase transition in the Dirac oscillatorrdquo PhysicalReview A vol 77 no 6 Article ID 063815 2008
[2] N Ferkous and A Bounames ldquoEnergy spectrum of a 2DDirac oscillator in the presence of the AharonovmdashBohm effectrdquoPhysics Letters A vol 325 no 1 pp 21ndash29 2004
[3] V M Villalba ldquoExact solution of the two-dimensional Diracoscillatorrdquo Physical Review A vol 49 no 1 586 pages 1994
[4] A Bermudez M A Martin-Delgado and E Solano ldquoExactmapping of the 2 + 1Dirac oscillator onto the Jaynes-Cummingsmodel Ion-trap experimental proposalrdquo Physical Review A vol76 no 4 Article ID 041801 2007
[5] M Moreno and A Zentella ldquoCovariance CPT and the Foldy-Wouthuysen transformation for the Dirac oscillatorrdquo Journal ofPhysics A Mathematical and General vol 22 no 17 L821 pages1989
[6] R de Lima Rodrigues ldquoOn the Dirac oscillatorrdquo Physics LettersA vol 372 no 15 pp 2587ndash2591 2008
[7] D Ito K Mori and E Carriere ldquoAn example of dynamicalsystems with linear trajectoryrdquo Il Nuovo Cimento A vol 51 no4 pp 1119ndash1121 1967
[8] M Moshinsky and A Szczepaniak ldquoThe Dirac oscillatorrdquoJournal of Physics A Mathematical and General vol 22 no 17L817 pages 1989
[9] E T Jaynes and FW Cummings ldquoComparison of quantum andsemiclassical radiation theories with application to the beammaserrdquo Proceedings of the IEEE vol 51 no 1 pp 89ndash109 1963
[10] J Larso ldquoDynamics of the JaynesndashCummings and Rabi modelsoldwine in newbottlesrdquoPhysica Scripta vol 76 no 2 146 pages2007
[11] L Allen and J H Eberly Optical Resonance and Two LevelAtoms Dover Publications Mineola New York USA 1987
[12] R P Martinez-y-Romero and A L Salas-Brito ldquoConformalinvariance in a Dirac oscillatorrdquo Journal of MathematicalPhysics vol 33 no 5 1831 pages 1992
[13] J Benitez R P Martnez y Romero H N Nuez-Yepez and AL Salas-Brito ldquoSolution and hidden supersymmetry of a Diracoscillatorrdquo Physical Review Letters vol 64 1643 no 14 1990
[14] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 no 09article 032 1999
[15] A Connes M R Douglas and A Schwarz ldquoNoncommutativegeometry and Matrix theoryrdquo Journal of High Energy Physicsvol 1998 003 no 02 1998
6 Advances in High Energy Physics
[16] M R Douglas and N A Nekrasov ldquoNoncommutative fieldtheoryrdquo Reviews of Modern Physics vol 73 no 4 977 pages2001
[17] C S Chu and P M Ho ldquoNon-commutative open string and D-branerdquo Nuclear Physics B vol 550 no 1-2 pp 151ndash168 1999
[18] C-S Chu and P-M Ho ldquoConstrained quantization of openstring in background B field and non-commutative D-branerdquoNuclear Physics B vol 568 no 1-2 pp 447ndash456 2000
[19] F Ardalan H Arfaei and M M Sheikh-Jabbari ldquoDiracquantization of open strings and noncommutativity in branesrdquoNuclear Physics B vol 576 no 1-3 pp 578ndash596 2000
[20] J Jing and Z-W Long ldquoOpen string in the constant B-fieldbackgroundrdquo Physical Review D vol 72 no 12 126002 pages2005
[21] I Hinchliffe N Kersting and Y L Ma ldquoReview of thephenomenology of noncommutative geometryrdquo InternationalJournal of Modern Physics A Particles and Fields GravitationCosmology vol 19 no 2 179 pages 2004
[22] SMinwalla M V Raamsdonk andN Seiberg ldquoNoncommuta-tive perturbative dynamicsrdquo Journal of High Energy Physics vol2000 020 no 02 2000
[23] M V Raamsdonk and N Seiberg ldquoComments on noncommu-tative perturbative dynamicsrdquo Journal of High Energy Physicsvol 2000 035 no 03 2000
[24] R Gopakumar S Minwalla and A Strominger ldquoNoncommu-tative solitonsrdquo Journal of High Energy Physics vol 2000 020no 05 2000
[25] S Cai T Jing G Guo and R Zhang ldquoDirac oscillator in non-commutative phase spacerdquo International Journal of TheoreticalPhysics vol 49 no 8 pp 1699ndash1705 2010
[26] O Bertolami and R Queiroz ldquoPhase-space noncommutativityand the Dirac equationrdquo Physics Letters A vol 375 no 46 pp4116ndash4119 2011
[27] Z-Y Luo Q Wang X Li and J Jing ldquoDirac oscillatorin noncommutative phase space and (anti)-Jaynes-Cummingsmodelsrdquo International Journal of Theoretical Physics vol 51 no7 pp 2143ndash2151 2012
[28] B P Mandal and S K Rai ldquoNoncommutative Dirac oscillatorin an external magnetic fieldrdquo Physics Letters A vol 376 no 36pp 2467ndash2470 2012
[29] C Bastos O Bertolami N C Dias and J N Prata ldquoNoncom-mutative graphenerdquo International Journal of Modern Physics Avol 28 no 16 Article ID 1350064 13 pages 2013
[30] O Panella and P Roy ldquoQuantum phase transitions in thenoncommutative Dirac oscillatorrdquo Physical Review A vol 90no 4 Article ID 042111 2014
[31] A Boumali and H Hassanabadi ldquoExact solutions of the (2 +1)-dimensional dirac oscillator under a magnetic field in thepresence of a minimal length in the non-commutative phasespacerdquo Zeitschrift fur Naturforschung A - A Journal of PhysicalSciences vol 70 no 8 619 pages 2015
[32] H Ciftci R L Hall and N Saad ldquoAsymptotic iteration methodfor eigenvalue problemsrdquo Journal of Physics A Mathematicaland General vol 36 no 47 11807 pages 2003
[33] S H Dong D Morales and J Garcia-Ravelo ldquoExact quanti-zation rule and its applications to physical potentialsrdquo Interna-tional Journal of Modern Physics E vol 16 no 1 p 189 2007
[34] G FWei and S H Dong ldquoAlgebraic approach to energy spectraof the Scarf type and generalized PoschlndashTeller potentialsrdquoCanadian Journal of Physics vol 89 no 12 pp 1225ndash1231 2011
[35] H Hassanabadi S Zarrinkamar and A A Rajabi ldquoExact solu-tions of D-dimensional Schrodinger equation for an energy-dependent potential by NUmethodrdquo Communications inTheo-retical Physics vol 55 no 4 pp 541ndash544 2011
[36] H Hassanabadi A A Rajabi and S Zarrinkamar ldquoCornelland kratzer potentials within the semirelativistic treatmentrdquoModern Physics Letters A vol 27 no 10 Article ID 12500572012
[37] A G Ushveridze ldquoLie-algebraic approach to the problem ofquasi-exact solubility in quantum mechanicsrdquo Modern PhysicsLetters A Particles and Fields Gravitation Cosmology andNuclear Physics vol 5 no 23 pp 1891ndash1899 1990
[38] A V Turbiner ldquoQuasi-exactly-solvable problems and sl(2)algebrardquo Communications in Mathematical Physics vol 118 no3 pp 467ndash474 1988
[39] A V Turbiner ldquoLie-algebras and linear operators with invariantsubspacesrdquo Contemporary Mathematics vol 160 p 263 1994
[40] D Gomez-Ullate N Kamran and R Milson ldquoQuasi-exactsolvability beyond the sl(2) algebraizationrdquo Physics of AtomicNuclei vol 70 no 3 pp 520ndash528 2007
[41] L Menculini O Panella and P Roy ldquoExact solutions of the (2+ 1) dimensional Dirac equation in a constant magnetic field inthe presence of aminimal lengthrdquo Physical ReviewD vol 87 no6 Article ID 065017 2013
[42] Y-Q Luo Y Cui Z-W Long and J Jing ldquo2 + 1 Dimensionalnoncommutative dirac oscillator and (anti)-jaynes-cummingsmodelsrdquo International Journal of Theoretical Physics vol 50 no10 pp 2992ndash3000 2011
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Advances in High Energy Physics 5
study here but one can see a similar study in [42] wherethe Dirac oscillator in 2 + 1-dimensional noncommutativespace has been studied for the cases of positive and negativenoncommutative parameter Hence it seems that the resultscan be used to study the connection with non-Hermitianquantum mechanics Also the possible case of imaginaryenergy gives useful clue regarding the lack of bound-states inthe spectroscopy of relativistic fermions
Therefore for a given 119899 the energy eigenvalue is obtainedfrom (31) and the corresponding unnormalized wave func-tion can be obtained from (6) as
120595119863 equiv 120595119899119898 (119901 120579) = (1205951 (119901 120579)1205952 (119901 120579))
= ( 1119888119901+119864 + 11989801198882)1205951 (119901 120579) (33)
where
1205951 (119901 120579) = 119901119898119890minus(1198962)1199012119890119894119898120579 119899sum119895=0
119886119895119903119895 (34)
with 119903 = 1198961199012 Now for clarifying that the wave functionobtained from this method is in good agreement with theresults of [31] we calculate it for 119899 = 2 According to (28) and(32) the three-dimensional invariant subspace is obtained as
119865 (119903) = 1198860 + 1198861119903 + 11988621199032= 1198860 [1 + (minus12058124119896 + (12) (119898 + 1))119898 + 1 (1198961199012) + (minus12058124119896 + (12) (119898 + 1)) (minus12058124119896 + (12) (119898 + 1) + 1)(119898 + 1) (2 + 2 (119898 + 1)) (1198961199012)2] (35)
where by using the definition of the confluent hypergeometricfunction for 119899 = 2 we have119865 (119903) = 1198860 11198651 (minus119899119898 + 1 1198961199012) (36)
The above equation is exactly identical to equation (32) in [31]and so for other values of 119899 and after some calculations onecan show that 119865(119903) can be written in terms of the confluenthypergeometric function
4 Conclusions
Using the Lie algebraic approach we have solved the Diracoscillator in (2 + 1) dimension in the framework of non-commutative phase space We have obtained the energyeigenvalues and the corresponding wave functions throughthe sl(2) Lie algebra representation We have also shown thatour results are in good agreement with the results in [31] It isseen that the Lie algebraic approach is a powerful method forreproducing the exact analytical results
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] A Bermudez M A Martin-Delgado and A Luis ldquoChiralityquantum phase transition in the Dirac oscillatorrdquo PhysicalReview A vol 77 no 6 Article ID 063815 2008
[2] N Ferkous and A Bounames ldquoEnergy spectrum of a 2DDirac oscillator in the presence of the AharonovmdashBohm effectrdquoPhysics Letters A vol 325 no 1 pp 21ndash29 2004
[3] V M Villalba ldquoExact solution of the two-dimensional Diracoscillatorrdquo Physical Review A vol 49 no 1 586 pages 1994
[4] A Bermudez M A Martin-Delgado and E Solano ldquoExactmapping of the 2 + 1Dirac oscillator onto the Jaynes-Cummingsmodel Ion-trap experimental proposalrdquo Physical Review A vol76 no 4 Article ID 041801 2007
[5] M Moreno and A Zentella ldquoCovariance CPT and the Foldy-Wouthuysen transformation for the Dirac oscillatorrdquo Journal ofPhysics A Mathematical and General vol 22 no 17 L821 pages1989
[6] R de Lima Rodrigues ldquoOn the Dirac oscillatorrdquo Physics LettersA vol 372 no 15 pp 2587ndash2591 2008
[7] D Ito K Mori and E Carriere ldquoAn example of dynamicalsystems with linear trajectoryrdquo Il Nuovo Cimento A vol 51 no4 pp 1119ndash1121 1967
[8] M Moshinsky and A Szczepaniak ldquoThe Dirac oscillatorrdquoJournal of Physics A Mathematical and General vol 22 no 17L817 pages 1989
[9] E T Jaynes and FW Cummings ldquoComparison of quantum andsemiclassical radiation theories with application to the beammaserrdquo Proceedings of the IEEE vol 51 no 1 pp 89ndash109 1963
[10] J Larso ldquoDynamics of the JaynesndashCummings and Rabi modelsoldwine in newbottlesrdquoPhysica Scripta vol 76 no 2 146 pages2007
[11] L Allen and J H Eberly Optical Resonance and Two LevelAtoms Dover Publications Mineola New York USA 1987
[12] R P Martinez-y-Romero and A L Salas-Brito ldquoConformalinvariance in a Dirac oscillatorrdquo Journal of MathematicalPhysics vol 33 no 5 1831 pages 1992
[13] J Benitez R P Martnez y Romero H N Nuez-Yepez and AL Salas-Brito ldquoSolution and hidden supersymmetry of a Diracoscillatorrdquo Physical Review Letters vol 64 1643 no 14 1990
[14] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 no 09article 032 1999
[15] A Connes M R Douglas and A Schwarz ldquoNoncommutativegeometry and Matrix theoryrdquo Journal of High Energy Physicsvol 1998 003 no 02 1998
6 Advances in High Energy Physics
[16] M R Douglas and N A Nekrasov ldquoNoncommutative fieldtheoryrdquo Reviews of Modern Physics vol 73 no 4 977 pages2001
[17] C S Chu and P M Ho ldquoNon-commutative open string and D-branerdquo Nuclear Physics B vol 550 no 1-2 pp 151ndash168 1999
[18] C-S Chu and P-M Ho ldquoConstrained quantization of openstring in background B field and non-commutative D-branerdquoNuclear Physics B vol 568 no 1-2 pp 447ndash456 2000
[19] F Ardalan H Arfaei and M M Sheikh-Jabbari ldquoDiracquantization of open strings and noncommutativity in branesrdquoNuclear Physics B vol 576 no 1-3 pp 578ndash596 2000
[20] J Jing and Z-W Long ldquoOpen string in the constant B-fieldbackgroundrdquo Physical Review D vol 72 no 12 126002 pages2005
[21] I Hinchliffe N Kersting and Y L Ma ldquoReview of thephenomenology of noncommutative geometryrdquo InternationalJournal of Modern Physics A Particles and Fields GravitationCosmology vol 19 no 2 179 pages 2004
[22] SMinwalla M V Raamsdonk andN Seiberg ldquoNoncommuta-tive perturbative dynamicsrdquo Journal of High Energy Physics vol2000 020 no 02 2000
[23] M V Raamsdonk and N Seiberg ldquoComments on noncommu-tative perturbative dynamicsrdquo Journal of High Energy Physicsvol 2000 035 no 03 2000
[24] R Gopakumar S Minwalla and A Strominger ldquoNoncommu-tative solitonsrdquo Journal of High Energy Physics vol 2000 020no 05 2000
[25] S Cai T Jing G Guo and R Zhang ldquoDirac oscillator in non-commutative phase spacerdquo International Journal of TheoreticalPhysics vol 49 no 8 pp 1699ndash1705 2010
[26] O Bertolami and R Queiroz ldquoPhase-space noncommutativityand the Dirac equationrdquo Physics Letters A vol 375 no 46 pp4116ndash4119 2011
[27] Z-Y Luo Q Wang X Li and J Jing ldquoDirac oscillatorin noncommutative phase space and (anti)-Jaynes-Cummingsmodelsrdquo International Journal of Theoretical Physics vol 51 no7 pp 2143ndash2151 2012
[28] B P Mandal and S K Rai ldquoNoncommutative Dirac oscillatorin an external magnetic fieldrdquo Physics Letters A vol 376 no 36pp 2467ndash2470 2012
[29] C Bastos O Bertolami N C Dias and J N Prata ldquoNoncom-mutative graphenerdquo International Journal of Modern Physics Avol 28 no 16 Article ID 1350064 13 pages 2013
[30] O Panella and P Roy ldquoQuantum phase transitions in thenoncommutative Dirac oscillatorrdquo Physical Review A vol 90no 4 Article ID 042111 2014
[31] A Boumali and H Hassanabadi ldquoExact solutions of the (2 +1)-dimensional dirac oscillator under a magnetic field in thepresence of a minimal length in the non-commutative phasespacerdquo Zeitschrift fur Naturforschung A - A Journal of PhysicalSciences vol 70 no 8 619 pages 2015
[32] H Ciftci R L Hall and N Saad ldquoAsymptotic iteration methodfor eigenvalue problemsrdquo Journal of Physics A Mathematicaland General vol 36 no 47 11807 pages 2003
[33] S H Dong D Morales and J Garcia-Ravelo ldquoExact quanti-zation rule and its applications to physical potentialsrdquo Interna-tional Journal of Modern Physics E vol 16 no 1 p 189 2007
[34] G FWei and S H Dong ldquoAlgebraic approach to energy spectraof the Scarf type and generalized PoschlndashTeller potentialsrdquoCanadian Journal of Physics vol 89 no 12 pp 1225ndash1231 2011
[35] H Hassanabadi S Zarrinkamar and A A Rajabi ldquoExact solu-tions of D-dimensional Schrodinger equation for an energy-dependent potential by NUmethodrdquo Communications inTheo-retical Physics vol 55 no 4 pp 541ndash544 2011
[36] H Hassanabadi A A Rajabi and S Zarrinkamar ldquoCornelland kratzer potentials within the semirelativistic treatmentrdquoModern Physics Letters A vol 27 no 10 Article ID 12500572012
[37] A G Ushveridze ldquoLie-algebraic approach to the problem ofquasi-exact solubility in quantum mechanicsrdquo Modern PhysicsLetters A Particles and Fields Gravitation Cosmology andNuclear Physics vol 5 no 23 pp 1891ndash1899 1990
[38] A V Turbiner ldquoQuasi-exactly-solvable problems and sl(2)algebrardquo Communications in Mathematical Physics vol 118 no3 pp 467ndash474 1988
[39] A V Turbiner ldquoLie-algebras and linear operators with invariantsubspacesrdquo Contemporary Mathematics vol 160 p 263 1994
[40] D Gomez-Ullate N Kamran and R Milson ldquoQuasi-exactsolvability beyond the sl(2) algebraizationrdquo Physics of AtomicNuclei vol 70 no 3 pp 520ndash528 2007
[41] L Menculini O Panella and P Roy ldquoExact solutions of the (2+ 1) dimensional Dirac equation in a constant magnetic field inthe presence of aminimal lengthrdquo Physical ReviewD vol 87 no6 Article ID 065017 2013
[42] Y-Q Luo Y Cui Z-W Long and J Jing ldquo2 + 1 Dimensionalnoncommutative dirac oscillator and (anti)-jaynes-cummingsmodelsrdquo International Journal of Theoretical Physics vol 50 no10 pp 2992ndash3000 2011
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
6 Advances in High Energy Physics
[16] M R Douglas and N A Nekrasov ldquoNoncommutative fieldtheoryrdquo Reviews of Modern Physics vol 73 no 4 977 pages2001
[17] C S Chu and P M Ho ldquoNon-commutative open string and D-branerdquo Nuclear Physics B vol 550 no 1-2 pp 151ndash168 1999
[18] C-S Chu and P-M Ho ldquoConstrained quantization of openstring in background B field and non-commutative D-branerdquoNuclear Physics B vol 568 no 1-2 pp 447ndash456 2000
[19] F Ardalan H Arfaei and M M Sheikh-Jabbari ldquoDiracquantization of open strings and noncommutativity in branesrdquoNuclear Physics B vol 576 no 1-3 pp 578ndash596 2000
[20] J Jing and Z-W Long ldquoOpen string in the constant B-fieldbackgroundrdquo Physical Review D vol 72 no 12 126002 pages2005
[21] I Hinchliffe N Kersting and Y L Ma ldquoReview of thephenomenology of noncommutative geometryrdquo InternationalJournal of Modern Physics A Particles and Fields GravitationCosmology vol 19 no 2 179 pages 2004
[22] SMinwalla M V Raamsdonk andN Seiberg ldquoNoncommuta-tive perturbative dynamicsrdquo Journal of High Energy Physics vol2000 020 no 02 2000
[23] M V Raamsdonk and N Seiberg ldquoComments on noncommu-tative perturbative dynamicsrdquo Journal of High Energy Physicsvol 2000 035 no 03 2000
[24] R Gopakumar S Minwalla and A Strominger ldquoNoncommu-tative solitonsrdquo Journal of High Energy Physics vol 2000 020no 05 2000
[25] S Cai T Jing G Guo and R Zhang ldquoDirac oscillator in non-commutative phase spacerdquo International Journal of TheoreticalPhysics vol 49 no 8 pp 1699ndash1705 2010
[26] O Bertolami and R Queiroz ldquoPhase-space noncommutativityand the Dirac equationrdquo Physics Letters A vol 375 no 46 pp4116ndash4119 2011
[27] Z-Y Luo Q Wang X Li and J Jing ldquoDirac oscillatorin noncommutative phase space and (anti)-Jaynes-Cummingsmodelsrdquo International Journal of Theoretical Physics vol 51 no7 pp 2143ndash2151 2012
[28] B P Mandal and S K Rai ldquoNoncommutative Dirac oscillatorin an external magnetic fieldrdquo Physics Letters A vol 376 no 36pp 2467ndash2470 2012
[29] C Bastos O Bertolami N C Dias and J N Prata ldquoNoncom-mutative graphenerdquo International Journal of Modern Physics Avol 28 no 16 Article ID 1350064 13 pages 2013
[30] O Panella and P Roy ldquoQuantum phase transitions in thenoncommutative Dirac oscillatorrdquo Physical Review A vol 90no 4 Article ID 042111 2014
[31] A Boumali and H Hassanabadi ldquoExact solutions of the (2 +1)-dimensional dirac oscillator under a magnetic field in thepresence of a minimal length in the non-commutative phasespacerdquo Zeitschrift fur Naturforschung A - A Journal of PhysicalSciences vol 70 no 8 619 pages 2015
[32] H Ciftci R L Hall and N Saad ldquoAsymptotic iteration methodfor eigenvalue problemsrdquo Journal of Physics A Mathematicaland General vol 36 no 47 11807 pages 2003
[33] S H Dong D Morales and J Garcia-Ravelo ldquoExact quanti-zation rule and its applications to physical potentialsrdquo Interna-tional Journal of Modern Physics E vol 16 no 1 p 189 2007
[34] G FWei and S H Dong ldquoAlgebraic approach to energy spectraof the Scarf type and generalized PoschlndashTeller potentialsrdquoCanadian Journal of Physics vol 89 no 12 pp 1225ndash1231 2011
[35] H Hassanabadi S Zarrinkamar and A A Rajabi ldquoExact solu-tions of D-dimensional Schrodinger equation for an energy-dependent potential by NUmethodrdquo Communications inTheo-retical Physics vol 55 no 4 pp 541ndash544 2011
[36] H Hassanabadi A A Rajabi and S Zarrinkamar ldquoCornelland kratzer potentials within the semirelativistic treatmentrdquoModern Physics Letters A vol 27 no 10 Article ID 12500572012
[37] A G Ushveridze ldquoLie-algebraic approach to the problem ofquasi-exact solubility in quantum mechanicsrdquo Modern PhysicsLetters A Particles and Fields Gravitation Cosmology andNuclear Physics vol 5 no 23 pp 1891ndash1899 1990
[38] A V Turbiner ldquoQuasi-exactly-solvable problems and sl(2)algebrardquo Communications in Mathematical Physics vol 118 no3 pp 467ndash474 1988
[39] A V Turbiner ldquoLie-algebras and linear operators with invariantsubspacesrdquo Contemporary Mathematics vol 160 p 263 1994
[40] D Gomez-Ullate N Kamran and R Milson ldquoQuasi-exactsolvability beyond the sl(2) algebraizationrdquo Physics of AtomicNuclei vol 70 no 3 pp 520ndash528 2007
[41] L Menculini O Panella and P Roy ldquoExact solutions of the (2+ 1) dimensional Dirac equation in a constant magnetic field inthe presence of aminimal lengthrdquo Physical ReviewD vol 87 no6 Article ID 065017 2013
[42] Y-Q Luo Y Cui Z-W Long and J Jing ldquo2 + 1 Dimensionalnoncommutative dirac oscillator and (anti)-jaynes-cummingsmodelsrdquo International Journal of Theoretical Physics vol 50 no10 pp 2992ndash3000 2011
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
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![KXUUø Ahep A6p ¤ù }Éh - Vikersund Utleiesenter · 2020. 3. 27. · KXUUø Ahep A6p ¤ù }Éh KXUU hp}Éh AühA¹ p} pAüÉhÁÉÉ""ùAh}]h 6üÉ}}Éh¤ 66p ÁÉh×}]h 6ü ¤ ¹°ü66üÉhÉ""Éh}]hh"](https://static.fdocuments.in/doc/165x107/60b970c6b410112e3e0590d2/kxuu-ahep-a6p-h-vikersund-utleiesenter-2020-3-27-kxuu-ahep-a6p.jpg)
