Analysis of the Dirac equation with the Killingbeck potential in non-commutative space
Lan Zhong, Hao Chen, Qi-Kang Ran, Chao-Yun Long, Zheng-Wen Long
AAnalysis of the Dirac equation with the Killingbeck potential innon-commutative space
Lan Zhong , Hao Chen , Qi-Kang Ran , Chao-Yun Long and Zheng-Wen Long ∗ College of Physics, Guizhou University, Guiyang, 550025, China. College of Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, China.
Abstract . In this paper, we investigate the Dirac equation with the Killingbeck potential underthe external magnetic field in non-commutative space. Corresponding to the expressions of theenergy level and wave functions in spin symmetry limit and pseudo-spin symmetry limit are derivedby using the Bethe ansatz method. The parameter B associated with the external magnetic fieldand non-commutative parameter θ make to modify the energy level for considered systems. Keywords:
Dirac equation, non-commutative space, Killingbeck potential.
I. INTRODUCTION
Killingbeck potential with form − r +2 λr +2 λ r was introduced by Killingbeck [1] in studying a polynomialperturbation problem about hydrogen atom. The term − r represents the gluon exchange potential while thelast two terms are for confinement. The Killingbeck potential rewritten as V ( r ) = ar + br − qr has widelystudied in various fields of physics, such as atomic and molecular physics [2], particle physics [3, 4]. Thereare some investigations about the Dirac equation for Killingbeck potential in Refs [5, 6]. In the relativisticquantum mechanics, the Dirac equation takes an important part to describe the motion of spin- particles.As one solvable model, there are many ways to study, including: Nikiforov-Uvarov (NU) method [7–9], theasymptotic iteration method (AIM) [10, 11] and supersymmetry quantum mechanics (SUSYQM) [12]. If doingmomentum transformation, it is well-known as Dirac oscillator [13], which is a typical system investigated inthe curved space [14–18], and the effect of generalized uncertainty principle on spin- Dirac oscillator andspin-0 or 1 DKP oscillator also are addressed [19–21].Noncommutativity is an interesting topic whatever in mathematics and physics. The first introduction ofnoncommutative coordinates appeared in the problem of ultraviolet divergence solved by Snyder [22], thiswork kept the Lorentz covariance better but it didn’t get concerns. With the development about string theory[23, 24], the studies about the low energy effective theory of D-brane with B field showed the noncommuta-tivity among the coordinates on D-brane worldvolume [25, 26]. In 2000, Shiraz Minwalla et al [27] studiedthe perturbative dynamics of noncommutative field theories and succeed to explain the appearance of UV/IRmixing in noncommutativity. The effect of the noncommutativity on quantum systems have been widely re-ported such as the spin Hall effect on noncommutative space [28], the noncommutative corrections on thepersistent charged current [29], the Landau problem [30] and the HMW effect [31]. The Dirac equation innoncommutative space for hydrogen atom was considered in Ref[32], this work showed the degeneracy of someenergy levels was completely lifted in the NC space. In addition, the Aharonov-Bohm effect for a relativisticspin-half particle have also been analyzed [33, 34]. Therefore, it seems interesting to study the Dirac equationwith the external magnetic in non-commutative space by considering the Killingbeck potential.The rest of essays are composed as follow. In the next section, we briefly review the property of non-commutative space. In Section 3, we first derive the general expression of the Dirac equation with scalarand vector potentials and then study the two cases in spin symmetry limit and pseudo-spin symmetry limit,finally discuss the energy levels for investigated systems. The conclusion is presented in Section 4.
II. NON-COMMUTATIVE SPACE
In non-commutative space, the commutative relations between coordinate operator and momentum operatorare given by( (cid:126) = c = 1) [ˆ x i , ˆ x j ] = iθ ij , [ˆ p i , ˆ p j ] = 0 , [ˆ x i , ˆ p j ] = iδ ij ( i, j = 1 , , (1) ∗ [email protected] a r X i v : . [ h e p - t h ] J a n where ˆ x i and ˆ p i are coordinate operator and momentum operator in non-commutative space, respectively. θ ij = θε ij represents the anti-symmetric matrix, θ is non-commutative parameter. The Moyal-Weyl productwhich is a way to deal with the problem in non-commutative quantum physics reads( f ∗ g ) = exp( i θ ij ∂ x i ∂ x j ) f ( x i ) g ( x j ) , (2)here f ( x ) and g ( x ) are two arbitrary functions. The Moyal-Weyl product can transform to usual product byBopp shift[35] and the corresponding expression isˆ x i = x i − θ ij p j , ˆ p i = p i . (3)with x i , p i are the coordinate operator and momentum operator in usual quantum mechanics, respectively. III. THE DIRAC EQUATION WITH SCALAR AND VECTOR POTENTIALS
The Dirac equation with the potentials is [36, 37][ (cid:126)α · (cid:126)p + β ( M + S (ˆ r ))]Ψ( (cid:126)r ) = [ E − V (ˆ r )]Ψ( (cid:126)r ) . (4)Note that S (ˆ r ) is scalar potential while V (ˆ r ) is vector potential. (cid:126)α and β are Dirac matrices (cid:126)α i = (cid:18) (cid:126)σ i (cid:126)σ i (cid:19) , (5) β = (cid:18) I − I (cid:19) . (6)with (cid:126)σ i are Pauli’s 2 × × (cid:126)r ) as wave function has formΨ( (cid:126)r ) = (cid:0) ϕ ( (cid:126)r ) χ ( (cid:126)r ) (cid:1) T . (7)Combination Eq. (4),
Eq. (5),
Eq. (6) and
Eq. (7), one obtains (cid:126)σ · (cid:126)pχ ( (cid:126)r ) = ( E − M − V (ˆ r ) − S (ˆ r )) ϕ ( (cid:126)r ) , (8) (cid:126)σ · (cid:126)pϕ ( (cid:126)r ) = ( E + M − V (ˆ r ) + S (ˆ r )) χ ( (cid:126)r ) . (9)Motivated by the Ref [38], taking the exact spin symmetry limit ∆(ˆ r ) = V (ˆ r ) − S (ˆ r ) = 0 and the exactpseudo-spin symmetry limit Σ(ˆ r ) = V (ˆ r ) + S (ˆ r ) = 0, we have (cid:126)σ · (cid:126)pχ ( (cid:126)r ) = ( E − M − V (ˆ r )) ϕ ( (cid:126)r ) ,(cid:126)σ · (cid:126)pϕ ( (cid:126)r ) = ( E + M ) χ ( (cid:126)r ) , (10) (cid:126)σ · (cid:126)pχ ( (cid:126)r ) = ( E − M ) ϕ ( (cid:126)r ) ,(cid:126)σ · (cid:126)pϕ ( (cid:126)r ) = ( E + M − V (ˆ r )) χ ( (cid:126)r ) . (11) A. The case of ∆(ˆ r ) = 0 Along the Z − axis , the Dirac equation with uniform magnetic filed B is as follow [39][( (cid:126)p − ec (cid:126)A ( NC ) ) + 2( E + M ) V (ˆ r )] ϕ ( (cid:126)r ) = [ E − M ] ϕ ( (cid:126)r ) . (12) (cid:126)A is vector potential can be shown (cid:126)A = (cid:126)A + (cid:126)A , (cid:126)B = B ˆ Z , (cid:126)p is replaced by (cid:126)p → (cid:126)p − ec (cid:126)A ( r ) with (cid:126)p = − i (cid:126) (cid:126) ∇ .The potential in the non-commutative space is given by [40–42] V (ˆ r ) = V ( r ) + 12 ( (cid:126)θ × (cid:126)p ) · (cid:126) ∇ V ( r ) + O ( θ ) = V ( r ) − (cid:126)L · (cid:126)θ r ∂V∂r + O ( θ ) . (13)So the vector potential under the additional magnetic flux AB [43] has became (cid:126)A NC ) = (cid:126)B × (cid:126)r B ˆ r ϕ = B r − L z θ r ) ˜ ϕ, (14) (cid:126)A NC ) = Φ AB π ˆ r (cid:126)ϕ = Φ AB π ( 1 r + L z θ r ) ˜ ϕ, (15) (cid:126)A ( NC ) = ( Br AB πr − BL z θ r + L z θ Φ AB πr ) ˜ ϕ. (16)As same way, in NC space, Killingbeck potential is given by V (ˆ r ) = ar + br − ( q + L z θb r − L z θq r − aL z θ + O ( θ ) . (17)Combination Eq. (12) and
Eq. (16), in cylindrical coordinates, there is {− [ ∂ ∂r + r ∂∂r + r ∂ ∂ϕ ] + e c ( Br + Φ AB πr − BL z θ r + L z θ Φ AB πr ) − ieBL z θ cr ∂∂ϕ + ieBc ∂∂ϕ + ie Φ AB πcr ∂∂ϕ + ieL z θ Φ AB πr ∂∂ϕ + 2( E + M ) V (ˆ r ) + M − E } ϕ ( (cid:126)r ) = 0 . (18)If we set ϕ ( (cid:126)r ) = e ( imϕ ) r − U ( r ) , (19)inserting Eq. (19) into
Eq. (18), and considering
Eq. (17), one obtains { d dr + ( − m − e Φ AB π c + e Φ AB mπc − eBL z θm c ) r + ( − e B c − E + M ) a ) r − E + M ) br + ( eL z θm Φ AB πc − e L z θ Φ AB c π ) r + ( E + M ) L z θq r + 2( E + M )( q + L z θb ) r + e B L z θ c − e B Φ AB c π + eBmc + 2( E + M ) aL z θ + E − M } U ( r )= 0 . (20)Simplifing Eq. (20) can get { d dr + ζ r + ζ r + ζ r + ζ r − ζ r − ζ r + ζ } U ( r ) = 0 , (21)with the notations ζ = 2( E + M )( q + L z θb ) ,ζ = − m − e Φ AB π c + e Φ AB mπc − eBL z θm c ,ζ = ( E + M ) L z θq,ζ = eL z θm Φ AB πc − e L z θ Φ AB c π ,ζ = 2( E + M ) b,ζ = e B c + 2( E + M ) a,ζ = e B L z θ c − e B Φ AB c π + eBmc + 2( E + M ) aL z θ + E − M . (22)Introducing the transformation U ( r ) = exp( a r + a r + a r ) υ ( r ) , (23)we put Eq. (23) into
Eq. (21) and using the Bethe ansatz method[44, 45], we obtain υ ( r ) = n (cid:89) i ( r − r i ) , υ ( r ) = 1 f or n = 1 , (24) E B n=1 n=2 n=3 Fig. 1.
Energy eigenfunctions E for different magnetic filed B with values of the quantum numbers n (1 , , e = c = L z = m = 1 , Φ AB = 2 , M = 5 , a = 0 . , b = 0 . , θ = 0 . . E B=1 B=1.2 B=1.4
Fig. 2.
Energy eigenfunctions E for different non-commutative parameter θ with values of the magnetic filed B (1,1.2,1.4), e = c = L z = n = m = 1 , Φ AB = 2 , M = 5 , a = 0 . , b = 0 .
007 . where a + ζ = 0 , a a − ζ = 0 , a − ζ = 0 . (25)Combination Eq. (22) and
Eq. (25), the expression of energy can be got − (cid:113) e B c + 2( E + M ) a ( n + ) + e B L z θ c − e B Φ AB c π + E − M + eBmc + 2( E + M ) aL z θ + ( E + M ) b e B c +2( E + M ) a = 0 . (26)This is the implicit energy level expression in the case of spin limit with the external magnetic in NC space.In order to analyze the above result, we consider to plot the positive energy eigenvalues E versus the magneticfiled B , the non-commutative parameter θ and potential function parameter a , respectively. In the Fig.1, itis about energy eigenvalues and magnetic filed B . It directly shows that the energy eigenvalues increase withthe increase of the magnetic filed B in the case of different values of quantum member n . The Fig.2 and Fig.3show the variable trend of E via θ and a , respectively. There is the decrease trend of the energy eigenvalueswith the large of non-commutative parameter in the Fig.2 while the E has increase trend with a in the Fig.3.And the corresponding wave function is given by ϕ ( (cid:126)r ) = exp ( imϕ ) r − exp( a r + a r + a r ) υ ( r ) . (27) E a n=1 n=2 n=3 Fig. 3.
Energy eigenfunctions E for different potential function parameter a with values of the quantum numbers n (1 , , B = e = c = L z = m = 1 , Φ AB = 2 , M = 5 , b = 0 . , θ = 0 .
001 .
B. The case of
Σ(ˆ r ) = 0 In this case, the pseudo-spin symmetry limit V (ˆ r ) = − S (ˆ r ) is considered. So Eq. (11) under the externalmagnetic reads [( (cid:126)p − ec (cid:126)A NC ) + 2( E − M ) V (ˆ r )] χ ( (cid:126)r ) = [ E − M ] χ ( (cid:126)r ) . (28)Combination Eq. (16) and the above equation in cylindrical coordinate, we obtain {− [ ∂ ∂r + r ∂∂r + r ∂ ∂ϕ ] + e c ( Br + Φ AB πr − BL z θ r + L z θ Φ AB πr ) − ieBL z θ cr ∂∂ϕ + ieBc ∂∂ϕ + ie Φ AB πcr ∂∂ϕ + ieL z θ Φ AB πr ∂∂ϕ + 2( E − M ) V (ˆ r ) + M − E } χ ( (cid:126)r ) = 0 . (29)Taking Eq. (17) into
Eq. (29) and doing the transformation χ ( (cid:126)r ) = e ( ilϕ ) r − τ ( r ), one obtains { d dr + ( − l − e Φ AB π c + e Φ AB lπc − eBL z θl c ) r + ( − e B c − E − M ) a ) r − E − M ) br + ( eL z θl Φ AB πc − e L z θ Φ AB c π ) r + ( E − M ) L z θq r + 2( E − M )( q + L z θb ) r + e B L z θ c − e B Φ AB c π + eBlc + 2( E − M ) aL z θ + E − M } τ ( r )= 0 . (30)The above equation is rewrriten { d dr + ξ r + ξ r + ξ r + ξ r − ξ r − ξ r + ξ } τ ( r ) = 0 , (31)with the new parameters ξ = 2( E − M )( q + L z θb ) ,ξ = − l − e Φ AB π c + e Φ AB lπc − eBL z θl c ,ξ = ( E − M ) L z θq,ξ = eL z θl Φ AB πc − e L z θ Φ AB c π ,ξ = 2( E − M ) b,ξ = e B c + 2( E − M ) a,ξ = e B L z θ c − e B Φ AB c π + eBmc + 2( E − M ) aL z θ + E − M . (32)We set the auxiliary function τ ( r ) = exp( b r + b r + b r ) ν ( r ) , (33) E B n=1 n=2 n=3 Fig. 4.
Energy eigenfunctions E for different magnetic filed B with values of the quantum numbers n (1 , , e = c = L z = l = 1 , Φ AB = 2 , M = 5 , a = 0 . , b = 0 . , θ = 0 . . . E B=1 B=1.2 B=1.4
Fig. 5.
Energy eigenfunctions E for different non-commutative parameter θ with values of the magnetic filed B (1,1.2,1.4), e = c = L z = n = l = 1 , Φ AB = 2 , M = 5 , a = 0 . , b = 0 .
007 . putting
Eq. (33) into
Eq. (31) and using the Bethe ansatz method, we obtain ν ( r ) = n (cid:89) i ( r − r i ) , ν ( r ) = 1 f or n = 1 , (34)with b + ξ = 0 , b b − ξ = 0 , b − ξ = 0 . (35)Combination Eq. (32) and
Eq. (35), the implicit expression of E is given by − (cid:113) e B c + 2( E − M ) a ( n + ) + e B L z θ c − e B Φ AB c π + eBlc +2( E − M ) aL z θ + E − M + ( E − M ) b e B c +2( E − M ) a = 0 . (36)Take similar ways, we respectively plot the positive energy eigenvalues E versus the magnetic filed B , thenon-commutative parameters θ and potential function parameter a under pseudo-spin symmetry. In the Fig.4and Fig.6, they both show the increase trend of the energy eigenvalues with the large of magnetic filed B and potential function parameter a , respectively. In the Fig.5, there is the energy decreases with the non-commutative parameter. And the wave function is given by χ ( (cid:126)r ) = exp ( ilϕ ) r − exp( b r + b r + b r ) ν ( r ) . (37) E a n=1 n=2 n=3 Fig. 6.
Energy eigenfunctions E for different potential function parameter a with values of the quantum numbers n (1 , , B = e = c = L z = l = 1 , Φ AB = 2 , M = 5 , b = 0 . , θ = 0 .
001 .
IV. CONCLUSION
In this work, we have investigated the Dirac equation in the presence of the external magnetic field with theKillingbeck potential in non-commutative space and obtain the implicit expression about energy. Some plotsof energy eigenvalues under different magnetic field B , potential function parameters a and non-commutativeparameters θ in spin symmetry limit and pseudo-spin symmetry limit have been analyzed in detail. On theone hand, in both cases, the energy eigenvalues respectively show increase trend with the increase of themagnetic field B and potential function parameter a while decrease trend with non-commutative parameter θ , on the other hand there are different values of energy between the case of spin symmetry limit and the caseof pseudo-spin symmetry limit. Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant nos. 11465006 and11565009) and the Major Research Project of innovative Group of Guizhou province (2018-013). [1] J. Killingbeck, Phys. Lett. A 65, 87 (1978).[2] J. Killingbeck, J. Phys. A: Math. Gen 13, L393 (1980).[3] E. R. Vrscay, Phys. Rev. A 31, 2054 (1985).[4] E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane, T. M. Yan, Phys. Rev. D 17, 3090 (1978).[5] Z. Sharifi, F. Tajic, M. Hamzavi, S. M. Ikhdair, Z. Naturforsch. 70a, 499 (2015).[6] M. Eshghi, H. Mehraban, S. M. Ikhdair, Eur. Phys. J. A. 52, 201 (2016).[7] E. Maghsoodi, H. Hassanabadi, S. Zarrinkamar, Chin. Phys. B 22, 030302 (2013).[8] H. Chen, Z. W. Long, Y. Yang, Z. L. Zhao, C. Y. Long, Int. J. Mod .Phys A 35, 2050107 (2020).[9] H. Hassanabadi, E. Maghsoodi, S. Zarrinkamar, Commun. Theor. Phys. 58, 807 (2012).[10] M. Hamzavi, A. A. Rajabi, H. Hassanabadi, Phys. Lett. A 374, 4303 (2010).[11] O. Aydo˘ g du, R. Sever, Ann. Phys. 325, 373 (2010).[12] H. Hassanabadi, E. Maghsoodi, S. Zarrinkamar, Eur. Phys. J. Plus 127, 31 (2012).[13] D. Itˆ o , K. Mori, E. Carriere, Nuovo Cimento A 51, 4 (1967).[14] M. Hosseinpour, H. Hassanabadi, M. de. Montigny, Eur. Phys. J. C 79, 311 (2019).[15] K. Bakke, H. Mota, Eur. Phys. J. Plus, 133, 409 (2018).[16] K. Bakke, Gen. Relat. Gravit. 45, 1847 (2013).[17] F. Ahmed, Ann. Phys. 415, 168113 (2020).[18] H. Chen, Z. W. Long, Y. Yang, C. Y. Long, Mod. Phys. Lett. A 35, 2050179 (2020).[19] K. Nouicer, J. Phys. A 39, 5125 (2006).[20] L. B. Castro, A. E. Obispo, J. Phys. A Math. Theor. 50, 285202 (2017).[21] H. Chen, Z. W. Long, Z. L. Zhao, C. Y. Long, Few-Body Syst. 61, 11 (2020).[22] H. S. Snyder, Phys. Rev. 71, 38 (1947).[23] N. Seiberg, E. Witten, JHEP 09, 032 (1999). [24] N. Seiberg, L. Susskind, N. Toumbas, JHEP 06, 021 (2000).[25] C. S. Chu, P. M. Ho, Nucl. Phys. B 550, 151 (1999).[26] C. S. Chu, P. M. Ho, Nucl. Phys. B 568, 447 (2000).[27] S. Minwalla, M. V. Raamsdonk, N. Seiberg, JHEP 02, 020 (2000).[28] K. Ma, S. Dulat, Phys. Rev. A 84, 012104 (2011).[29] K. Ma, Y. J. Ren, Y. H. Wang, Phys. Rev. D 97, 115011 (2018).[30] J. Gamboa, F. M´ ee