ResearchArticle The DKP Oscillator in Spinning Cosmic ... · The DKP Oscillator in Spinning Cosmic String Background MansourehHosseinpour andHassanHassanabadi...

Research ArticleThe DKP Oscillator in Spinning Cosmic String Background

Mansoureh Hosseinpour and Hassan Hassanabadi

Faculty of Physics Shahrood University of Technology Shahrood PO Box 3619995161-316 Iran

Correspondence should be addressed to Mansoureh Hosseinpour hosseinpourmansourehgmailcom

Received 11 June 2018 Revised 3 September 2018 Accepted 7 November 2018 Published 30 December 2018

Academic Editor Diego Saez-Chillon Gomez

Copyright copy 2018 Mansoureh Hosseinpour and Hassan Hassanabadi This is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited The publication of this article was funded by SCOAP3

In this article we investigate the behaviour of relativistic spin-zero bosons in the space-time generated by a spinning cosmic stringWe obtain the generalized beta-matrices in terms of the flat space-time ones and rewrite the covariant form of Duffin-Kemmer-Petiau (DKP) equation in spinning cosmic string space-time We find the solution of DKP oscillator and determine the energylevels We also discuss the influence of the topology of the cosmic string on the energy levels and the DKP spinors

1 Introduction

The Duffin-Kemmer-Petiau (DKP) equation has been usedto describe relativistic spin-0 and spin-1 bosons [1ndash4] TheDKP equation has five- and ten-dimensional representationrespectively for spin-0 and spin-1 bosons [5] This equationis compared to the Dirac equation for fermions [6] TheDKP equation has been widely investigated in many areasof physics The DKP equation has been investigated in themomentum space with the presence of minimal length [7 8]and for spins 0 and 1 in a noncommutative space [9ndash12]Also the DKP oscillator has been studied in the presenceof topological defects [13] Recently there has been growinginterest in the so-called DKP oscillator [14ndash23] in particularin the background of a magnetic cosmic string [13] Thecosmic strings and other topological defects can form at acosmological phase transition [24 25] The conical natureof the space-time around the string causes a number ofinteresting physical effects Until now some problems havebeen investigated in the gravitational fields of topologicaldefects including the one-electron atom problem [26ndash28]Spinning cosmic strings are similar usual cosmic stringcharacterized by an angular parameter 120572 that depends ontheir linearmass density120583TheDKPoscillator is described byperforming the nonminimal coupling with a linear potentialThe name distinguishes it from the system called a DKPoscillator with Lorentz tensor couplings of [7ndash12 14ndash16] TheDKP oscillator for spin-0 bosons has been investigated by

Guo et al in [10] in noncommutative phase space The DKPoscillator with spin-0 has been studied by Yang et al [11]Exact solution of DKP oscillator in the momentum spacewith the presence of minimal length has been analysed in[8] De Melo et al construct the Galilean DKP equationfor the harmonic oscillator in a noncommutative phasespace [29] Falek and Merad investigated the DKP oscillatorof spins 0 and 1 bosons in noncommutative space [9]Recently there has been an increasing interest on the DKPoscillator [13ndash16 22 29ndash31] The nonrelativistic limit ofparticle dynamics in curved space-time is considered in [32ndash36] Also the dynamics of relativistic bosons and fermions incurved space-time is considered in [17 20 31]

The influence of topological defect in the dynamics ofbosons via DKP formalism has not been established forspinning cosmic strings In thiswaywe consider the quantumdynamics of scalar bosons via DKP formalism embedded inthe background of a spinning cosmic string We solve DKPequation in presence of the spinning cosmic string space-timewhose metric has off diagonal terms which involves time andspace The influence of this topological defect in the energyspectrum and DKP spinor presented graphically

The structure of this paper is as follows Section 2describes the covariant form of DKP equation in a spinningcosmic string background In Section 3 we introduce theDKP oscillator by performing the nonminimal coupling inthis space-time and we obtain the radial equations that aresolvedWe plotted theDKP spinor density of probability and

HindawiAdvances in High Energy PhysicsVolume 2018 Article ID 2959354 8 pageshttpsdoiorg10115520182959354

2 Advances in High Energy Physics

the energy spectrum for different conditions involving thedeficit angle and the oscillator frequency In the Section 4 wepresent our conclusions

2 Covariant Form of the DKP Equation in theSpinning Cosmic String Background

We choose the cosmic string space-time background wherethe line element is given by1198891199042 = minus1198891198792 + 1198891198832 + 1198891198842 + 1198891198852 (1)

The space-time generated by a spinning cosmic stringwithoutinternal structure which is termed ideal spinning cosmicstring can be obtain by coordinate transformation as119879 = 119905 + 119886120572minus1120593119883 = 119903 cos (120593)119884 = 119903 sin (120593)120593 = 120572120593耠

(2)

With this transformation the line element ((1)) becomes[37ndash43]1198891199042 = minus (119889119905 + 119886119889120593)2 + 1198891199032 + 120572211990321198891205932 + 1198891199112= minus1198891199052 + 1198891199032 minus 2119886119889119905119889120593 + (12057221199032 minus 1198862) 1198891205932 + 1198891199112 (3)

with minusinfin lt 119911 lt infin 120588 ge 0 and 0 le 120593 le 2120587 From thispoint on we will take 119888 = 1 The angular parameter 120572 whichruns in the interval (0 1] is related to the linear mass density120583 of the string as 120572 = 1 minus 4120583 and corresponds to a deficitangle 120574 = 2120587(1minus120572) We take 119886 = 4119866119895where119866 is the universalgravitation constant and 119895 is the angular momentum of thespinning string thus 119886 is a length that represents the rotationof the cosmic string Note that in this case the source ofthe gravitational field relative to a spinning cosmic stringpossesses angular momentum and the metric (1) has an offdiagonal term involving time and space

The DKP equation in the cosmic string space-time (1)reads [13 17 31] (i120573휇 (119909) nabla휇 minus119872)Ψ (119909) = 0 (4)

The covariant derivative in (4) isnabla휇 = 120597휇 + Γ휇 (119909) (5)

where Γ휇 are the spinorial affine connections given by

Γ휇 = 12120596휇푎푏 [120573푎 120573푏] (6)

The matrices 120573푎 are the standard Kemmer matrices inMinkowski space-time 120573휇 = 119890휇푎120573푎 (7)

The Kemmer matrices are an analogous to Dirac matrices inDirac equation There has been an increasing interest Diracequation for spin half particles [44ndash47] The matrices 120573푎satisfies the DKP algebra120573휇120573]120573휆 + 120573휆120573]120573휇 = 119892휇]120573휇 + 119892휆]120573휇 (8)

The conserved four-current is given by119869휇 = 12Ψ120573휇Ψ (9)

and the conservation law for 119869휇 takes the formnabla휇119869휇 + 1198942Ψ (119880 minus 1205780119880dagger1205780)Ψ = 12Ψ (nabla휇120573휇)Ψ (10)

The adjoint spinorΨ is defined asΨ = Ψdagger1205780with 1205780 = 212057301205730minus1 in such a way that (1205780120573휇)dagger = 1205780120573휇 The factor 12 whichmultiplies Ψ120573휇Ψ is of no importance for the conservationlaw and ensures the charge density is compatible with the oneused in the Klein-Gordon theory and its nonrelativistic limitThus if 119880 is Hermitian with respect to 1205780 and the curved-space beta-matrices are covariantly constant then the four-current will be conserved if [30]nabla휇120573휇 = 0 (11)

The algebra expressed by these matrices generates a set of126 independent matrices whose irreducible representationscomprise a trivial representation a five-dimensional rep-resentation describing the spin-zero particles and a ten-dimensional representation associated with spin-one parti-cles We choose the 5 times 5 beta-matrices as follows [31]

1205730 = ( 120579 02times303times2 03times3) 997888rarr120573 = (02times2 997888rarr120591minus997888rarr120591 푇 03times3) 120579 = (0 11 0) 1205911 = (minus1 0 00 0 0) 1205912 = (0 minus1 00 0 0) 1205913 = (0 0 minus10 0 0 )

(12)

In (7) 119890휇푎 denote the tetrad basis that we can choose as

119890휇푎 =(((

1 119886 sin (120593)119903120572 minus119886 cos (120593)119903120572 00 cos (120593) sin (120593) 00 minus sin (120593)119903120572 cos (120593)119903120572 00 0 0 1

)))

(13)

Advances in High Energy Physics 3

For the specific tetrad basis given by (13) we find from (7)that the curved-space beta-matrices read

120573(0) = 119890푡푎120573푎 = 1205730 minus 119886119903120572120573휑120573(1) = 119890푟푎120573푎 = 120573푟120573푟 = cos1205931205731 + sin1205931205732120573(2) = 1198902푎120573푎 = 120573휑119903120572120573휑 = minus sin1205931205731 + cos1205931205732120573(3) = 119890푧푎120573푎 = 1205733 = 120573푧

(14a)

and the spin connections are given by

Γ휑 = (1 minus 120572) [1205731 1205732] (15)

We consider only the radial component in the nonminimalsubstitution Since the interaction is time-independent onecan write Ψ(119903 119905) prop 119890푖푚휑119890푖푘119911푧119890minus푖퐸푡Φ(119903) where 119864 is the energyof the scalar boson 119898 is the magnetic quantum numberand 119896푧 is the wave number The five-component DKP spinorcan be written as Φ푇 = (Φ1 Φ2 Φ3 Φ4 Φ5) and the DKPequation (4) leads to the five

(119903120572 (minusMΦ1 (119903) + EΦ2 (119903) + 119896119911Φ5 (119903)) + cos120593 ((119886E+ 119898)Φ4 (119903) minus i ((minus1 + 120572)Φ3 (119903) + 119903120572Φ耠3 (119903)))minus sin120593 ((119886E + 119898)Φ3 (119903)+ i ((minus1 + 120572)Φ4 (119903) + 119903120572Φ耠4 (119903)))) = 0(EΦ1 (119903) minusMΦ2 (119903)) = 0((119886E + 119898) sin120593Φ1 (119903) + 119903120572 (minusMΦ3 (119903) + isdot cos120601Φ耠1 (119903))) = 0(minus (119886E + 119898) cos120593Φ1 (119903) + 119903120572 (minusMΦ4 (119903) + isdot sin120593Φ耠1 (119903))) = 0(119896119911 +M120595Φ5 (119903)) = 0

(16)

Then we obtain the following equation of motion for the firstcomponentΦ1 of the DKP spinor

(minus(119886E + 119898)212057221199032 + E2 minus kz2 minusM2)Φ1 (119903)+ (120572 minus 1)Φ耠1 (119903)120572119903 + Φ耠耠1 (119903) = 0 (17)

Φ1 = 11990312훼119877푛ℓ (119903) (18)

Then (17) changes to

119877耠耠푛푚 (119903) + 119877耠푛푚 (119903)119903+ (1198642 minus 119896푧2 minus1198722 minus 1 + 4 (119886119864 + 119898)2411990321205722 )119877 (119903)

= 0(19)

By the change of variable 119903 = 119909120578 we can write (19) in theform

119877耠耠푛푚 (119909) + 119877耠푛푚 (119909)119909 + (1 minus 12058221199092)119877 (119909) = 0 (20)

where 120582 = ((1 + 4(119886119864 + 119898)2)41205722)12 and 120578 = (1198642 minus 119896푧2 minus1198722)minus12 The physical solution of (20) is 119861119890119904119904119890119897 119869 functionTherefore the general solution to (20) is given by

119877 (119903) = 119860휆휂119869휆 (119903120578) + 119861휆휂119884휆 (119903120578) (21)

where 119884휆(119903120578) is the Bessel function of the second kindSometimes this family of functions is also called Neumannfunctions or Weber functions 119869휆(119903120578) is the Bessel functionof the first kind given by

119869] (119909) = infinsum푘=0

(minus1)푘119896Γ (119896 + ] + 1) (1199092)]+2푘 (22)

and 119884휆(119903120578) is the Bessel function of the second kind givenby

119884] (119909) = 119869] (119909) cos (]120587) minus 119869minus] (119909)sin (]120587) (23)

By considering the boundary condition for (21) such that119861휆휂 = 0 we find119877 (119903) = 119860휆휂119869휆 (119903120578) (24)

3 The DKP Oscillator in Spinning CosmicString Background

TheDKP oscillator is introduced via the nonminimal substi-tution [17 30 31] 1119894 997888rarrnabla훼 997888rarr 1119894 997888rarrnabla훼 minus 1198941198721205961205780997888rarr119903 (25)

where120596 is the oscillator frequency119872 is themass of the bosonalready found in (4) and 997888rarrnabla is defined in (5) We consider

4 Advances in High Energy Physics

only the radial component in the nonminimal substitutionThe DKP equation (4) leads to the five equations(119903120572 (minus119872Φ1 (119903) + 119864Φ2 (119903) + 119896119911Φ5 (119903)) + (cos120593 (119886119864+ 119898)Φ4 (119903)) + (i cos120593 (Φ3 (119903)+ 120572 (minus1 + 1199032M120596)Φ3 (119903) minus 119903120572Φ耠3 (119903)))minus sin120593 ((119886119864 + 119898)Φ3 (119903)minus i (1 minus 120572 + 1199032120572M120596)Φ4 (119903) + i119903120572Φ耠4 (119903))) = 0((119886119864 + 119898) sin120593Φ1 (119903) minus 119872119903120572Φ3 (119903) + i119903120572

sdot cos120593 (119903M120596Φ1 (119903) + Φ耠1 (119903))) = 0(minus (119886119864 + 119898) cos120593Φ1 (119903) minus 119872119903120572Φ4 (119903) + i119903120572sdot sin120593 (119903M120596Φ1 (119903) + Φ耠1 (119903))) = 0(119864Φ1 (119903) minus 119872Φ2 (119903)) = 0(119896119911Φ1 (119903) +MΦ4 (119903)) = 0

(26)

By solving the above system of (26) in favour of Φ1 we getΦ2 (119903) = 119864119872Φ1 (119903)Φ5 (119903) = minus 119896푧119872Φ1 (119903)Φ4 (119903) = minus119886119864 cos120593Φ1 (119903) minus 119898 sin120593Φ1 (119903) + i (1199032120572M120596 sin120593Φ1 (119903) + 119903120572 sin120593Φ耠1 (119903))119872119903120572Φ3 (119903) = 119886119864 sin120593Φ1 (119903) + 119898 sin120593Φ1 (119903) + i (1199032120572M120596 cos120593Φ1 (119903) + 119903120572 cos120593Φ耠1 (119903))119872119903120572

(27)

Combining these results we obtain (23) of motion for thefirst component of the DKP spinor

Φ耠耠1 (119903) + (minus1 + 120572)Φ耠1 (119903)119903120572 + (1198642 minus 1198961199112 minus1198722+ 2119872120596 minus 119872120596120572 minus (119886119864 + 119898)211990321205722 minus 1199032M21205962)Φ1 (119903)= 0

(28)

Let us takeΦ1 as Φ1 = 11990312훼119877푛ℓ (119903) (29)

Then (28) changes to

119877耠耠푛푚 (119903) + 119877耠푛푚 (119903)119903 + (1198642 minus 119896푍2 minus1198722 + 2119872120596minus 119872120596120572 minus 1 + 4 (119886119864 + 119898)2411990321205722 minus 1199032M21205962)119877푛푚 (119903) = 0 (30)

In order to solve the above equation we employ the changeof variable 119904 = 1199032 thus we rewrite the radial equation (34) inthe form119877푛푚耠耠 (119904) + 1119904 119877푛푚耠 (119904) + 11199042 (minus12058511199042 + 1205852119904 minus 1205853) 119877푛푚 (119903)= 0 (31)

If we compare with this second-order differential equationwith the Nikiforov-Uvarov (NU) form given in (A1) ofAppendix we see that

1205851 = 1198722120596241205852 = 14 (1198642 minus 1198961199112 minus1198722 + 2119872120596 minus 119872120596120572 ) 1205853 = 1 minus 1205722 + 4 (119886119864 + 119898)2161205722

(32)

which gives the energy levels of the relativistic DKP equationfrom (2119899 + 1)radic1205851 minus 1205852 + 2radic12058531205851 = 0 (33)

where 1205721 = 11205722 = 1205723 = 1205724 = 1205725 = 01205726 = 12058511205727 = minus12058521205728 = 12058531205729 = 120585112057210 = 1 + 2radic1205853

Advances in High Energy Physics 5

2 4 6 8 10r

02

04

06

n=1m=1 E+n=1m=1 Eminus

n=2m=1 Eminus

n=2m=2Eminus

minus02

minus04

minus06

Φ1(r)

Figure 1 The wave function Φ1 for 119899 = 1 2 and 00 le 119903 le 100GeV minus1 with the parameters119872 = 1 GeV 120572 = 09 120596 = 025 and119898 = 119896푧 = 119886 = 1

12057211 = 2radic120585112057212 = radic120585312057213 = minusradic1205851

(34)

As the final step it should be mentioned that the correspond-ing wave function is

119877푛푚 (119903) = 1198731199032훼12119890훼13푟2119871훼10minus1푛 (120572111199032) (35)

where 119873 is the normalization constant In limit a997888rarr 0we have the usual metric in cylindrical coordinates wheredescribed by the line element1198891199042 = minus1198891199052 + 1198891199032 + 120572211990321198891205932 + 1198891199112 (36)

as pointed out by authors in [17] dynamic of DKP oscillatorin the presence of this metric describe by

120593耠耠1 (119903) + 120572 minus 1120572119903 120593耠1 (119903) + (1198642 minus1198722 minus 1198961199112+ (2120572 minus 1)119872120596120572 minus 119898212057221199032 minus119872212059621199032)1205931 (119903) = 0

(37)

and the corresponding wave function is

120593 (119903) = 1198731199032퐴119890퐵푟2119871퐶minus1푛 (1198631199032) (38)

where ABC and D are constant and 119871퐶minus1푛 denotes thegeneralized Laguerre polynomial In Figure 1Φ1(119903) is plottedversus 119903 for different quantum number with the parameterslisted under it The density of probability |Φ1|2 is shown in

n=1m=1 E+n=1m=1 Eminus

n=2m=1 Eminus

n=2m=2Eminus

2 4 6 8 10r

01

02

03

04

Φ21(r)

Figure 2 Density of probability |Φ1|2 for 119899 = 1 2 and 00 le 119903 le 100GeV minus1 with the parameters119872 = 1 GeV 120572 = 09 120596 = 025 and119898 = 119896푧 = 119886 = 1

5

10

15

20

25Enm

102 04 06 08

n = 1

n = 5

n = 10

Figure 3 The energy as a function of 120572 with119872 = 119898 = 119886 = 120596 =119896푧 = 1Figure 2 The negative and positive solution of energy versus120572 is shown in Figures 3 and 4 for 119899 = 1 5 and 10 Asin Figures 3 and 4 we observe that the absolute value ofenergy decreases with 120572 Also in Figure 5 energy is plottedversus 120596 for quantum numbers We see that absolute value ofenergy increases with120596The negative and positive solution ofenergy versus 119899 is shown in Figure 6 for different parameter120572 We obtained the energy levels of the DKP oscillator inthat background and observed that the energy increases withthe level number In Figure 7 energy is plotted versus 119886 fordifferent quantum numbers We see energy increases withparameter 119886 Also we observed that the energy levels of the

6 Advances in High Energy Physics

102 04 06 08

n = 1

n = 5

n = 10

Enm

minus5

minus10

minus15

minus20

Figure 4 The energy as a function of 120572 with119872 = 119898 = 119886 = 120596 =119896푧 = 1

0 1

0

5

10

minus5

minus10

02 04 06 08

Enm

n = 2

n = 4

n = 6

n = 10

n = 8

Figure 5 The energy as a function of 120596 with119872 = 1 119896푧 = 1119898 = 1119886 = 1 and 120572 = 05DKP oscillator in that background increases with the levelnumber

4 Conclusion

The overall objective of this paper is the study of therelativistic quantum dynamics of a DKP oscillator field forspin-0 particle in the spinning cosmic string space-time Theline element in this background is obtained by coordinate

1 2 3 4 5

15

0

5

10

minus5

minus10

Enm

n

= 02

= 03

= 04

= 05

Figure 6 The energy as a function of 119899 with119872 = 119896푧 = 119898 = 1 =120596 = 1 and 120572 = 05

0 1

4

5

6

7

Enm

02 04 06 08a

n = 1

n = 2

n = 3

n = 4

Figure 7 The energy as a function of 119886 with119872 = 119898 = 120596 = 119896푧 = 1and 120572 = 05transformation of Cartesian coordinate The metric has offdiagonal terms which involves time and space We consid-ered the covariant form of DKP equation in the spinningcosmic string background and obtained the solutions of DKPequation for spin-0 bosons Second we introduced DKPoscillator via the nonminimal substitution and consideredDKP oscillator in that background From the correspondingDKP equation we obtained a system of five equations By

Advances in High Energy Physics 7

combining the results of this system we obtained a second-order differential equation for first component of DKP spinorthat the solutions are Laguerre polynomials We see that theresults are dependent on the linearmass density of the cosmicstring In the limit case of 119886 = 0 and 120572 = 1 ie in the absenceof a topological defect recover the general solution for flatspace- time We plotted Φ1(119903) for 119899 = 1 2 We examinedthe behaviour of the density of probability |Φ1|2We observedthat |Φ1|2 for any parameter by increasing 119903 have a very smallpeak at beginning and then have a taller peak and then byincreasing 119903 it tends to zero We obtained the behaviour ofenergy spectrum as a function of 120572 We see that the absolutevalue of energy decreases as 120572 increasingAppendix

Nikiforov-Uvarov (NU) Method

The Nikiforov-Uvarov method is helpful in order to findeigenvalues and eigenfunctions of the Schrodinger equationas well as other second-order differential equations of physi-cal interest More details can be found in [48 49] Accordingto this method the eigenfunctions and eigenvalues of asecond- order differential equation with potential are

Φ耠耠 (119904) + 1205721 minus 1205722119904119904 (1 minus 1205723119904)Φ耠 (119904)+ 1(119904 (1 minus 1205723119904))2 (minus12058511199042 + 1205852119904 minus 1205853)Φ (119904) = 0(A1)

According to the NU method the eigenfunctions andeigenenergies respectively areΦ (119904) = 119904훼12 (1 minus 1205723119904)minus훼12minus(훼13훼3)sdot 119875(훼10minus1(훼11훼3)minus훼10minus1)푛 (1 minus 21205723119904) (A2)

and 1205722119899 minus (2119899 + 1) 1205725 + (2119899 + 1) (radic1205729 + 1205723radic1205728)+ 119899 (119899 minus 1) 1205723 + 1205727 + 212057231205728 + 2radic12057281205729 = 0 (A3)

where 1205724 = 12 (1 minus 1205721) 1205725 = 12 (1205722 minus 21205723) 1205726 = 12057225 + 12058511205727 = 212057241205725 minus 12058521205728 = 12057224 + 12058531205729 = 12057231205727 + 120572231205728 + 120572612057210 = 1205721 + 21205724 + 2radic120572812057211 = 1205722 minus 21205725 + 2 (radic1205729 + 1205723radic1205728)

12057212 = 1205724 + radic120572812057213 = 1205725 minus (radic1205729 + 1205723radic1205728)(A4)

In the rather more special case of 120572 = 0lim훼3㨀rarr0

119875(훼10minus1(훼11훼3)minus훼10minus1)푛 (1 minus 21205723119904) = 119871훼10minus1푛 (12057211119904)lim훼3㨀rarr0

(1 minus 1205723119904)minus훼12minus(훼13훼3) = 119890훼13푠 (A5)

and from (14a) we find for the wave functionΦ (119904) = 119904훼12119890훼13푠119871훼10minus1푛 (12057211119904) (A6)

where 119871훼10minus1푛 denotes the generalized Laguerre polynomial

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] N Kemmer ldquoQuantum theory of einstein-bose particles andnuclear interactionrdquo Proceedings of the Royal Society A Mathe-matical Physical and Engineering Sciences vol 166 no 924 pp127ndash153 1938

[2] R J Duffin ldquoOn the characteristic matrices of covariantsystemsrdquo Physical Review A Atomic Molecular and OpticalPhysics vol 54 no 12 article no 1114 1938

[3] N Kemmer ldquoThe particle aspect of meson theoryrdquo Proceedingsof the Royal Society A Mathematical Physical and EngineeringSciences vol 173 no 952 pp 91ndash116 1939

[4] G Petiau University of Paris Thesis (1936) Published in AcadR Belg Cl Sci Mem Collect 8 16 (2) (1936)

[5] E M Corson Introduction to Tensors Spinors Relativistic WaveEquations Chelsea Publishing 1953

[6] W Greiner Relativistic Quantum Mechanics Springer BerlinGermany 2000

[7] M Falek and M Merad ldquoBosonic oscillator in the presence ofminimal lengthrdquo Journal of Mathematical Physics vol 50 no 2Article ID 023508 2009

[8] M Falek andMMerad ldquoA generalized bosonic oscillator in thepresence of a minimal lengthrdquo Journal of Mathematical Physicsvol 51 no 3 Article ID 033516 2010

[9] M Falek and M Merad ldquoDKP oscillator in a noncommutativespacerdquo Communications inTheoretical Physics vol 50 no 3 pp587ndash592 2008

[10] G Guo C Long Z Yang and S Qin ldquoDKP oscillator innoncommutative phase spacerdquoCanadian Journal of Physics vol87 no 9 pp 989ndash993 2009

[11] Z-H Yang C-Y Long S-J Qin and Z-W Long ldquoDKPoscillator with spin-0 in three-dimensional noncommutativephase spacerdquo International Journal ofTheoretical Physics vol 49no 3 pp 644ndash651 2010

8 Advances in High Energy Physics

[12] H Hassanabadi Z Molaee and S Zarrinkamar ldquoDKP oscil-lator in the presence of magnetic field in (1+2)-dimensionsfor spin-zero and spin-one particles in noncommutative phasespacerdquo The European Physical Journal C vol 72 no 11 articleno 2217 2012

[13] L B Castro ldquoQuantum dynamics of scalar bosons in a cosmicstring backgroundrdquoTheEuropean Physical Journal C vol 75 no6 article no 287 2015

[14] N Debergh J Ndimubandi and D Strivay ldquoOn relativisticscalar and vector mesons with harmonic oscillator - likeinteractionsrdquo Zeitschrift fur Physik C Particles and Fields vol56 pp 421ndash425 1992

[15] Y Nedjadi and R C Barrett ldquoThe Duffin-Kemmer-Petiauoscillatorrdquo Journal of Physics A Mathematical and General vol27 no 12 pp 4301ndash4315 1994

[16] Y Nedjadi S Ait-Tahar and R C Barrett ldquoAn extendedrelativistic quantum oscillator for S = 1 particlesrdquo Journal ofPhysics A Mathematical and General vol 31 no 16 pp 3867ndash3874 1998

[17] MHosseinpourHHassanabadi and FMAndrade ldquoTheDKPoscillator with a linear interaction in the cosmic string space-timerdquoTheEuropean Physical Journal C vol 78 no 2 p 93 2018

[18] A Boumali and L Chetouani ldquoExact solutions of the Kemmerequation for a Dirac oscillatorrdquo Physics Letters A vol 346 no 4pp 261ndash268 2005

[19] I BoztosunM Karakoc F Yasuk and A Durmus ldquoAsymptoticiteration method solutions to the relativistic Duffin-Kemmer-Petiau equationrdquo Journal of Mathematical Physics vol 47 no 6Article ID 062301 2006

[20] M de Montigny M Hosseinpour and H Hassanabadi ldquoThespin-zero Duffin-Kemmer-Petiau equation in a cosmic-stringspace-time with the Cornell interactionrdquo International Journalof Modern Physics A vol 31 no 36 Article ID 1650191 2016

[21] F Yasuk M Karakoc and I Boztosun ldquoThe relativisticDuffinndashKemmerndashPetiau sextic oscillatorrdquo Physica Scripta vol78 no 4 Article ID 045010 2008

[22] A Boumali ldquoOn the eigensolutions of the one-dimensionalDuffin-Kemmer-Petiau oscillatorrdquo Journal of MathematicalPhysics vol 49 no 2 Article ID 022302 2008

[23] Y Kasri and L Chetouani ldquoEnergy spectrum of the relativisticDuffin-Kemmer-Petiau equationrdquo International Journal of The-oretical Physics vol 47 no 9 pp 2249ndash2258 2008

[24] A Vilenkin and E P Shellard Cosmic Strings and OtherTopological Defects Cambridge University Press CambridgeUK 1994

[25] A Vilenkin ldquoCosmic strings and domain wallsrdquo PhysicsReports vol 121 no 5 pp 263ndash315 1985

[26] NGMarchuknmarchuk ldquoDirac equation inRiemannian spacewithout tetradsrdquo Il Nuovo Cimento B vol 115 no 11 2000

[27] L D Landau and E M Lifshitz Quantum Mechanics Non-Relativistic Theory Pergamon New York USA 1977

[28] G de A Marques and V B Bezerra ldquoHydrogen atom in thegravitational fields of topological defectsrdquo Physical Review DCovering Particles Fields Gravitation and Cosmology vol 66no 10 Article ID 105011 2002

[29] G R de Melo M de Montigny and E S Santos ldquoSpin-less Duffin-Kemmer-Petiau oscillator in a Galilean non-commutative phase spacerdquo Journal of Physics Conference Seriesvol 343 Article ID 012028 2012

[30] L B Castro ldquoNoninertial effects on the quantum dynamics ofscalar bosonsrdquo The European Physical Journal C vol 76 no 2article no 61 2016

[31] HHassanabadiMHosseinpour andM deMontigny ldquoDuffin-Kemmer-Petiau equation in curved space-time with scalarlinear interactionrdquoThe European Physical Journal Plus vol 132no 12 p 541 2017

[32] R Bausch R Schmitz and L A Turski ldquoSingle-ParticleQuantum States in a Crystal with Topological Defectsrdquo PhysicalReview Letters vol 80 no 11 pp 2257ndash2260 1998

[33] E Aurell ldquoTorsion and electron motion in quantum dots withcrystal lattice dislocationsrdquo Journal of Physics A Mathematicaland General vol 32 no 4 article no 571 1999

[34] C R Muniz V B Bezerra andM S Cunha ldquoLandau quantiza-tion in the spinning cosmic string spacetimerdquoAnnals of Physicsvol 350 pp 105ndash111 2014

[35] V B Bezerra ldquoGlobal effects due to a chiral conerdquo Journal ofMathematical Physics vol 38 no 5 pp 2553ndash2564 1997

[36] C Furtado V B Bezerra and F Moraes ldquoQuantum scatteringby amagnetic flux screw dislocationrdquo Physics Letters A vol 289no 3 pp 160ndash166 2001

[37] M S Cunha C R Muniz H R Christiansen and V BBezerra ldquoRelativistic Landau levels in the rotating cosmic stringspacetimerdquoTheEuropeanPhysical Journal C vol 76 no 9 p 5122016

[38] G Clement ldquoRotating string sources in three-dimensionalgravityrdquo Annals of Physics vol 201 no 2 pp 241ndash257 1990

[39] C Furtado F Moraes and V B Bezerra ldquoGlobal effects dueto cosmic defects in Kaluza-Klein theoryrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 59 no 10Article ID 107504 1999

[40] R A Puntigam and H H Soleng ldquoVolterra distortionsspinning strings and cosmic defectsrdquo Classical and QuantumGravity vol 14 no 5 article no 1129 1997

[41] P S Letelier ldquoSpinning strings as torsion line spacetime defectsrdquoClassical and Quantum Gravity vol 12 no 2 1995

[42] P O Mazur ldquoSpinning cosmic strings and quantization ofenergyrdquo Physical Review Letters vol 57 no 8 pp 929ndash932 1986

[43] J R Gott and M Alpert ldquoGeneral relativity in a (2+1)-dimensional space-timerdquo General Relativity and Gravitationvol 16 no 3 pp 243ndash247 1984

[44] K Bakke and C Furtado ldquoOn the interaction of the Diracoscillator with the AharonovndashCasher system in topologicaldefect backgroundsrdquo Annals of Physics vol 336 pp 489ndash5042013

[45] G Q Garcia J R de S Oliveira K Bakke and C FurtadoldquoFermions in Godel-type background space-times with torsionand the Landau quantizationrdquo The European Physical JournalPlus vol 132 no 3 article no 123 2017

[46] J Carvalho C Furtado and F Moraes ldquoDirac oscillator inter-acting with a topological defectrdquo Physical Review A AtomicMolecular and Optical Physics and Quantum Information vol84 no 3 Article ID 032109 2011

[47] E R F Medeiros and E R B de Mello ldquoRelativistic quantumdynamics of a charged particle in cosmic string spacetime in thepresence of magnetic field and scalar potentialrdquo The EuropeanPhysical Journal C vol 72 no 6 article no 2051 2012

[48] A F Nikiforov andV B Uvarov Special Functions ofMathemat-ical Physics Birkhaauser Basel Switzerland 1988

[49] C Tezcan and R Sever ldquoA general approach for the exactsolution of the Schrodinger equationrdquo International Journal ofTheoretical Physics vol 48 no 2 pp 337ndash350 2009

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom