Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2013 Article ID 562959 11 pageshttpdxdoiorg1011552013562959

Research ArticleRelativistic Two-Dimensional Harmonic Oscillator Plus CornellPotentials in External Magnetic and AB Fields

Sameer M Ikhdair12 and Ramazan Sever3

1 Department of Physics Faculty of Science An-Najah National University PO Box 7 Nablus 400 West Bank Palestine2 Department of Electrical and Electronic Engineering Near East University 922022 Nicosia Northern Cyprus Turkey3 Department of Physics Middle East Technical University 06531 Ankara Turkey

Correspondence should be addressed to Sameer M Ikhdair sameerikhdairnajahedu

Received 13 June 2013 Accepted 7 September 2013

Academic Editor Shi-Hai Dong

Copyright copy 2013 S M Ikhdair and R Sever 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

TheKlein-Gordon (KG) equation for the two-dimensional scalar-vector harmonic oscillator plus Cornell potentials in the presenceof external magnetic and Aharonov-Bohm (AB) flux fields is solved using the wave function ansatz method The exact energyeigenvalues and the wave functions are obtained in terms of potential parameters magnetic field strength AB flux field andmagnetic quantumnumberThe results obtained by using different Larmor frequencies are comparedwith the results in the absenceof both magnetic field (120596

119871= 0) and AB flux field (120585 = 0) cases Effect of external fields on the nonrelativistic energy eigenvalues

and wave function solutions is also precisely presented Some special cases like harmonic oscillator and Coulombic fields are alsostudied

1 Introduction

The exact solution of Schrodinger equation (SE) and therelativistic wave equations for some physical potentials arevery important in many fields of physics and chemistry sincethey contain all the necessary information for the quantumsystem under investigation The hydrogen atom and theharmonic oscillator are usually given in textbooks as twoof several exactly solvable problems in both classical andquantum physics [1] The exact 119897-state solutions of the SE arepossible only for a few potentials and hence approximationmethods are used to obtain their solutions According to theSchrodinger formulation of quantummechanics a total wavefunction provides implicitly all relevant information aboutthe behaviour of a physical system Hence if it is exactlysolvable for a given potential the wave function can describesuch a system completely Until now many efforts have beenmade to solve the stationary SE with anharmonic potentialsin three dimensions (3D) and two dimensions (2D) [2ndash7]with many applications to molecular and chemical physicsHowever the study of SE with some of these potentials inarbitrary dimension D is also solved in the following (cf

[8] and the references therein) The study of bound states isfundamental in the understanding of molecular spectrum ofa diatomic molecule and provides us with insight into thephysical problemunder consideration in quantummechanics[9]

Recently some authors have studied the bound statesof the 119897-wave equations with some typical potentials inthe presence of an equal scalar potential 119878(119903) and a vectorpotential 119881(119903) These potentials include the harmonic oscil-lator potential [10 11] ring-shaped Kratzer-type potential[12] pseudo-harmonic oscillator potential [13] double ring-shaped harmonic oscillator potential [14] and ring-shapedpseudo-harmonic oscillator potential [15ndash17]

It is well-known that the non-relativistic quantummechanics is an approximate theory of the relativistic oneHowever when a particle moves in a strong potential fieldthe relativistic effects must be considered which give thecorrections for non-relativistic quantum mechanics [18]Therefore themotion of spin-0 and spin-12 particles satisfiesthe KG and the Dirac equations respectively In particularsolutions to relativistic equations play an important role inmany aspects of modern physics For instance the Dirac

2 Advances in High Energy Physics

equation has been used to explain the antinucleon bound ina nucleus [19] deformed nuclei [20] and super deformation[21] and to establish an effective nuclear shell model scheme[22ndash24] while the KG equation has been used in describinga wide variety of phenomena which include classical wavesystems such as the displacement of a string attached to anelastic bed [25] and quantum system based on scalar fieldtheories [26]

Recently the Schrodinger equation is solved exactly forits bound states (energy spectrum and wave functions) [27ndash29] to study the spectral properties in a 2D charged particle(electron or hole) confined by a harmonic oscillator in thepresence of external strong uniform magnetic field 997888rarr119861 alongthe 119911 direction and Aharonov-Bohm (AB) flux field createdby a solenoid The energy levels and the wave functions ofan electron confined by 2D harmonic and pseudo-harmonicoscillators have been studied in presence of external fields [3031] using theNikiforov-Uvarov (NU)method Overmore it isnatural to study the relativistic effects of the externalmagneticand AB flux fields on the KG equation for pseudo-harmonicoscillator potential especially for a strong coupling by meansof the NU method [32] The 2D solution of Schrodingerequation for the Kratzer potential with and without thepresence of a constant magnetic field is studied within theframework of the asymptotic iteration method [33] Theenergy eigenvalues are obtained analytically (numerically)for the absence (presence) of uniform magnetic field caseThese results have been obtained by using different Larmorfrequencies (120596

119871= 0) and potential parameters are compared

with the results in the absence of magnetic field case (120596119871=

0) Overmore the spectral properties of the 2D Schrodingerequation for the pseudo-harmonic-Coulomb-linear potentialare studied using the analytical iteration method [34] Underspin symmetry the energy states and wave functions of theDirac equation for the Killingbeck (harmonic oscillator plusCornell) potential have been carried out using the wave func-tion ansatz method [35] The spin and pseudospin symmetryinDirac equation have been studied in the Killingbeck poten-tial within the context of quasi-exact solution [36] Furtherthe energy eigenvalues and normalized eigenfunctions of theradial Schrodinger equation in119873-dimensional Hilbert spacefor the quark-antiquark interaction Killingbeck potentialhave been obtained using the power series technique via asuitable choice of ansatz to the wave function [37 38]

Very recently we studied the scalar charged particleexposed to relativistic scalar-vector Killingbeck potentials inpresence of magnetic and Aharonov-Bohm flux fields andobtained its energy eigenvalues and wave functions usingthe analytical exact iteration method [39] Therefore thebehavior of a spinless relativistic particle moving under theKillingbeck potential in a static magnetic and AB flux fieldshas not been investigated yet and in thisworkwe aim to solvethe KG equation in 2D for equal mixture of scalar and vectorKillingbeck potentials with and without constant magneticand AB flux fields for the first time We present the exactenergy eigenvalues and wave functions of the Killingbeckpotential for any 119899 and 119898 quantum numbers in a constantmagnetic field with the different Larmor frequencies using

the wave function ansatz method [36] The non-relativisticenergy eigenvalues and wave functions of our solution arepresented by making an appropriate mapping of parametersOvermore special cases of KG for equal scalar-vector Killing-beck potentials are also presented in the presence (120596

119871= 0

120585 = 0) and absence (120596119871= 0 120585 = 0) of uniform fields

The structure of this paper is as follows We studyeffect of external uniform magnetic and AB flux fields on arelativistic spinless particle (antiparticle) under equalmixtureof scalar and vector Killingbeck potentials in Section 2 Wediscuss some special cases in Section 3 Finally we give ourconcluding remarks in Section 4

2 The Klein-Gordon Atom

The Klein-Gordon atom for the spinless particle with mass119898119890and charge minus119890 moving in external electromagnetic field

and AB flux field given by potentials 119881(119903) 119878(119903) and 997888rarr119860 reads[40 41]

[1198882(997888rarr119901 +

119890

119888

997888rarr

119860)

2

minus (119864 minus 119881 (119903))2+ (1198981198901198882+ 119878 (119903))

2

]

times 120595 (119903 120601) = 0

(1)

The scalar and vector potentials are chosen in the followingform [27ndash30]

119881119870 (119903) = 120582119903

2+ 120590119903 minus

120581

119903

997888rarr

119860 =

1

2

997888rarr

119861 times997888rarr119903 +

ΦAB2120587119903

120601 (2)

where 119881119870(119903) is the Killingbeck potential that is harmonic

oscillator potential plus Cornell potential [35ndash38] which isextensively used in particle physics [42 43] Moreover thevector potential in the symmetric gauge is defined by 997888rarr119860 =

997888rarr

1198601+

997888rarr

1198602such that 997888rarrnabla times

997888rarr

1198601=

997888rarr

119861 and 997888rarrnabla times

997888rarr

1198602= 0 where the

applied magnetic field 997888rarr119861 = (0 0 119861) is perpendicular to theplane of transversal motion of the particle and 997888rarr119860

2describes

the additional AB flux field ΦAB created by a solenoid incylindrical coordinates [32] The wave function in (1) isdefined by

120595 (119903 120601) =

1

radic2120587

119890119894119898120601119877 (119903)

radic119903

119898 = 0 plusmn1 plusmn2 (3)

where 119898 is the eigenvalue of angular momentum Therelationship between the attractive scalar and repulsive vectorpotentials is given by 119878(119903) = 120573119881(119903) where minus1 le 120573 le

1 is arbitrary constant and hence the KG equation couldbe reduced to a Schrodinger-type second-order differentialequation as follows

[1198882(997888rarr119901 +

119890

119888

997888rarr

119860)

2

+ 2 (119864119881 (119903) + 1198981198901198882119878 (119903))

+1198782(119903) minus 119881

2(119903) + 119898

2

1198901198884minus 1198642]120595 (119903 120601) = 0

(4)

Now we will treat the bound-state solutions of the two casesin (4) as follows

Advances in High Energy Physics 3

21 The Positive Energy Solution The positive energy statesrequire that 119878(119903) = 119881(119903) (ie 120573 = 1 case) which in thenon-relativistic limit corresponds to the solution of the waveequation

1

2120583

[minus119894ℎ

997888rarr

nabla +

119890

119888

(

119861119903

2

+

ΦAB2120587119903

)120601]

2

+ 2119881119870 (119903) minus 119864

times 120595 (119903 120601) = 0

(5)

where120595(119903 120601) stands for non-relativistic wave functionThusthe choice 119878(119903) = 119881(119903) produces a non-relativistic limit with apotential function 2119881(119903) and not 119881(119903) Accordingly it wouldbe natural to scale the potential term in (4) and (5) so thatin the non-relativistic limit the interaction potential becomes119881(119903) not 2119881(119903)Thus we need to recast (4) for the 119878(119903) = 119881(119903)

as [32 39 40 44]

[1198882(minus119894ℎ

997888rarr

nabla +

119890

119888

997888rarr

119860)

2

+ (119864 + 1198981198901198882)119881 (119903)]120595 (119903 120601)

= (1198642minus 1198982

1198901198884) 120595 (119903 120601)

(6)

where

nabla2=

1

120588

120597

120597120588

+

1205972

1205971205882+

1

1205882

1205972

1205971206012+

1205972

1205971199112 (7)

and in order to simplify (6) we introduce new parameters1205721= 119864 +119898

1198901198882 and 120572

2= 119864 minus119898

1198901198882 so that it can be reduced to

the following form

1198882[minus119894ℎ

997888rarr

nabla +

119890

119888

(

119861119903

2

+

ΦAB2120587119903

)120601]

2

minus 1205721(1205722minus 119881119870 (119903))

times 120595 (119903 120601) = 0

(8)

By inserting (2) and (3) into (8) we obtain

1198892119877 (119903)

1198891199032

+

1205721

ℎ21198882[1205722minus 119880eff (119903 120596119871 120585)] 119877 (119903) = 0 (9a)

119880eff (119903 120596119871 120585) = 119881119870 (119903) +

1198982

11989011988821205962

119871

1205721

1199032

+

ℎ21198882

1205721

(11989810158402minus 14)

1199032

+

2ℎ1205961198711198981198901198882

1205721

1198981015840

(9b)

120596119871=

Ω

2

Ω =

|119890| 119861

119898119890119888

1198981015840= 119898 + 120585

1198981015840= 1 2 120585 =

ΦABΦ0

(9c)

where the effective potentials depending on the magnitudesof two fields strength with 120596

119871and 119898

1015840 are the Larmorfrequency and a new eigenvalue of angularmomentum (mag-netic quantum number) respectively It is worthy to mentionthat the frequency Ω is called the cyclotron frequency [45]This is the frequency of rotation corresponding to the classical

motion of a charged particle in a uniform magnetic field andΩ2 is the Larmor frequency in units of Hz [45] Moreoverwe take 120585 as integer with the flux quantum Φ

0= ℎ119888119890 Here

119881119870(119903) is a pure Killingbeck potential (2) the second term is

the harmonic oscillator-type potential and the other termsare the rotational potential creating the rotational energylevels Equations (9a) (9b) and (9c) can be alternativelyexpressed as

1198892119877 (119903)

1198891199032

+ (

1198621

1199032+

1198622

119903

minus 1198623minus 1198624119903 minus 11986251199032)119877 (119903) = 0 (10a)

1198621= minus(119898

10158402minus

1

4

) 1198622=

1205721120581

ℎ21198882

1198623=

21198981198901205961198711198981015840

ℎ

minus

12057211205722

ℎ21198882

(10b)

1198624=

1205721120590

ℎ21198882 119862

5=

1205721120582

ℎ21198882+ (

119898119890120596119871

ℎ

)

2

(10c)

with the asymptotic behaviors 119877(0) = 0 and 119877(infin) rarr 0 Itis interesting to look at the results obtained from (10a) for aspecial case 119862

2= 1198624= 0 that is we have

1198892119877 (119903)

1198891199032

+ (

1198621

1199032minus 11986251199032)119877 (119903) = 119862

3119877 (119903) (11)

which corresponds to the differential equation of a harmonicoscillator with a centrifugal term It is a form similar to theone given by (7) in [46]with the equivalence119862

1rarr minus119897

1015840(1198971015840+1)

1198625rarr 1205722 and119862

3rarr minus120582

119899(120572 119897 and120582 are the parameters used

in [46]) So by putting the parameter values

1198971015840= 1198981015840minus

1

2

120572 =radic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

120582119899=

1198642minus 1198982

1198901198884

ℎ21198882

minus

21198981198901205961198711198981015840

ℎ

(12)

into (14) of [45] we obtain the energy equation as

1198642minus 1198982

1198901198884= 21198981198901198882ℎ1205961198711198981015840

+ 2ℎ1198882(2119899 + 119898

1015840+ 1)

timesradic(119864 + 119898

1198901198882) 120582

1198882

+ (119898119890120596119871)2

(13)

Now we find a solution to (10a) by making the followingchoice of the wave function [37 38 47 48]

119877119899119898 (

119903) = exp(12

1199011199032+ 119902119903)

infin

sum

119899=0

119886119899119903119899+120575

(14)

where 119901 and 119902 are parameters whose values are to bedetermined in terms of the potential parameters 120582 120590 and 120581Substituting (14) into (10a) we obtain the series [36]infin

sum

119899=0

119886119899119878119899119903119899+120575minus2

+

infin

sum

119899=1

119886119899minus1

119879119899minus1

119903119899+120575minus2

+

infin

sum

119899=2

119886119899minus2

119882119899minus2

119903119899+120575minus2

= 0

1199012= 1198625 2119901119902 = 119862

4

(15)

4 Advances in High Energy Physics

with119878119899= (119899 + 120575) (119899 + 120575 minus 1) + 1198621

119879119899minus1

= 2119902 (119899 + 120575 minus 1) + 1198622

119882119899minus2

= 1199022+ 2119901(119899 + 120575 minus

3

2

) minus 1198623

(16)

where the two parameters 119901 and 119902 are determined in orderfor the radial wave functions 119877

119899119898(119903) to be finite everywhere

and vanish at 119903 = 0 and as 119903 rarr infin To obtain the recurrencerelation which can connect various expansion coefficients 119886

119899

we make identical powers of 119903 in (15) that is equate thecoefficients of 119903119899+120575minus2 to zero Thus the relations become

1198860[120575 (120575 minus 1) + 1198621

] = 0 997904rArr 1205752+ 1198621= 120575

1198860

= 0 997904rArr 120575 = plusmn1198981015840+

1

2

1198861= minus

(2119902120575 + 1198622)

2120575

1198860

1198862= minus

[119901 (2120575 + 1) + 1199022minus 1198623] 1198860+ [2119902 (120575 + 1) + 1198622

] 1198861

2 (2120575 + 1)

1198863= minus(

(2119901119902 minus 1198624) 1198860+ [119901 (2120575 + 3) + 119902

2minus 1198623] 1198861

6 (120575 + 1)

+

[2119902 (120575 + 2) + 1198622] 1198862

6 (120575 + 1)

)

119886119899= minus(

(1199012minus 1198625) 119886119899minus4

+ (2119901119902 minus 1198624) 119886119899minus3

119899 (2120575 + 119899 minus 1)

+

[119901 (2120575 + 2119899 minus 3) + 1199022minus 1198623] 119886119899minus2

119899 (2120575 + 119899 minus 1)

+

[2119902 (120575 + 119899 minus 1) + 1198622] 119886119899minus1

119899 (2120575 + 119899 minus 1)

)

(17)

where 119899 = 0 1 2 with 1198860

= 0 Here the positive sign ofparameter 120575 = 119898

1015840+ 12 has been selected The power series

for large values of 120575 or1198981015840 is convergent For convenience wetake the ratio of two successive terms that is 119886

119899+1119886119899 which

becomes1198861

1198860

= minus(119902 +

1198622

2120575

) 997888rarr minus119902 when 120575 997888rarr infin

1198862

1198860

=

1199022(120575 + 119862

22119902) [120575 + 119862

22119902 + 1]

120575 (2120575 + 1)

minus

[119901 (2120575 + 1) + 1199022minus 1198623]

2 (2120575 + 1)

997888rarr

1199022minus 119901

2

when 120575 997888rarr infin

(18)

It is apparent from the above relations that the power seriesconverges to zero when 120575 rarr infin Hence the series mustbe truncated (bounded) for 119899 = 119899max At this value of 119899 weobtain the following equations

119901 = plusmnradic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

997904rArr 119901 = minusradic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

119901 = 0

(19a)

2119901119902 = 1198624

997904rArr 119902 = minus

(119864 + 1198981198901198882) 120590

2ℎ21198882radic((119864 + 119898

1198901198882) 120582ℎ21198882) + (119898

119890120596119871ℎ)2

(19b)

(119864 + 1198981198901198882) 120581

ℎ21198882

= minus2119902 (119899 + 1198981015840minus

1

2

) (19c)

(119864 + 1198981198901198882) (119864 minus 119898

1198901198882)

ℎ21198882

=

21198981198901205961198711198981015840

ℎ

+ 2radic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

times (1198981015840+ 119899 minus 1) minus 119902

2

(19d)

where the negative sign of the coefficient 119901 has been chosenin (19a) It is worth noting that the potential parameter120582mustbe positive when 120596

119871= 0 Hence (19c) gives a restriction on

the potential parameters 120582 120590 and 120581 as follows

120590 =

120581

(119899 + 1198981015840minus 12)

radic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

(20)

and it follows that (19d) gives the energy formula as

1198642minus 1198982

1198901198884= 21198981198901198882ℎ1205961198711198981015840+ 2ℎ21198882(119899 + 119898

1015840minus 1)

timesradic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

minus

(119864 + 1198981198901198882)

2

1205812

4ℎ21198882(119899 + 119898

1015840minus 12)

2

(21)

where 119899 = 0 1 2 We may find a solution to theabove transcendental equation as 119864 = 119864

(+)

KG = 1198641198991198981015840(120596119871 120585)

Overmore the wave function (14) with the help of (17) (19a)and (19b) becomes

Advances in High Energy Physics 5

120595(+)

119899119898(119903 120601) = 119862

119899119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(12)radic(119864+119898

1198901198882)120582ℎ21198882+(119898119890120596119871ℎ)2[1199032+((119864+119898

1198901198882)120590((119864+119898

1198901198882)1198892+ℎ21198882(119898119890120596119871ℎ)2))119903]

119899max

sum

119899=0

119886119899119903119899

(22a)

1198861= minus1198860[119902 +

(119864 + 1198981198901198882) 120581

2ℎ21198882(1198981015840+ 12)

] (22b)

1198862= minus

1

2

1198860[119901 minus 119902

2minus

1198981198901205961198711198981015840

ℎ (1198981015840+ 1)

minus

(119864 + 1198981198901198882) 120581

ℎ21198882(1198981015840+ 12)

(119902 +

(119864 + 1198981198901198882) 120581

4ℎ21198882(1198981015840+ 1)

)] (22c)

where 119862119899119898

is the normalization constantIn the non-relativistic limit when 119864 + 119898

1198901198882

rarr 2120583119864 + 119898

1198901198882rarr 1198641198991198981015840 and 119888 = 1 (21) and (22a) give the energy

formula

1198641198991198981015840 (120596119871 120585) = ℎ120596

1198711198981015840+ ℎ (119899 + 119898

1015840minus 1)

times radic

2120582

120583

+ 1205962

119871minus

1205831205812

2ℎ2(119899 + 119898

1015840minus 12)

2

(23)

and wave function

120595(+)

119899119898(119903 120601) = 119862

119899119898

1

radic2120587

119890119894119898120601

119903119898+120585

times 119890minus(1205832ℎ)radic2120582120583+120596

2

119871[1199032+(2120590120583(2119889

2120583+1205962

119871))119903]

119899max

sum

119899=0

119886119899119903119899

(24)

respectively for the Schrodinger-Killingbeck system with thefollowing restriction on the potential parameters 120582 120590 and 120581given by

120590 =

120583120581

ℎ (119899 + 1198981015840minus 12)

radic

2120582

120583

+ 1205962

119871 (25)

Notice that the present model has been solved in 2D spacewith an external uniform magnetic field since it is perpen-dicular to the plane where the vector and scalare Cornellpotentials have the dimensions of 119903 = 120588 and 120601 that is119881 = 119878 = 119881(120588 120601) [32] However without the magnetic fieldthe model can be solved in any desired dimensional spaceby considering the change 119898 rarr 119897 + (119863 minus 2)2 [49 50]where119863 refers to spatial dimension and also the existence ofinterdimensional degeneracy

As shown in Figure 1(a) and (9b) the effective potentialfunction changes in shape when the magnetic field strength

increases say 120596119871= 8 and in absence of AB flux field The

energy levels are raised when the strength of the magneticfield increases and in absence of AB flux field 120585 = 0 Wesee that the effective potential changes gradually from thepure pseudo-harmonic oscillator potential when 120596

119871= 0 to

a pure harmonic oscillator type behavior in short potentialrange when120596

119871= 8 In Figure 1(b) the effective potential (9b)

which is pseudo-harmonic oscillator when 120596119871= 0 becomes

sensitive to the increasing AB flux field 120585 = 8 in the shortrange region that is 0 lt 119903 lt 4 au

22 The Bound States for Negative Energy In this case of119878(119903) = minus119881(119903) the first inspection of (4) shows that thefollowing changes 119881(119903) rarr minus119881(119903) (ie 120582 rarr minus120582 120590 rarr minus120590120581 rarr minus120581) 120595(+)

119899119898(119903 120601) rarr 120595

(minus)

119899119898(119903 120601) and 119864 rarr minus119864 give the

negative energy solution for antiparticles as

1198642minus 1198982

1198901198884minus 21198981198901198882ℎ1205961198711198981015840

= 2ℎ21198882(119899 + 119898

1015840minus 1)

radic(119864 minus 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

minus

(119864 minus 1198981198901198882)

2

1205812

4ℎ21198882(119899 + 119898

1015840minus 12)

2

(26)

with restriction on potential parameters

120590 =

120581

(119899 + 1198981015840minus 12)

radic(119864 minus 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

(27)

We may find solution to the above transcendental equationas 119864 = 119864

(minus)

119870119866= 1198641198991198981015840(120596119871 120585) Overmore the wave function for

antiparticle reads

6 Advances in High Energy Physics

0 1 2 3 4 5 6 70

2

4

6

8

10

12

14

16

18

20

120596L = 0

120596L = 1

120596L = 5

120596L = 8

r (au)

Ueff(r120596

L)

(a)

0 1 2 3 4 50

2

4

6

8

10

12

14

16

18

20

r (au)

120585 = 0

120585 = 1

120585 = 5

120585 = 8

Ueff(r120585)

(b)

Figure 1 The KG effective potential function for (a) 120596119871= 0 1 5 8 and with 120585 = 0 (b) 120585 = 0 1 5 8 and with 120596

119871= 0 (colour online) Here

119898119890= ℎ = 119888 = 1

120595(minus)

119899119898(119903 120601) = 119863

119899119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(12)radic(119864minus119898

1198901198882)120582ℎ21198882+(119898119890120596119871ℎ)2[1199032+((119864minus119898

1198901198882)120590((119864minus119898

1198901198882)1198892+ℎ21198882(119898119890120596119871ℎ)2))119903]

119899max

sum

119899=0

119886119899119903119899 (28)

1198861= minus1198860[119902 +

(119864 minus 1198981198901198882) 120581

2ℎ21198882(1198981015840+ 12)

] (29)

1198862= minus

1

2

1198860[119901 minus 119902

2minus

1198981198901205961198711198981015840

ℎ (1198981015840+ 1)

minus

(119864 minus 1198981198901198882) 120581

ℎ21198882(1198981015840+ 12)

(119902 +

(119864 minus 1198981198901198882) 120581

4ℎ21198882(1198981015840+ 1)

)] (30)

where119863119899119898

is the normalization constant

3 Discussions

In this section we discuss some special cases of interest fromour general solution

(i) If we set 120590 = 120581 = 0 the effective Killingbeck potentialturns to effective harmonic oscillator potential in relativisticcase as

119880eff (119903 120596119871 120585) = 1205821199032+

1198982

11989011988821205962

119871

(119864 + 1198981198901198882)

1199032

+

ℎ21198882

(119864 + 1198981198901198882)

(11989810158402minus 14)

1199032

+

2ℎ12059611987111989811989011988821198981015840

(119864 + 1198981198901198882)

120582 =

1

2

1205831205962

(31)

The bound state solutions (with the change 119899 = 2(119899119903+ 1)) are

[32 36]

1198642minus 1198982

1198901198884= 21198981198901198882ℎ1205961198711198981015840+ 4ℎ1198882(119899119903+

1198981015840+ 1

2

)

timesradic(119864 + 119898

1198901198882) 120582

1198882

+ (119898119890120596119871)2

(32)

120595(+)

119899119903119898(119903 120601) = 119860

119899119903119898

1

radic2120587

119890119894119898120601

119903119898+120585

times 119890minus(12)radic(119864+119898

1198901198882)120582ℎ21198882+(119898119890120596119871ℎ)21199032

119899max

sum

119899=0

1198861198991199032(119899119903+1)

(33a)

1198861= 0

1198862=

1

2

1198860[

[

radic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

+

1198981198901205961198711198981015840

ℎ (1198981015840+ 1)

]

]

(33b)

Advances in High Energy Physics 7

where 119860119899119903119898

is the normalization constant On the otherhand the nonrelativistic effective harmonic oscillator poten-tial

119880eff (119903 120596119871 120585) = 1205821199032+

1

2

1205962

1198711199032+

(11989810158402minus 14) ℎ

2

21198981198901199032

+ ℎ1205961198711198981015840

(34)

has bound states as [32]

1198641198991198981015840 (120596119871 120585) = ℎ120596

1198711198981015840

+ 2ℎΩ1015840(119899119903+

10038161003816100381610038161003816119898101584010038161003816100381610038161003816+ 1

2

) Ω1015840= radic120596

2

119871+ 1205962

(35)

120595(+)

119899119903119898(119903 120601) = 119862

119899119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(1205832ℎ)Ω

10158401199032

times

119899max

sum

119899=0

1198861198991199032(119899119903+1) 119899119903= 0 1 2

=

1

radic2120587

119890119894119898120601

radic21198871198981015840+1119899

(119899 + 1198981015840)

1199031198981015840

119890minus11988711990322119871(1198981015840)

119899(1198871199032)

119887 =

120583

ℎ

Ω

1015840

(36a)

1198861= 0 119886

2=

120583

2ℎ

Ω10158401198860 1198860

= 0 (36b)

(ii) Setting 120582 = 120590 = 0 in (21) the killingbeck potentialturns to the Coulomb potential case (ie 119881(119903) = minus119860119903 and119899 = 119899119903+ 1) Thus we obtain the relativistic energy formula

1198642minus 1198982

1198901198884= 21198981198901198882(119899119903+ 21198981015840) ℎ120596119871

minus

(119864 + 1198981198901198882)

2

1205812

4ℎ21198882(119899119903+ 1198981015840+ 12)

2 120581 = 119885119890

2

(37)

and the wave function

120595(+)

119899119903119898(119903 120601) = 119862

119899119903119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(119898119890120596119871)2ℎ1199032

times

119899max

sum

119899=0

119886119899119903119899119903+1

(38a)

1198861= minus

(119864 + 1198981198901198882) 120581

2ℎ21198882(1198981015840+ 12)

1198860 (38b)

1198862= minus

1

2

[

[

119898119890120596119871

ℎ

minus

1198981198901205961198711198981015840

ℎ (1198981015840+ 1)

minus

(119864 + 1198981198901198882)

2

1205812

4ℎ41198884(1198981015840+ 1) (119898

1015840+ 12)

]

]

1198860

(38c)

Overmore in the nonrelativistic limit we have

119864 = ℎ1205961198711198981015840+ ℎ (119899

119903+ 1198981015840) 120596119871minus

11989811989011988521198904

2ℎ2(119899119903+ 1198981015840+ 12)

2 (39)

and the wave function becomes

120595(+)

119899119903119898(119903 120601) = 119862

119899119903119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(1205831205961198712ℎ)1199032

119899max

sum

119899=0

119886119899119903119899119903+1

(40a)

1198861= minus

119898119890119860

ℎ2(1198981015840+ 12)

1198860 (40b)

1198862= minus

1

2

[

120583120596119871

ℎ

minus

1205831205961198711198981015840

ℎ (1198981015840+ 1)

minus

120583211988521198904

ℎ4(1198981015840+ 1) (119898

1015840+ 12)

] 1198860

(40c)

Notice that in the absence of external fields 120596119871= 120585 = 0

the problem can be solved in any dimension Thus applyingthe transformation |119898| rarr 119897 + 12 our previous results arereduced to the well-known three-dimensional bound statesolutions for the harmonic oscillator and the hydrogen-likeatoms in the Coulomb fields [40]

119864plusmn= plusmn1198981198901198882[[

[

1 +

2ℎ119888

1198981198901198882(2119899119903+ 119897 +

3

2

)radic

(119864 + 1198981198901198882) 120582

1198982

1198901198884

]]

]

12

119864plusmn= plusmn1198981198901198882[1 minus

11988521198904

4ℎ21198882(119899119903+ 119897 + 1)

2(1 +

119864

1198981198901198882)

2

]

12

(41)

respectively According to (41) the energy spectrum can befound for the scalar particle and antiparticle

In Figure 2(a) we plot the effective potential for the caseof low vibrational (119899 = 0 1 2 3) and rotational (119898 = 1)

levels for various Larmor frequencies 120596119871

= 0 1 5 8 and120585 = 0 case As shown in Figure 2(a) and (31) the effectivepotential function changes in shape as well as the bound stateenergy eigenvalues increase when 120596

119871= 8 It is shown that the

energy levels are raised when the strength of the magneticfield increases and in the absence of AB flux field It is alsoobvious that the effective potential changes gradually fromthe pure pseudo-harmonic oscillator potential which is anonmagnetic (120596

119871= 0) and AB flux (120585 = 0) fields case to

a pure harmonic oscillator type behavior in short potentialrange when the strength of the applied magnetic field isincreased to 120596

119871= 8 If we consider a strong magnetic field

case 120596119871= 8 which has the shape of pure harmonic oscillator

potential function the energy difference between adjacentenergy levels is nearly equal which is a known characteristicof the pure harmonic oscillator potential In Figure 2(b) weplot the effective potential (34) in the absence of themagneticfield and in the presence of AB flux field in the short rangeregion

In Tables 1 and 2 we show the effect of magnetic fieldand AB flux field respectively on the low vibrational 119899

8 Advances in High Energy Physics

120596L = 0

120596L = 1

120596L = 5

120596L = 8

E01E11

E21

E31

E01

E11

E21E31

E01E11

E21

E31

0 1 2 3 4 50

10

20

30

40

50

60

70

80

r (au)

Ueff(r120596

L)

(a)

0 1 2 3 4 5 6 7 80

10

20

30

40

50

60

70

80

r (au)

120585 = 0

120585 = 3

120585 = 5

120585 = 8

Ueff(r120585)

(b)

Figure 2 The Schrodinger effective potential function and corresponding bound state energy levels (119864119899119898) in low vibrational (119899 = 0 1 2 3)

and rotational (119898 = 1) levels for (a) 120596119871= 0 1 5 8 with 120585 = 0 (b) 120585 = 0 1 5 8 and with 120596

119871= 0 (color online) Here 120583 = ℎ = 1

and rotational 119898 relativistic energy states of the harmonicoscillator potential As shown in Table 1 when the magneticfield is not applied and without AB flux field (120596

119871= 0

120585 = 0) the spacing between the energy levels of the effectivepotential is narrow and decreases with increasing 119899 Butwhen the magnetic field strength increases the energy levelsof the effective potential increase and the spacings betweenstates also increase In Table 2 when the AB flux field isapplied and without magnetic field the energy states becomedegenerate for various values of 119899 and 119898 and for variousAB flux field strength values In Tables 3 and 4 we showthe effect of magnetic field and AB flux field respectivelyon the low vibrational 119899 and rotational 119898 nonrelativisticenergy states of the harmonic oscillator potential As shownin Table 3 when themagnetic field is not applied and withoutAB flux field (120596

119871= 0 120585 = 0) the energy states are equally

spaced (the pure harmonic oscillator case) But when themagnetic field strength is raised the energy levels of theeffective potential increase and the spacings between statesalso increase In Table 4 when the AB flux field is applied andwithout magnetic field the energy states become degenerateand equally spaced for various values of 119899 and 119898 and forvarious AB flux field strength values

4 Concluding Remarks

To sum up in this paper we have studied the solutionof two-dimensional KG and Schrodinger equations withthe Killingbeck potential for low vibrational and rotationalenergy levels without and with a constant magnetic fieldhaving arbitrary Larmor frequeny and AB flux field Wehave applied the wave function ansatz method for 120596

119871= 0

(with magnetic field) and 120585 = 0 (with AB flux field) toobtain analytical expressions in closed form for boundstate energies and wave functions of the spinless relativisticparticle subject to a Killingbeck interaction expressed interms of external uniform magnetic and AB flux fields inany vibrational 119899 and rotational 119898 states The above resultsshow that the problems of relativistic quantum mechanicscan be also solved exactly as in the non-relativistic onesWe considered the solution of both positive (particle) andnegative (antiparticle) KG energy states It is noticed that thesolution with equal mixture of scalar-vector potentials can beeasily reduced into the well-known Schrodinger solution fora particle with an interaction potential field and a free fieldrespectively We have also studied the bound-state solutionsfor some special cases including the non-relativistic limits(Schrodinger equation for harmonic oscillator and Coulombpotentials under external magnetic and AB flux fields) andalso the KG equation for harmonic oscillator and CoulombinteractionsThe results show that the splitting is not constantand is mainly dependent on the strength of the externalmagnetic field and AB flux field In order to show the effectof constant magnetic and AB flux fields on the vibrationaland rotational energy levels of the harmonic oscillator weplot the effective potential and corresponding energy levelswith increasing Larmor frequency and flux field for specialpotential parametersWehave seen that the effective potentialfunction and corresponding energy levels are raised in energywhen magnetic and AB flux field strengths increase Theeffective potential function behavior gradually changes fromthe pure pseudo-harmonic oscillator to a pure harmonicoscillator shape in short potential range as the magnetic andAB flux fields strengths increase

Advances in High Energy Physics 9

Table 1 The KG energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120596119871values and 120585 = 0 Here ℎ = 119888 =

119898119890= 119896 = 1

119898 119899

119864119899119898

120596119871= 0 120596

119871= 1 120596

119871= 2 120596

119871= 3 120596

119871= 4 120596

119871= 5 120596

119871= 6 120596

119871= 7 120596

119871= 8

0

0 183929 204353 240325 275615 308137 338066 365815 391752 4161661 309625 34289 403986 465113 521808 574094 622602 667941 7106092 412383 451488 526359 603138 675146 74193 804087 862301 9171583 503104 545806 630283 718744 802542 880667 953605 102205 108663

1

0 250976 316597 389307 455265 514478 568284 617802 663858 707061 362919 427262 512444 593706 668087 736315 799437 858337 9137072 458916 522885 616824 709549 795634 87516 949036 101815 1083233 545354 609216 709891 811893 90768 996689 107966 115745 123081

Table 2 The KG energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120585 values and 120596119871= 0

119898 119899

119864119899119898

120585 = 0 120585 = 1 120585 = 2 120585 = 3 120585 = 4 120585 = 5 120585 = 6 120585 = 7 120585 = 8

0

0 183929 250976 309625 362919 412383 458916 503104 545354 5859661 309625 362919 412383 458916 503104 545354 585966 625166 663132 412383 458916 503104 545354 585966 625166 6631370 70 7358923 503104 545354 585966 625166 66313 70 735892 770901 805108

1

0 250976 309625 362919 412383 458916 503104 545354 585966 6251661 362919 412383 458916 503104 545354 585966 625166 66313 702 458916 503104 545354 585966 625166 66313 70 735892 7709013 545354 585966 625166 66313 70 735892 770901 805108 838582

Table 3 The nonrelativistic energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120596119871values and 120585 = 0

119898 119899

119864119899119898

120596119871= 0 120596

119871= 1 120596

119871= 2 120596

119871= 3 120596

119871= 4 120596

119871= 5 120596

119871= 6 120596

119871= 7 120596

119871= 8

0

0 10 141421 223607 316228 412311 509902 608276 707107 8062261 30 424264 67082 948683 123693 152971 182483 212132 2418682 50 707107 111803 158114 206155 254951 304138 353553 4031133 70 989949 156525 221359 288617 356931 425793 494972 564358

1

0 20 382843 647214 932456 122462 15198 181655 211421 2412451 40 665685 109443 156491 204924 253961 303311 352843 402492 60 948528 154164 219737 287386 355941 424966 494264 5637353 80 123137 198885 282982 369848 457922 546621 635685 724981

Table 4 The nonrelativistic energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120585 values and 120596119871= 0

119898 119899

119864119899119898

120585 = 0 120585 = 1 120585 = 2 120585 = 3 120585 = 4 120585 = 5 120585 = 6 120585 = 7 120585 = 8

0

0 10 20 30 40 50 60 70 80 901 30 40 50 60 70 80 90 100 1102 50 60 70 80 90 100 110 120 1303 70 80 90 100 110 120 130 140 150

1

0 20 30 40 50 60 70 80 90 1001 40 50 60 70 80 90 100 110 1202 60 70 80 90 100 110 120 130 1403 80 90 100 110 120 130 140 150 160

10 Advances in High Energy Physics

Acknowledgments

One of the authors (Sameer M Ikhdair) would like to thankthe president of An-Najah National University (ANU) Pro-fessor Rami Alhamdallah and the acting president ProfessorMaher Natsheh for their continuous support during hispresent work at ANU The authors acknowledge the partialsupport of the Scientific and Technological Research Councilof Turkey

References

[1] WGreiner andBMullerQuantumMechanics An IntroductionSpringer Berlin Germany 1994

[2] S Ozcelik and M Simsek ldquoExact solutions of the radialSchrodinger equation for inverse-power potentialsrdquo PhysicsLetters A vol 152 no 3-4 pp 145ndash150 1991

[3] S-H Dong ldquoSchrodinger equation with the potential 119881(119903) =119860119903minus4+ 119861119903minus3+ 119862119903minus2+ 119863119903minus1rdquo Physica Scripta vol 64 no 4 pp

273ndash276 2001[4] S-H Dong and Z-Q Ma ldquoExact solutions to the Schrodinger

equation for the potential 119881(119903) = 1198861199032+ 119887119903minus4

+ 119888119903minus6 in two

dimensionsrdquo Journal of Physics A vol 31 no 49 pp 9855ndash98591998

[5] M S Child S-H Dong and X-GWang ldquoQuantum states of asextic potential hidden symmetry and quantum monodromyrdquoJournal of Physics A vol 33 no 32 pp 5653ndash5661 2000

[6] S-H Dong ldquoExact solutions of the two-dimensional Schrodin-ger equation with certain central potentialsrdquo InternationalJournal of Theoretical Physics vol 39 no 4 pp 1119ndash1128 2000

[7] S-H Dong ldquoA new approach to the relativistic schrodingerequation with central potential Ansatz methodrdquo InternationalJournal of Theoretical Physics vol 40 no 2 pp 559ndash567 2001

[8] S-H Dong ldquoOn the solutions of the Schrodinger equation withsome anharmonic potentials wave function ansatzrdquo PhysicaScripta vol 65 no 4 pp 289ndash295 2002

[9] S Flugge Practical Quantum Mechanics 1 Springer BerlinGermany 1994

[10] X-Q Zhao C-S Jia and Q-B Yang ldquoBound states of rel-ativistic particles in the generalized symmetrical double-wellpotentialrdquo Physics Letters A vol 337 no 3 pp 189ndash196 2005

[11] S M Ikhdair ldquoExact solution of Dirac equation with chargedharmonic oscillator in electric field bound statesrdquo Journal ofModern Physics vol 3 no 2 pp 170ndash179 2012

[12] W C Qiang ldquoBound states of the Klein-Gordon equation forring-shaped Kratzer-type potentialrdquo Chinese Physics vol 13 no5 pp 575ndash578 2004

[13] S Ikhdair and R Sever ldquoExact polynomial eigensolutions ofthe Schrodinger equation for the pseudoharmonic potentialrdquoJournal of Molecular Structure vol 806 no 1ndash3 pp 155ndash1582007

[14] F-L Lu C-Y Chen and D-S Sun ldquoBound states of Klein-Gordon equation for double ring-shaped oscillator scalar andvector potentialsrdquo Chinese Physics vol 14 no 3 pp 463ndash4672005

[15] S M Ikhdair and R Sever ldquoExact solutions of the D-dimensional Schrodinger equation for a ring-shaped pseudo-harmonic potentialrdquo Central European Journal of Physics vol 6no 3 pp 685ndash696 2008

[16] S M Ikhdair and R Sever ldquoExact bound states of the D-dimensional Klein-Gordon equation with equal scalar and

vector ring-shaped pseudoharmonic potentialrdquo InternationalJournal of Modern Physics C vol 19 no 9 pp 1425ndash1442 2008

[17] S M Ikhdair and R Sever ldquoExactly solvable effective massd-dimensional schrodinger equation for pseudoharmonic andmodified kratzer problemsrdquo International Journal of ModernPhysics C vol 20 no 3 pp 361ndash372 2009

[18] R-CWang andC-YWong ldquoFinite-size effect in the Schwingerparticle-productionmechanismrdquo Physical ReviewD vol 38 no1 pp 348ndash359 1988

[19] J N Ginocchio ldquoA relativistic symmetry in nucleirdquo PhysicsReport vol 315 no 1ndash4 pp 231ndash240 1999

[20] A Bohr I Hamarnoto and B R Motelson ldquoPseudospln inrotating nuclear potentialsrdquo Physica Scripta vol 26 no 4 pp267ndash272 1982

[21] J Dudek W Nazarewicz Z Szymanski and G A LeanderldquoAbundance and systematics of nuclear superdeformed statesrelation to the pseudospin and pseudo-SU(3) symmetriesrdquoPhysical Review Letters vol 59 no 13 pp 1405ndash1408 1987

[22] A ArimaM Harvey and K Shimizu ldquoPseudo LS coupling andpseudo SU3 coupling schemesrdquo Physics Letters B vol 30 no 8pp 517ndash522 1969

[23] K T Hecht and A Adler ldquoGeneralized seniority for favored J= 0 pairs in mixed configurationsrdquo Nuclear Physics A vol 137

no 1 pp 129ndash143 1969[24] P R Page T Goldman and J N Ginocchio ldquoRelativistic sym-

metry suppresses quark spin-orbit splittingrdquo Physical ReviewLetters vol 86 no 2 pp 204ndash207 2001

[25] S Weinberg The Quantum Theory of Fields vol 1 CambridgeUniversity Press Cambridge UK 1995

[26] D Boyanovsky C Destri and H J de Vega ldquoApproach tothermalization in the classical 1205934 theory in 1 + 1 dimensionsenergy cascades and universal scalingrdquo Physical Review D vol69 no 4 Article ID 045003 2004

[27] R Khordad ldquoEffects of magnetic field and geometrical size onthe interband light absorption in a quantumpseudodot systemrdquoSolid State Sciences vol 12 no 7 pp 1253ndash1256 2010

[28] R Khordad ldquoSimultaneous effects of temperature and pressureon the donor binding energy in a V-groove quantum wirerdquoSuperlattices and Microstructures vol 47 no 3 pp 422ndash4312010

[29] A Cetin ldquoA quantum pseudodot system with a two-dimensional pseudoharmonic potentialrdquo Physics LettersA vol 372 no 21 pp 3852ndash3856 2008

[30] S M Ikhdair M Hamzavi and R Sever ldquoSpectra of cylindricalquantum dots the effect of electrical and magnetic fieldstogether with AB flux fieldrdquo Physica B vol 407 pp 4523ndash45292012

[31] S M Ikhdair and M Hamzavi ldquoA quantum pseudodot systemwith two-dimensional pseudoharmonic oscillator in externalmagnetic and Aharonov-Bohm fieldsrdquo Physica B vol 407 pp4198ndash4207 2012

[32] S M Ikhdair and M Hamzavi ldquoEffects of extermal fieldson a two-dimensional Klein-Gordon particle under pseudo-harmonic oscillator interactionrdquo Chinese Physics B vol 21 no11 pp 110302ndash110306 2012

[33] M Aygun O Bayrak I Boztosun and Y Sahin ldquoThe energyeigenvalues of the Kratzer potential in the presence of amagnetic fieldrdquo European Physical Journal D vol 66 no 2Article ID 20319 2012

[34] S M Ikhdair andM Hamzavi ldquoSpectral properties of quantumdots influenced by a confining potential modelrdquo Physica B vol407 pp 4797ndash4803 2012

Advances in High Energy Physics 11

[35] M Hamzavi and A A Rajabi ldquoSolution of Dirac equation withKillingbeck potential by using wave function ansatz methodunder spin symmetry limitrdquo Communications in TheoreticalPhysics vol 55 no 1 pp 35ndash37 2011

[36] M Hamzavi S M Ikhdair and K E Thylwe ldquoPseudospinsymmetry in the relativistic Killingbeck potential Quasi-exactsolutionrdquo Zeitschrift fur Naturforschung A vol 67 pp 567ndash5712012

[37] R Kumar and F Chand ldquoSeries solution to the N-dimensionalradial Schrodinger equation for the quark-antiquark interac-tionrdquo Physica Scripta vol 85 no 5 pp 055008ndash055004 2012

[38] R Kumar and F Chand ldquoReply to comments on seriessolution to the N-dimensionalradial Schrodinger equation forthe quark-antiquark interactionrdquo Physica Scripta vol 86 no 2p 027001 2012

[39] S M Ikhdair ldquoA scalar charged particle in presence of magneticandAharonov-Bohmfield plus scalar-vectorKillingbeck poten-tialsrdquo Few-Body Systems vol 54 no 11 pp 1987ndash1995 2013

[40] W Greiner Relativistic Quantum Mechanics Wave EquationsSpringer Berlin Germany 2000

[41] A D Alhaidari H Bahlouli and A Al-Hasan ldquoDirac andKlein-Gordon equations with equal scalar and vector poten-tialsrdquo Physics Letters A vol 349 no 1ndash4 pp 87ndash97 2006

[42] G Plante and A F Antippa ldquoAnalytic solution of theSchrodinger equation for the Coulomb-plus-linear potentialrdquoJournal of Mathematical Physics vol 46 no 6 pp 1ndash20 2005

[43] J Killingbeck ldquoPerturbation theory without wavefunctionsrdquoPhysics Letters A vol 65 no 2 pp 87ndash88 1978

[44] Y Xu S He and C-S Jia ldquoApproximate analytical solutionsof the Klein-Gordon equation with the Poschl-Teller potentialincluding the centrifugal termrdquo Physica Scripta vol 81 no 4Article ID 045001 2010

[45] R L Liboff Introductory Quantum Mechanics PearsonEducation-Addison Wesley San Francisco Calif USA 2003

[46] GChen Z-DChen andZ-M Lou ldquoBound states of theKlein-Gordon and Dirac equation for scalar and vector pseudohar-monic oscillator potentialsrdquo Chinese Physics vol 13 no 3 pp279ndash282 2004

[47] S H Dong Z Ma and G Espozito ldquoExact solutions of theSchrodinger equation with inversepowerpotentialrdquo Founda-tions of Physics Letters vol 12 no 5 pp 465ndash474 1999

[48] A Arda and R Sever ldquoEffective-mass Klein-Gordon equationfor non-PTnon-Hermitian generalizedMorse potentialrdquo Phys-ica Scripta vol 82 no 6 Article ID 065007 2010

[49] S M Ikhdair and J Abu-Hasna ldquoQuantization rule solutionto the Hulthen potential in arbitrary dimension with a newapproximate scheme for the centrifugal termrdquo Physica Scriptavol 83 no 2 Article ID 025002 2011

[50] S M Ikhdair and R Sever ldquoExact quantization rule tothe Kratzer-type potentials An application to the diatomicmoleculesrdquo Journal of Mathematical Chemistry vol 45 no 4pp 1137ndash1152 2009

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2 Advances in High Energy Physics

equation has been used to explain the antinucleon bound ina nucleus [19] deformed nuclei [20] and super deformation[21] and to establish an effective nuclear shell model scheme[22ndash24] while the KG equation has been used in describinga wide variety of phenomena which include classical wavesystems such as the displacement of a string attached to anelastic bed [25] and quantum system based on scalar fieldtheories [26]

Recently the Schrodinger equation is solved exactly forits bound states (energy spectrum and wave functions) [27ndash29] to study the spectral properties in a 2D charged particle(electron or hole) confined by a harmonic oscillator in thepresence of external strong uniform magnetic field 997888rarr119861 alongthe 119911 direction and Aharonov-Bohm (AB) flux field createdby a solenoid The energy levels and the wave functions ofan electron confined by 2D harmonic and pseudo-harmonicoscillators have been studied in presence of external fields [3031] using theNikiforov-Uvarov (NU)method Overmore it isnatural to study the relativistic effects of the externalmagneticand AB flux fields on the KG equation for pseudo-harmonicoscillator potential especially for a strong coupling by meansof the NU method [32] The 2D solution of Schrodingerequation for the Kratzer potential with and without thepresence of a constant magnetic field is studied within theframework of the asymptotic iteration method [33] Theenergy eigenvalues are obtained analytically (numerically)for the absence (presence) of uniform magnetic field caseThese results have been obtained by using different Larmorfrequencies (120596

119871= 0) and potential parameters are compared

with the results in the absence of magnetic field case (120596119871=

0) Overmore the spectral properties of the 2D Schrodingerequation for the pseudo-harmonic-Coulomb-linear potentialare studied using the analytical iteration method [34] Underspin symmetry the energy states and wave functions of theDirac equation for the Killingbeck (harmonic oscillator plusCornell) potential have been carried out using the wave func-tion ansatz method [35] The spin and pseudospin symmetryinDirac equation have been studied in the Killingbeck poten-tial within the context of quasi-exact solution [36] Furtherthe energy eigenvalues and normalized eigenfunctions of theradial Schrodinger equation in119873-dimensional Hilbert spacefor the quark-antiquark interaction Killingbeck potentialhave been obtained using the power series technique via asuitable choice of ansatz to the wave function [37 38]

Very recently we studied the scalar charged particleexposed to relativistic scalar-vector Killingbeck potentials inpresence of magnetic and Aharonov-Bohm flux fields andobtained its energy eigenvalues and wave functions usingthe analytical exact iteration method [39] Therefore thebehavior of a spinless relativistic particle moving under theKillingbeck potential in a static magnetic and AB flux fieldshas not been investigated yet and in thisworkwe aim to solvethe KG equation in 2D for equal mixture of scalar and vectorKillingbeck potentials with and without constant magneticand AB flux fields for the first time We present the exactenergy eigenvalues and wave functions of the Killingbeckpotential for any 119899 and 119898 quantum numbers in a constantmagnetic field with the different Larmor frequencies using

the wave function ansatz method [36] The non-relativisticenergy eigenvalues and wave functions of our solution arepresented by making an appropriate mapping of parametersOvermore special cases of KG for equal scalar-vector Killing-beck potentials are also presented in the presence (120596

119871= 0

120585 = 0) and absence (120596119871= 0 120585 = 0) of uniform fields

The structure of this paper is as follows We studyeffect of external uniform magnetic and AB flux fields on arelativistic spinless particle (antiparticle) under equalmixtureof scalar and vector Killingbeck potentials in Section 2 Wediscuss some special cases in Section 3 Finally we give ourconcluding remarks in Section 4

2 The Klein-Gordon Atom

The Klein-Gordon atom for the spinless particle with mass119898119890and charge minus119890 moving in external electromagnetic field

and AB flux field given by potentials 119881(119903) 119878(119903) and 997888rarr119860 reads[40 41]

[1198882(997888rarr119901 +

119890

119888

997888rarr

119860)

2

minus (119864 minus 119881 (119903))2+ (1198981198901198882+ 119878 (119903))

2

]

times 120595 (119903 120601) = 0

(1)

The scalar and vector potentials are chosen in the followingform [27ndash30]

119881119870 (119903) = 120582119903

2+ 120590119903 minus

120581

119903

997888rarr

119860 =

1

2

997888rarr

119861 times997888rarr119903 +

ΦAB2120587119903

120601 (2)

where 119881119870(119903) is the Killingbeck potential that is harmonic

oscillator potential plus Cornell potential [35ndash38] which isextensively used in particle physics [42 43] Moreover thevector potential in the symmetric gauge is defined by 997888rarr119860 =

997888rarr

1198601+

997888rarr

1198602such that 997888rarrnabla times

997888rarr

1198601=

997888rarr

119861 and 997888rarrnabla times

997888rarr

1198602= 0 where the

applied magnetic field 997888rarr119861 = (0 0 119861) is perpendicular to theplane of transversal motion of the particle and 997888rarr119860

2describes

the additional AB flux field ΦAB created by a solenoid incylindrical coordinates [32] The wave function in (1) isdefined by

120595 (119903 120601) =

1

radic2120587

119890119894119898120601119877 (119903)

radic119903

119898 = 0 plusmn1 plusmn2 (3)

where 119898 is the eigenvalue of angular momentum Therelationship between the attractive scalar and repulsive vectorpotentials is given by 119878(119903) = 120573119881(119903) where minus1 le 120573 le

1 is arbitrary constant and hence the KG equation couldbe reduced to a Schrodinger-type second-order differentialequation as follows

[1198882(997888rarr119901 +

119890

119888

997888rarr

119860)

2

+ 2 (119864119881 (119903) + 1198981198901198882119878 (119903))

+1198782(119903) minus 119881

2(119903) + 119898

2

1198901198884minus 1198642]120595 (119903 120601) = 0

(4)

Now we will treat the bound-state solutions of the two casesin (4) as follows

Advances in High Energy Physics 3

21 The Positive Energy Solution The positive energy statesrequire that 119878(119903) = 119881(119903) (ie 120573 = 1 case) which in thenon-relativistic limit corresponds to the solution of the waveequation

1

2120583

[minus119894ℎ

997888rarr

nabla +

119890

119888

(

119861119903

2

+

ΦAB2120587119903

)120601]

2

+ 2119881119870 (119903) minus 119864

times 120595 (119903 120601) = 0

(5)

where120595(119903 120601) stands for non-relativistic wave functionThusthe choice 119878(119903) = 119881(119903) produces a non-relativistic limit with apotential function 2119881(119903) and not 119881(119903) Accordingly it wouldbe natural to scale the potential term in (4) and (5) so thatin the non-relativistic limit the interaction potential becomes119881(119903) not 2119881(119903)Thus we need to recast (4) for the 119878(119903) = 119881(119903)

as [32 39 40 44]

[1198882(minus119894ℎ

997888rarr

nabla +

119890

119888

997888rarr

119860)

2

+ (119864 + 1198981198901198882)119881 (119903)]120595 (119903 120601)

= (1198642minus 1198982

1198901198884) 120595 (119903 120601)

(6)

where

nabla2=

1

120588

120597

120597120588

+

1205972

1205971205882+

1

1205882

1205972

1205971206012+

1205972

1205971199112 (7)

and in order to simplify (6) we introduce new parameters1205721= 119864 +119898

1198901198882 and 120572

2= 119864 minus119898

1198901198882 so that it can be reduced to

the following form

1198882[minus119894ℎ

997888rarr

nabla +

119890

119888

(

119861119903

2

+

ΦAB2120587119903

)120601]

2

minus 1205721(1205722minus 119881119870 (119903))

times 120595 (119903 120601) = 0

(8)

By inserting (2) and (3) into (8) we obtain

1198892119877 (119903)

1198891199032

+

1205721

ℎ21198882[1205722minus 119880eff (119903 120596119871 120585)] 119877 (119903) = 0 (9a)

119880eff (119903 120596119871 120585) = 119881119870 (119903) +

1198982

11989011988821205962

119871

1205721

1199032

+

ℎ21198882

1205721

(11989810158402minus 14)

1199032

+

2ℎ1205961198711198981198901198882

1205721

1198981015840

(9b)

120596119871=

Ω

2

Ω =

|119890| 119861

119898119890119888

1198981015840= 119898 + 120585

1198981015840= 1 2 120585 =

ΦABΦ0

(9c)

where the effective potentials depending on the magnitudesof two fields strength with 120596

119871and 119898

1015840 are the Larmorfrequency and a new eigenvalue of angularmomentum (mag-netic quantum number) respectively It is worthy to mentionthat the frequency Ω is called the cyclotron frequency [45]This is the frequency of rotation corresponding to the classical

motion of a charged particle in a uniform magnetic field andΩ2 is the Larmor frequency in units of Hz [45] Moreoverwe take 120585 as integer with the flux quantum Φ

0= ℎ119888119890 Here

119881119870(119903) is a pure Killingbeck potential (2) the second term is

the harmonic oscillator-type potential and the other termsare the rotational potential creating the rotational energylevels Equations (9a) (9b) and (9c) can be alternativelyexpressed as

1198892119877 (119903)

1198891199032

+ (

1198621

1199032+

1198622

119903

minus 1198623minus 1198624119903 minus 11986251199032)119877 (119903) = 0 (10a)

1198621= minus(119898

10158402minus

1

4

) 1198622=

1205721120581

ℎ21198882

1198623=

21198981198901205961198711198981015840

ℎ

minus

12057211205722

ℎ21198882

(10b)

1198624=

1205721120590

ℎ21198882 119862

5=

1205721120582

ℎ21198882+ (

119898119890120596119871

ℎ

)

2

(10c)

with the asymptotic behaviors 119877(0) = 0 and 119877(infin) rarr 0 Itis interesting to look at the results obtained from (10a) for aspecial case 119862

2= 1198624= 0 that is we have

1198892119877 (119903)

1198891199032

+ (

1198621

1199032minus 11986251199032)119877 (119903) = 119862

3119877 (119903) (11)

which corresponds to the differential equation of a harmonicoscillator with a centrifugal term It is a form similar to theone given by (7) in [46]with the equivalence119862

1rarr minus119897

1015840(1198971015840+1)

1198625rarr 1205722 and119862

3rarr minus120582

119899(120572 119897 and120582 are the parameters used

in [46]) So by putting the parameter values

1198971015840= 1198981015840minus

1

2

120572 =radic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

120582119899=

1198642minus 1198982

1198901198884

ℎ21198882

minus

21198981198901205961198711198981015840

ℎ

(12)

into (14) of [45] we obtain the energy equation as

1198642minus 1198982

1198901198884= 21198981198901198882ℎ1205961198711198981015840

+ 2ℎ1198882(2119899 + 119898

1015840+ 1)

timesradic(119864 + 119898

1198901198882) 120582

1198882

+ (119898119890120596119871)2

(13)

Now we find a solution to (10a) by making the followingchoice of the wave function [37 38 47 48]

119877119899119898 (

119903) = exp(12

1199011199032+ 119902119903)

infin

sum

119899=0

119886119899119903119899+120575

(14)

where 119901 and 119902 are parameters whose values are to bedetermined in terms of the potential parameters 120582 120590 and 120581Substituting (14) into (10a) we obtain the series [36]infin

sum

119899=0

119886119899119878119899119903119899+120575minus2

+

infin

sum

119899=1

119886119899minus1

119879119899minus1

119903119899+120575minus2

+

infin

sum

119899=2

119886119899minus2

119882119899minus2

119903119899+120575minus2

= 0

1199012= 1198625 2119901119902 = 119862

4

(15)

4 Advances in High Energy Physics

with119878119899= (119899 + 120575) (119899 + 120575 minus 1) + 1198621

119879119899minus1

= 2119902 (119899 + 120575 minus 1) + 1198622

119882119899minus2

= 1199022+ 2119901(119899 + 120575 minus

3

2

) minus 1198623

(16)

where the two parameters 119901 and 119902 are determined in orderfor the radial wave functions 119877

119899119898(119903) to be finite everywhere

and vanish at 119903 = 0 and as 119903 rarr infin To obtain the recurrencerelation which can connect various expansion coefficients 119886

119899

we make identical powers of 119903 in (15) that is equate thecoefficients of 119903119899+120575minus2 to zero Thus the relations become

1198860[120575 (120575 minus 1) + 1198621

] = 0 997904rArr 1205752+ 1198621= 120575

1198860

= 0 997904rArr 120575 = plusmn1198981015840+

1

2

1198861= minus

(2119902120575 + 1198622)

2120575

1198860

1198862= minus

[119901 (2120575 + 1) + 1199022minus 1198623] 1198860+ [2119902 (120575 + 1) + 1198622

] 1198861

2 (2120575 + 1)

1198863= minus(

(2119901119902 minus 1198624) 1198860+ [119901 (2120575 + 3) + 119902

2minus 1198623] 1198861

6 (120575 + 1)

+

[2119902 (120575 + 2) + 1198622] 1198862

6 (120575 + 1)

)

119886119899= minus(

(1199012minus 1198625) 119886119899minus4

+ (2119901119902 minus 1198624) 119886119899minus3

119899 (2120575 + 119899 minus 1)

+

[119901 (2120575 + 2119899 minus 3) + 1199022minus 1198623] 119886119899minus2

119899 (2120575 + 119899 minus 1)

+

[2119902 (120575 + 119899 minus 1) + 1198622] 119886119899minus1

119899 (2120575 + 119899 minus 1)

)

(17)

where 119899 = 0 1 2 with 1198860

= 0 Here the positive sign ofparameter 120575 = 119898

1015840+ 12 has been selected The power series

for large values of 120575 or1198981015840 is convergent For convenience wetake the ratio of two successive terms that is 119886

119899+1119886119899 which

becomes1198861

1198860

= minus(119902 +

1198622

2120575

) 997888rarr minus119902 when 120575 997888rarr infin

1198862

1198860

=

1199022(120575 + 119862

22119902) [120575 + 119862

22119902 + 1]

120575 (2120575 + 1)

minus

[119901 (2120575 + 1) + 1199022minus 1198623]

2 (2120575 + 1)

997888rarr

1199022minus 119901

2

when 120575 997888rarr infin

(18)

It is apparent from the above relations that the power seriesconverges to zero when 120575 rarr infin Hence the series mustbe truncated (bounded) for 119899 = 119899max At this value of 119899 weobtain the following equations

119901 = plusmnradic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

997904rArr 119901 = minusradic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

119901 = 0

(19a)

2119901119902 = 1198624

997904rArr 119902 = minus

(119864 + 1198981198901198882) 120590

2ℎ21198882radic((119864 + 119898

1198901198882) 120582ℎ21198882) + (119898

119890120596119871ℎ)2

(19b)

(119864 + 1198981198901198882) 120581

ℎ21198882

= minus2119902 (119899 + 1198981015840minus

1

2

) (19c)

(119864 + 1198981198901198882) (119864 minus 119898

1198901198882)

ℎ21198882

=

21198981198901205961198711198981015840

ℎ

+ 2radic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

times (1198981015840+ 119899 minus 1) minus 119902

2

(19d)

where the negative sign of the coefficient 119901 has been chosenin (19a) It is worth noting that the potential parameter120582mustbe positive when 120596

119871= 0 Hence (19c) gives a restriction on

the potential parameters 120582 120590 and 120581 as follows

120590 =

120581

(119899 + 1198981015840minus 12)

radic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

(20)

and it follows that (19d) gives the energy formula as

1198642minus 1198982

1198901198884= 21198981198901198882ℎ1205961198711198981015840+ 2ℎ21198882(119899 + 119898

1015840minus 1)

timesradic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

minus

(119864 + 1198981198901198882)

2

1205812

4ℎ21198882(119899 + 119898

1015840minus 12)

2

(21)

where 119899 = 0 1 2 We may find a solution to theabove transcendental equation as 119864 = 119864

(+)

KG = 1198641198991198981015840(120596119871 120585)

Overmore the wave function (14) with the help of (17) (19a)and (19b) becomes

Advances in High Energy Physics 5

120595(+)

119899119898(119903 120601) = 119862

119899119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(12)radic(119864+119898

1198901198882)120582ℎ21198882+(119898119890120596119871ℎ)2[1199032+((119864+119898

1198901198882)120590((119864+119898

1198901198882)1198892+ℎ21198882(119898119890120596119871ℎ)2))119903]

119899max

sum

119899=0

119886119899119903119899

(22a)

1198861= minus1198860[119902 +

(119864 + 1198981198901198882) 120581

2ℎ21198882(1198981015840+ 12)

] (22b)

1198862= minus

1

2

1198860[119901 minus 119902

2minus

1198981198901205961198711198981015840

ℎ (1198981015840+ 1)

minus

(119864 + 1198981198901198882) 120581

ℎ21198882(1198981015840+ 12)

(119902 +

(119864 + 1198981198901198882) 120581

4ℎ21198882(1198981015840+ 1)

)] (22c)

where 119862119899119898

is the normalization constantIn the non-relativistic limit when 119864 + 119898

1198901198882

rarr 2120583119864 + 119898

1198901198882rarr 1198641198991198981015840 and 119888 = 1 (21) and (22a) give the energy

formula

1198641198991198981015840 (120596119871 120585) = ℎ120596

1198711198981015840+ ℎ (119899 + 119898

1015840minus 1)

times radic

2120582

120583

+ 1205962

119871minus

1205831205812

2ℎ2(119899 + 119898

1015840minus 12)

2

(23)

and wave function

120595(+)

119899119898(119903 120601) = 119862

119899119898

1

radic2120587

119890119894119898120601

119903119898+120585

times 119890minus(1205832ℎ)radic2120582120583+120596

2

119871[1199032+(2120590120583(2119889

2120583+1205962

119871))119903]

119899max

sum

119899=0

119886119899119903119899

(24)

respectively for the Schrodinger-Killingbeck system with thefollowing restriction on the potential parameters 120582 120590 and 120581given by

120590 =

120583120581

ℎ (119899 + 1198981015840minus 12)

radic

2120582

120583

+ 1205962

119871 (25)

Notice that the present model has been solved in 2D spacewith an external uniform magnetic field since it is perpen-dicular to the plane where the vector and scalare Cornellpotentials have the dimensions of 119903 = 120588 and 120601 that is119881 = 119878 = 119881(120588 120601) [32] However without the magnetic fieldthe model can be solved in any desired dimensional spaceby considering the change 119898 rarr 119897 + (119863 minus 2)2 [49 50]where119863 refers to spatial dimension and also the existence ofinterdimensional degeneracy

As shown in Figure 1(a) and (9b) the effective potentialfunction changes in shape when the magnetic field strength

increases say 120596119871= 8 and in absence of AB flux field The

energy levels are raised when the strength of the magneticfield increases and in absence of AB flux field 120585 = 0 Wesee that the effective potential changes gradually from thepure pseudo-harmonic oscillator potential when 120596

119871= 0 to

a pure harmonic oscillator type behavior in short potentialrange when120596

119871= 8 In Figure 1(b) the effective potential (9b)

which is pseudo-harmonic oscillator when 120596119871= 0 becomes

sensitive to the increasing AB flux field 120585 = 8 in the shortrange region that is 0 lt 119903 lt 4 au

22 The Bound States for Negative Energy In this case of119878(119903) = minus119881(119903) the first inspection of (4) shows that thefollowing changes 119881(119903) rarr minus119881(119903) (ie 120582 rarr minus120582 120590 rarr minus120590120581 rarr minus120581) 120595(+)

119899119898(119903 120601) rarr 120595

(minus)

119899119898(119903 120601) and 119864 rarr minus119864 give the

negative energy solution for antiparticles as

1198642minus 1198982

1198901198884minus 21198981198901198882ℎ1205961198711198981015840

= 2ℎ21198882(119899 + 119898

1015840minus 1)

radic(119864 minus 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

minus

(119864 minus 1198981198901198882)

2

1205812

4ℎ21198882(119899 + 119898

1015840minus 12)

2

(26)

with restriction on potential parameters

120590 =

120581

(119899 + 1198981015840minus 12)

radic(119864 minus 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

(27)

We may find solution to the above transcendental equationas 119864 = 119864

(minus)

119870119866= 1198641198991198981015840(120596119871 120585) Overmore the wave function for

antiparticle reads

6 Advances in High Energy Physics

0 1 2 3 4 5 6 70

2

4

6

8

10

12

14

16

18

20

120596L = 0

120596L = 1

120596L = 5

120596L = 8

r (au)

Ueff(r120596

L)

(a)

0 1 2 3 4 50

2

4

6

8

10

12

14

16

18

20

r (au)

120585 = 0

120585 = 1

120585 = 5

120585 = 8

Ueff(r120585)

(b)

Figure 1 The KG effective potential function for (a) 120596119871= 0 1 5 8 and with 120585 = 0 (b) 120585 = 0 1 5 8 and with 120596

119871= 0 (colour online) Here

119898119890= ℎ = 119888 = 1

120595(minus)

119899119898(119903 120601) = 119863

119899119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(12)radic(119864minus119898

1198901198882)120582ℎ21198882+(119898119890120596119871ℎ)2[1199032+((119864minus119898

1198901198882)120590((119864minus119898

1198901198882)1198892+ℎ21198882(119898119890120596119871ℎ)2))119903]

119899max

sum

119899=0

119886119899119903119899 (28)

1198861= minus1198860[119902 +

(119864 minus 1198981198901198882) 120581

2ℎ21198882(1198981015840+ 12)

] (29)

1198862= minus

1

2

1198860[119901 minus 119902

2minus

1198981198901205961198711198981015840

ℎ (1198981015840+ 1)

minus

(119864 minus 1198981198901198882) 120581

ℎ21198882(1198981015840+ 12)

(119902 +

(119864 minus 1198981198901198882) 120581

4ℎ21198882(1198981015840+ 1)

)] (30)

where119863119899119898

is the normalization constant

3 Discussions

In this section we discuss some special cases of interest fromour general solution

(i) If we set 120590 = 120581 = 0 the effective Killingbeck potentialturns to effective harmonic oscillator potential in relativisticcase as

119880eff (119903 120596119871 120585) = 1205821199032+

1198982

11989011988821205962

119871

(119864 + 1198981198901198882)

1199032

+

ℎ21198882

(119864 + 1198981198901198882)

(11989810158402minus 14)

1199032

+

2ℎ12059611987111989811989011988821198981015840

(119864 + 1198981198901198882)

120582 =

1

2

1205831205962

(31)

The bound state solutions (with the change 119899 = 2(119899119903+ 1)) are

[32 36]

1198642minus 1198982

1198901198884= 21198981198901198882ℎ1205961198711198981015840+ 4ℎ1198882(119899119903+

1198981015840+ 1

2

)

timesradic(119864 + 119898

1198901198882) 120582

1198882

+ (119898119890120596119871)2

(32)

120595(+)

119899119903119898(119903 120601) = 119860

119899119903119898

1

radic2120587

119890119894119898120601

119903119898+120585

times 119890minus(12)radic(119864+119898

1198901198882)120582ℎ21198882+(119898119890120596119871ℎ)21199032

119899max

sum

119899=0

1198861198991199032(119899119903+1)

(33a)

1198861= 0

1198862=

1

2

1198860[

[

radic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

+

1198981198901205961198711198981015840

ℎ (1198981015840+ 1)

]

]

(33b)

Advances in High Energy Physics 7

where 119860119899119903119898

is the normalization constant On the otherhand the nonrelativistic effective harmonic oscillator poten-tial

119880eff (119903 120596119871 120585) = 1205821199032+

1

2

1205962

1198711199032+

(11989810158402minus 14) ℎ

2

21198981198901199032

+ ℎ1205961198711198981015840

(34)

has bound states as [32]

1198641198991198981015840 (120596119871 120585) = ℎ120596

1198711198981015840

+ 2ℎΩ1015840(119899119903+

10038161003816100381610038161003816119898101584010038161003816100381610038161003816+ 1

2

) Ω1015840= radic120596

2

119871+ 1205962

(35)

120595(+)

119899119903119898(119903 120601) = 119862

119899119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(1205832ℎ)Ω

10158401199032

times

119899max

sum

119899=0

1198861198991199032(119899119903+1) 119899119903= 0 1 2

=

1

radic2120587

119890119894119898120601

radic21198871198981015840+1119899

(119899 + 1198981015840)

1199031198981015840

119890minus11988711990322119871(1198981015840)

119899(1198871199032)

119887 =

120583

ℎ

Ω

1015840

(36a)

1198861= 0 119886

2=

120583

2ℎ

Ω10158401198860 1198860

= 0 (36b)

(ii) Setting 120582 = 120590 = 0 in (21) the killingbeck potentialturns to the Coulomb potential case (ie 119881(119903) = minus119860119903 and119899 = 119899119903+ 1) Thus we obtain the relativistic energy formula

1198642minus 1198982

1198901198884= 21198981198901198882(119899119903+ 21198981015840) ℎ120596119871

minus

(119864 + 1198981198901198882)

2

1205812

4ℎ21198882(119899119903+ 1198981015840+ 12)

2 120581 = 119885119890

2

(37)

and the wave function

120595(+)

119899119903119898(119903 120601) = 119862

119899119903119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(119898119890120596119871)2ℎ1199032

times

119899max

sum

119899=0

119886119899119903119899119903+1

(38a)

1198861= minus

(119864 + 1198981198901198882) 120581

2ℎ21198882(1198981015840+ 12)

1198860 (38b)

1198862= minus

1

2

[

[

119898119890120596119871

ℎ

minus

1198981198901205961198711198981015840

ℎ (1198981015840+ 1)

minus

(119864 + 1198981198901198882)

2

1205812

4ℎ41198884(1198981015840+ 1) (119898

1015840+ 12)

]

]

1198860

(38c)

Overmore in the nonrelativistic limit we have

119864 = ℎ1205961198711198981015840+ ℎ (119899

119903+ 1198981015840) 120596119871minus

11989811989011988521198904

2ℎ2(119899119903+ 1198981015840+ 12)

2 (39)

and the wave function becomes

120595(+)

119899119903119898(119903 120601) = 119862

119899119903119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(1205831205961198712ℎ)1199032

119899max

sum

119899=0

119886119899119903119899119903+1

(40a)

1198861= minus

119898119890119860

ℎ2(1198981015840+ 12)

1198860 (40b)

1198862= minus

1

2

[

120583120596119871

ℎ

minus

1205831205961198711198981015840

ℎ (1198981015840+ 1)

minus

120583211988521198904

ℎ4(1198981015840+ 1) (119898

1015840+ 12)

] 1198860

(40c)

Notice that in the absence of external fields 120596119871= 120585 = 0

the problem can be solved in any dimension Thus applyingthe transformation |119898| rarr 119897 + 12 our previous results arereduced to the well-known three-dimensional bound statesolutions for the harmonic oscillator and the hydrogen-likeatoms in the Coulomb fields [40]

119864plusmn= plusmn1198981198901198882[[

[

1 +

2ℎ119888

1198981198901198882(2119899119903+ 119897 +

3

2

)radic

(119864 + 1198981198901198882) 120582

1198982

1198901198884

]]

]

12

119864plusmn= plusmn1198981198901198882[1 minus

11988521198904

4ℎ21198882(119899119903+ 119897 + 1)

2(1 +

119864

1198981198901198882)

2

]

12

(41)

respectively According to (41) the energy spectrum can befound for the scalar particle and antiparticle

In Figure 2(a) we plot the effective potential for the caseof low vibrational (119899 = 0 1 2 3) and rotational (119898 = 1)

levels for various Larmor frequencies 120596119871

= 0 1 5 8 and120585 = 0 case As shown in Figure 2(a) and (31) the effectivepotential function changes in shape as well as the bound stateenergy eigenvalues increase when 120596

119871= 8 It is shown that the

energy levels are raised when the strength of the magneticfield increases and in the absence of AB flux field It is alsoobvious that the effective potential changes gradually fromthe pure pseudo-harmonic oscillator potential which is anonmagnetic (120596

119871= 0) and AB flux (120585 = 0) fields case to

a pure harmonic oscillator type behavior in short potentialrange when the strength of the applied magnetic field isincreased to 120596

119871= 8 If we consider a strong magnetic field

case 120596119871= 8 which has the shape of pure harmonic oscillator

potential function the energy difference between adjacentenergy levels is nearly equal which is a known characteristicof the pure harmonic oscillator potential In Figure 2(b) weplot the effective potential (34) in the absence of themagneticfield and in the presence of AB flux field in the short rangeregion

In Tables 1 and 2 we show the effect of magnetic fieldand AB flux field respectively on the low vibrational 119899

8 Advances in High Energy Physics

120596L = 0

120596L = 1

120596L = 5

120596L = 8

E01E11

E21

E31

E01

E11

E21E31

E01E11

E21

E31

0 1 2 3 4 50

10

20

30

40

50

60

70

80

r (au)

Ueff(r120596

L)

(a)

0 1 2 3 4 5 6 7 80

10

20

30

40

50

60

70

80

r (au)

120585 = 0

120585 = 3

120585 = 5

120585 = 8

Ueff(r120585)

(b)

Figure 2 The Schrodinger effective potential function and corresponding bound state energy levels (119864119899119898) in low vibrational (119899 = 0 1 2 3)

and rotational (119898 = 1) levels for (a) 120596119871= 0 1 5 8 with 120585 = 0 (b) 120585 = 0 1 5 8 and with 120596

119871= 0 (color online) Here 120583 = ℎ = 1

and rotational 119898 relativistic energy states of the harmonicoscillator potential As shown in Table 1 when the magneticfield is not applied and without AB flux field (120596

119871= 0

120585 = 0) the spacing between the energy levels of the effectivepotential is narrow and decreases with increasing 119899 Butwhen the magnetic field strength increases the energy levelsof the effective potential increase and the spacings betweenstates also increase In Table 2 when the AB flux field isapplied and without magnetic field the energy states becomedegenerate for various values of 119899 and 119898 and for variousAB flux field strength values In Tables 3 and 4 we showthe effect of magnetic field and AB flux field respectivelyon the low vibrational 119899 and rotational 119898 nonrelativisticenergy states of the harmonic oscillator potential As shownin Table 3 when themagnetic field is not applied and withoutAB flux field (120596

119871= 0 120585 = 0) the energy states are equally

spaced (the pure harmonic oscillator case) But when themagnetic field strength is raised the energy levels of theeffective potential increase and the spacings between statesalso increase In Table 4 when the AB flux field is applied andwithout magnetic field the energy states become degenerateand equally spaced for various values of 119899 and 119898 and forvarious AB flux field strength values

4 Concluding Remarks

To sum up in this paper we have studied the solutionof two-dimensional KG and Schrodinger equations withthe Killingbeck potential for low vibrational and rotationalenergy levels without and with a constant magnetic fieldhaving arbitrary Larmor frequeny and AB flux field Wehave applied the wave function ansatz method for 120596

119871= 0

(with magnetic field) and 120585 = 0 (with AB flux field) toobtain analytical expressions in closed form for boundstate energies and wave functions of the spinless relativisticparticle subject to a Killingbeck interaction expressed interms of external uniform magnetic and AB flux fields inany vibrational 119899 and rotational 119898 states The above resultsshow that the problems of relativistic quantum mechanicscan be also solved exactly as in the non-relativistic onesWe considered the solution of both positive (particle) andnegative (antiparticle) KG energy states It is noticed that thesolution with equal mixture of scalar-vector potentials can beeasily reduced into the well-known Schrodinger solution fora particle with an interaction potential field and a free fieldrespectively We have also studied the bound-state solutionsfor some special cases including the non-relativistic limits(Schrodinger equation for harmonic oscillator and Coulombpotentials under external magnetic and AB flux fields) andalso the KG equation for harmonic oscillator and CoulombinteractionsThe results show that the splitting is not constantand is mainly dependent on the strength of the externalmagnetic field and AB flux field In order to show the effectof constant magnetic and AB flux fields on the vibrationaland rotational energy levels of the harmonic oscillator weplot the effective potential and corresponding energy levelswith increasing Larmor frequency and flux field for specialpotential parametersWehave seen that the effective potentialfunction and corresponding energy levels are raised in energywhen magnetic and AB flux field strengths increase Theeffective potential function behavior gradually changes fromthe pure pseudo-harmonic oscillator to a pure harmonicoscillator shape in short potential range as the magnetic andAB flux fields strengths increase

Advances in High Energy Physics 9

Table 1 The KG energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120596119871values and 120585 = 0 Here ℎ = 119888 =

119898119890= 119896 = 1

119898 119899

119864119899119898

120596119871= 0 120596

119871= 1 120596

119871= 2 120596

119871= 3 120596

119871= 4 120596

119871= 5 120596

119871= 6 120596

119871= 7 120596

119871= 8

0

0 183929 204353 240325 275615 308137 338066 365815 391752 4161661 309625 34289 403986 465113 521808 574094 622602 667941 7106092 412383 451488 526359 603138 675146 74193 804087 862301 9171583 503104 545806 630283 718744 802542 880667 953605 102205 108663

1

0 250976 316597 389307 455265 514478 568284 617802 663858 707061 362919 427262 512444 593706 668087 736315 799437 858337 9137072 458916 522885 616824 709549 795634 87516 949036 101815 1083233 545354 609216 709891 811893 90768 996689 107966 115745 123081

Table 2 The KG energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120585 values and 120596119871= 0

119898 119899

119864119899119898

120585 = 0 120585 = 1 120585 = 2 120585 = 3 120585 = 4 120585 = 5 120585 = 6 120585 = 7 120585 = 8

0

0 183929 250976 309625 362919 412383 458916 503104 545354 5859661 309625 362919 412383 458916 503104 545354 585966 625166 663132 412383 458916 503104 545354 585966 625166 6631370 70 7358923 503104 545354 585966 625166 66313 70 735892 770901 805108

1

0 250976 309625 362919 412383 458916 503104 545354 585966 6251661 362919 412383 458916 503104 545354 585966 625166 66313 702 458916 503104 545354 585966 625166 66313 70 735892 7709013 545354 585966 625166 66313 70 735892 770901 805108 838582

Table 3 The nonrelativistic energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120596119871values and 120585 = 0

119898 119899

119864119899119898

120596119871= 0 120596

119871= 1 120596

119871= 2 120596

119871= 3 120596

119871= 4 120596

119871= 5 120596

119871= 6 120596

119871= 7 120596

119871= 8

0

0 10 141421 223607 316228 412311 509902 608276 707107 8062261 30 424264 67082 948683 123693 152971 182483 212132 2418682 50 707107 111803 158114 206155 254951 304138 353553 4031133 70 989949 156525 221359 288617 356931 425793 494972 564358

1

0 20 382843 647214 932456 122462 15198 181655 211421 2412451 40 665685 109443 156491 204924 253961 303311 352843 402492 60 948528 154164 219737 287386 355941 424966 494264 5637353 80 123137 198885 282982 369848 457922 546621 635685 724981

Table 4 The nonrelativistic energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120585 values and 120596119871= 0

119898 119899

119864119899119898

120585 = 0 120585 = 1 120585 = 2 120585 = 3 120585 = 4 120585 = 5 120585 = 6 120585 = 7 120585 = 8

0

0 10 20 30 40 50 60 70 80 901 30 40 50 60 70 80 90 100 1102 50 60 70 80 90 100 110 120 1303 70 80 90 100 110 120 130 140 150

1

0 20 30 40 50 60 70 80 90 1001 40 50 60 70 80 90 100 110 1202 60 70 80 90 100 110 120 130 1403 80 90 100 110 120 130 140 150 160

10 Advances in High Energy Physics

Acknowledgments

One of the authors (Sameer M Ikhdair) would like to thankthe president of An-Najah National University (ANU) Pro-fessor Rami Alhamdallah and the acting president ProfessorMaher Natsheh for their continuous support during hispresent work at ANU The authors acknowledge the partialsupport of the Scientific and Technological Research Councilof Turkey

References

[1] WGreiner andBMullerQuantumMechanics An IntroductionSpringer Berlin Germany 1994

[2] S Ozcelik and M Simsek ldquoExact solutions of the radialSchrodinger equation for inverse-power potentialsrdquo PhysicsLetters A vol 152 no 3-4 pp 145ndash150 1991

[3] S-H Dong ldquoSchrodinger equation with the potential 119881(119903) =119860119903minus4+ 119861119903minus3+ 119862119903minus2+ 119863119903minus1rdquo Physica Scripta vol 64 no 4 pp

273ndash276 2001[4] S-H Dong and Z-Q Ma ldquoExact solutions to the Schrodinger

equation for the potential 119881(119903) = 1198861199032+ 119887119903minus4

+ 119888119903minus6 in two

dimensionsrdquo Journal of Physics A vol 31 no 49 pp 9855ndash98591998

[5] M S Child S-H Dong and X-GWang ldquoQuantum states of asextic potential hidden symmetry and quantum monodromyrdquoJournal of Physics A vol 33 no 32 pp 5653ndash5661 2000

[6] S-H Dong ldquoExact solutions of the two-dimensional Schrodin-ger equation with certain central potentialsrdquo InternationalJournal of Theoretical Physics vol 39 no 4 pp 1119ndash1128 2000

[7] S-H Dong ldquoA new approach to the relativistic schrodingerequation with central potential Ansatz methodrdquo InternationalJournal of Theoretical Physics vol 40 no 2 pp 559ndash567 2001

[8] S-H Dong ldquoOn the solutions of the Schrodinger equation withsome anharmonic potentials wave function ansatzrdquo PhysicaScripta vol 65 no 4 pp 289ndash295 2002

[9] S Flugge Practical Quantum Mechanics 1 Springer BerlinGermany 1994

[10] X-Q Zhao C-S Jia and Q-B Yang ldquoBound states of rel-ativistic particles in the generalized symmetrical double-wellpotentialrdquo Physics Letters A vol 337 no 3 pp 189ndash196 2005

[11] S M Ikhdair ldquoExact solution of Dirac equation with chargedharmonic oscillator in electric field bound statesrdquo Journal ofModern Physics vol 3 no 2 pp 170ndash179 2012

[12] W C Qiang ldquoBound states of the Klein-Gordon equation forring-shaped Kratzer-type potentialrdquo Chinese Physics vol 13 no5 pp 575ndash578 2004

[13] S Ikhdair and R Sever ldquoExact polynomial eigensolutions ofthe Schrodinger equation for the pseudoharmonic potentialrdquoJournal of Molecular Structure vol 806 no 1ndash3 pp 155ndash1582007

[14] F-L Lu C-Y Chen and D-S Sun ldquoBound states of Klein-Gordon equation for double ring-shaped oscillator scalar andvector potentialsrdquo Chinese Physics vol 14 no 3 pp 463ndash4672005

[15] S M Ikhdair and R Sever ldquoExact solutions of the D-dimensional Schrodinger equation for a ring-shaped pseudo-harmonic potentialrdquo Central European Journal of Physics vol 6no 3 pp 685ndash696 2008

[16] S M Ikhdair and R Sever ldquoExact bound states of the D-dimensional Klein-Gordon equation with equal scalar and

vector ring-shaped pseudoharmonic potentialrdquo InternationalJournal of Modern Physics C vol 19 no 9 pp 1425ndash1442 2008

[17] S M Ikhdair and R Sever ldquoExactly solvable effective massd-dimensional schrodinger equation for pseudoharmonic andmodified kratzer problemsrdquo International Journal of ModernPhysics C vol 20 no 3 pp 361ndash372 2009

[18] R-CWang andC-YWong ldquoFinite-size effect in the Schwingerparticle-productionmechanismrdquo Physical ReviewD vol 38 no1 pp 348ndash359 1988

[19] J N Ginocchio ldquoA relativistic symmetry in nucleirdquo PhysicsReport vol 315 no 1ndash4 pp 231ndash240 1999

[20] A Bohr I Hamarnoto and B R Motelson ldquoPseudospln inrotating nuclear potentialsrdquo Physica Scripta vol 26 no 4 pp267ndash272 1982

[21] J Dudek W Nazarewicz Z Szymanski and G A LeanderldquoAbundance and systematics of nuclear superdeformed statesrelation to the pseudospin and pseudo-SU(3) symmetriesrdquoPhysical Review Letters vol 59 no 13 pp 1405ndash1408 1987

[22] A ArimaM Harvey and K Shimizu ldquoPseudo LS coupling andpseudo SU3 coupling schemesrdquo Physics Letters B vol 30 no 8pp 517ndash522 1969

[23] K T Hecht and A Adler ldquoGeneralized seniority for favored J= 0 pairs in mixed configurationsrdquo Nuclear Physics A vol 137

no 1 pp 129ndash143 1969[24] P R Page T Goldman and J N Ginocchio ldquoRelativistic sym-

metry suppresses quark spin-orbit splittingrdquo Physical ReviewLetters vol 86 no 2 pp 204ndash207 2001

[25] S Weinberg The Quantum Theory of Fields vol 1 CambridgeUniversity Press Cambridge UK 1995

[26] D Boyanovsky C Destri and H J de Vega ldquoApproach tothermalization in the classical 1205934 theory in 1 + 1 dimensionsenergy cascades and universal scalingrdquo Physical Review D vol69 no 4 Article ID 045003 2004

[27] R Khordad ldquoEffects of magnetic field and geometrical size onthe interband light absorption in a quantumpseudodot systemrdquoSolid State Sciences vol 12 no 7 pp 1253ndash1256 2010

[28] R Khordad ldquoSimultaneous effects of temperature and pressureon the donor binding energy in a V-groove quantum wirerdquoSuperlattices and Microstructures vol 47 no 3 pp 422ndash4312010

[29] A Cetin ldquoA quantum pseudodot system with a two-dimensional pseudoharmonic potentialrdquo Physics LettersA vol 372 no 21 pp 3852ndash3856 2008

[30] S M Ikhdair M Hamzavi and R Sever ldquoSpectra of cylindricalquantum dots the effect of electrical and magnetic fieldstogether with AB flux fieldrdquo Physica B vol 407 pp 4523ndash45292012

[31] S M Ikhdair and M Hamzavi ldquoA quantum pseudodot systemwith two-dimensional pseudoharmonic oscillator in externalmagnetic and Aharonov-Bohm fieldsrdquo Physica B vol 407 pp4198ndash4207 2012

[32] S M Ikhdair and M Hamzavi ldquoEffects of extermal fieldson a two-dimensional Klein-Gordon particle under pseudo-harmonic oscillator interactionrdquo Chinese Physics B vol 21 no11 pp 110302ndash110306 2012

[33] M Aygun O Bayrak I Boztosun and Y Sahin ldquoThe energyeigenvalues of the Kratzer potential in the presence of amagnetic fieldrdquo European Physical Journal D vol 66 no 2Article ID 20319 2012

[34] S M Ikhdair andM Hamzavi ldquoSpectral properties of quantumdots influenced by a confining potential modelrdquo Physica B vol407 pp 4797ndash4803 2012

Advances in High Energy Physics 11

[35] M Hamzavi and A A Rajabi ldquoSolution of Dirac equation withKillingbeck potential by using wave function ansatz methodunder spin symmetry limitrdquo Communications in TheoreticalPhysics vol 55 no 1 pp 35ndash37 2011

[36] M Hamzavi S M Ikhdair and K E Thylwe ldquoPseudospinsymmetry in the relativistic Killingbeck potential Quasi-exactsolutionrdquo Zeitschrift fur Naturforschung A vol 67 pp 567ndash5712012

[37] R Kumar and F Chand ldquoSeries solution to the N-dimensionalradial Schrodinger equation for the quark-antiquark interac-tionrdquo Physica Scripta vol 85 no 5 pp 055008ndash055004 2012

[38] R Kumar and F Chand ldquoReply to comments on seriessolution to the N-dimensionalradial Schrodinger equation forthe quark-antiquark interactionrdquo Physica Scripta vol 86 no 2p 027001 2012

[39] S M Ikhdair ldquoA scalar charged particle in presence of magneticandAharonov-Bohmfield plus scalar-vectorKillingbeck poten-tialsrdquo Few-Body Systems vol 54 no 11 pp 1987ndash1995 2013

[40] W Greiner Relativistic Quantum Mechanics Wave EquationsSpringer Berlin Germany 2000

[41] A D Alhaidari H Bahlouli and A Al-Hasan ldquoDirac andKlein-Gordon equations with equal scalar and vector poten-tialsrdquo Physics Letters A vol 349 no 1ndash4 pp 87ndash97 2006

[42] G Plante and A F Antippa ldquoAnalytic solution of theSchrodinger equation for the Coulomb-plus-linear potentialrdquoJournal of Mathematical Physics vol 46 no 6 pp 1ndash20 2005

[43] J Killingbeck ldquoPerturbation theory without wavefunctionsrdquoPhysics Letters A vol 65 no 2 pp 87ndash88 1978

[44] Y Xu S He and C-S Jia ldquoApproximate analytical solutionsof the Klein-Gordon equation with the Poschl-Teller potentialincluding the centrifugal termrdquo Physica Scripta vol 81 no 4Article ID 045001 2010

[45] R L Liboff Introductory Quantum Mechanics PearsonEducation-Addison Wesley San Francisco Calif USA 2003

[46] GChen Z-DChen andZ-M Lou ldquoBound states of theKlein-Gordon and Dirac equation for scalar and vector pseudohar-monic oscillator potentialsrdquo Chinese Physics vol 13 no 3 pp279ndash282 2004

[47] S H Dong Z Ma and G Espozito ldquoExact solutions of theSchrodinger equation with inversepowerpotentialrdquo Founda-tions of Physics Letters vol 12 no 5 pp 465ndash474 1999

[48] A Arda and R Sever ldquoEffective-mass Klein-Gordon equationfor non-PTnon-Hermitian generalizedMorse potentialrdquo Phys-ica Scripta vol 82 no 6 Article ID 065007 2010

[49] S M Ikhdair and J Abu-Hasna ldquoQuantization rule solutionto the Hulthen potential in arbitrary dimension with a newapproximate scheme for the centrifugal termrdquo Physica Scriptavol 83 no 2 Article ID 025002 2011

[50] S M Ikhdair and R Sever ldquoExact quantization rule tothe Kratzer-type potentials An application to the diatomicmoleculesrdquo Journal of Mathematical Chemistry vol 45 no 4pp 1137ndash1152 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

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AstronomyAdvances in

International Journal of

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Advances in High Energy Physics 3

21 The Positive Energy Solution The positive energy statesrequire that 119878(119903) = 119881(119903) (ie 120573 = 1 case) which in thenon-relativistic limit corresponds to the solution of the waveequation

1

2120583

[minus119894ℎ

997888rarr

nabla +

119890

119888

(

119861119903

2

+

ΦAB2120587119903

)120601]

2

+ 2119881119870 (119903) minus 119864

times 120595 (119903 120601) = 0

(5)

where120595(119903 120601) stands for non-relativistic wave functionThusthe choice 119878(119903) = 119881(119903) produces a non-relativistic limit with apotential function 2119881(119903) and not 119881(119903) Accordingly it wouldbe natural to scale the potential term in (4) and (5) so thatin the non-relativistic limit the interaction potential becomes119881(119903) not 2119881(119903)Thus we need to recast (4) for the 119878(119903) = 119881(119903)

as [32 39 40 44]

[1198882(minus119894ℎ

997888rarr

nabla +

119890

119888

997888rarr

119860)

2

+ (119864 + 1198981198901198882)119881 (119903)]120595 (119903 120601)

= (1198642minus 1198982

1198901198884) 120595 (119903 120601)

(6)

where

nabla2=

1

120588

120597

120597120588

+

1205972

1205971205882+

1

1205882

1205972

1205971206012+

1205972

1205971199112 (7)

and in order to simplify (6) we introduce new parameters1205721= 119864 +119898

1198901198882 and 120572

2= 119864 minus119898

1198901198882 so that it can be reduced to

the following form

1198882[minus119894ℎ

997888rarr

nabla +

119890

119888

(

119861119903

2

+

ΦAB2120587119903

)120601]

2

minus 1205721(1205722minus 119881119870 (119903))

times 120595 (119903 120601) = 0

(8)

By inserting (2) and (3) into (8) we obtain

1198892119877 (119903)

1198891199032

+

1205721

ℎ21198882[1205722minus 119880eff (119903 120596119871 120585)] 119877 (119903) = 0 (9a)

119880eff (119903 120596119871 120585) = 119881119870 (119903) +

1198982

11989011988821205962

119871

1205721

1199032

+

ℎ21198882

1205721

(11989810158402minus 14)

1199032

+

2ℎ1205961198711198981198901198882

1205721

1198981015840

(9b)

120596119871=

Ω

2

Ω =

|119890| 119861

119898119890119888

1198981015840= 119898 + 120585

1198981015840= 1 2 120585 =

ΦABΦ0

(9c)

where the effective potentials depending on the magnitudesof two fields strength with 120596

119871and 119898

1015840 are the Larmorfrequency and a new eigenvalue of angularmomentum (mag-netic quantum number) respectively It is worthy to mentionthat the frequency Ω is called the cyclotron frequency [45]This is the frequency of rotation corresponding to the classical

motion of a charged particle in a uniform magnetic field andΩ2 is the Larmor frequency in units of Hz [45] Moreoverwe take 120585 as integer with the flux quantum Φ

0= ℎ119888119890 Here

119881119870(119903) is a pure Killingbeck potential (2) the second term is

the harmonic oscillator-type potential and the other termsare the rotational potential creating the rotational energylevels Equations (9a) (9b) and (9c) can be alternativelyexpressed as

1198892119877 (119903)

1198891199032

+ (

1198621

1199032+

1198622

119903

minus 1198623minus 1198624119903 minus 11986251199032)119877 (119903) = 0 (10a)

1198621= minus(119898

10158402minus

1

4

) 1198622=

1205721120581

ℎ21198882

1198623=

21198981198901205961198711198981015840

ℎ

minus

12057211205722

ℎ21198882

(10b)

1198624=

1205721120590

ℎ21198882 119862

5=

1205721120582

ℎ21198882+ (

119898119890120596119871

ℎ

)

2

(10c)

with the asymptotic behaviors 119877(0) = 0 and 119877(infin) rarr 0 Itis interesting to look at the results obtained from (10a) for aspecial case 119862

2= 1198624= 0 that is we have

1198892119877 (119903)

1198891199032

+ (

1198621

1199032minus 11986251199032)119877 (119903) = 119862

3119877 (119903) (11)

which corresponds to the differential equation of a harmonicoscillator with a centrifugal term It is a form similar to theone given by (7) in [46]with the equivalence119862

1rarr minus119897

1015840(1198971015840+1)

1198625rarr 1205722 and119862

3rarr minus120582

119899(120572 119897 and120582 are the parameters used

in [46]) So by putting the parameter values

1198971015840= 1198981015840minus

1

2

120572 =radic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

120582119899=

1198642minus 1198982

1198901198884

ℎ21198882

minus

21198981198901205961198711198981015840

ℎ

(12)

into (14) of [45] we obtain the energy equation as

1198642minus 1198982

1198901198884= 21198981198901198882ℎ1205961198711198981015840

+ 2ℎ1198882(2119899 + 119898

1015840+ 1)

timesradic(119864 + 119898

1198901198882) 120582

1198882

+ (119898119890120596119871)2

(13)

Now we find a solution to (10a) by making the followingchoice of the wave function [37 38 47 48]

119877119899119898 (

119903) = exp(12

1199011199032+ 119902119903)

infin

sum

119899=0

119886119899119903119899+120575

(14)

where 119901 and 119902 are parameters whose values are to bedetermined in terms of the potential parameters 120582 120590 and 120581Substituting (14) into (10a) we obtain the series [36]infin

sum

119899=0

119886119899119878119899119903119899+120575minus2

+

infin

sum

119899=1

119886119899minus1

119879119899minus1

119903119899+120575minus2

+

infin

sum

119899=2

119886119899minus2

119882119899minus2

119903119899+120575minus2

= 0

1199012= 1198625 2119901119902 = 119862

4

(15)

4 Advances in High Energy Physics

with119878119899= (119899 + 120575) (119899 + 120575 minus 1) + 1198621

119879119899minus1

= 2119902 (119899 + 120575 minus 1) + 1198622

119882119899minus2

= 1199022+ 2119901(119899 + 120575 minus

3

2

) minus 1198623

(16)

where the two parameters 119901 and 119902 are determined in orderfor the radial wave functions 119877

119899119898(119903) to be finite everywhere

and vanish at 119903 = 0 and as 119903 rarr infin To obtain the recurrencerelation which can connect various expansion coefficients 119886

119899

we make identical powers of 119903 in (15) that is equate thecoefficients of 119903119899+120575minus2 to zero Thus the relations become

1198860[120575 (120575 minus 1) + 1198621

] = 0 997904rArr 1205752+ 1198621= 120575

1198860

= 0 997904rArr 120575 = plusmn1198981015840+

1

2

1198861= minus

(2119902120575 + 1198622)

2120575

1198860

1198862= minus

[119901 (2120575 + 1) + 1199022minus 1198623] 1198860+ [2119902 (120575 + 1) + 1198622

] 1198861

2 (2120575 + 1)

1198863= minus(

(2119901119902 minus 1198624) 1198860+ [119901 (2120575 + 3) + 119902

2minus 1198623] 1198861

6 (120575 + 1)

+

[2119902 (120575 + 2) + 1198622] 1198862

6 (120575 + 1)

)

119886119899= minus(

(1199012minus 1198625) 119886119899minus4

+ (2119901119902 minus 1198624) 119886119899minus3

119899 (2120575 + 119899 minus 1)

+

[119901 (2120575 + 2119899 minus 3) + 1199022minus 1198623] 119886119899minus2

119899 (2120575 + 119899 minus 1)

+

[2119902 (120575 + 119899 minus 1) + 1198622] 119886119899minus1

119899 (2120575 + 119899 minus 1)

)

(17)

where 119899 = 0 1 2 with 1198860

= 0 Here the positive sign ofparameter 120575 = 119898

1015840+ 12 has been selected The power series

for large values of 120575 or1198981015840 is convergent For convenience wetake the ratio of two successive terms that is 119886

119899+1119886119899 which

becomes1198861

1198860

= minus(119902 +

1198622

2120575

) 997888rarr minus119902 when 120575 997888rarr infin

1198862

1198860

=

1199022(120575 + 119862

22119902) [120575 + 119862

22119902 + 1]

120575 (2120575 + 1)

minus

[119901 (2120575 + 1) + 1199022minus 1198623]

2 (2120575 + 1)

997888rarr

1199022minus 119901

2

when 120575 997888rarr infin

(18)

It is apparent from the above relations that the power seriesconverges to zero when 120575 rarr infin Hence the series mustbe truncated (bounded) for 119899 = 119899max At this value of 119899 weobtain the following equations

119901 = plusmnradic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

997904rArr 119901 = minusradic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

119901 = 0

(19a)

2119901119902 = 1198624

997904rArr 119902 = minus

(119864 + 1198981198901198882) 120590

2ℎ21198882radic((119864 + 119898

1198901198882) 120582ℎ21198882) + (119898

119890120596119871ℎ)2

(19b)

(119864 + 1198981198901198882) 120581

ℎ21198882

= minus2119902 (119899 + 1198981015840minus

1

2

) (19c)

(119864 + 1198981198901198882) (119864 minus 119898

1198901198882)

ℎ21198882

=

21198981198901205961198711198981015840

ℎ

+ 2radic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

times (1198981015840+ 119899 minus 1) minus 119902

2

(19d)

where the negative sign of the coefficient 119901 has been chosenin (19a) It is worth noting that the potential parameter120582mustbe positive when 120596

119871= 0 Hence (19c) gives a restriction on

the potential parameters 120582 120590 and 120581 as follows

120590 =

120581

(119899 + 1198981015840minus 12)

radic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

(20)

and it follows that (19d) gives the energy formula as

1198642minus 1198982

1198901198884= 21198981198901198882ℎ1205961198711198981015840+ 2ℎ21198882(119899 + 119898

1015840minus 1)

timesradic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

minus

(119864 + 1198981198901198882)

2

1205812

4ℎ21198882(119899 + 119898

1015840minus 12)

2

(21)

where 119899 = 0 1 2 We may find a solution to theabove transcendental equation as 119864 = 119864

(+)

KG = 1198641198991198981015840(120596119871 120585)

Overmore the wave function (14) with the help of (17) (19a)and (19b) becomes

Advances in High Energy Physics 5

120595(+)

119899119898(119903 120601) = 119862

119899119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(12)radic(119864+119898

1198901198882)120582ℎ21198882+(119898119890120596119871ℎ)2[1199032+((119864+119898

1198901198882)120590((119864+119898

1198901198882)1198892+ℎ21198882(119898119890120596119871ℎ)2))119903]

119899max

sum

119899=0

119886119899119903119899

(22a)

1198861= minus1198860[119902 +

(119864 + 1198981198901198882) 120581

2ℎ21198882(1198981015840+ 12)

] (22b)

1198862= minus

1

2

1198860[119901 minus 119902

2minus

1198981198901205961198711198981015840

ℎ (1198981015840+ 1)

minus

(119864 + 1198981198901198882) 120581

ℎ21198882(1198981015840+ 12)

(119902 +

(119864 + 1198981198901198882) 120581

4ℎ21198882(1198981015840+ 1)

)] (22c)

where 119862119899119898

is the normalization constantIn the non-relativistic limit when 119864 + 119898

1198901198882

rarr 2120583119864 + 119898

1198901198882rarr 1198641198991198981015840 and 119888 = 1 (21) and (22a) give the energy

formula

1198641198991198981015840 (120596119871 120585) = ℎ120596

1198711198981015840+ ℎ (119899 + 119898

1015840minus 1)

times radic

2120582

120583

+ 1205962

119871minus

1205831205812

2ℎ2(119899 + 119898

1015840minus 12)

2

(23)

and wave function

120595(+)

119899119898(119903 120601) = 119862

119899119898

1

radic2120587

119890119894119898120601

119903119898+120585

times 119890minus(1205832ℎ)radic2120582120583+120596

2

119871[1199032+(2120590120583(2119889

2120583+1205962

119871))119903]

119899max

sum

119899=0

119886119899119903119899

(24)

respectively for the Schrodinger-Killingbeck system with thefollowing restriction on the potential parameters 120582 120590 and 120581given by

120590 =

120583120581

ℎ (119899 + 1198981015840minus 12)

radic

2120582

120583

+ 1205962

119871 (25)

Notice that the present model has been solved in 2D spacewith an external uniform magnetic field since it is perpen-dicular to the plane where the vector and scalare Cornellpotentials have the dimensions of 119903 = 120588 and 120601 that is119881 = 119878 = 119881(120588 120601) [32] However without the magnetic fieldthe model can be solved in any desired dimensional spaceby considering the change 119898 rarr 119897 + (119863 minus 2)2 [49 50]where119863 refers to spatial dimension and also the existence ofinterdimensional degeneracy

As shown in Figure 1(a) and (9b) the effective potentialfunction changes in shape when the magnetic field strength

increases say 120596119871= 8 and in absence of AB flux field The

energy levels are raised when the strength of the magneticfield increases and in absence of AB flux field 120585 = 0 Wesee that the effective potential changes gradually from thepure pseudo-harmonic oscillator potential when 120596

119871= 0 to

a pure harmonic oscillator type behavior in short potentialrange when120596

119871= 8 In Figure 1(b) the effective potential (9b)

which is pseudo-harmonic oscillator when 120596119871= 0 becomes

sensitive to the increasing AB flux field 120585 = 8 in the shortrange region that is 0 lt 119903 lt 4 au

22 The Bound States for Negative Energy In this case of119878(119903) = minus119881(119903) the first inspection of (4) shows that thefollowing changes 119881(119903) rarr minus119881(119903) (ie 120582 rarr minus120582 120590 rarr minus120590120581 rarr minus120581) 120595(+)

119899119898(119903 120601) rarr 120595

(minus)

119899119898(119903 120601) and 119864 rarr minus119864 give the

negative energy solution for antiparticles as

1198642minus 1198982

1198901198884minus 21198981198901198882ℎ1205961198711198981015840

= 2ℎ21198882(119899 + 119898

1015840minus 1)

radic(119864 minus 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

minus

(119864 minus 1198981198901198882)

2

1205812

4ℎ21198882(119899 + 119898

1015840minus 12)

2

(26)

with restriction on potential parameters

120590 =

120581

(119899 + 1198981015840minus 12)

radic(119864 minus 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

(27)

We may find solution to the above transcendental equationas 119864 = 119864

(minus)

119870119866= 1198641198991198981015840(120596119871 120585) Overmore the wave function for

antiparticle reads

6 Advances in High Energy Physics

0 1 2 3 4 5 6 70

2

4

6

8

10

12

14

16

18

20

120596L = 0

120596L = 1

120596L = 5

120596L = 8

r (au)

Ueff(r120596

L)

(a)

0 1 2 3 4 50

2

4

6

8

10

12

14

16

18

20

r (au)

120585 = 0

120585 = 1

120585 = 5

120585 = 8

Ueff(r120585)

(b)

Figure 1 The KG effective potential function for (a) 120596119871= 0 1 5 8 and with 120585 = 0 (b) 120585 = 0 1 5 8 and with 120596

119871= 0 (colour online) Here

119898119890= ℎ = 119888 = 1

120595(minus)

119899119898(119903 120601) = 119863

119899119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(12)radic(119864minus119898

1198901198882)120582ℎ21198882+(119898119890120596119871ℎ)2[1199032+((119864minus119898

1198901198882)120590((119864minus119898

1198901198882)1198892+ℎ21198882(119898119890120596119871ℎ)2))119903]

119899max

sum

119899=0

119886119899119903119899 (28)

1198861= minus1198860[119902 +

(119864 minus 1198981198901198882) 120581

2ℎ21198882(1198981015840+ 12)

] (29)

1198862= minus

1

2

1198860[119901 minus 119902

2minus

1198981198901205961198711198981015840

ℎ (1198981015840+ 1)

minus

(119864 minus 1198981198901198882) 120581

ℎ21198882(1198981015840+ 12)

(119902 +

(119864 minus 1198981198901198882) 120581

4ℎ21198882(1198981015840+ 1)

)] (30)

where119863119899119898

is the normalization constant

3 Discussions

In this section we discuss some special cases of interest fromour general solution

(i) If we set 120590 = 120581 = 0 the effective Killingbeck potentialturns to effective harmonic oscillator potential in relativisticcase as

119880eff (119903 120596119871 120585) = 1205821199032+

1198982

11989011988821205962

119871

(119864 + 1198981198901198882)

1199032

+

ℎ21198882

(119864 + 1198981198901198882)

(11989810158402minus 14)

1199032

+

2ℎ12059611987111989811989011988821198981015840

(119864 + 1198981198901198882)

120582 =

1

2

1205831205962

(31)

The bound state solutions (with the change 119899 = 2(119899119903+ 1)) are

[32 36]

1198642minus 1198982

1198901198884= 21198981198901198882ℎ1205961198711198981015840+ 4ℎ1198882(119899119903+

1198981015840+ 1

2

)

timesradic(119864 + 119898

1198901198882) 120582

1198882

+ (119898119890120596119871)2

(32)

120595(+)

119899119903119898(119903 120601) = 119860

119899119903119898

1

radic2120587

119890119894119898120601

119903119898+120585

times 119890minus(12)radic(119864+119898

1198901198882)120582ℎ21198882+(119898119890120596119871ℎ)21199032

119899max

sum

119899=0

1198861198991199032(119899119903+1)

(33a)

1198861= 0

1198862=

1

2

1198860[

[

radic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

+

1198981198901205961198711198981015840

ℎ (1198981015840+ 1)

]

]

(33b)

Advances in High Energy Physics 7

where 119860119899119903119898

is the normalization constant On the otherhand the nonrelativistic effective harmonic oscillator poten-tial

119880eff (119903 120596119871 120585) = 1205821199032+

1

2

1205962

1198711199032+

(11989810158402minus 14) ℎ

2

21198981198901199032

+ ℎ1205961198711198981015840

(34)

has bound states as [32]

1198641198991198981015840 (120596119871 120585) = ℎ120596

1198711198981015840

+ 2ℎΩ1015840(119899119903+

10038161003816100381610038161003816119898101584010038161003816100381610038161003816+ 1

2

) Ω1015840= radic120596

2

119871+ 1205962

(35)

120595(+)

119899119903119898(119903 120601) = 119862

119899119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(1205832ℎ)Ω

10158401199032

times

119899max

sum

119899=0

1198861198991199032(119899119903+1) 119899119903= 0 1 2

=

1

radic2120587

119890119894119898120601

radic21198871198981015840+1119899

(119899 + 1198981015840)

1199031198981015840

119890minus11988711990322119871(1198981015840)

119899(1198871199032)

119887 =

120583

ℎ

Ω

1015840

(36a)

1198861= 0 119886

2=

120583

2ℎ

Ω10158401198860 1198860

= 0 (36b)

(ii) Setting 120582 = 120590 = 0 in (21) the killingbeck potentialturns to the Coulomb potential case (ie 119881(119903) = minus119860119903 and119899 = 119899119903+ 1) Thus we obtain the relativistic energy formula

1198642minus 1198982

1198901198884= 21198981198901198882(119899119903+ 21198981015840) ℎ120596119871

minus

(119864 + 1198981198901198882)

2

1205812

4ℎ21198882(119899119903+ 1198981015840+ 12)

2 120581 = 119885119890

2

(37)

and the wave function

120595(+)

119899119903119898(119903 120601) = 119862

119899119903119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(119898119890120596119871)2ℎ1199032

times

119899max

sum

119899=0

119886119899119903119899119903+1

(38a)

1198861= minus

(119864 + 1198981198901198882) 120581

2ℎ21198882(1198981015840+ 12)

1198860 (38b)

1198862= minus

1

2

[

[

119898119890120596119871

ℎ

minus

1198981198901205961198711198981015840

ℎ (1198981015840+ 1)

minus

(119864 + 1198981198901198882)

2

1205812

4ℎ41198884(1198981015840+ 1) (119898

1015840+ 12)

]

]

1198860

(38c)

Overmore in the nonrelativistic limit we have

119864 = ℎ1205961198711198981015840+ ℎ (119899

119903+ 1198981015840) 120596119871minus

11989811989011988521198904

2ℎ2(119899119903+ 1198981015840+ 12)

2 (39)

and the wave function becomes

120595(+)

119899119903119898(119903 120601) = 119862

119899119903119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(1205831205961198712ℎ)1199032

119899max

sum

119899=0

119886119899119903119899119903+1

(40a)

1198861= minus

119898119890119860

ℎ2(1198981015840+ 12)

1198860 (40b)

1198862= minus

1

2

[

120583120596119871

ℎ

minus

1205831205961198711198981015840

ℎ (1198981015840+ 1)

minus

120583211988521198904

ℎ4(1198981015840+ 1) (119898

1015840+ 12)

] 1198860

(40c)

Notice that in the absence of external fields 120596119871= 120585 = 0

the problem can be solved in any dimension Thus applyingthe transformation |119898| rarr 119897 + 12 our previous results arereduced to the well-known three-dimensional bound statesolutions for the harmonic oscillator and the hydrogen-likeatoms in the Coulomb fields [40]

119864plusmn= plusmn1198981198901198882[[

[

1 +

2ℎ119888

1198981198901198882(2119899119903+ 119897 +

3

2

)radic

(119864 + 1198981198901198882) 120582

1198982

1198901198884

]]

]

12

119864plusmn= plusmn1198981198901198882[1 minus

11988521198904

4ℎ21198882(119899119903+ 119897 + 1)

2(1 +

119864

1198981198901198882)

2

]

12

(41)

respectively According to (41) the energy spectrum can befound for the scalar particle and antiparticle

In Figure 2(a) we plot the effective potential for the caseof low vibrational (119899 = 0 1 2 3) and rotational (119898 = 1)

levels for various Larmor frequencies 120596119871

= 0 1 5 8 and120585 = 0 case As shown in Figure 2(a) and (31) the effectivepotential function changes in shape as well as the bound stateenergy eigenvalues increase when 120596

119871= 8 It is shown that the

energy levels are raised when the strength of the magneticfield increases and in the absence of AB flux field It is alsoobvious that the effective potential changes gradually fromthe pure pseudo-harmonic oscillator potential which is anonmagnetic (120596

119871= 0) and AB flux (120585 = 0) fields case to

a pure harmonic oscillator type behavior in short potentialrange when the strength of the applied magnetic field isincreased to 120596

119871= 8 If we consider a strong magnetic field

case 120596119871= 8 which has the shape of pure harmonic oscillator

potential function the energy difference between adjacentenergy levels is nearly equal which is a known characteristicof the pure harmonic oscillator potential In Figure 2(b) weplot the effective potential (34) in the absence of themagneticfield and in the presence of AB flux field in the short rangeregion

In Tables 1 and 2 we show the effect of magnetic fieldand AB flux field respectively on the low vibrational 119899

8 Advances in High Energy Physics

120596L = 0

120596L = 1

120596L = 5

120596L = 8

E01E11

E21

E31

E01

E11

E21E31

E01E11

E21

E31

0 1 2 3 4 50

10

20

30

40

50

60

70

80

r (au)

Ueff(r120596

L)

(a)

0 1 2 3 4 5 6 7 80

10

20

30

40

50

60

70

80

r (au)

120585 = 0

120585 = 3

120585 = 5

120585 = 8

Ueff(r120585)

(b)

Figure 2 The Schrodinger effective potential function and corresponding bound state energy levels (119864119899119898) in low vibrational (119899 = 0 1 2 3)

and rotational (119898 = 1) levels for (a) 120596119871= 0 1 5 8 with 120585 = 0 (b) 120585 = 0 1 5 8 and with 120596

119871= 0 (color online) Here 120583 = ℎ = 1

and rotational 119898 relativistic energy states of the harmonicoscillator potential As shown in Table 1 when the magneticfield is not applied and without AB flux field (120596

119871= 0

120585 = 0) the spacing between the energy levels of the effectivepotential is narrow and decreases with increasing 119899 Butwhen the magnetic field strength increases the energy levelsof the effective potential increase and the spacings betweenstates also increase In Table 2 when the AB flux field isapplied and without magnetic field the energy states becomedegenerate for various values of 119899 and 119898 and for variousAB flux field strength values In Tables 3 and 4 we showthe effect of magnetic field and AB flux field respectivelyon the low vibrational 119899 and rotational 119898 nonrelativisticenergy states of the harmonic oscillator potential As shownin Table 3 when themagnetic field is not applied and withoutAB flux field (120596

119871= 0 120585 = 0) the energy states are equally

spaced (the pure harmonic oscillator case) But when themagnetic field strength is raised the energy levels of theeffective potential increase and the spacings between statesalso increase In Table 4 when the AB flux field is applied andwithout magnetic field the energy states become degenerateand equally spaced for various values of 119899 and 119898 and forvarious AB flux field strength values

4 Concluding Remarks

To sum up in this paper we have studied the solutionof two-dimensional KG and Schrodinger equations withthe Killingbeck potential for low vibrational and rotationalenergy levels without and with a constant magnetic fieldhaving arbitrary Larmor frequeny and AB flux field Wehave applied the wave function ansatz method for 120596

119871= 0

(with magnetic field) and 120585 = 0 (with AB flux field) toobtain analytical expressions in closed form for boundstate energies and wave functions of the spinless relativisticparticle subject to a Killingbeck interaction expressed interms of external uniform magnetic and AB flux fields inany vibrational 119899 and rotational 119898 states The above resultsshow that the problems of relativistic quantum mechanicscan be also solved exactly as in the non-relativistic onesWe considered the solution of both positive (particle) andnegative (antiparticle) KG energy states It is noticed that thesolution with equal mixture of scalar-vector potentials can beeasily reduced into the well-known Schrodinger solution fora particle with an interaction potential field and a free fieldrespectively We have also studied the bound-state solutionsfor some special cases including the non-relativistic limits(Schrodinger equation for harmonic oscillator and Coulombpotentials under external magnetic and AB flux fields) andalso the KG equation for harmonic oscillator and CoulombinteractionsThe results show that the splitting is not constantand is mainly dependent on the strength of the externalmagnetic field and AB flux field In order to show the effectof constant magnetic and AB flux fields on the vibrationaland rotational energy levels of the harmonic oscillator weplot the effective potential and corresponding energy levelswith increasing Larmor frequency and flux field for specialpotential parametersWehave seen that the effective potentialfunction and corresponding energy levels are raised in energywhen magnetic and AB flux field strengths increase Theeffective potential function behavior gradually changes fromthe pure pseudo-harmonic oscillator to a pure harmonicoscillator shape in short potential range as the magnetic andAB flux fields strengths increase

Advances in High Energy Physics 9

Table 1 The KG energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120596119871values and 120585 = 0 Here ℎ = 119888 =

119898119890= 119896 = 1

119898 119899

119864119899119898

120596119871= 0 120596

119871= 1 120596

119871= 2 120596

119871= 3 120596

119871= 4 120596

119871= 5 120596

119871= 6 120596

119871= 7 120596

119871= 8

0

0 183929 204353 240325 275615 308137 338066 365815 391752 4161661 309625 34289 403986 465113 521808 574094 622602 667941 7106092 412383 451488 526359 603138 675146 74193 804087 862301 9171583 503104 545806 630283 718744 802542 880667 953605 102205 108663

1

0 250976 316597 389307 455265 514478 568284 617802 663858 707061 362919 427262 512444 593706 668087 736315 799437 858337 9137072 458916 522885 616824 709549 795634 87516 949036 101815 1083233 545354 609216 709891 811893 90768 996689 107966 115745 123081

Table 2 The KG energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120585 values and 120596119871= 0

119898 119899

119864119899119898

120585 = 0 120585 = 1 120585 = 2 120585 = 3 120585 = 4 120585 = 5 120585 = 6 120585 = 7 120585 = 8

0

0 183929 250976 309625 362919 412383 458916 503104 545354 5859661 309625 362919 412383 458916 503104 545354 585966 625166 663132 412383 458916 503104 545354 585966 625166 6631370 70 7358923 503104 545354 585966 625166 66313 70 735892 770901 805108

1

0 250976 309625 362919 412383 458916 503104 545354 585966 6251661 362919 412383 458916 503104 545354 585966 625166 66313 702 458916 503104 545354 585966 625166 66313 70 735892 7709013 545354 585966 625166 66313 70 735892 770901 805108 838582

Table 3 The nonrelativistic energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120596119871values and 120585 = 0

119898 119899

119864119899119898

120596119871= 0 120596

119871= 1 120596

119871= 2 120596

119871= 3 120596

119871= 4 120596

119871= 5 120596

119871= 6 120596

119871= 7 120596

119871= 8

0

0 10 141421 223607 316228 412311 509902 608276 707107 8062261 30 424264 67082 948683 123693 152971 182483 212132 2418682 50 707107 111803 158114 206155 254951 304138 353553 4031133 70 989949 156525 221359 288617 356931 425793 494972 564358

1

0 20 382843 647214 932456 122462 15198 181655 211421 2412451 40 665685 109443 156491 204924 253961 303311 352843 402492 60 948528 154164 219737 287386 355941 424966 494264 5637353 80 123137 198885 282982 369848 457922 546621 635685 724981

Table 4 The nonrelativistic energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120585 values and 120596119871= 0

119898 119899

119864119899119898

120585 = 0 120585 = 1 120585 = 2 120585 = 3 120585 = 4 120585 = 5 120585 = 6 120585 = 7 120585 = 8

0

0 10 20 30 40 50 60 70 80 901 30 40 50 60 70 80 90 100 1102 50 60 70 80 90 100 110 120 1303 70 80 90 100 110 120 130 140 150

1

0 20 30 40 50 60 70 80 90 1001 40 50 60 70 80 90 100 110 1202 60 70 80 90 100 110 120 130 1403 80 90 100 110 120 130 140 150 160

10 Advances in High Energy Physics

Acknowledgments

One of the authors (Sameer M Ikhdair) would like to thankthe president of An-Najah National University (ANU) Pro-fessor Rami Alhamdallah and the acting president ProfessorMaher Natsheh for their continuous support during hispresent work at ANU The authors acknowledge the partialsupport of the Scientific and Technological Research Councilof Turkey

References

[1] WGreiner andBMullerQuantumMechanics An IntroductionSpringer Berlin Germany 1994

[2] S Ozcelik and M Simsek ldquoExact solutions of the radialSchrodinger equation for inverse-power potentialsrdquo PhysicsLetters A vol 152 no 3-4 pp 145ndash150 1991

[3] S-H Dong ldquoSchrodinger equation with the potential 119881(119903) =119860119903minus4+ 119861119903minus3+ 119862119903minus2+ 119863119903minus1rdquo Physica Scripta vol 64 no 4 pp

273ndash276 2001[4] S-H Dong and Z-Q Ma ldquoExact solutions to the Schrodinger

equation for the potential 119881(119903) = 1198861199032+ 119887119903minus4

+ 119888119903minus6 in two

dimensionsrdquo Journal of Physics A vol 31 no 49 pp 9855ndash98591998

[5] M S Child S-H Dong and X-GWang ldquoQuantum states of asextic potential hidden symmetry and quantum monodromyrdquoJournal of Physics A vol 33 no 32 pp 5653ndash5661 2000

[6] S-H Dong ldquoExact solutions of the two-dimensional Schrodin-ger equation with certain central potentialsrdquo InternationalJournal of Theoretical Physics vol 39 no 4 pp 1119ndash1128 2000

[7] S-H Dong ldquoA new approach to the relativistic schrodingerequation with central potential Ansatz methodrdquo InternationalJournal of Theoretical Physics vol 40 no 2 pp 559ndash567 2001

[8] S-H Dong ldquoOn the solutions of the Schrodinger equation withsome anharmonic potentials wave function ansatzrdquo PhysicaScripta vol 65 no 4 pp 289ndash295 2002

[9] S Flugge Practical Quantum Mechanics 1 Springer BerlinGermany 1994

[10] X-Q Zhao C-S Jia and Q-B Yang ldquoBound states of rel-ativistic particles in the generalized symmetrical double-wellpotentialrdquo Physics Letters A vol 337 no 3 pp 189ndash196 2005

[11] S M Ikhdair ldquoExact solution of Dirac equation with chargedharmonic oscillator in electric field bound statesrdquo Journal ofModern Physics vol 3 no 2 pp 170ndash179 2012

[12] W C Qiang ldquoBound states of the Klein-Gordon equation forring-shaped Kratzer-type potentialrdquo Chinese Physics vol 13 no5 pp 575ndash578 2004

[13] S Ikhdair and R Sever ldquoExact polynomial eigensolutions ofthe Schrodinger equation for the pseudoharmonic potentialrdquoJournal of Molecular Structure vol 806 no 1ndash3 pp 155ndash1582007

[14] F-L Lu C-Y Chen and D-S Sun ldquoBound states of Klein-Gordon equation for double ring-shaped oscillator scalar andvector potentialsrdquo Chinese Physics vol 14 no 3 pp 463ndash4672005

[15] S M Ikhdair and R Sever ldquoExact solutions of the D-dimensional Schrodinger equation for a ring-shaped pseudo-harmonic potentialrdquo Central European Journal of Physics vol 6no 3 pp 685ndash696 2008

[16] S M Ikhdair and R Sever ldquoExact bound states of the D-dimensional Klein-Gordon equation with equal scalar and

vector ring-shaped pseudoharmonic potentialrdquo InternationalJournal of Modern Physics C vol 19 no 9 pp 1425ndash1442 2008

[17] S M Ikhdair and R Sever ldquoExactly solvable effective massd-dimensional schrodinger equation for pseudoharmonic andmodified kratzer problemsrdquo International Journal of ModernPhysics C vol 20 no 3 pp 361ndash372 2009

[18] R-CWang andC-YWong ldquoFinite-size effect in the Schwingerparticle-productionmechanismrdquo Physical ReviewD vol 38 no1 pp 348ndash359 1988

[19] J N Ginocchio ldquoA relativistic symmetry in nucleirdquo PhysicsReport vol 315 no 1ndash4 pp 231ndash240 1999

[20] A Bohr I Hamarnoto and B R Motelson ldquoPseudospln inrotating nuclear potentialsrdquo Physica Scripta vol 26 no 4 pp267ndash272 1982

[21] J Dudek W Nazarewicz Z Szymanski and G A LeanderldquoAbundance and systematics of nuclear superdeformed statesrelation to the pseudospin and pseudo-SU(3) symmetriesrdquoPhysical Review Letters vol 59 no 13 pp 1405ndash1408 1987

[22] A ArimaM Harvey and K Shimizu ldquoPseudo LS coupling andpseudo SU3 coupling schemesrdquo Physics Letters B vol 30 no 8pp 517ndash522 1969

[23] K T Hecht and A Adler ldquoGeneralized seniority for favored J= 0 pairs in mixed configurationsrdquo Nuclear Physics A vol 137

no 1 pp 129ndash143 1969[24] P R Page T Goldman and J N Ginocchio ldquoRelativistic sym-

metry suppresses quark spin-orbit splittingrdquo Physical ReviewLetters vol 86 no 2 pp 204ndash207 2001

[25] S Weinberg The Quantum Theory of Fields vol 1 CambridgeUniversity Press Cambridge UK 1995

[26] D Boyanovsky C Destri and H J de Vega ldquoApproach tothermalization in the classical 1205934 theory in 1 + 1 dimensionsenergy cascades and universal scalingrdquo Physical Review D vol69 no 4 Article ID 045003 2004

[27] R Khordad ldquoEffects of magnetic field and geometrical size onthe interband light absorption in a quantumpseudodot systemrdquoSolid State Sciences vol 12 no 7 pp 1253ndash1256 2010

[28] R Khordad ldquoSimultaneous effects of temperature and pressureon the donor binding energy in a V-groove quantum wirerdquoSuperlattices and Microstructures vol 47 no 3 pp 422ndash4312010

[29] A Cetin ldquoA quantum pseudodot system with a two-dimensional pseudoharmonic potentialrdquo Physics LettersA vol 372 no 21 pp 3852ndash3856 2008

[30] S M Ikhdair M Hamzavi and R Sever ldquoSpectra of cylindricalquantum dots the effect of electrical and magnetic fieldstogether with AB flux fieldrdquo Physica B vol 407 pp 4523ndash45292012

[31] S M Ikhdair and M Hamzavi ldquoA quantum pseudodot systemwith two-dimensional pseudoharmonic oscillator in externalmagnetic and Aharonov-Bohm fieldsrdquo Physica B vol 407 pp4198ndash4207 2012

[32] S M Ikhdair and M Hamzavi ldquoEffects of extermal fieldson a two-dimensional Klein-Gordon particle under pseudo-harmonic oscillator interactionrdquo Chinese Physics B vol 21 no11 pp 110302ndash110306 2012

[33] M Aygun O Bayrak I Boztosun and Y Sahin ldquoThe energyeigenvalues of the Kratzer potential in the presence of amagnetic fieldrdquo European Physical Journal D vol 66 no 2Article ID 20319 2012

[34] S M Ikhdair andM Hamzavi ldquoSpectral properties of quantumdots influenced by a confining potential modelrdquo Physica B vol407 pp 4797ndash4803 2012

Advances in High Energy Physics 11

[35] M Hamzavi and A A Rajabi ldquoSolution of Dirac equation withKillingbeck potential by using wave function ansatz methodunder spin symmetry limitrdquo Communications in TheoreticalPhysics vol 55 no 1 pp 35ndash37 2011

[36] M Hamzavi S M Ikhdair and K E Thylwe ldquoPseudospinsymmetry in the relativistic Killingbeck potential Quasi-exactsolutionrdquo Zeitschrift fur Naturforschung A vol 67 pp 567ndash5712012

[37] R Kumar and F Chand ldquoSeries solution to the N-dimensionalradial Schrodinger equation for the quark-antiquark interac-tionrdquo Physica Scripta vol 85 no 5 pp 055008ndash055004 2012

[38] R Kumar and F Chand ldquoReply to comments on seriessolution to the N-dimensionalradial Schrodinger equation forthe quark-antiquark interactionrdquo Physica Scripta vol 86 no 2p 027001 2012

[39] S M Ikhdair ldquoA scalar charged particle in presence of magneticandAharonov-Bohmfield plus scalar-vectorKillingbeck poten-tialsrdquo Few-Body Systems vol 54 no 11 pp 1987ndash1995 2013

[40] W Greiner Relativistic Quantum Mechanics Wave EquationsSpringer Berlin Germany 2000

[41] A D Alhaidari H Bahlouli and A Al-Hasan ldquoDirac andKlein-Gordon equations with equal scalar and vector poten-tialsrdquo Physics Letters A vol 349 no 1ndash4 pp 87ndash97 2006

[42] G Plante and A F Antippa ldquoAnalytic solution of theSchrodinger equation for the Coulomb-plus-linear potentialrdquoJournal of Mathematical Physics vol 46 no 6 pp 1ndash20 2005

[43] J Killingbeck ldquoPerturbation theory without wavefunctionsrdquoPhysics Letters A vol 65 no 2 pp 87ndash88 1978

[44] Y Xu S He and C-S Jia ldquoApproximate analytical solutionsof the Klein-Gordon equation with the Poschl-Teller potentialincluding the centrifugal termrdquo Physica Scripta vol 81 no 4Article ID 045001 2010

[45] R L Liboff Introductory Quantum Mechanics PearsonEducation-Addison Wesley San Francisco Calif USA 2003

[46] GChen Z-DChen andZ-M Lou ldquoBound states of theKlein-Gordon and Dirac equation for scalar and vector pseudohar-monic oscillator potentialsrdquo Chinese Physics vol 13 no 3 pp279ndash282 2004

[47] S H Dong Z Ma and G Espozito ldquoExact solutions of theSchrodinger equation with inversepowerpotentialrdquo Founda-tions of Physics Letters vol 12 no 5 pp 465ndash474 1999

[48] A Arda and R Sever ldquoEffective-mass Klein-Gordon equationfor non-PTnon-Hermitian generalizedMorse potentialrdquo Phys-ica Scripta vol 82 no 6 Article ID 065007 2010

[49] S M Ikhdair and J Abu-Hasna ldquoQuantization rule solutionto the Hulthen potential in arbitrary dimension with a newapproximate scheme for the centrifugal termrdquo Physica Scriptavol 83 no 2 Article ID 025002 2011

[50] S M Ikhdair and R Sever ldquoExact quantization rule tothe Kratzer-type potentials An application to the diatomicmoleculesrdquo Journal of Mathematical Chemistry vol 45 no 4pp 1137ndash1152 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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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

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AstronomyAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

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GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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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

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AerodynamicsJournal of

Volume 2014

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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

with119878119899= (119899 + 120575) (119899 + 120575 minus 1) + 1198621

119879119899minus1

= 2119902 (119899 + 120575 minus 1) + 1198622

119882119899minus2

= 1199022+ 2119901(119899 + 120575 minus

3

2

) minus 1198623

(16)

where the two parameters 119901 and 119902 are determined in orderfor the radial wave functions 119877

119899119898(119903) to be finite everywhere

and vanish at 119903 = 0 and as 119903 rarr infin To obtain the recurrencerelation which can connect various expansion coefficients 119886

119899

we make identical powers of 119903 in (15) that is equate thecoefficients of 119903119899+120575minus2 to zero Thus the relations become

1198860[120575 (120575 minus 1) + 1198621

] = 0 997904rArr 1205752+ 1198621= 120575

1198860

= 0 997904rArr 120575 = plusmn1198981015840+

1

2

1198861= minus

(2119902120575 + 1198622)

2120575

1198860

1198862= minus

[119901 (2120575 + 1) + 1199022minus 1198623] 1198860+ [2119902 (120575 + 1) + 1198622

] 1198861

2 (2120575 + 1)

1198863= minus(

(2119901119902 minus 1198624) 1198860+ [119901 (2120575 + 3) + 119902

2minus 1198623] 1198861

6 (120575 + 1)

+

[2119902 (120575 + 2) + 1198622] 1198862

6 (120575 + 1)

)

119886119899= minus(

(1199012minus 1198625) 119886119899minus4

+ (2119901119902 minus 1198624) 119886119899minus3

119899 (2120575 + 119899 minus 1)

+

[119901 (2120575 + 2119899 minus 3) + 1199022minus 1198623] 119886119899minus2

119899 (2120575 + 119899 minus 1)

+

[2119902 (120575 + 119899 minus 1) + 1198622] 119886119899minus1

119899 (2120575 + 119899 minus 1)

)

(17)

where 119899 = 0 1 2 with 1198860

= 0 Here the positive sign ofparameter 120575 = 119898

1015840+ 12 has been selected The power series

for large values of 120575 or1198981015840 is convergent For convenience wetake the ratio of two successive terms that is 119886

119899+1119886119899 which

becomes1198861

1198860

= minus(119902 +

1198622

2120575

) 997888rarr minus119902 when 120575 997888rarr infin

1198862

1198860

=

1199022(120575 + 119862

22119902) [120575 + 119862

22119902 + 1]

120575 (2120575 + 1)

minus

[119901 (2120575 + 1) + 1199022minus 1198623]

2 (2120575 + 1)

997888rarr

1199022minus 119901

2

when 120575 997888rarr infin

(18)

It is apparent from the above relations that the power seriesconverges to zero when 120575 rarr infin Hence the series mustbe truncated (bounded) for 119899 = 119899max At this value of 119899 weobtain the following equations

119901 = plusmnradic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

997904rArr 119901 = minusradic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

119901 = 0

(19a)

2119901119902 = 1198624

997904rArr 119902 = minus

(119864 + 1198981198901198882) 120590

2ℎ21198882radic((119864 + 119898

1198901198882) 120582ℎ21198882) + (119898

119890120596119871ℎ)2

(19b)

(119864 + 1198981198901198882) 120581

ℎ21198882

= minus2119902 (119899 + 1198981015840minus

1

2

) (19c)

(119864 + 1198981198901198882) (119864 minus 119898

1198901198882)

ℎ21198882

=

21198981198901205961198711198981015840

ℎ

+ 2radic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

times (1198981015840+ 119899 minus 1) minus 119902

2

(19d)

where the negative sign of the coefficient 119901 has been chosenin (19a) It is worth noting that the potential parameter120582mustbe positive when 120596

119871= 0 Hence (19c) gives a restriction on

the potential parameters 120582 120590 and 120581 as follows

120590 =

120581

(119899 + 1198981015840minus 12)

radic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

(20)

and it follows that (19d) gives the energy formula as

1198642minus 1198982

1198901198884= 21198981198901198882ℎ1205961198711198981015840+ 2ℎ21198882(119899 + 119898

1015840minus 1)

timesradic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

minus

(119864 + 1198981198901198882)

2

1205812

4ℎ21198882(119899 + 119898

1015840minus 12)

2

(21)

where 119899 = 0 1 2 We may find a solution to theabove transcendental equation as 119864 = 119864

(+)

KG = 1198641198991198981015840(120596119871 120585)

Overmore the wave function (14) with the help of (17) (19a)and (19b) becomes

Advances in High Energy Physics 5

120595(+)

119899119898(119903 120601) = 119862

119899119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(12)radic(119864+119898

1198901198882)120582ℎ21198882+(119898119890120596119871ℎ)2[1199032+((119864+119898

1198901198882)120590((119864+119898

1198901198882)1198892+ℎ21198882(119898119890120596119871ℎ)2))119903]

119899max

sum

119899=0

119886119899119903119899

(22a)

1198861= minus1198860[119902 +

(119864 + 1198981198901198882) 120581

2ℎ21198882(1198981015840+ 12)

] (22b)

1198862= minus

1

2

1198860[119901 minus 119902

2minus

1198981198901205961198711198981015840

ℎ (1198981015840+ 1)

minus

(119864 + 1198981198901198882) 120581

ℎ21198882(1198981015840+ 12)

(119902 +

(119864 + 1198981198901198882) 120581

4ℎ21198882(1198981015840+ 1)

)] (22c)

where 119862119899119898

is the normalization constantIn the non-relativistic limit when 119864 + 119898

1198901198882

rarr 2120583119864 + 119898

1198901198882rarr 1198641198991198981015840 and 119888 = 1 (21) and (22a) give the energy

formula

1198641198991198981015840 (120596119871 120585) = ℎ120596

1198711198981015840+ ℎ (119899 + 119898

1015840minus 1)

times radic

2120582

120583

+ 1205962

119871minus

1205831205812

2ℎ2(119899 + 119898

1015840minus 12)

2

(23)

and wave function

120595(+)

119899119898(119903 120601) = 119862

119899119898

1

radic2120587

119890119894119898120601

119903119898+120585

times 119890minus(1205832ℎ)radic2120582120583+120596

2

119871[1199032+(2120590120583(2119889

2120583+1205962

119871))119903]

119899max

sum

119899=0

119886119899119903119899

(24)

respectively for the Schrodinger-Killingbeck system with thefollowing restriction on the potential parameters 120582 120590 and 120581given by

120590 =

120583120581

ℎ (119899 + 1198981015840minus 12)

radic

2120582

120583

+ 1205962

119871 (25)

Notice that the present model has been solved in 2D spacewith an external uniform magnetic field since it is perpen-dicular to the plane where the vector and scalare Cornellpotentials have the dimensions of 119903 = 120588 and 120601 that is119881 = 119878 = 119881(120588 120601) [32] However without the magnetic fieldthe model can be solved in any desired dimensional spaceby considering the change 119898 rarr 119897 + (119863 minus 2)2 [49 50]where119863 refers to spatial dimension and also the existence ofinterdimensional degeneracy

As shown in Figure 1(a) and (9b) the effective potentialfunction changes in shape when the magnetic field strength

increases say 120596119871= 8 and in absence of AB flux field The

energy levels are raised when the strength of the magneticfield increases and in absence of AB flux field 120585 = 0 Wesee that the effective potential changes gradually from thepure pseudo-harmonic oscillator potential when 120596

119871= 0 to

a pure harmonic oscillator type behavior in short potentialrange when120596

119871= 8 In Figure 1(b) the effective potential (9b)

which is pseudo-harmonic oscillator when 120596119871= 0 becomes

sensitive to the increasing AB flux field 120585 = 8 in the shortrange region that is 0 lt 119903 lt 4 au

22 The Bound States for Negative Energy In this case of119878(119903) = minus119881(119903) the first inspection of (4) shows that thefollowing changes 119881(119903) rarr minus119881(119903) (ie 120582 rarr minus120582 120590 rarr minus120590120581 rarr minus120581) 120595(+)

119899119898(119903 120601) rarr 120595

(minus)

119899119898(119903 120601) and 119864 rarr minus119864 give the

negative energy solution for antiparticles as

1198642minus 1198982

1198901198884minus 21198981198901198882ℎ1205961198711198981015840

= 2ℎ21198882(119899 + 119898

1015840minus 1)

radic(119864 minus 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

minus

(119864 minus 1198981198901198882)

2

1205812

4ℎ21198882(119899 + 119898

1015840minus 12)

2

(26)

with restriction on potential parameters

120590 =

120581

(119899 + 1198981015840minus 12)

radic(119864 minus 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

(27)

We may find solution to the above transcendental equationas 119864 = 119864

(minus)

119870119866= 1198641198991198981015840(120596119871 120585) Overmore the wave function for

antiparticle reads

6 Advances in High Energy Physics

0 1 2 3 4 5 6 70

2

4

6

8

10

12

14

16

18

20

120596L = 0

120596L = 1

120596L = 5

120596L = 8

r (au)

Ueff(r120596

L)

(a)

0 1 2 3 4 50

2

4

6

8

10

12

14

16

18

20

r (au)

120585 = 0

120585 = 1

120585 = 5

120585 = 8

Ueff(r120585)

(b)

Figure 1 The KG effective potential function for (a) 120596119871= 0 1 5 8 and with 120585 = 0 (b) 120585 = 0 1 5 8 and with 120596

119871= 0 (colour online) Here

119898119890= ℎ = 119888 = 1

120595(minus)

119899119898(119903 120601) = 119863

119899119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(12)radic(119864minus119898

1198901198882)120582ℎ21198882+(119898119890120596119871ℎ)2[1199032+((119864minus119898

1198901198882)120590((119864minus119898

1198901198882)1198892+ℎ21198882(119898119890120596119871ℎ)2))119903]

119899max

sum

119899=0

119886119899119903119899 (28)

1198861= minus1198860[119902 +

(119864 minus 1198981198901198882) 120581

2ℎ21198882(1198981015840+ 12)

] (29)

1198862= minus

1

2

1198860[119901 minus 119902

2minus

1198981198901205961198711198981015840

ℎ (1198981015840+ 1)

minus

(119864 minus 1198981198901198882) 120581

ℎ21198882(1198981015840+ 12)

(119902 +

(119864 minus 1198981198901198882) 120581

4ℎ21198882(1198981015840+ 1)

)] (30)

where119863119899119898

is the normalization constant

3 Discussions

In this section we discuss some special cases of interest fromour general solution

(i) If we set 120590 = 120581 = 0 the effective Killingbeck potentialturns to effective harmonic oscillator potential in relativisticcase as

119880eff (119903 120596119871 120585) = 1205821199032+

1198982

11989011988821205962

119871

(119864 + 1198981198901198882)

1199032

+

ℎ21198882

(119864 + 1198981198901198882)

(11989810158402minus 14)

1199032

+

2ℎ12059611987111989811989011988821198981015840

(119864 + 1198981198901198882)

120582 =

1

2

1205831205962

(31)

The bound state solutions (with the change 119899 = 2(119899119903+ 1)) are

[32 36]

1198642minus 1198982

1198901198884= 21198981198901198882ℎ1205961198711198981015840+ 4ℎ1198882(119899119903+

1198981015840+ 1

2

)

timesradic(119864 + 119898

1198901198882) 120582

1198882

+ (119898119890120596119871)2

(32)

120595(+)

119899119903119898(119903 120601) = 119860

119899119903119898

1

radic2120587

119890119894119898120601

119903119898+120585

times 119890minus(12)radic(119864+119898

1198901198882)120582ℎ21198882+(119898119890120596119871ℎ)21199032

119899max

sum

119899=0

1198861198991199032(119899119903+1)

(33a)

1198861= 0

1198862=

1

2

1198860[

[

radic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

+

1198981198901205961198711198981015840

ℎ (1198981015840+ 1)

]

]

(33b)

Advances in High Energy Physics 7

where 119860119899119903119898

is the normalization constant On the otherhand the nonrelativistic effective harmonic oscillator poten-tial

119880eff (119903 120596119871 120585) = 1205821199032+

1

2

1205962

1198711199032+

(11989810158402minus 14) ℎ

2

21198981198901199032

+ ℎ1205961198711198981015840

(34)

has bound states as [32]

1198641198991198981015840 (120596119871 120585) = ℎ120596

1198711198981015840

+ 2ℎΩ1015840(119899119903+

10038161003816100381610038161003816119898101584010038161003816100381610038161003816+ 1

2

) Ω1015840= radic120596

2

119871+ 1205962

(35)

120595(+)

119899119903119898(119903 120601) = 119862

119899119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(1205832ℎ)Ω

10158401199032

times

119899max

sum

119899=0

1198861198991199032(119899119903+1) 119899119903= 0 1 2

=

1

radic2120587

119890119894119898120601

radic21198871198981015840+1119899

(119899 + 1198981015840)

1199031198981015840

119890minus11988711990322119871(1198981015840)

119899(1198871199032)

119887 =

120583

ℎ

Ω

1015840

(36a)

1198861= 0 119886

2=

120583

2ℎ

Ω10158401198860 1198860

= 0 (36b)

(ii) Setting 120582 = 120590 = 0 in (21) the killingbeck potentialturns to the Coulomb potential case (ie 119881(119903) = minus119860119903 and119899 = 119899119903+ 1) Thus we obtain the relativistic energy formula

1198642minus 1198982

1198901198884= 21198981198901198882(119899119903+ 21198981015840) ℎ120596119871

minus

(119864 + 1198981198901198882)

2

1205812

4ℎ21198882(119899119903+ 1198981015840+ 12)

2 120581 = 119885119890

2

(37)

and the wave function

120595(+)

119899119903119898(119903 120601) = 119862

119899119903119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(119898119890120596119871)2ℎ1199032

times

119899max

sum

119899=0

119886119899119903119899119903+1

(38a)

1198861= minus

(119864 + 1198981198901198882) 120581

2ℎ21198882(1198981015840+ 12)

1198860 (38b)

1198862= minus

1

2

[

[

119898119890120596119871

ℎ

minus

1198981198901205961198711198981015840

ℎ (1198981015840+ 1)

minus

(119864 + 1198981198901198882)

2

1205812

4ℎ41198884(1198981015840+ 1) (119898

1015840+ 12)

]

]

1198860

(38c)

Overmore in the nonrelativistic limit we have

119864 = ℎ1205961198711198981015840+ ℎ (119899

119903+ 1198981015840) 120596119871minus

11989811989011988521198904

2ℎ2(119899119903+ 1198981015840+ 12)

2 (39)

and the wave function becomes

120595(+)

119899119903119898(119903 120601) = 119862

119899119903119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(1205831205961198712ℎ)1199032

119899max

sum

119899=0

119886119899119903119899119903+1

(40a)

1198861= minus

119898119890119860

ℎ2(1198981015840+ 12)

1198860 (40b)

1198862= minus

1

2

[

120583120596119871

ℎ

minus

1205831205961198711198981015840

ℎ (1198981015840+ 1)

minus

120583211988521198904

ℎ4(1198981015840+ 1) (119898

1015840+ 12)

] 1198860

(40c)

Notice that in the absence of external fields 120596119871= 120585 = 0

the problem can be solved in any dimension Thus applyingthe transformation |119898| rarr 119897 + 12 our previous results arereduced to the well-known three-dimensional bound statesolutions for the harmonic oscillator and the hydrogen-likeatoms in the Coulomb fields [40]

119864plusmn= plusmn1198981198901198882[[

[

1 +

2ℎ119888

1198981198901198882(2119899119903+ 119897 +

3

2

)radic

(119864 + 1198981198901198882) 120582

1198982

1198901198884

]]

]

12

119864plusmn= plusmn1198981198901198882[1 minus

11988521198904

4ℎ21198882(119899119903+ 119897 + 1)

2(1 +

119864

1198981198901198882)

2

]

12

(41)

respectively According to (41) the energy spectrum can befound for the scalar particle and antiparticle

In Figure 2(a) we plot the effective potential for the caseof low vibrational (119899 = 0 1 2 3) and rotational (119898 = 1)

levels for various Larmor frequencies 120596119871

= 0 1 5 8 and120585 = 0 case As shown in Figure 2(a) and (31) the effectivepotential function changes in shape as well as the bound stateenergy eigenvalues increase when 120596

119871= 8 It is shown that the

energy levels are raised when the strength of the magneticfield increases and in the absence of AB flux field It is alsoobvious that the effective potential changes gradually fromthe pure pseudo-harmonic oscillator potential which is anonmagnetic (120596

119871= 0) and AB flux (120585 = 0) fields case to

a pure harmonic oscillator type behavior in short potentialrange when the strength of the applied magnetic field isincreased to 120596

119871= 8 If we consider a strong magnetic field

case 120596119871= 8 which has the shape of pure harmonic oscillator

potential function the energy difference between adjacentenergy levels is nearly equal which is a known characteristicof the pure harmonic oscillator potential In Figure 2(b) weplot the effective potential (34) in the absence of themagneticfield and in the presence of AB flux field in the short rangeregion

In Tables 1 and 2 we show the effect of magnetic fieldand AB flux field respectively on the low vibrational 119899

8 Advances in High Energy Physics

120596L = 0

120596L = 1

120596L = 5

120596L = 8

E01E11

E21

E31

E01

E11

E21E31

E01E11

E21

E31

0 1 2 3 4 50

10

20

30

40

50

60

70

80

r (au)

Ueff(r120596

L)

(a)

0 1 2 3 4 5 6 7 80

10

20

30

40

50

60

70

80

r (au)

120585 = 0

120585 = 3

120585 = 5

120585 = 8

Ueff(r120585)

(b)

Figure 2 The Schrodinger effective potential function and corresponding bound state energy levels (119864119899119898) in low vibrational (119899 = 0 1 2 3)

and rotational (119898 = 1) levels for (a) 120596119871= 0 1 5 8 with 120585 = 0 (b) 120585 = 0 1 5 8 and with 120596

119871= 0 (color online) Here 120583 = ℎ = 1

and rotational 119898 relativistic energy states of the harmonicoscillator potential As shown in Table 1 when the magneticfield is not applied and without AB flux field (120596

119871= 0

120585 = 0) the spacing between the energy levels of the effectivepotential is narrow and decreases with increasing 119899 Butwhen the magnetic field strength increases the energy levelsof the effective potential increase and the spacings betweenstates also increase In Table 2 when the AB flux field isapplied and without magnetic field the energy states becomedegenerate for various values of 119899 and 119898 and for variousAB flux field strength values In Tables 3 and 4 we showthe effect of magnetic field and AB flux field respectivelyon the low vibrational 119899 and rotational 119898 nonrelativisticenergy states of the harmonic oscillator potential As shownin Table 3 when themagnetic field is not applied and withoutAB flux field (120596

119871= 0 120585 = 0) the energy states are equally

spaced (the pure harmonic oscillator case) But when themagnetic field strength is raised the energy levels of theeffective potential increase and the spacings between statesalso increase In Table 4 when the AB flux field is applied andwithout magnetic field the energy states become degenerateand equally spaced for various values of 119899 and 119898 and forvarious AB flux field strength values

4 Concluding Remarks

To sum up in this paper we have studied the solutionof two-dimensional KG and Schrodinger equations withthe Killingbeck potential for low vibrational and rotationalenergy levels without and with a constant magnetic fieldhaving arbitrary Larmor frequeny and AB flux field Wehave applied the wave function ansatz method for 120596

119871= 0

(with magnetic field) and 120585 = 0 (with AB flux field) toobtain analytical expressions in closed form for boundstate energies and wave functions of the spinless relativisticparticle subject to a Killingbeck interaction expressed interms of external uniform magnetic and AB flux fields inany vibrational 119899 and rotational 119898 states The above resultsshow that the problems of relativistic quantum mechanicscan be also solved exactly as in the non-relativistic onesWe considered the solution of both positive (particle) andnegative (antiparticle) KG energy states It is noticed that thesolution with equal mixture of scalar-vector potentials can beeasily reduced into the well-known Schrodinger solution fora particle with an interaction potential field and a free fieldrespectively We have also studied the bound-state solutionsfor some special cases including the non-relativistic limits(Schrodinger equation for harmonic oscillator and Coulombpotentials under external magnetic and AB flux fields) andalso the KG equation for harmonic oscillator and CoulombinteractionsThe results show that the splitting is not constantand is mainly dependent on the strength of the externalmagnetic field and AB flux field In order to show the effectof constant magnetic and AB flux fields on the vibrationaland rotational energy levels of the harmonic oscillator weplot the effective potential and corresponding energy levelswith increasing Larmor frequency and flux field for specialpotential parametersWehave seen that the effective potentialfunction and corresponding energy levels are raised in energywhen magnetic and AB flux field strengths increase Theeffective potential function behavior gradually changes fromthe pure pseudo-harmonic oscillator to a pure harmonicoscillator shape in short potential range as the magnetic andAB flux fields strengths increase

Advances in High Energy Physics 9

Table 1 The KG energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120596119871values and 120585 = 0 Here ℎ = 119888 =

119898119890= 119896 = 1

119898 119899

119864119899119898

120596119871= 0 120596

119871= 1 120596

119871= 2 120596

119871= 3 120596

119871= 4 120596

119871= 5 120596

119871= 6 120596

119871= 7 120596

119871= 8

0

0 183929 204353 240325 275615 308137 338066 365815 391752 4161661 309625 34289 403986 465113 521808 574094 622602 667941 7106092 412383 451488 526359 603138 675146 74193 804087 862301 9171583 503104 545806 630283 718744 802542 880667 953605 102205 108663

1

0 250976 316597 389307 455265 514478 568284 617802 663858 707061 362919 427262 512444 593706 668087 736315 799437 858337 9137072 458916 522885 616824 709549 795634 87516 949036 101815 1083233 545354 609216 709891 811893 90768 996689 107966 115745 123081

Table 2 The KG energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120585 values and 120596119871= 0

119898 119899

119864119899119898

120585 = 0 120585 = 1 120585 = 2 120585 = 3 120585 = 4 120585 = 5 120585 = 6 120585 = 7 120585 = 8

0

0 183929 250976 309625 362919 412383 458916 503104 545354 5859661 309625 362919 412383 458916 503104 545354 585966 625166 663132 412383 458916 503104 545354 585966 625166 6631370 70 7358923 503104 545354 585966 625166 66313 70 735892 770901 805108

1

0 250976 309625 362919 412383 458916 503104 545354 585966 6251661 362919 412383 458916 503104 545354 585966 625166 66313 702 458916 503104 545354 585966 625166 66313 70 735892 7709013 545354 585966 625166 66313 70 735892 770901 805108 838582

Table 3 The nonrelativistic energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120596119871values and 120585 = 0

119898 119899

119864119899119898

120596119871= 0 120596

119871= 1 120596

119871= 2 120596

119871= 3 120596

119871= 4 120596

119871= 5 120596

119871= 6 120596

119871= 7 120596

119871= 8

0

0 10 141421 223607 316228 412311 509902 608276 707107 8062261 30 424264 67082 948683 123693 152971 182483 212132 2418682 50 707107 111803 158114 206155 254951 304138 353553 4031133 70 989949 156525 221359 288617 356931 425793 494972 564358

1

0 20 382843 647214 932456 122462 15198 181655 211421 2412451 40 665685 109443 156491 204924 253961 303311 352843 402492 60 948528 154164 219737 287386 355941 424966 494264 5637353 80 123137 198885 282982 369848 457922 546621 635685 724981

Table 4 The nonrelativistic energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120585 values and 120596119871= 0

119898 119899

119864119899119898

120585 = 0 120585 = 1 120585 = 2 120585 = 3 120585 = 4 120585 = 5 120585 = 6 120585 = 7 120585 = 8

0

0 10 20 30 40 50 60 70 80 901 30 40 50 60 70 80 90 100 1102 50 60 70 80 90 100 110 120 1303 70 80 90 100 110 120 130 140 150

1

0 20 30 40 50 60 70 80 90 1001 40 50 60 70 80 90 100 110 1202 60 70 80 90 100 110 120 130 1403 80 90 100 110 120 130 140 150 160

10 Advances in High Energy Physics

Acknowledgments

One of the authors (Sameer M Ikhdair) would like to thankthe president of An-Najah National University (ANU) Pro-fessor Rami Alhamdallah and the acting president ProfessorMaher Natsheh for their continuous support during hispresent work at ANU The authors acknowledge the partialsupport of the Scientific and Technological Research Councilof Turkey

References

[1] WGreiner andBMullerQuantumMechanics An IntroductionSpringer Berlin Germany 1994

[2] S Ozcelik and M Simsek ldquoExact solutions of the radialSchrodinger equation for inverse-power potentialsrdquo PhysicsLetters A vol 152 no 3-4 pp 145ndash150 1991

[3] S-H Dong ldquoSchrodinger equation with the potential 119881(119903) =119860119903minus4+ 119861119903minus3+ 119862119903minus2+ 119863119903minus1rdquo Physica Scripta vol 64 no 4 pp

273ndash276 2001[4] S-H Dong and Z-Q Ma ldquoExact solutions to the Schrodinger

equation for the potential 119881(119903) = 1198861199032+ 119887119903minus4

+ 119888119903minus6 in two

dimensionsrdquo Journal of Physics A vol 31 no 49 pp 9855ndash98591998

[5] M S Child S-H Dong and X-GWang ldquoQuantum states of asextic potential hidden symmetry and quantum monodromyrdquoJournal of Physics A vol 33 no 32 pp 5653ndash5661 2000

[6] S-H Dong ldquoExact solutions of the two-dimensional Schrodin-ger equation with certain central potentialsrdquo InternationalJournal of Theoretical Physics vol 39 no 4 pp 1119ndash1128 2000

[7] S-H Dong ldquoA new approach to the relativistic schrodingerequation with central potential Ansatz methodrdquo InternationalJournal of Theoretical Physics vol 40 no 2 pp 559ndash567 2001

[8] S-H Dong ldquoOn the solutions of the Schrodinger equation withsome anharmonic potentials wave function ansatzrdquo PhysicaScripta vol 65 no 4 pp 289ndash295 2002

[9] S Flugge Practical Quantum Mechanics 1 Springer BerlinGermany 1994

[10] X-Q Zhao C-S Jia and Q-B Yang ldquoBound states of rel-ativistic particles in the generalized symmetrical double-wellpotentialrdquo Physics Letters A vol 337 no 3 pp 189ndash196 2005

[11] S M Ikhdair ldquoExact solution of Dirac equation with chargedharmonic oscillator in electric field bound statesrdquo Journal ofModern Physics vol 3 no 2 pp 170ndash179 2012

[12] W C Qiang ldquoBound states of the Klein-Gordon equation forring-shaped Kratzer-type potentialrdquo Chinese Physics vol 13 no5 pp 575ndash578 2004

[13] S Ikhdair and R Sever ldquoExact polynomial eigensolutions ofthe Schrodinger equation for the pseudoharmonic potentialrdquoJournal of Molecular Structure vol 806 no 1ndash3 pp 155ndash1582007

[14] F-L Lu C-Y Chen and D-S Sun ldquoBound states of Klein-Gordon equation for double ring-shaped oscillator scalar andvector potentialsrdquo Chinese Physics vol 14 no 3 pp 463ndash4672005

[15] S M Ikhdair and R Sever ldquoExact solutions of the D-dimensional Schrodinger equation for a ring-shaped pseudo-harmonic potentialrdquo Central European Journal of Physics vol 6no 3 pp 685ndash696 2008

[16] S M Ikhdair and R Sever ldquoExact bound states of the D-dimensional Klein-Gordon equation with equal scalar and

vector ring-shaped pseudoharmonic potentialrdquo InternationalJournal of Modern Physics C vol 19 no 9 pp 1425ndash1442 2008

[17] S M Ikhdair and R Sever ldquoExactly solvable effective massd-dimensional schrodinger equation for pseudoharmonic andmodified kratzer problemsrdquo International Journal of ModernPhysics C vol 20 no 3 pp 361ndash372 2009

[18] R-CWang andC-YWong ldquoFinite-size effect in the Schwingerparticle-productionmechanismrdquo Physical ReviewD vol 38 no1 pp 348ndash359 1988

[19] J N Ginocchio ldquoA relativistic symmetry in nucleirdquo PhysicsReport vol 315 no 1ndash4 pp 231ndash240 1999

[20] A Bohr I Hamarnoto and B R Motelson ldquoPseudospln inrotating nuclear potentialsrdquo Physica Scripta vol 26 no 4 pp267ndash272 1982

[21] J Dudek W Nazarewicz Z Szymanski and G A LeanderldquoAbundance and systematics of nuclear superdeformed statesrelation to the pseudospin and pseudo-SU(3) symmetriesrdquoPhysical Review Letters vol 59 no 13 pp 1405ndash1408 1987

[22] A ArimaM Harvey and K Shimizu ldquoPseudo LS coupling andpseudo SU3 coupling schemesrdquo Physics Letters B vol 30 no 8pp 517ndash522 1969

[23] K T Hecht and A Adler ldquoGeneralized seniority for favored J= 0 pairs in mixed configurationsrdquo Nuclear Physics A vol 137

no 1 pp 129ndash143 1969[24] P R Page T Goldman and J N Ginocchio ldquoRelativistic sym-

metry suppresses quark spin-orbit splittingrdquo Physical ReviewLetters vol 86 no 2 pp 204ndash207 2001

[25] S Weinberg The Quantum Theory of Fields vol 1 CambridgeUniversity Press Cambridge UK 1995

[26] D Boyanovsky C Destri and H J de Vega ldquoApproach tothermalization in the classical 1205934 theory in 1 + 1 dimensionsenergy cascades and universal scalingrdquo Physical Review D vol69 no 4 Article ID 045003 2004

[27] R Khordad ldquoEffects of magnetic field and geometrical size onthe interband light absorption in a quantumpseudodot systemrdquoSolid State Sciences vol 12 no 7 pp 1253ndash1256 2010

[28] R Khordad ldquoSimultaneous effects of temperature and pressureon the donor binding energy in a V-groove quantum wirerdquoSuperlattices and Microstructures vol 47 no 3 pp 422ndash4312010

[29] A Cetin ldquoA quantum pseudodot system with a two-dimensional pseudoharmonic potentialrdquo Physics LettersA vol 372 no 21 pp 3852ndash3856 2008

[30] S M Ikhdair M Hamzavi and R Sever ldquoSpectra of cylindricalquantum dots the effect of electrical and magnetic fieldstogether with AB flux fieldrdquo Physica B vol 407 pp 4523ndash45292012

[31] S M Ikhdair and M Hamzavi ldquoA quantum pseudodot systemwith two-dimensional pseudoharmonic oscillator in externalmagnetic and Aharonov-Bohm fieldsrdquo Physica B vol 407 pp4198ndash4207 2012

[32] S M Ikhdair and M Hamzavi ldquoEffects of extermal fieldson a two-dimensional Klein-Gordon particle under pseudo-harmonic oscillator interactionrdquo Chinese Physics B vol 21 no11 pp 110302ndash110306 2012

[33] M Aygun O Bayrak I Boztosun and Y Sahin ldquoThe energyeigenvalues of the Kratzer potential in the presence of amagnetic fieldrdquo European Physical Journal D vol 66 no 2Article ID 20319 2012

[34] S M Ikhdair andM Hamzavi ldquoSpectral properties of quantumdots influenced by a confining potential modelrdquo Physica B vol407 pp 4797ndash4803 2012

Advances in High Energy Physics 11

[35] M Hamzavi and A A Rajabi ldquoSolution of Dirac equation withKillingbeck potential by using wave function ansatz methodunder spin symmetry limitrdquo Communications in TheoreticalPhysics vol 55 no 1 pp 35ndash37 2011

[36] M Hamzavi S M Ikhdair and K E Thylwe ldquoPseudospinsymmetry in the relativistic Killingbeck potential Quasi-exactsolutionrdquo Zeitschrift fur Naturforschung A vol 67 pp 567ndash5712012

[37] R Kumar and F Chand ldquoSeries solution to the N-dimensionalradial Schrodinger equation for the quark-antiquark interac-tionrdquo Physica Scripta vol 85 no 5 pp 055008ndash055004 2012

[38] R Kumar and F Chand ldquoReply to comments on seriessolution to the N-dimensionalradial Schrodinger equation forthe quark-antiquark interactionrdquo Physica Scripta vol 86 no 2p 027001 2012

[39] S M Ikhdair ldquoA scalar charged particle in presence of magneticandAharonov-Bohmfield plus scalar-vectorKillingbeck poten-tialsrdquo Few-Body Systems vol 54 no 11 pp 1987ndash1995 2013

[40] W Greiner Relativistic Quantum Mechanics Wave EquationsSpringer Berlin Germany 2000

[41] A D Alhaidari H Bahlouli and A Al-Hasan ldquoDirac andKlein-Gordon equations with equal scalar and vector poten-tialsrdquo Physics Letters A vol 349 no 1ndash4 pp 87ndash97 2006

[42] G Plante and A F Antippa ldquoAnalytic solution of theSchrodinger equation for the Coulomb-plus-linear potentialrdquoJournal of Mathematical Physics vol 46 no 6 pp 1ndash20 2005

[43] J Killingbeck ldquoPerturbation theory without wavefunctionsrdquoPhysics Letters A vol 65 no 2 pp 87ndash88 1978

[44] Y Xu S He and C-S Jia ldquoApproximate analytical solutionsof the Klein-Gordon equation with the Poschl-Teller potentialincluding the centrifugal termrdquo Physica Scripta vol 81 no 4Article ID 045001 2010

[45] R L Liboff Introductory Quantum Mechanics PearsonEducation-Addison Wesley San Francisco Calif USA 2003

[46] GChen Z-DChen andZ-M Lou ldquoBound states of theKlein-Gordon and Dirac equation for scalar and vector pseudohar-monic oscillator potentialsrdquo Chinese Physics vol 13 no 3 pp279ndash282 2004

[47] S H Dong Z Ma and G Espozito ldquoExact solutions of theSchrodinger equation with inversepowerpotentialrdquo Founda-tions of Physics Letters vol 12 no 5 pp 465ndash474 1999

[48] A Arda and R Sever ldquoEffective-mass Klein-Gordon equationfor non-PTnon-Hermitian generalizedMorse potentialrdquo Phys-ica Scripta vol 82 no 6 Article ID 065007 2010

[49] S M Ikhdair and J Abu-Hasna ldquoQuantization rule solutionto the Hulthen potential in arbitrary dimension with a newapproximate scheme for the centrifugal termrdquo Physica Scriptavol 83 no 2 Article ID 025002 2011

[50] S M Ikhdair and R Sever ldquoExact quantization rule tothe Kratzer-type potentials An application to the diatomicmoleculesrdquo Journal of Mathematical Chemistry vol 45 no 4pp 1137ndash1152 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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ThermodynamicsJournal of

Advances in High Energy Physics 5

120595(+)

119899119898(119903 120601) = 119862

119899119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(12)radic(119864+119898

1198901198882)120582ℎ21198882+(119898119890120596119871ℎ)2[1199032+((119864+119898

1198901198882)120590((119864+119898

1198901198882)1198892+ℎ21198882(119898119890120596119871ℎ)2))119903]

119899max

sum

119899=0

119886119899119903119899

(22a)

1198861= minus1198860[119902 +

(119864 + 1198981198901198882) 120581

2ℎ21198882(1198981015840+ 12)

] (22b)

1198862= minus

1

2

1198860[119901 minus 119902

2minus

1198981198901205961198711198981015840

ℎ (1198981015840+ 1)

minus

(119864 + 1198981198901198882) 120581

ℎ21198882(1198981015840+ 12)

(119902 +

(119864 + 1198981198901198882) 120581

4ℎ21198882(1198981015840+ 1)

)] (22c)

where 119862119899119898

is the normalization constantIn the non-relativistic limit when 119864 + 119898

1198901198882

rarr 2120583119864 + 119898

1198901198882rarr 1198641198991198981015840 and 119888 = 1 (21) and (22a) give the energy

formula

1198641198991198981015840 (120596119871 120585) = ℎ120596

1198711198981015840+ ℎ (119899 + 119898

1015840minus 1)

times radic

2120582

120583

+ 1205962

119871minus

1205831205812

2ℎ2(119899 + 119898

1015840minus 12)

2

(23)

and wave function

120595(+)

119899119898(119903 120601) = 119862

119899119898

1

radic2120587

119890119894119898120601

119903119898+120585

times 119890minus(1205832ℎ)radic2120582120583+120596

2

119871[1199032+(2120590120583(2119889

2120583+1205962

119871))119903]

119899max

sum

119899=0

119886119899119903119899

(24)

respectively for the Schrodinger-Killingbeck system with thefollowing restriction on the potential parameters 120582 120590 and 120581given by

120590 =

120583120581

ℎ (119899 + 1198981015840minus 12)

radic

2120582

120583

+ 1205962

119871 (25)

Notice that the present model has been solved in 2D spacewith an external uniform magnetic field since it is perpen-dicular to the plane where the vector and scalare Cornellpotentials have the dimensions of 119903 = 120588 and 120601 that is119881 = 119878 = 119881(120588 120601) [32] However without the magnetic fieldthe model can be solved in any desired dimensional spaceby considering the change 119898 rarr 119897 + (119863 minus 2)2 [49 50]where119863 refers to spatial dimension and also the existence ofinterdimensional degeneracy

As shown in Figure 1(a) and (9b) the effective potentialfunction changes in shape when the magnetic field strength

increases say 120596119871= 8 and in absence of AB flux field The

energy levels are raised when the strength of the magneticfield increases and in absence of AB flux field 120585 = 0 Wesee that the effective potential changes gradually from thepure pseudo-harmonic oscillator potential when 120596

119871= 0 to

a pure harmonic oscillator type behavior in short potentialrange when120596

119871= 8 In Figure 1(b) the effective potential (9b)

which is pseudo-harmonic oscillator when 120596119871= 0 becomes

sensitive to the increasing AB flux field 120585 = 8 in the shortrange region that is 0 lt 119903 lt 4 au

22 The Bound States for Negative Energy In this case of119878(119903) = minus119881(119903) the first inspection of (4) shows that thefollowing changes 119881(119903) rarr minus119881(119903) (ie 120582 rarr minus120582 120590 rarr minus120590120581 rarr minus120581) 120595(+)

119899119898(119903 120601) rarr 120595

(minus)

119899119898(119903 120601) and 119864 rarr minus119864 give the

negative energy solution for antiparticles as

1198642minus 1198982

1198901198884minus 21198981198901198882ℎ1205961198711198981015840

= 2ℎ21198882(119899 + 119898

1015840minus 1)

radic(119864 minus 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

minus

(119864 minus 1198981198901198882)

2

1205812

4ℎ21198882(119899 + 119898

1015840minus 12)

2

(26)

with restriction on potential parameters

120590 =

120581

(119899 + 1198981015840minus 12)

radic(119864 minus 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

(27)

We may find solution to the above transcendental equationas 119864 = 119864

(minus)

119870119866= 1198641198991198981015840(120596119871 120585) Overmore the wave function for

antiparticle reads

6 Advances in High Energy Physics

0 1 2 3 4 5 6 70

2

4

6

8

10

12

14

16

18

20

120596L = 0

120596L = 1

120596L = 5

120596L = 8

r (au)

Ueff(r120596

L)

(a)

0 1 2 3 4 50

2

4

6

8

10

12

14

16

18

20

r (au)

120585 = 0

120585 = 1

120585 = 5

120585 = 8

Ueff(r120585)

(b)

Figure 1 The KG effective potential function for (a) 120596119871= 0 1 5 8 and with 120585 = 0 (b) 120585 = 0 1 5 8 and with 120596

119871= 0 (colour online) Here

119898119890= ℎ = 119888 = 1

120595(minus)

119899119898(119903 120601) = 119863

119899119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(12)radic(119864minus119898

1198901198882)120582ℎ21198882+(119898119890120596119871ℎ)2[1199032+((119864minus119898

1198901198882)120590((119864minus119898

1198901198882)1198892+ℎ21198882(119898119890120596119871ℎ)2))119903]

119899max

sum

119899=0

119886119899119903119899 (28)

1198861= minus1198860[119902 +

(119864 minus 1198981198901198882) 120581

2ℎ21198882(1198981015840+ 12)

] (29)

1198862= minus

1

2

1198860[119901 minus 119902

2minus

1198981198901205961198711198981015840

ℎ (1198981015840+ 1)

minus

(119864 minus 1198981198901198882) 120581

ℎ21198882(1198981015840+ 12)

(119902 +

(119864 minus 1198981198901198882) 120581

4ℎ21198882(1198981015840+ 1)

)] (30)

where119863119899119898

is the normalization constant

3 Discussions

In this section we discuss some special cases of interest fromour general solution

(i) If we set 120590 = 120581 = 0 the effective Killingbeck potentialturns to effective harmonic oscillator potential in relativisticcase as

119880eff (119903 120596119871 120585) = 1205821199032+

1198982

11989011988821205962

119871

(119864 + 1198981198901198882)

1199032

+

ℎ21198882

(119864 + 1198981198901198882)

(11989810158402minus 14)

1199032

+

2ℎ12059611987111989811989011988821198981015840

(119864 + 1198981198901198882)

120582 =

1

2

1205831205962

(31)

The bound state solutions (with the change 119899 = 2(119899119903+ 1)) are

[32 36]

1198642minus 1198982

1198901198884= 21198981198901198882ℎ1205961198711198981015840+ 4ℎ1198882(119899119903+

1198981015840+ 1

2

)

timesradic(119864 + 119898

1198901198882) 120582

1198882

+ (119898119890120596119871)2

(32)

120595(+)

119899119903119898(119903 120601) = 119860

119899119903119898

1

radic2120587

119890119894119898120601

119903119898+120585

times 119890minus(12)radic(119864+119898

1198901198882)120582ℎ21198882+(119898119890120596119871ℎ)21199032

119899max

sum

119899=0

1198861198991199032(119899119903+1)

(33a)

1198861= 0

1198862=

1

2

1198860[

[

radic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

+

1198981198901205961198711198981015840

ℎ (1198981015840+ 1)

]

]

(33b)

Advances in High Energy Physics 7

where 119860119899119903119898

is the normalization constant On the otherhand the nonrelativistic effective harmonic oscillator poten-tial

119880eff (119903 120596119871 120585) = 1205821199032+

1

2

1205962

1198711199032+

(11989810158402minus 14) ℎ

2

21198981198901199032

+ ℎ1205961198711198981015840

(34)

has bound states as [32]

1198641198991198981015840 (120596119871 120585) = ℎ120596

1198711198981015840

+ 2ℎΩ1015840(119899119903+

10038161003816100381610038161003816119898101584010038161003816100381610038161003816+ 1

2

) Ω1015840= radic120596

2

119871+ 1205962

(35)

120595(+)

119899119903119898(119903 120601) = 119862

119899119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(1205832ℎ)Ω

10158401199032

times

119899max

sum

119899=0

1198861198991199032(119899119903+1) 119899119903= 0 1 2

=

1

radic2120587

119890119894119898120601

radic21198871198981015840+1119899

(119899 + 1198981015840)

1199031198981015840

119890minus11988711990322119871(1198981015840)

119899(1198871199032)

119887 =

120583

ℎ

Ω

1015840

(36a)

1198861= 0 119886

2=

120583

2ℎ

Ω10158401198860 1198860

= 0 (36b)

(ii) Setting 120582 = 120590 = 0 in (21) the killingbeck potentialturns to the Coulomb potential case (ie 119881(119903) = minus119860119903 and119899 = 119899119903+ 1) Thus we obtain the relativistic energy formula

1198642minus 1198982

1198901198884= 21198981198901198882(119899119903+ 21198981015840) ℎ120596119871

minus

(119864 + 1198981198901198882)

2

1205812

4ℎ21198882(119899119903+ 1198981015840+ 12)

2 120581 = 119885119890

2

(37)

and the wave function

120595(+)

119899119903119898(119903 120601) = 119862

119899119903119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(119898119890120596119871)2ℎ1199032

times

119899max

sum

119899=0

119886119899119903119899119903+1

(38a)

1198861= minus

(119864 + 1198981198901198882) 120581

2ℎ21198882(1198981015840+ 12)

1198860 (38b)

1198862= minus

1

2

[

[

119898119890120596119871

ℎ

minus

1198981198901205961198711198981015840

ℎ (1198981015840+ 1)

minus

(119864 + 1198981198901198882)

2

1205812

4ℎ41198884(1198981015840+ 1) (119898

1015840+ 12)

]

]

1198860

(38c)

Overmore in the nonrelativistic limit we have

119864 = ℎ1205961198711198981015840+ ℎ (119899

119903+ 1198981015840) 120596119871minus

11989811989011988521198904

2ℎ2(119899119903+ 1198981015840+ 12)

2 (39)

and the wave function becomes

120595(+)

119899119903119898(119903 120601) = 119862

119899119903119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(1205831205961198712ℎ)1199032

119899max

sum

119899=0

119886119899119903119899119903+1

(40a)

1198861= minus

119898119890119860

ℎ2(1198981015840+ 12)

1198860 (40b)

1198862= minus

1

2

[

120583120596119871

ℎ

minus

1205831205961198711198981015840

ℎ (1198981015840+ 1)

minus

120583211988521198904

ℎ4(1198981015840+ 1) (119898

1015840+ 12)

] 1198860

(40c)

Notice that in the absence of external fields 120596119871= 120585 = 0

the problem can be solved in any dimension Thus applyingthe transformation |119898| rarr 119897 + 12 our previous results arereduced to the well-known three-dimensional bound statesolutions for the harmonic oscillator and the hydrogen-likeatoms in the Coulomb fields [40]

119864plusmn= plusmn1198981198901198882[[

[

1 +

2ℎ119888

1198981198901198882(2119899119903+ 119897 +

3

2

)radic

(119864 + 1198981198901198882) 120582

1198982

1198901198884

]]

]

12

119864plusmn= plusmn1198981198901198882[1 minus

11988521198904

4ℎ21198882(119899119903+ 119897 + 1)

2(1 +

119864

1198981198901198882)

2

]

12

(41)

respectively According to (41) the energy spectrum can befound for the scalar particle and antiparticle

In Figure 2(a) we plot the effective potential for the caseof low vibrational (119899 = 0 1 2 3) and rotational (119898 = 1)

levels for various Larmor frequencies 120596119871

= 0 1 5 8 and120585 = 0 case As shown in Figure 2(a) and (31) the effectivepotential function changes in shape as well as the bound stateenergy eigenvalues increase when 120596

119871= 8 It is shown that the

energy levels are raised when the strength of the magneticfield increases and in the absence of AB flux field It is alsoobvious that the effective potential changes gradually fromthe pure pseudo-harmonic oscillator potential which is anonmagnetic (120596

119871= 0) and AB flux (120585 = 0) fields case to

a pure harmonic oscillator type behavior in short potentialrange when the strength of the applied magnetic field isincreased to 120596

119871= 8 If we consider a strong magnetic field

case 120596119871= 8 which has the shape of pure harmonic oscillator

potential function the energy difference between adjacentenergy levels is nearly equal which is a known characteristicof the pure harmonic oscillator potential In Figure 2(b) weplot the effective potential (34) in the absence of themagneticfield and in the presence of AB flux field in the short rangeregion

In Tables 1 and 2 we show the effect of magnetic fieldand AB flux field respectively on the low vibrational 119899

8 Advances in High Energy Physics

120596L = 0

120596L = 1

120596L = 5

120596L = 8

E01E11

E21

E31

E01

E11

E21E31

E01E11

E21

E31

0 1 2 3 4 50

10

20

30

40

50

60

70

80

r (au)

Ueff(r120596

L)

(a)

0 1 2 3 4 5 6 7 80

10

20

30

40

50

60

70

80

r (au)

120585 = 0

120585 = 3

120585 = 5

120585 = 8

Ueff(r120585)

(b)

Figure 2 The Schrodinger effective potential function and corresponding bound state energy levels (119864119899119898) in low vibrational (119899 = 0 1 2 3)

and rotational (119898 = 1) levels for (a) 120596119871= 0 1 5 8 with 120585 = 0 (b) 120585 = 0 1 5 8 and with 120596

119871= 0 (color online) Here 120583 = ℎ = 1

and rotational 119898 relativistic energy states of the harmonicoscillator potential As shown in Table 1 when the magneticfield is not applied and without AB flux field (120596

119871= 0

120585 = 0) the spacing between the energy levels of the effectivepotential is narrow and decreases with increasing 119899 Butwhen the magnetic field strength increases the energy levelsof the effective potential increase and the spacings betweenstates also increase In Table 2 when the AB flux field isapplied and without magnetic field the energy states becomedegenerate for various values of 119899 and 119898 and for variousAB flux field strength values In Tables 3 and 4 we showthe effect of magnetic field and AB flux field respectivelyon the low vibrational 119899 and rotational 119898 nonrelativisticenergy states of the harmonic oscillator potential As shownin Table 3 when themagnetic field is not applied and withoutAB flux field (120596

119871= 0 120585 = 0) the energy states are equally

spaced (the pure harmonic oscillator case) But when themagnetic field strength is raised the energy levels of theeffective potential increase and the spacings between statesalso increase In Table 4 when the AB flux field is applied andwithout magnetic field the energy states become degenerateand equally spaced for various values of 119899 and 119898 and forvarious AB flux field strength values

4 Concluding Remarks

To sum up in this paper we have studied the solutionof two-dimensional KG and Schrodinger equations withthe Killingbeck potential for low vibrational and rotationalenergy levels without and with a constant magnetic fieldhaving arbitrary Larmor frequeny and AB flux field Wehave applied the wave function ansatz method for 120596

119871= 0

(with magnetic field) and 120585 = 0 (with AB flux field) toobtain analytical expressions in closed form for boundstate energies and wave functions of the spinless relativisticparticle subject to a Killingbeck interaction expressed interms of external uniform magnetic and AB flux fields inany vibrational 119899 and rotational 119898 states The above resultsshow that the problems of relativistic quantum mechanicscan be also solved exactly as in the non-relativistic onesWe considered the solution of both positive (particle) andnegative (antiparticle) KG energy states It is noticed that thesolution with equal mixture of scalar-vector potentials can beeasily reduced into the well-known Schrodinger solution fora particle with an interaction potential field and a free fieldrespectively We have also studied the bound-state solutionsfor some special cases including the non-relativistic limits(Schrodinger equation for harmonic oscillator and Coulombpotentials under external magnetic and AB flux fields) andalso the KG equation for harmonic oscillator and CoulombinteractionsThe results show that the splitting is not constantand is mainly dependent on the strength of the externalmagnetic field and AB flux field In order to show the effectof constant magnetic and AB flux fields on the vibrationaland rotational energy levels of the harmonic oscillator weplot the effective potential and corresponding energy levelswith increasing Larmor frequency and flux field for specialpotential parametersWehave seen that the effective potentialfunction and corresponding energy levels are raised in energywhen magnetic and AB flux field strengths increase Theeffective potential function behavior gradually changes fromthe pure pseudo-harmonic oscillator to a pure harmonicoscillator shape in short potential range as the magnetic andAB flux fields strengths increase

Advances in High Energy Physics 9

Table 1 The KG energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120596119871values and 120585 = 0 Here ℎ = 119888 =

119898119890= 119896 = 1

119898 119899

119864119899119898

120596119871= 0 120596

119871= 1 120596

119871= 2 120596

119871= 3 120596

119871= 4 120596

119871= 5 120596

119871= 6 120596

119871= 7 120596

119871= 8

0

0 183929 204353 240325 275615 308137 338066 365815 391752 4161661 309625 34289 403986 465113 521808 574094 622602 667941 7106092 412383 451488 526359 603138 675146 74193 804087 862301 9171583 503104 545806 630283 718744 802542 880667 953605 102205 108663

1

0 250976 316597 389307 455265 514478 568284 617802 663858 707061 362919 427262 512444 593706 668087 736315 799437 858337 9137072 458916 522885 616824 709549 795634 87516 949036 101815 1083233 545354 609216 709891 811893 90768 996689 107966 115745 123081

Table 2 The KG energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120585 values and 120596119871= 0

119898 119899

119864119899119898

120585 = 0 120585 = 1 120585 = 2 120585 = 3 120585 = 4 120585 = 5 120585 = 6 120585 = 7 120585 = 8

0

0 183929 250976 309625 362919 412383 458916 503104 545354 5859661 309625 362919 412383 458916 503104 545354 585966 625166 663132 412383 458916 503104 545354 585966 625166 6631370 70 7358923 503104 545354 585966 625166 66313 70 735892 770901 805108

1

0 250976 309625 362919 412383 458916 503104 545354 585966 6251661 362919 412383 458916 503104 545354 585966 625166 66313 702 458916 503104 545354 585966 625166 66313 70 735892 7709013 545354 585966 625166 66313 70 735892 770901 805108 838582

Table 3 The nonrelativistic energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120596119871values and 120585 = 0

119898 119899

119864119899119898

120596119871= 0 120596

119871= 1 120596

119871= 2 120596

119871= 3 120596

119871= 4 120596

119871= 5 120596

119871= 6 120596

119871= 7 120596

119871= 8

0

0 10 141421 223607 316228 412311 509902 608276 707107 8062261 30 424264 67082 948683 123693 152971 182483 212132 2418682 50 707107 111803 158114 206155 254951 304138 353553 4031133 70 989949 156525 221359 288617 356931 425793 494972 564358

1

0 20 382843 647214 932456 122462 15198 181655 211421 2412451 40 665685 109443 156491 204924 253961 303311 352843 402492 60 948528 154164 219737 287386 355941 424966 494264 5637353 80 123137 198885 282982 369848 457922 546621 635685 724981

Table 4 The nonrelativistic energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120585 values and 120596119871= 0

119898 119899

119864119899119898

120585 = 0 120585 = 1 120585 = 2 120585 = 3 120585 = 4 120585 = 5 120585 = 6 120585 = 7 120585 = 8

0

0 10 20 30 40 50 60 70 80 901 30 40 50 60 70 80 90 100 1102 50 60 70 80 90 100 110 120 1303 70 80 90 100 110 120 130 140 150

1

0 20 30 40 50 60 70 80 90 1001 40 50 60 70 80 90 100 110 1202 60 70 80 90 100 110 120 130 1403 80 90 100 110 120 130 140 150 160

10 Advances in High Energy Physics

Acknowledgments

One of the authors (Sameer M Ikhdair) would like to thankthe president of An-Najah National University (ANU) Pro-fessor Rami Alhamdallah and the acting president ProfessorMaher Natsheh for their continuous support during hispresent work at ANU The authors acknowledge the partialsupport of the Scientific and Technological Research Councilof Turkey

References

[1] WGreiner andBMullerQuantumMechanics An IntroductionSpringer Berlin Germany 1994

[2] S Ozcelik and M Simsek ldquoExact solutions of the radialSchrodinger equation for inverse-power potentialsrdquo PhysicsLetters A vol 152 no 3-4 pp 145ndash150 1991

[3] S-H Dong ldquoSchrodinger equation with the potential 119881(119903) =119860119903minus4+ 119861119903minus3+ 119862119903minus2+ 119863119903minus1rdquo Physica Scripta vol 64 no 4 pp

273ndash276 2001[4] S-H Dong and Z-Q Ma ldquoExact solutions to the Schrodinger

equation for the potential 119881(119903) = 1198861199032+ 119887119903minus4

+ 119888119903minus6 in two

dimensionsrdquo Journal of Physics A vol 31 no 49 pp 9855ndash98591998

[5] M S Child S-H Dong and X-GWang ldquoQuantum states of asextic potential hidden symmetry and quantum monodromyrdquoJournal of Physics A vol 33 no 32 pp 5653ndash5661 2000

[6] S-H Dong ldquoExact solutions of the two-dimensional Schrodin-ger equation with certain central potentialsrdquo InternationalJournal of Theoretical Physics vol 39 no 4 pp 1119ndash1128 2000

[7] S-H Dong ldquoA new approach to the relativistic schrodingerequation with central potential Ansatz methodrdquo InternationalJournal of Theoretical Physics vol 40 no 2 pp 559ndash567 2001

[8] S-H Dong ldquoOn the solutions of the Schrodinger equation withsome anharmonic potentials wave function ansatzrdquo PhysicaScripta vol 65 no 4 pp 289ndash295 2002

[9] S Flugge Practical Quantum Mechanics 1 Springer BerlinGermany 1994

[10] X-Q Zhao C-S Jia and Q-B Yang ldquoBound states of rel-ativistic particles in the generalized symmetrical double-wellpotentialrdquo Physics Letters A vol 337 no 3 pp 189ndash196 2005

[11] S M Ikhdair ldquoExact solution of Dirac equation with chargedharmonic oscillator in electric field bound statesrdquo Journal ofModern Physics vol 3 no 2 pp 170ndash179 2012

[12] W C Qiang ldquoBound states of the Klein-Gordon equation forring-shaped Kratzer-type potentialrdquo Chinese Physics vol 13 no5 pp 575ndash578 2004

[13] S Ikhdair and R Sever ldquoExact polynomial eigensolutions ofthe Schrodinger equation for the pseudoharmonic potentialrdquoJournal of Molecular Structure vol 806 no 1ndash3 pp 155ndash1582007

[14] F-L Lu C-Y Chen and D-S Sun ldquoBound states of Klein-Gordon equation for double ring-shaped oscillator scalar andvector potentialsrdquo Chinese Physics vol 14 no 3 pp 463ndash4672005

[15] S M Ikhdair and R Sever ldquoExact solutions of the D-dimensional Schrodinger equation for a ring-shaped pseudo-harmonic potentialrdquo Central European Journal of Physics vol 6no 3 pp 685ndash696 2008

[16] S M Ikhdair and R Sever ldquoExact bound states of the D-dimensional Klein-Gordon equation with equal scalar and

vector ring-shaped pseudoharmonic potentialrdquo InternationalJournal of Modern Physics C vol 19 no 9 pp 1425ndash1442 2008

[17] S M Ikhdair and R Sever ldquoExactly solvable effective massd-dimensional schrodinger equation for pseudoharmonic andmodified kratzer problemsrdquo International Journal of ModernPhysics C vol 20 no 3 pp 361ndash372 2009

[18] R-CWang andC-YWong ldquoFinite-size effect in the Schwingerparticle-productionmechanismrdquo Physical ReviewD vol 38 no1 pp 348ndash359 1988

[19] J N Ginocchio ldquoA relativistic symmetry in nucleirdquo PhysicsReport vol 315 no 1ndash4 pp 231ndash240 1999

[20] A Bohr I Hamarnoto and B R Motelson ldquoPseudospln inrotating nuclear potentialsrdquo Physica Scripta vol 26 no 4 pp267ndash272 1982

[21] J Dudek W Nazarewicz Z Szymanski and G A LeanderldquoAbundance and systematics of nuclear superdeformed statesrelation to the pseudospin and pseudo-SU(3) symmetriesrdquoPhysical Review Letters vol 59 no 13 pp 1405ndash1408 1987

[22] A ArimaM Harvey and K Shimizu ldquoPseudo LS coupling andpseudo SU3 coupling schemesrdquo Physics Letters B vol 30 no 8pp 517ndash522 1969

[23] K T Hecht and A Adler ldquoGeneralized seniority for favored J= 0 pairs in mixed configurationsrdquo Nuclear Physics A vol 137

no 1 pp 129ndash143 1969[24] P R Page T Goldman and J N Ginocchio ldquoRelativistic sym-

metry suppresses quark spin-orbit splittingrdquo Physical ReviewLetters vol 86 no 2 pp 204ndash207 2001

[25] S Weinberg The Quantum Theory of Fields vol 1 CambridgeUniversity Press Cambridge UK 1995

[26] D Boyanovsky C Destri and H J de Vega ldquoApproach tothermalization in the classical 1205934 theory in 1 + 1 dimensionsenergy cascades and universal scalingrdquo Physical Review D vol69 no 4 Article ID 045003 2004

[27] R Khordad ldquoEffects of magnetic field and geometrical size onthe interband light absorption in a quantumpseudodot systemrdquoSolid State Sciences vol 12 no 7 pp 1253ndash1256 2010

[28] R Khordad ldquoSimultaneous effects of temperature and pressureon the donor binding energy in a V-groove quantum wirerdquoSuperlattices and Microstructures vol 47 no 3 pp 422ndash4312010

[29] A Cetin ldquoA quantum pseudodot system with a two-dimensional pseudoharmonic potentialrdquo Physics LettersA vol 372 no 21 pp 3852ndash3856 2008

[30] S M Ikhdair M Hamzavi and R Sever ldquoSpectra of cylindricalquantum dots the effect of electrical and magnetic fieldstogether with AB flux fieldrdquo Physica B vol 407 pp 4523ndash45292012

[31] S M Ikhdair and M Hamzavi ldquoA quantum pseudodot systemwith two-dimensional pseudoharmonic oscillator in externalmagnetic and Aharonov-Bohm fieldsrdquo Physica B vol 407 pp4198ndash4207 2012

[32] S M Ikhdair and M Hamzavi ldquoEffects of extermal fieldson a two-dimensional Klein-Gordon particle under pseudo-harmonic oscillator interactionrdquo Chinese Physics B vol 21 no11 pp 110302ndash110306 2012

[33] M Aygun O Bayrak I Boztosun and Y Sahin ldquoThe energyeigenvalues of the Kratzer potential in the presence of amagnetic fieldrdquo European Physical Journal D vol 66 no 2Article ID 20319 2012

[34] S M Ikhdair andM Hamzavi ldquoSpectral properties of quantumdots influenced by a confining potential modelrdquo Physica B vol407 pp 4797ndash4803 2012

Advances in High Energy Physics 11

[35] M Hamzavi and A A Rajabi ldquoSolution of Dirac equation withKillingbeck potential by using wave function ansatz methodunder spin symmetry limitrdquo Communications in TheoreticalPhysics vol 55 no 1 pp 35ndash37 2011

[36] M Hamzavi S M Ikhdair and K E Thylwe ldquoPseudospinsymmetry in the relativistic Killingbeck potential Quasi-exactsolutionrdquo Zeitschrift fur Naturforschung A vol 67 pp 567ndash5712012

[37] R Kumar and F Chand ldquoSeries solution to the N-dimensionalradial Schrodinger equation for the quark-antiquark interac-tionrdquo Physica Scripta vol 85 no 5 pp 055008ndash055004 2012

[38] R Kumar and F Chand ldquoReply to comments on seriessolution to the N-dimensionalradial Schrodinger equation forthe quark-antiquark interactionrdquo Physica Scripta vol 86 no 2p 027001 2012

[39] S M Ikhdair ldquoA scalar charged particle in presence of magneticandAharonov-Bohmfield plus scalar-vectorKillingbeck poten-tialsrdquo Few-Body Systems vol 54 no 11 pp 1987ndash1995 2013

[40] W Greiner Relativistic Quantum Mechanics Wave EquationsSpringer Berlin Germany 2000

[41] A D Alhaidari H Bahlouli and A Al-Hasan ldquoDirac andKlein-Gordon equations with equal scalar and vector poten-tialsrdquo Physics Letters A vol 349 no 1ndash4 pp 87ndash97 2006

[42] G Plante and A F Antippa ldquoAnalytic solution of theSchrodinger equation for the Coulomb-plus-linear potentialrdquoJournal of Mathematical Physics vol 46 no 6 pp 1ndash20 2005

[43] J Killingbeck ldquoPerturbation theory without wavefunctionsrdquoPhysics Letters A vol 65 no 2 pp 87ndash88 1978

[44] Y Xu S He and C-S Jia ldquoApproximate analytical solutionsof the Klein-Gordon equation with the Poschl-Teller potentialincluding the centrifugal termrdquo Physica Scripta vol 81 no 4Article ID 045001 2010

[45] R L Liboff Introductory Quantum Mechanics PearsonEducation-Addison Wesley San Francisco Calif USA 2003

[46] GChen Z-DChen andZ-M Lou ldquoBound states of theKlein-Gordon and Dirac equation for scalar and vector pseudohar-monic oscillator potentialsrdquo Chinese Physics vol 13 no 3 pp279ndash282 2004

[47] S H Dong Z Ma and G Espozito ldquoExact solutions of theSchrodinger equation with inversepowerpotentialrdquo Founda-tions of Physics Letters vol 12 no 5 pp 465ndash474 1999

[48] A Arda and R Sever ldquoEffective-mass Klein-Gordon equationfor non-PTnon-Hermitian generalizedMorse potentialrdquo Phys-ica Scripta vol 82 no 6 Article ID 065007 2010

[49] S M Ikhdair and J Abu-Hasna ldquoQuantization rule solutionto the Hulthen potential in arbitrary dimension with a newapproximate scheme for the centrifugal termrdquo Physica Scriptavol 83 no 2 Article ID 025002 2011

[50] S M Ikhdair and R Sever ldquoExact quantization rule tothe Kratzer-type potentials An application to the diatomicmoleculesrdquo Journal of Mathematical Chemistry vol 45 no 4pp 1137ndash1152 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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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

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AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

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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

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Soft MatterJournal of

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AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

6 Advances in High Energy Physics

0 1 2 3 4 5 6 70

2

4

6

8

10

12

14

16

18

20

120596L = 0

120596L = 1

120596L = 5

120596L = 8

r (au)

Ueff(r120596

L)

(a)

0 1 2 3 4 50

2

4

6

8

10

12

14

16

18

20

r (au)

120585 = 0

120585 = 1

120585 = 5

120585 = 8

Ueff(r120585)

(b)

Figure 1 The KG effective potential function for (a) 120596119871= 0 1 5 8 and with 120585 = 0 (b) 120585 = 0 1 5 8 and with 120596

119871= 0 (colour online) Here

119898119890= ℎ = 119888 = 1

120595(minus)

119899119898(119903 120601) = 119863

119899119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(12)radic(119864minus119898

1198901198882)120582ℎ21198882+(119898119890120596119871ℎ)2[1199032+((119864minus119898

1198901198882)120590((119864minus119898

1198901198882)1198892+ℎ21198882(119898119890120596119871ℎ)2))119903]

119899max

sum

119899=0

119886119899119903119899 (28)

1198861= minus1198860[119902 +

(119864 minus 1198981198901198882) 120581

2ℎ21198882(1198981015840+ 12)

] (29)

1198862= minus

1

2

1198860[119901 minus 119902

2minus

1198981198901205961198711198981015840

ℎ (1198981015840+ 1)

minus

(119864 minus 1198981198901198882) 120581

ℎ21198882(1198981015840+ 12)

(119902 +

(119864 minus 1198981198901198882) 120581

4ℎ21198882(1198981015840+ 1)

)] (30)

where119863119899119898

is the normalization constant

3 Discussions

In this section we discuss some special cases of interest fromour general solution

(i) If we set 120590 = 120581 = 0 the effective Killingbeck potentialturns to effective harmonic oscillator potential in relativisticcase as

119880eff (119903 120596119871 120585) = 1205821199032+

1198982

11989011988821205962

119871

(119864 + 1198981198901198882)

1199032

+

ℎ21198882

(119864 + 1198981198901198882)

(11989810158402minus 14)

1199032

+

2ℎ12059611987111989811989011988821198981015840

(119864 + 1198981198901198882)

120582 =

1

2

1205831205962

(31)

The bound state solutions (with the change 119899 = 2(119899119903+ 1)) are

[32 36]

1198642minus 1198982

1198901198884= 21198981198901198882ℎ1205961198711198981015840+ 4ℎ1198882(119899119903+

1198981015840+ 1

2

)

timesradic(119864 + 119898

1198901198882) 120582

1198882

+ (119898119890120596119871)2

(32)

120595(+)

119899119903119898(119903 120601) = 119860

119899119903119898

1

radic2120587

119890119894119898120601

119903119898+120585

times 119890minus(12)radic(119864+119898

1198901198882)120582ℎ21198882+(119898119890120596119871ℎ)21199032

119899max

sum

119899=0

1198861198991199032(119899119903+1)

(33a)

1198861= 0

1198862=

1

2

1198860[

[

radic(119864 + 119898

1198901198882) 120582

ℎ21198882

+ (

119898119890120596119871

ℎ

)

2

+

1198981198901205961198711198981015840

ℎ (1198981015840+ 1)

]

]

(33b)

Advances in High Energy Physics 7

where 119860119899119903119898

is the normalization constant On the otherhand the nonrelativistic effective harmonic oscillator poten-tial

119880eff (119903 120596119871 120585) = 1205821199032+

1

2

1205962

1198711199032+

(11989810158402minus 14) ℎ

2

21198981198901199032

+ ℎ1205961198711198981015840

(34)

has bound states as [32]

1198641198991198981015840 (120596119871 120585) = ℎ120596

1198711198981015840

+ 2ℎΩ1015840(119899119903+

10038161003816100381610038161003816119898101584010038161003816100381610038161003816+ 1

2

) Ω1015840= radic120596

2

119871+ 1205962

(35)

120595(+)

119899119903119898(119903 120601) = 119862

119899119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(1205832ℎ)Ω

10158401199032

times

119899max

sum

119899=0

1198861198991199032(119899119903+1) 119899119903= 0 1 2

=

1

radic2120587

119890119894119898120601

radic21198871198981015840+1119899

(119899 + 1198981015840)

1199031198981015840

119890minus11988711990322119871(1198981015840)

119899(1198871199032)

119887 =

120583

ℎ

Ω

1015840

(36a)

1198861= 0 119886

2=

120583

2ℎ

Ω10158401198860 1198860

= 0 (36b)

(ii) Setting 120582 = 120590 = 0 in (21) the killingbeck potentialturns to the Coulomb potential case (ie 119881(119903) = minus119860119903 and119899 = 119899119903+ 1) Thus we obtain the relativistic energy formula

1198642minus 1198982

1198901198884= 21198981198901198882(119899119903+ 21198981015840) ℎ120596119871

minus

(119864 + 1198981198901198882)

2

1205812

4ℎ21198882(119899119903+ 1198981015840+ 12)

2 120581 = 119885119890

2

(37)

and the wave function

120595(+)

119899119903119898(119903 120601) = 119862

119899119903119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(119898119890120596119871)2ℎ1199032

times

119899max

sum

119899=0

119886119899119903119899119903+1

(38a)

1198861= minus

(119864 + 1198981198901198882) 120581

2ℎ21198882(1198981015840+ 12)

1198860 (38b)

1198862= minus

1

2

[

[

119898119890120596119871

ℎ

minus

1198981198901205961198711198981015840

ℎ (1198981015840+ 1)

minus

(119864 + 1198981198901198882)

2

1205812

4ℎ41198884(1198981015840+ 1) (119898

1015840+ 12)

]

]

1198860

(38c)

Overmore in the nonrelativistic limit we have

119864 = ℎ1205961198711198981015840+ ℎ (119899

119903+ 1198981015840) 120596119871minus

11989811989011988521198904

2ℎ2(119899119903+ 1198981015840+ 12)

2 (39)

and the wave function becomes

120595(+)

119899119903119898(119903 120601) = 119862

119899119903119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(1205831205961198712ℎ)1199032

119899max

sum

119899=0

119886119899119903119899119903+1

(40a)

1198861= minus

119898119890119860

ℎ2(1198981015840+ 12)

1198860 (40b)

1198862= minus

1

2

[

120583120596119871

ℎ

minus

1205831205961198711198981015840

ℎ (1198981015840+ 1)

minus

120583211988521198904

ℎ4(1198981015840+ 1) (119898

1015840+ 12)

] 1198860

(40c)

Notice that in the absence of external fields 120596119871= 120585 = 0

the problem can be solved in any dimension Thus applyingthe transformation |119898| rarr 119897 + 12 our previous results arereduced to the well-known three-dimensional bound statesolutions for the harmonic oscillator and the hydrogen-likeatoms in the Coulomb fields [40]

119864plusmn= plusmn1198981198901198882[[

[

1 +

2ℎ119888

1198981198901198882(2119899119903+ 119897 +

3

2

)radic

(119864 + 1198981198901198882) 120582

1198982

1198901198884

]]

]

12

119864plusmn= plusmn1198981198901198882[1 minus

11988521198904

4ℎ21198882(119899119903+ 119897 + 1)

2(1 +

119864

1198981198901198882)

2

]

12

(41)

respectively According to (41) the energy spectrum can befound for the scalar particle and antiparticle

In Figure 2(a) we plot the effective potential for the caseof low vibrational (119899 = 0 1 2 3) and rotational (119898 = 1)

levels for various Larmor frequencies 120596119871

= 0 1 5 8 and120585 = 0 case As shown in Figure 2(a) and (31) the effectivepotential function changes in shape as well as the bound stateenergy eigenvalues increase when 120596

119871= 8 It is shown that the

energy levels are raised when the strength of the magneticfield increases and in the absence of AB flux field It is alsoobvious that the effective potential changes gradually fromthe pure pseudo-harmonic oscillator potential which is anonmagnetic (120596

119871= 0) and AB flux (120585 = 0) fields case to

a pure harmonic oscillator type behavior in short potentialrange when the strength of the applied magnetic field isincreased to 120596

119871= 8 If we consider a strong magnetic field

case 120596119871= 8 which has the shape of pure harmonic oscillator

potential function the energy difference between adjacentenergy levels is nearly equal which is a known characteristicof the pure harmonic oscillator potential In Figure 2(b) weplot the effective potential (34) in the absence of themagneticfield and in the presence of AB flux field in the short rangeregion

In Tables 1 and 2 we show the effect of magnetic fieldand AB flux field respectively on the low vibrational 119899

8 Advances in High Energy Physics

120596L = 0

120596L = 1

120596L = 5

120596L = 8

E01E11

E21

E31

E01

E11

E21E31

E01E11

E21

E31

0 1 2 3 4 50

10

20

30

40

50

60

70

80

r (au)

Ueff(r120596

L)

(a)

0 1 2 3 4 5 6 7 80

10

20

30

40

50

60

70

80

r (au)

120585 = 0

120585 = 3

120585 = 5

120585 = 8

Ueff(r120585)

(b)

Figure 2 The Schrodinger effective potential function and corresponding bound state energy levels (119864119899119898) in low vibrational (119899 = 0 1 2 3)

and rotational (119898 = 1) levels for (a) 120596119871= 0 1 5 8 with 120585 = 0 (b) 120585 = 0 1 5 8 and with 120596

119871= 0 (color online) Here 120583 = ℎ = 1

and rotational 119898 relativistic energy states of the harmonicoscillator potential As shown in Table 1 when the magneticfield is not applied and without AB flux field (120596

119871= 0

120585 = 0) the spacing between the energy levels of the effectivepotential is narrow and decreases with increasing 119899 Butwhen the magnetic field strength increases the energy levelsof the effective potential increase and the spacings betweenstates also increase In Table 2 when the AB flux field isapplied and without magnetic field the energy states becomedegenerate for various values of 119899 and 119898 and for variousAB flux field strength values In Tables 3 and 4 we showthe effect of magnetic field and AB flux field respectivelyon the low vibrational 119899 and rotational 119898 nonrelativisticenergy states of the harmonic oscillator potential As shownin Table 3 when themagnetic field is not applied and withoutAB flux field (120596

119871= 0 120585 = 0) the energy states are equally

spaced (the pure harmonic oscillator case) But when themagnetic field strength is raised the energy levels of theeffective potential increase and the spacings between statesalso increase In Table 4 when the AB flux field is applied andwithout magnetic field the energy states become degenerateand equally spaced for various values of 119899 and 119898 and forvarious AB flux field strength values

4 Concluding Remarks

To sum up in this paper we have studied the solutionof two-dimensional KG and Schrodinger equations withthe Killingbeck potential for low vibrational and rotationalenergy levels without and with a constant magnetic fieldhaving arbitrary Larmor frequeny and AB flux field Wehave applied the wave function ansatz method for 120596

119871= 0

(with magnetic field) and 120585 = 0 (with AB flux field) toobtain analytical expressions in closed form for boundstate energies and wave functions of the spinless relativisticparticle subject to a Killingbeck interaction expressed interms of external uniform magnetic and AB flux fields inany vibrational 119899 and rotational 119898 states The above resultsshow that the problems of relativistic quantum mechanicscan be also solved exactly as in the non-relativistic onesWe considered the solution of both positive (particle) andnegative (antiparticle) KG energy states It is noticed that thesolution with equal mixture of scalar-vector potentials can beeasily reduced into the well-known Schrodinger solution fora particle with an interaction potential field and a free fieldrespectively We have also studied the bound-state solutionsfor some special cases including the non-relativistic limits(Schrodinger equation for harmonic oscillator and Coulombpotentials under external magnetic and AB flux fields) andalso the KG equation for harmonic oscillator and CoulombinteractionsThe results show that the splitting is not constantand is mainly dependent on the strength of the externalmagnetic field and AB flux field In order to show the effectof constant magnetic and AB flux fields on the vibrationaland rotational energy levels of the harmonic oscillator weplot the effective potential and corresponding energy levelswith increasing Larmor frequency and flux field for specialpotential parametersWehave seen that the effective potentialfunction and corresponding energy levels are raised in energywhen magnetic and AB flux field strengths increase Theeffective potential function behavior gradually changes fromthe pure pseudo-harmonic oscillator to a pure harmonicoscillator shape in short potential range as the magnetic andAB flux fields strengths increase

Advances in High Energy Physics 9

Table 1 The KG energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120596119871values and 120585 = 0 Here ℎ = 119888 =

119898119890= 119896 = 1

119898 119899

119864119899119898

120596119871= 0 120596

119871= 1 120596

119871= 2 120596

119871= 3 120596

119871= 4 120596

119871= 5 120596

119871= 6 120596

119871= 7 120596

119871= 8

0

0 183929 204353 240325 275615 308137 338066 365815 391752 4161661 309625 34289 403986 465113 521808 574094 622602 667941 7106092 412383 451488 526359 603138 675146 74193 804087 862301 9171583 503104 545806 630283 718744 802542 880667 953605 102205 108663

1

0 250976 316597 389307 455265 514478 568284 617802 663858 707061 362919 427262 512444 593706 668087 736315 799437 858337 9137072 458916 522885 616824 709549 795634 87516 949036 101815 1083233 545354 609216 709891 811893 90768 996689 107966 115745 123081

Table 2 The KG energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120585 values and 120596119871= 0

119898 119899

119864119899119898

120585 = 0 120585 = 1 120585 = 2 120585 = 3 120585 = 4 120585 = 5 120585 = 6 120585 = 7 120585 = 8

0

0 183929 250976 309625 362919 412383 458916 503104 545354 5859661 309625 362919 412383 458916 503104 545354 585966 625166 663132 412383 458916 503104 545354 585966 625166 6631370 70 7358923 503104 545354 585966 625166 66313 70 735892 770901 805108

1

0 250976 309625 362919 412383 458916 503104 545354 585966 6251661 362919 412383 458916 503104 545354 585966 625166 66313 702 458916 503104 545354 585966 625166 66313 70 735892 7709013 545354 585966 625166 66313 70 735892 770901 805108 838582

Table 3 The nonrelativistic energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120596119871values and 120585 = 0

119898 119899

119864119899119898

120596119871= 0 120596

119871= 1 120596

119871= 2 120596

119871= 3 120596

119871= 4 120596

119871= 5 120596

119871= 6 120596

119871= 7 120596

119871= 8

0

0 10 141421 223607 316228 412311 509902 608276 707107 8062261 30 424264 67082 948683 123693 152971 182483 212132 2418682 50 707107 111803 158114 206155 254951 304138 353553 4031133 70 989949 156525 221359 288617 356931 425793 494972 564358

1

0 20 382843 647214 932456 122462 15198 181655 211421 2412451 40 665685 109443 156491 204924 253961 303311 352843 402492 60 948528 154164 219737 287386 355941 424966 494264 5637353 80 123137 198885 282982 369848 457922 546621 635685 724981

Table 4 The nonrelativistic energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120585 values and 120596119871= 0

119898 119899

119864119899119898

120585 = 0 120585 = 1 120585 = 2 120585 = 3 120585 = 4 120585 = 5 120585 = 6 120585 = 7 120585 = 8

0

0 10 20 30 40 50 60 70 80 901 30 40 50 60 70 80 90 100 1102 50 60 70 80 90 100 110 120 1303 70 80 90 100 110 120 130 140 150

1

0 20 30 40 50 60 70 80 90 1001 40 50 60 70 80 90 100 110 1202 60 70 80 90 100 110 120 130 1403 80 90 100 110 120 130 140 150 160

10 Advances in High Energy Physics

Acknowledgments

One of the authors (Sameer M Ikhdair) would like to thankthe president of An-Najah National University (ANU) Pro-fessor Rami Alhamdallah and the acting president ProfessorMaher Natsheh for their continuous support during hispresent work at ANU The authors acknowledge the partialsupport of the Scientific and Technological Research Councilof Turkey

References

[1] WGreiner andBMullerQuantumMechanics An IntroductionSpringer Berlin Germany 1994

[2] S Ozcelik and M Simsek ldquoExact solutions of the radialSchrodinger equation for inverse-power potentialsrdquo PhysicsLetters A vol 152 no 3-4 pp 145ndash150 1991

[3] S-H Dong ldquoSchrodinger equation with the potential 119881(119903) =119860119903minus4+ 119861119903minus3+ 119862119903minus2+ 119863119903minus1rdquo Physica Scripta vol 64 no 4 pp

273ndash276 2001[4] S-H Dong and Z-Q Ma ldquoExact solutions to the Schrodinger

equation for the potential 119881(119903) = 1198861199032+ 119887119903minus4

+ 119888119903minus6 in two

dimensionsrdquo Journal of Physics A vol 31 no 49 pp 9855ndash98591998

[5] M S Child S-H Dong and X-GWang ldquoQuantum states of asextic potential hidden symmetry and quantum monodromyrdquoJournal of Physics A vol 33 no 32 pp 5653ndash5661 2000

[6] S-H Dong ldquoExact solutions of the two-dimensional Schrodin-ger equation with certain central potentialsrdquo InternationalJournal of Theoretical Physics vol 39 no 4 pp 1119ndash1128 2000

[7] S-H Dong ldquoA new approach to the relativistic schrodingerequation with central potential Ansatz methodrdquo InternationalJournal of Theoretical Physics vol 40 no 2 pp 559ndash567 2001

[8] S-H Dong ldquoOn the solutions of the Schrodinger equation withsome anharmonic potentials wave function ansatzrdquo PhysicaScripta vol 65 no 4 pp 289ndash295 2002

[9] S Flugge Practical Quantum Mechanics 1 Springer BerlinGermany 1994

[10] X-Q Zhao C-S Jia and Q-B Yang ldquoBound states of rel-ativistic particles in the generalized symmetrical double-wellpotentialrdquo Physics Letters A vol 337 no 3 pp 189ndash196 2005

[11] S M Ikhdair ldquoExact solution of Dirac equation with chargedharmonic oscillator in electric field bound statesrdquo Journal ofModern Physics vol 3 no 2 pp 170ndash179 2012

[12] W C Qiang ldquoBound states of the Klein-Gordon equation forring-shaped Kratzer-type potentialrdquo Chinese Physics vol 13 no5 pp 575ndash578 2004

[13] S Ikhdair and R Sever ldquoExact polynomial eigensolutions ofthe Schrodinger equation for the pseudoharmonic potentialrdquoJournal of Molecular Structure vol 806 no 1ndash3 pp 155ndash1582007

[14] F-L Lu C-Y Chen and D-S Sun ldquoBound states of Klein-Gordon equation for double ring-shaped oscillator scalar andvector potentialsrdquo Chinese Physics vol 14 no 3 pp 463ndash4672005

[15] S M Ikhdair and R Sever ldquoExact solutions of the D-dimensional Schrodinger equation for a ring-shaped pseudo-harmonic potentialrdquo Central European Journal of Physics vol 6no 3 pp 685ndash696 2008

[16] S M Ikhdair and R Sever ldquoExact bound states of the D-dimensional Klein-Gordon equation with equal scalar and

vector ring-shaped pseudoharmonic potentialrdquo InternationalJournal of Modern Physics C vol 19 no 9 pp 1425ndash1442 2008

[17] S M Ikhdair and R Sever ldquoExactly solvable effective massd-dimensional schrodinger equation for pseudoharmonic andmodified kratzer problemsrdquo International Journal of ModernPhysics C vol 20 no 3 pp 361ndash372 2009

[18] R-CWang andC-YWong ldquoFinite-size effect in the Schwingerparticle-productionmechanismrdquo Physical ReviewD vol 38 no1 pp 348ndash359 1988

[19] J N Ginocchio ldquoA relativistic symmetry in nucleirdquo PhysicsReport vol 315 no 1ndash4 pp 231ndash240 1999

[20] A Bohr I Hamarnoto and B R Motelson ldquoPseudospln inrotating nuclear potentialsrdquo Physica Scripta vol 26 no 4 pp267ndash272 1982

[21] J Dudek W Nazarewicz Z Szymanski and G A LeanderldquoAbundance and systematics of nuclear superdeformed statesrelation to the pseudospin and pseudo-SU(3) symmetriesrdquoPhysical Review Letters vol 59 no 13 pp 1405ndash1408 1987

[22] A ArimaM Harvey and K Shimizu ldquoPseudo LS coupling andpseudo SU3 coupling schemesrdquo Physics Letters B vol 30 no 8pp 517ndash522 1969

[23] K T Hecht and A Adler ldquoGeneralized seniority for favored J= 0 pairs in mixed configurationsrdquo Nuclear Physics A vol 137

no 1 pp 129ndash143 1969[24] P R Page T Goldman and J N Ginocchio ldquoRelativistic sym-

metry suppresses quark spin-orbit splittingrdquo Physical ReviewLetters vol 86 no 2 pp 204ndash207 2001

[25] S Weinberg The Quantum Theory of Fields vol 1 CambridgeUniversity Press Cambridge UK 1995

[26] D Boyanovsky C Destri and H J de Vega ldquoApproach tothermalization in the classical 1205934 theory in 1 + 1 dimensionsenergy cascades and universal scalingrdquo Physical Review D vol69 no 4 Article ID 045003 2004

[27] R Khordad ldquoEffects of magnetic field and geometrical size onthe interband light absorption in a quantumpseudodot systemrdquoSolid State Sciences vol 12 no 7 pp 1253ndash1256 2010

[28] R Khordad ldquoSimultaneous effects of temperature and pressureon the donor binding energy in a V-groove quantum wirerdquoSuperlattices and Microstructures vol 47 no 3 pp 422ndash4312010

[29] A Cetin ldquoA quantum pseudodot system with a two-dimensional pseudoharmonic potentialrdquo Physics LettersA vol 372 no 21 pp 3852ndash3856 2008

[30] S M Ikhdair M Hamzavi and R Sever ldquoSpectra of cylindricalquantum dots the effect of electrical and magnetic fieldstogether with AB flux fieldrdquo Physica B vol 407 pp 4523ndash45292012

[31] S M Ikhdair and M Hamzavi ldquoA quantum pseudodot systemwith two-dimensional pseudoharmonic oscillator in externalmagnetic and Aharonov-Bohm fieldsrdquo Physica B vol 407 pp4198ndash4207 2012

[32] S M Ikhdair and M Hamzavi ldquoEffects of extermal fieldson a two-dimensional Klein-Gordon particle under pseudo-harmonic oscillator interactionrdquo Chinese Physics B vol 21 no11 pp 110302ndash110306 2012

[33] M Aygun O Bayrak I Boztosun and Y Sahin ldquoThe energyeigenvalues of the Kratzer potential in the presence of amagnetic fieldrdquo European Physical Journal D vol 66 no 2Article ID 20319 2012

[34] S M Ikhdair andM Hamzavi ldquoSpectral properties of quantumdots influenced by a confining potential modelrdquo Physica B vol407 pp 4797ndash4803 2012

Advances in High Energy Physics 11

[35] M Hamzavi and A A Rajabi ldquoSolution of Dirac equation withKillingbeck potential by using wave function ansatz methodunder spin symmetry limitrdquo Communications in TheoreticalPhysics vol 55 no 1 pp 35ndash37 2011

[36] M Hamzavi S M Ikhdair and K E Thylwe ldquoPseudospinsymmetry in the relativistic Killingbeck potential Quasi-exactsolutionrdquo Zeitschrift fur Naturforschung A vol 67 pp 567ndash5712012

[37] R Kumar and F Chand ldquoSeries solution to the N-dimensionalradial Schrodinger equation for the quark-antiquark interac-tionrdquo Physica Scripta vol 85 no 5 pp 055008ndash055004 2012

[38] R Kumar and F Chand ldquoReply to comments on seriessolution to the N-dimensionalradial Schrodinger equation forthe quark-antiquark interactionrdquo Physica Scripta vol 86 no 2p 027001 2012

[39] S M Ikhdair ldquoA scalar charged particle in presence of magneticandAharonov-Bohmfield plus scalar-vectorKillingbeck poten-tialsrdquo Few-Body Systems vol 54 no 11 pp 1987ndash1995 2013

[40] W Greiner Relativistic Quantum Mechanics Wave EquationsSpringer Berlin Germany 2000

[41] A D Alhaidari H Bahlouli and A Al-Hasan ldquoDirac andKlein-Gordon equations with equal scalar and vector poten-tialsrdquo Physics Letters A vol 349 no 1ndash4 pp 87ndash97 2006

[42] G Plante and A F Antippa ldquoAnalytic solution of theSchrodinger equation for the Coulomb-plus-linear potentialrdquoJournal of Mathematical Physics vol 46 no 6 pp 1ndash20 2005

[43] J Killingbeck ldquoPerturbation theory without wavefunctionsrdquoPhysics Letters A vol 65 no 2 pp 87ndash88 1978

[44] Y Xu S He and C-S Jia ldquoApproximate analytical solutionsof the Klein-Gordon equation with the Poschl-Teller potentialincluding the centrifugal termrdquo Physica Scripta vol 81 no 4Article ID 045001 2010

[45] R L Liboff Introductory Quantum Mechanics PearsonEducation-Addison Wesley San Francisco Calif USA 2003

[46] GChen Z-DChen andZ-M Lou ldquoBound states of theKlein-Gordon and Dirac equation for scalar and vector pseudohar-monic oscillator potentialsrdquo Chinese Physics vol 13 no 3 pp279ndash282 2004

[47] S H Dong Z Ma and G Espozito ldquoExact solutions of theSchrodinger equation with inversepowerpotentialrdquo Founda-tions of Physics Letters vol 12 no 5 pp 465ndash474 1999

[48] A Arda and R Sever ldquoEffective-mass Klein-Gordon equationfor non-PTnon-Hermitian generalizedMorse potentialrdquo Phys-ica Scripta vol 82 no 6 Article ID 065007 2010

[49] S M Ikhdair and J Abu-Hasna ldquoQuantization rule solutionto the Hulthen potential in arbitrary dimension with a newapproximate scheme for the centrifugal termrdquo Physica Scriptavol 83 no 2 Article ID 025002 2011

[50] S M Ikhdair and R Sever ldquoExact quantization rule tothe Kratzer-type potentials An application to the diatomicmoleculesrdquo Journal of Mathematical Chemistry vol 45 no 4pp 1137ndash1152 2009

Submit your manuscripts athttpwwwhindawicom

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

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GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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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

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AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

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Journal of

Biophysics

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ThermodynamicsJournal of

Advances in High Energy Physics 7

where 119860119899119903119898

is the normalization constant On the otherhand the nonrelativistic effective harmonic oscillator poten-tial

119880eff (119903 120596119871 120585) = 1205821199032+

1

2

1205962

1198711199032+

(11989810158402minus 14) ℎ

2

21198981198901199032

+ ℎ1205961198711198981015840

(34)

has bound states as [32]

1198641198991198981015840 (120596119871 120585) = ℎ120596

1198711198981015840

+ 2ℎΩ1015840(119899119903+

10038161003816100381610038161003816119898101584010038161003816100381610038161003816+ 1

2

) Ω1015840= radic120596

2

119871+ 1205962

(35)

120595(+)

119899119903119898(119903 120601) = 119862

119899119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(1205832ℎ)Ω

10158401199032

times

119899max

sum

119899=0

1198861198991199032(119899119903+1) 119899119903= 0 1 2

=

1

radic2120587

119890119894119898120601

radic21198871198981015840+1119899

(119899 + 1198981015840)

1199031198981015840

119890minus11988711990322119871(1198981015840)

119899(1198871199032)

119887 =

120583

ℎ

Ω

1015840

(36a)

1198861= 0 119886

2=

120583

2ℎ

Ω10158401198860 1198860

= 0 (36b)

(ii) Setting 120582 = 120590 = 0 in (21) the killingbeck potentialturns to the Coulomb potential case (ie 119881(119903) = minus119860119903 and119899 = 119899119903+ 1) Thus we obtain the relativistic energy formula

1198642minus 1198982

1198901198884= 21198981198901198882(119899119903+ 21198981015840) ℎ120596119871

minus

(119864 + 1198981198901198882)

2

1205812

4ℎ21198882(119899119903+ 1198981015840+ 12)

2 120581 = 119885119890

2

(37)

and the wave function

120595(+)

119899119903119898(119903 120601) = 119862

119899119903119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(119898119890120596119871)2ℎ1199032

times

119899max

sum

119899=0

119886119899119903119899119903+1

(38a)

1198861= minus

(119864 + 1198981198901198882) 120581

2ℎ21198882(1198981015840+ 12)

1198860 (38b)

1198862= minus

1

2

[

[

119898119890120596119871

ℎ

minus

1198981198901205961198711198981015840

ℎ (1198981015840+ 1)

minus

(119864 + 1198981198901198882)

2

1205812

4ℎ41198884(1198981015840+ 1) (119898

1015840+ 12)

]

]

1198860

(38c)

Overmore in the nonrelativistic limit we have

119864 = ℎ1205961198711198981015840+ ℎ (119899

119903+ 1198981015840) 120596119871minus

11989811989011988521198904

2ℎ2(119899119903+ 1198981015840+ 12)

2 (39)

and the wave function becomes

120595(+)

119899119903119898(119903 120601) = 119862

119899119903119898

1

radic2120587

119890119894119898120601

119903119898+120585

119890minus(1205831205961198712ℎ)1199032

119899max

sum

119899=0

119886119899119903119899119903+1

(40a)

1198861= minus

119898119890119860

ℎ2(1198981015840+ 12)

1198860 (40b)

1198862= minus

1

2

[

120583120596119871

ℎ

minus

1205831205961198711198981015840

ℎ (1198981015840+ 1)

minus

120583211988521198904

ℎ4(1198981015840+ 1) (119898

1015840+ 12)

] 1198860

(40c)

Notice that in the absence of external fields 120596119871= 120585 = 0

the problem can be solved in any dimension Thus applyingthe transformation |119898| rarr 119897 + 12 our previous results arereduced to the well-known three-dimensional bound statesolutions for the harmonic oscillator and the hydrogen-likeatoms in the Coulomb fields [40]

119864plusmn= plusmn1198981198901198882[[

[

1 +

2ℎ119888

1198981198901198882(2119899119903+ 119897 +

3

2

)radic

(119864 + 1198981198901198882) 120582

1198982

1198901198884

]]

]

12

119864plusmn= plusmn1198981198901198882[1 minus

11988521198904

4ℎ21198882(119899119903+ 119897 + 1)

2(1 +

119864

1198981198901198882)

2

]

12

(41)

respectively According to (41) the energy spectrum can befound for the scalar particle and antiparticle

In Figure 2(a) we plot the effective potential for the caseof low vibrational (119899 = 0 1 2 3) and rotational (119898 = 1)

levels for various Larmor frequencies 120596119871

= 0 1 5 8 and120585 = 0 case As shown in Figure 2(a) and (31) the effectivepotential function changes in shape as well as the bound stateenergy eigenvalues increase when 120596

119871= 8 It is shown that the

energy levels are raised when the strength of the magneticfield increases and in the absence of AB flux field It is alsoobvious that the effective potential changes gradually fromthe pure pseudo-harmonic oscillator potential which is anonmagnetic (120596

119871= 0) and AB flux (120585 = 0) fields case to

a pure harmonic oscillator type behavior in short potentialrange when the strength of the applied magnetic field isincreased to 120596

119871= 8 If we consider a strong magnetic field

case 120596119871= 8 which has the shape of pure harmonic oscillator

potential function the energy difference between adjacentenergy levels is nearly equal which is a known characteristicof the pure harmonic oscillator potential In Figure 2(b) weplot the effective potential (34) in the absence of themagneticfield and in the presence of AB flux field in the short rangeregion

In Tables 1 and 2 we show the effect of magnetic fieldand AB flux field respectively on the low vibrational 119899

8 Advances in High Energy Physics

120596L = 0

120596L = 1

120596L = 5

120596L = 8

E01E11

E21

E31

E01

E11

E21E31

E01E11

E21

E31

0 1 2 3 4 50

10

20

30

40

50

60

70

80

r (au)

Ueff(r120596

L)

(a)

0 1 2 3 4 5 6 7 80

10

20

30

40

50

60

70

80

r (au)

120585 = 0

120585 = 3

120585 = 5

120585 = 8

Ueff(r120585)

(b)

Figure 2 The Schrodinger effective potential function and corresponding bound state energy levels (119864119899119898) in low vibrational (119899 = 0 1 2 3)

and rotational (119898 = 1) levels for (a) 120596119871= 0 1 5 8 with 120585 = 0 (b) 120585 = 0 1 5 8 and with 120596

119871= 0 (color online) Here 120583 = ℎ = 1

and rotational 119898 relativistic energy states of the harmonicoscillator potential As shown in Table 1 when the magneticfield is not applied and without AB flux field (120596

119871= 0

120585 = 0) the spacing between the energy levels of the effectivepotential is narrow and decreases with increasing 119899 Butwhen the magnetic field strength increases the energy levelsof the effective potential increase and the spacings betweenstates also increase In Table 2 when the AB flux field isapplied and without magnetic field the energy states becomedegenerate for various values of 119899 and 119898 and for variousAB flux field strength values In Tables 3 and 4 we showthe effect of magnetic field and AB flux field respectivelyon the low vibrational 119899 and rotational 119898 nonrelativisticenergy states of the harmonic oscillator potential As shownin Table 3 when themagnetic field is not applied and withoutAB flux field (120596

119871= 0 120585 = 0) the energy states are equally

spaced (the pure harmonic oscillator case) But when themagnetic field strength is raised the energy levels of theeffective potential increase and the spacings between statesalso increase In Table 4 when the AB flux field is applied andwithout magnetic field the energy states become degenerateand equally spaced for various values of 119899 and 119898 and forvarious AB flux field strength values

4 Concluding Remarks

To sum up in this paper we have studied the solutionof two-dimensional KG and Schrodinger equations withthe Killingbeck potential for low vibrational and rotationalenergy levels without and with a constant magnetic fieldhaving arbitrary Larmor frequeny and AB flux field Wehave applied the wave function ansatz method for 120596

119871= 0

(with magnetic field) and 120585 = 0 (with AB flux field) toobtain analytical expressions in closed form for boundstate energies and wave functions of the spinless relativisticparticle subject to a Killingbeck interaction expressed interms of external uniform magnetic and AB flux fields inany vibrational 119899 and rotational 119898 states The above resultsshow that the problems of relativistic quantum mechanicscan be also solved exactly as in the non-relativistic onesWe considered the solution of both positive (particle) andnegative (antiparticle) KG energy states It is noticed that thesolution with equal mixture of scalar-vector potentials can beeasily reduced into the well-known Schrodinger solution fora particle with an interaction potential field and a free fieldrespectively We have also studied the bound-state solutionsfor some special cases including the non-relativistic limits(Schrodinger equation for harmonic oscillator and Coulombpotentials under external magnetic and AB flux fields) andalso the KG equation for harmonic oscillator and CoulombinteractionsThe results show that the splitting is not constantand is mainly dependent on the strength of the externalmagnetic field and AB flux field In order to show the effectof constant magnetic and AB flux fields on the vibrationaland rotational energy levels of the harmonic oscillator weplot the effective potential and corresponding energy levelswith increasing Larmor frequency and flux field for specialpotential parametersWehave seen that the effective potentialfunction and corresponding energy levels are raised in energywhen magnetic and AB flux field strengths increase Theeffective potential function behavior gradually changes fromthe pure pseudo-harmonic oscillator to a pure harmonicoscillator shape in short potential range as the magnetic andAB flux fields strengths increase

Advances in High Energy Physics 9

Table 1 The KG energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120596119871values and 120585 = 0 Here ℎ = 119888 =

119898119890= 119896 = 1

119898 119899

119864119899119898

120596119871= 0 120596

119871= 1 120596

119871= 2 120596

119871= 3 120596

119871= 4 120596

119871= 5 120596

119871= 6 120596

119871= 7 120596

119871= 8

0

0 183929 204353 240325 275615 308137 338066 365815 391752 4161661 309625 34289 403986 465113 521808 574094 622602 667941 7106092 412383 451488 526359 603138 675146 74193 804087 862301 9171583 503104 545806 630283 718744 802542 880667 953605 102205 108663

1

0 250976 316597 389307 455265 514478 568284 617802 663858 707061 362919 427262 512444 593706 668087 736315 799437 858337 9137072 458916 522885 616824 709549 795634 87516 949036 101815 1083233 545354 609216 709891 811893 90768 996689 107966 115745 123081

Table 2 The KG energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120585 values and 120596119871= 0

119898 119899

119864119899119898

120585 = 0 120585 = 1 120585 = 2 120585 = 3 120585 = 4 120585 = 5 120585 = 6 120585 = 7 120585 = 8

0

0 183929 250976 309625 362919 412383 458916 503104 545354 5859661 309625 362919 412383 458916 503104 545354 585966 625166 663132 412383 458916 503104 545354 585966 625166 6631370 70 7358923 503104 545354 585966 625166 66313 70 735892 770901 805108

1

0 250976 309625 362919 412383 458916 503104 545354 585966 6251661 362919 412383 458916 503104 545354 585966 625166 66313 702 458916 503104 545354 585966 625166 66313 70 735892 7709013 545354 585966 625166 66313 70 735892 770901 805108 838582

Table 3 The nonrelativistic energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120596119871values and 120585 = 0

119898 119899

119864119899119898

120596119871= 0 120596

119871= 1 120596

119871= 2 120596

119871= 3 120596

119871= 4 120596

119871= 5 120596

119871= 6 120596

119871= 7 120596

119871= 8

0

0 10 141421 223607 316228 412311 509902 608276 707107 8062261 30 424264 67082 948683 123693 152971 182483 212132 2418682 50 707107 111803 158114 206155 254951 304138 353553 4031133 70 989949 156525 221359 288617 356931 425793 494972 564358

1

0 20 382843 647214 932456 122462 15198 181655 211421 2412451 40 665685 109443 156491 204924 253961 303311 352843 402492 60 948528 154164 219737 287386 355941 424966 494264 5637353 80 123137 198885 282982 369848 457922 546621 635685 724981

Table 4 The nonrelativistic energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120585 values and 120596119871= 0

119898 119899

119864119899119898

120585 = 0 120585 = 1 120585 = 2 120585 = 3 120585 = 4 120585 = 5 120585 = 6 120585 = 7 120585 = 8

0

0 10 20 30 40 50 60 70 80 901 30 40 50 60 70 80 90 100 1102 50 60 70 80 90 100 110 120 1303 70 80 90 100 110 120 130 140 150

1

0 20 30 40 50 60 70 80 90 1001 40 50 60 70 80 90 100 110 1202 60 70 80 90 100 110 120 130 1403 80 90 100 110 120 130 140 150 160

10 Advances in High Energy Physics

Acknowledgments

One of the authors (Sameer M Ikhdair) would like to thankthe president of An-Najah National University (ANU) Pro-fessor Rami Alhamdallah and the acting president ProfessorMaher Natsheh for their continuous support during hispresent work at ANU The authors acknowledge the partialsupport of the Scientific and Technological Research Councilof Turkey

References

[1] WGreiner andBMullerQuantumMechanics An IntroductionSpringer Berlin Germany 1994

[2] S Ozcelik and M Simsek ldquoExact solutions of the radialSchrodinger equation for inverse-power potentialsrdquo PhysicsLetters A vol 152 no 3-4 pp 145ndash150 1991

[3] S-H Dong ldquoSchrodinger equation with the potential 119881(119903) =119860119903minus4+ 119861119903minus3+ 119862119903minus2+ 119863119903minus1rdquo Physica Scripta vol 64 no 4 pp

273ndash276 2001[4] S-H Dong and Z-Q Ma ldquoExact solutions to the Schrodinger

equation for the potential 119881(119903) = 1198861199032+ 119887119903minus4

+ 119888119903minus6 in two

dimensionsrdquo Journal of Physics A vol 31 no 49 pp 9855ndash98591998

[5] M S Child S-H Dong and X-GWang ldquoQuantum states of asextic potential hidden symmetry and quantum monodromyrdquoJournal of Physics A vol 33 no 32 pp 5653ndash5661 2000

[6] S-H Dong ldquoExact solutions of the two-dimensional Schrodin-ger equation with certain central potentialsrdquo InternationalJournal of Theoretical Physics vol 39 no 4 pp 1119ndash1128 2000

[7] S-H Dong ldquoA new approach to the relativistic schrodingerequation with central potential Ansatz methodrdquo InternationalJournal of Theoretical Physics vol 40 no 2 pp 559ndash567 2001

[8] S-H Dong ldquoOn the solutions of the Schrodinger equation withsome anharmonic potentials wave function ansatzrdquo PhysicaScripta vol 65 no 4 pp 289ndash295 2002

[9] S Flugge Practical Quantum Mechanics 1 Springer BerlinGermany 1994

[10] X-Q Zhao C-S Jia and Q-B Yang ldquoBound states of rel-ativistic particles in the generalized symmetrical double-wellpotentialrdquo Physics Letters A vol 337 no 3 pp 189ndash196 2005

[11] S M Ikhdair ldquoExact solution of Dirac equation with chargedharmonic oscillator in electric field bound statesrdquo Journal ofModern Physics vol 3 no 2 pp 170ndash179 2012

[12] W C Qiang ldquoBound states of the Klein-Gordon equation forring-shaped Kratzer-type potentialrdquo Chinese Physics vol 13 no5 pp 575ndash578 2004

[13] S Ikhdair and R Sever ldquoExact polynomial eigensolutions ofthe Schrodinger equation for the pseudoharmonic potentialrdquoJournal of Molecular Structure vol 806 no 1ndash3 pp 155ndash1582007

[14] F-L Lu C-Y Chen and D-S Sun ldquoBound states of Klein-Gordon equation for double ring-shaped oscillator scalar andvector potentialsrdquo Chinese Physics vol 14 no 3 pp 463ndash4672005

[15] S M Ikhdair and R Sever ldquoExact solutions of the D-dimensional Schrodinger equation for a ring-shaped pseudo-harmonic potentialrdquo Central European Journal of Physics vol 6no 3 pp 685ndash696 2008

[16] S M Ikhdair and R Sever ldquoExact bound states of the D-dimensional Klein-Gordon equation with equal scalar and

vector ring-shaped pseudoharmonic potentialrdquo InternationalJournal of Modern Physics C vol 19 no 9 pp 1425ndash1442 2008

[17] S M Ikhdair and R Sever ldquoExactly solvable effective massd-dimensional schrodinger equation for pseudoharmonic andmodified kratzer problemsrdquo International Journal of ModernPhysics C vol 20 no 3 pp 361ndash372 2009

[18] R-CWang andC-YWong ldquoFinite-size effect in the Schwingerparticle-productionmechanismrdquo Physical ReviewD vol 38 no1 pp 348ndash359 1988

[19] J N Ginocchio ldquoA relativistic symmetry in nucleirdquo PhysicsReport vol 315 no 1ndash4 pp 231ndash240 1999

[20] A Bohr I Hamarnoto and B R Motelson ldquoPseudospln inrotating nuclear potentialsrdquo Physica Scripta vol 26 no 4 pp267ndash272 1982

[21] J Dudek W Nazarewicz Z Szymanski and G A LeanderldquoAbundance and systematics of nuclear superdeformed statesrelation to the pseudospin and pseudo-SU(3) symmetriesrdquoPhysical Review Letters vol 59 no 13 pp 1405ndash1408 1987

[22] A ArimaM Harvey and K Shimizu ldquoPseudo LS coupling andpseudo SU3 coupling schemesrdquo Physics Letters B vol 30 no 8pp 517ndash522 1969

[23] K T Hecht and A Adler ldquoGeneralized seniority for favored J= 0 pairs in mixed configurationsrdquo Nuclear Physics A vol 137

no 1 pp 129ndash143 1969[24] P R Page T Goldman and J N Ginocchio ldquoRelativistic sym-

metry suppresses quark spin-orbit splittingrdquo Physical ReviewLetters vol 86 no 2 pp 204ndash207 2001

[25] S Weinberg The Quantum Theory of Fields vol 1 CambridgeUniversity Press Cambridge UK 1995

[26] D Boyanovsky C Destri and H J de Vega ldquoApproach tothermalization in the classical 1205934 theory in 1 + 1 dimensionsenergy cascades and universal scalingrdquo Physical Review D vol69 no 4 Article ID 045003 2004

[27] R Khordad ldquoEffects of magnetic field and geometrical size onthe interband light absorption in a quantumpseudodot systemrdquoSolid State Sciences vol 12 no 7 pp 1253ndash1256 2010

[28] R Khordad ldquoSimultaneous effects of temperature and pressureon the donor binding energy in a V-groove quantum wirerdquoSuperlattices and Microstructures vol 47 no 3 pp 422ndash4312010

[29] A Cetin ldquoA quantum pseudodot system with a two-dimensional pseudoharmonic potentialrdquo Physics LettersA vol 372 no 21 pp 3852ndash3856 2008

[30] S M Ikhdair M Hamzavi and R Sever ldquoSpectra of cylindricalquantum dots the effect of electrical and magnetic fieldstogether with AB flux fieldrdquo Physica B vol 407 pp 4523ndash45292012

[31] S M Ikhdair and M Hamzavi ldquoA quantum pseudodot systemwith two-dimensional pseudoharmonic oscillator in externalmagnetic and Aharonov-Bohm fieldsrdquo Physica B vol 407 pp4198ndash4207 2012

[32] S M Ikhdair and M Hamzavi ldquoEffects of extermal fieldson a two-dimensional Klein-Gordon particle under pseudo-harmonic oscillator interactionrdquo Chinese Physics B vol 21 no11 pp 110302ndash110306 2012

[33] M Aygun O Bayrak I Boztosun and Y Sahin ldquoThe energyeigenvalues of the Kratzer potential in the presence of amagnetic fieldrdquo European Physical Journal D vol 66 no 2Article ID 20319 2012

[34] S M Ikhdair andM Hamzavi ldquoSpectral properties of quantumdots influenced by a confining potential modelrdquo Physica B vol407 pp 4797ndash4803 2012

Advances in High Energy Physics 11

[35] M Hamzavi and A A Rajabi ldquoSolution of Dirac equation withKillingbeck potential by using wave function ansatz methodunder spin symmetry limitrdquo Communications in TheoreticalPhysics vol 55 no 1 pp 35ndash37 2011

[36] M Hamzavi S M Ikhdair and K E Thylwe ldquoPseudospinsymmetry in the relativistic Killingbeck potential Quasi-exactsolutionrdquo Zeitschrift fur Naturforschung A vol 67 pp 567ndash5712012

[37] R Kumar and F Chand ldquoSeries solution to the N-dimensionalradial Schrodinger equation for the quark-antiquark interac-tionrdquo Physica Scripta vol 85 no 5 pp 055008ndash055004 2012

[38] R Kumar and F Chand ldquoReply to comments on seriessolution to the N-dimensionalradial Schrodinger equation forthe quark-antiquark interactionrdquo Physica Scripta vol 86 no 2p 027001 2012

[39] S M Ikhdair ldquoA scalar charged particle in presence of magneticandAharonov-Bohmfield plus scalar-vectorKillingbeck poten-tialsrdquo Few-Body Systems vol 54 no 11 pp 1987ndash1995 2013

[40] W Greiner Relativistic Quantum Mechanics Wave EquationsSpringer Berlin Germany 2000

[41] A D Alhaidari H Bahlouli and A Al-Hasan ldquoDirac andKlein-Gordon equations with equal scalar and vector poten-tialsrdquo Physics Letters A vol 349 no 1ndash4 pp 87ndash97 2006

[42] G Plante and A F Antippa ldquoAnalytic solution of theSchrodinger equation for the Coulomb-plus-linear potentialrdquoJournal of Mathematical Physics vol 46 no 6 pp 1ndash20 2005

[43] J Killingbeck ldquoPerturbation theory without wavefunctionsrdquoPhysics Letters A vol 65 no 2 pp 87ndash88 1978

[44] Y Xu S He and C-S Jia ldquoApproximate analytical solutionsof the Klein-Gordon equation with the Poschl-Teller potentialincluding the centrifugal termrdquo Physica Scripta vol 81 no 4Article ID 045001 2010

[45] R L Liboff Introductory Quantum Mechanics PearsonEducation-Addison Wesley San Francisco Calif USA 2003

[46] GChen Z-DChen andZ-M Lou ldquoBound states of theKlein-Gordon and Dirac equation for scalar and vector pseudohar-monic oscillator potentialsrdquo Chinese Physics vol 13 no 3 pp279ndash282 2004

[47] S H Dong Z Ma and G Espozito ldquoExact solutions of theSchrodinger equation with inversepowerpotentialrdquo Founda-tions of Physics Letters vol 12 no 5 pp 465ndash474 1999

[48] A Arda and R Sever ldquoEffective-mass Klein-Gordon equationfor non-PTnon-Hermitian generalizedMorse potentialrdquo Phys-ica Scripta vol 82 no 6 Article ID 065007 2010

[49] S M Ikhdair and J Abu-Hasna ldquoQuantization rule solutionto the Hulthen potential in arbitrary dimension with a newapproximate scheme for the centrifugal termrdquo Physica Scriptavol 83 no 2 Article ID 025002 2011

[50] S M Ikhdair and R Sever ldquoExact quantization rule tothe Kratzer-type potentials An application to the diatomicmoleculesrdquo Journal of Mathematical Chemistry vol 45 no 4pp 1137ndash1152 2009

Submit your manuscripts athttpwwwhindawicom

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

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AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

8 Advances in High Energy Physics

120596L = 0

120596L = 1

120596L = 5

120596L = 8

E01E11

E21

E31

E01

E11

E21E31

E01E11

E21

E31

0 1 2 3 4 50

10

20

30

40

50

60

70

80

r (au)

Ueff(r120596

L)

(a)

0 1 2 3 4 5 6 7 80

10

20

30

40

50

60

70

80

r (au)

120585 = 0

120585 = 3

120585 = 5

120585 = 8

Ueff(r120585)

(b)

Figure 2 The Schrodinger effective potential function and corresponding bound state energy levels (119864119899119898) in low vibrational (119899 = 0 1 2 3)

and rotational (119898 = 1) levels for (a) 120596119871= 0 1 5 8 with 120585 = 0 (b) 120585 = 0 1 5 8 and with 120596

119871= 0 (color online) Here 120583 = ℎ = 1

and rotational 119898 relativistic energy states of the harmonicoscillator potential As shown in Table 1 when the magneticfield is not applied and without AB flux field (120596

119871= 0

120585 = 0) the spacing between the energy levels of the effectivepotential is narrow and decreases with increasing 119899 Butwhen the magnetic field strength increases the energy levelsof the effective potential increase and the spacings betweenstates also increase In Table 2 when the AB flux field isapplied and without magnetic field the energy states becomedegenerate for various values of 119899 and 119898 and for variousAB flux field strength values In Tables 3 and 4 we showthe effect of magnetic field and AB flux field respectivelyon the low vibrational 119899 and rotational 119898 nonrelativisticenergy states of the harmonic oscillator potential As shownin Table 3 when themagnetic field is not applied and withoutAB flux field (120596

119871= 0 120585 = 0) the energy states are equally

spaced (the pure harmonic oscillator case) But when themagnetic field strength is raised the energy levels of theeffective potential increase and the spacings between statesalso increase In Table 4 when the AB flux field is applied andwithout magnetic field the energy states become degenerateand equally spaced for various values of 119899 and 119898 and forvarious AB flux field strength values

4 Concluding Remarks

To sum up in this paper we have studied the solutionof two-dimensional KG and Schrodinger equations withthe Killingbeck potential for low vibrational and rotationalenergy levels without and with a constant magnetic fieldhaving arbitrary Larmor frequeny and AB flux field Wehave applied the wave function ansatz method for 120596

119871= 0

(with magnetic field) and 120585 = 0 (with AB flux field) toobtain analytical expressions in closed form for boundstate energies and wave functions of the spinless relativisticparticle subject to a Killingbeck interaction expressed interms of external uniform magnetic and AB flux fields inany vibrational 119899 and rotational 119898 states The above resultsshow that the problems of relativistic quantum mechanicscan be also solved exactly as in the non-relativistic onesWe considered the solution of both positive (particle) andnegative (antiparticle) KG energy states It is noticed that thesolution with equal mixture of scalar-vector potentials can beeasily reduced into the well-known Schrodinger solution fora particle with an interaction potential field and a free fieldrespectively We have also studied the bound-state solutionsfor some special cases including the non-relativistic limits(Schrodinger equation for harmonic oscillator and Coulombpotentials under external magnetic and AB flux fields) andalso the KG equation for harmonic oscillator and CoulombinteractionsThe results show that the splitting is not constantand is mainly dependent on the strength of the externalmagnetic field and AB flux field In order to show the effectof constant magnetic and AB flux fields on the vibrationaland rotational energy levels of the harmonic oscillator weplot the effective potential and corresponding energy levelswith increasing Larmor frequency and flux field for specialpotential parametersWehave seen that the effective potentialfunction and corresponding energy levels are raised in energywhen magnetic and AB flux field strengths increase Theeffective potential function behavior gradually changes fromthe pure pseudo-harmonic oscillator to a pure harmonicoscillator shape in short potential range as the magnetic andAB flux fields strengths increase

Advances in High Energy Physics 9

Table 1 The KG energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120596119871values and 120585 = 0 Here ℎ = 119888 =

119898119890= 119896 = 1

119898 119899

119864119899119898

120596119871= 0 120596

119871= 1 120596

119871= 2 120596

119871= 3 120596

119871= 4 120596

119871= 5 120596

119871= 6 120596

119871= 7 120596

119871= 8

0

0 183929 204353 240325 275615 308137 338066 365815 391752 4161661 309625 34289 403986 465113 521808 574094 622602 667941 7106092 412383 451488 526359 603138 675146 74193 804087 862301 9171583 503104 545806 630283 718744 802542 880667 953605 102205 108663

1

0 250976 316597 389307 455265 514478 568284 617802 663858 707061 362919 427262 512444 593706 668087 736315 799437 858337 9137072 458916 522885 616824 709549 795634 87516 949036 101815 1083233 545354 609216 709891 811893 90768 996689 107966 115745 123081

Table 2 The KG energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120585 values and 120596119871= 0

119898 119899

119864119899119898

120585 = 0 120585 = 1 120585 = 2 120585 = 3 120585 = 4 120585 = 5 120585 = 6 120585 = 7 120585 = 8

0

0 183929 250976 309625 362919 412383 458916 503104 545354 5859661 309625 362919 412383 458916 503104 545354 585966 625166 663132 412383 458916 503104 545354 585966 625166 6631370 70 7358923 503104 545354 585966 625166 66313 70 735892 770901 805108

1

0 250976 309625 362919 412383 458916 503104 545354 585966 6251661 362919 412383 458916 503104 545354 585966 625166 66313 702 458916 503104 545354 585966 625166 66313 70 735892 7709013 545354 585966 625166 66313 70 735892 770901 805108 838582

Table 3 The nonrelativistic energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120596119871values and 120585 = 0

119898 119899

119864119899119898

120596119871= 0 120596

119871= 1 120596

119871= 2 120596

119871= 3 120596

119871= 4 120596

119871= 5 120596

119871= 6 120596

119871= 7 120596

119871= 8

0

0 10 141421 223607 316228 412311 509902 608276 707107 8062261 30 424264 67082 948683 123693 152971 182483 212132 2418682 50 707107 111803 158114 206155 254951 304138 353553 4031133 70 989949 156525 221359 288617 356931 425793 494972 564358

1

0 20 382843 647214 932456 122462 15198 181655 211421 2412451 40 665685 109443 156491 204924 253961 303311 352843 402492 60 948528 154164 219737 287386 355941 424966 494264 5637353 80 123137 198885 282982 369848 457922 546621 635685 724981

Table 4 The nonrelativistic energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120585 values and 120596119871= 0

119898 119899

119864119899119898

120585 = 0 120585 = 1 120585 = 2 120585 = 3 120585 = 4 120585 = 5 120585 = 6 120585 = 7 120585 = 8

0

0 10 20 30 40 50 60 70 80 901 30 40 50 60 70 80 90 100 1102 50 60 70 80 90 100 110 120 1303 70 80 90 100 110 120 130 140 150

1

0 20 30 40 50 60 70 80 90 1001 40 50 60 70 80 90 100 110 1202 60 70 80 90 100 110 120 130 1403 80 90 100 110 120 130 140 150 160

10 Advances in High Energy Physics

Acknowledgments

One of the authors (Sameer M Ikhdair) would like to thankthe president of An-Najah National University (ANU) Pro-fessor Rami Alhamdallah and the acting president ProfessorMaher Natsheh for their continuous support during hispresent work at ANU The authors acknowledge the partialsupport of the Scientific and Technological Research Councilof Turkey

References

[1] WGreiner andBMullerQuantumMechanics An IntroductionSpringer Berlin Germany 1994

[2] S Ozcelik and M Simsek ldquoExact solutions of the radialSchrodinger equation for inverse-power potentialsrdquo PhysicsLetters A vol 152 no 3-4 pp 145ndash150 1991

[3] S-H Dong ldquoSchrodinger equation with the potential 119881(119903) =119860119903minus4+ 119861119903minus3+ 119862119903minus2+ 119863119903minus1rdquo Physica Scripta vol 64 no 4 pp

273ndash276 2001[4] S-H Dong and Z-Q Ma ldquoExact solutions to the Schrodinger

equation for the potential 119881(119903) = 1198861199032+ 119887119903minus4

+ 119888119903minus6 in two

dimensionsrdquo Journal of Physics A vol 31 no 49 pp 9855ndash98591998

[5] M S Child S-H Dong and X-GWang ldquoQuantum states of asextic potential hidden symmetry and quantum monodromyrdquoJournal of Physics A vol 33 no 32 pp 5653ndash5661 2000

[6] S-H Dong ldquoExact solutions of the two-dimensional Schrodin-ger equation with certain central potentialsrdquo InternationalJournal of Theoretical Physics vol 39 no 4 pp 1119ndash1128 2000

[7] S-H Dong ldquoA new approach to the relativistic schrodingerequation with central potential Ansatz methodrdquo InternationalJournal of Theoretical Physics vol 40 no 2 pp 559ndash567 2001

[8] S-H Dong ldquoOn the solutions of the Schrodinger equation withsome anharmonic potentials wave function ansatzrdquo PhysicaScripta vol 65 no 4 pp 289ndash295 2002

[9] S Flugge Practical Quantum Mechanics 1 Springer BerlinGermany 1994

[10] X-Q Zhao C-S Jia and Q-B Yang ldquoBound states of rel-ativistic particles in the generalized symmetrical double-wellpotentialrdquo Physics Letters A vol 337 no 3 pp 189ndash196 2005

[11] S M Ikhdair ldquoExact solution of Dirac equation with chargedharmonic oscillator in electric field bound statesrdquo Journal ofModern Physics vol 3 no 2 pp 170ndash179 2012

[12] W C Qiang ldquoBound states of the Klein-Gordon equation forring-shaped Kratzer-type potentialrdquo Chinese Physics vol 13 no5 pp 575ndash578 2004

[13] S Ikhdair and R Sever ldquoExact polynomial eigensolutions ofthe Schrodinger equation for the pseudoharmonic potentialrdquoJournal of Molecular Structure vol 806 no 1ndash3 pp 155ndash1582007

[14] F-L Lu C-Y Chen and D-S Sun ldquoBound states of Klein-Gordon equation for double ring-shaped oscillator scalar andvector potentialsrdquo Chinese Physics vol 14 no 3 pp 463ndash4672005

[15] S M Ikhdair and R Sever ldquoExact solutions of the D-dimensional Schrodinger equation for a ring-shaped pseudo-harmonic potentialrdquo Central European Journal of Physics vol 6no 3 pp 685ndash696 2008

[16] S M Ikhdair and R Sever ldquoExact bound states of the D-dimensional Klein-Gordon equation with equal scalar and

vector ring-shaped pseudoharmonic potentialrdquo InternationalJournal of Modern Physics C vol 19 no 9 pp 1425ndash1442 2008

[17] S M Ikhdair and R Sever ldquoExactly solvable effective massd-dimensional schrodinger equation for pseudoharmonic andmodified kratzer problemsrdquo International Journal of ModernPhysics C vol 20 no 3 pp 361ndash372 2009

[18] R-CWang andC-YWong ldquoFinite-size effect in the Schwingerparticle-productionmechanismrdquo Physical ReviewD vol 38 no1 pp 348ndash359 1988

[19] J N Ginocchio ldquoA relativistic symmetry in nucleirdquo PhysicsReport vol 315 no 1ndash4 pp 231ndash240 1999

[20] A Bohr I Hamarnoto and B R Motelson ldquoPseudospln inrotating nuclear potentialsrdquo Physica Scripta vol 26 no 4 pp267ndash272 1982

[21] J Dudek W Nazarewicz Z Szymanski and G A LeanderldquoAbundance and systematics of nuclear superdeformed statesrelation to the pseudospin and pseudo-SU(3) symmetriesrdquoPhysical Review Letters vol 59 no 13 pp 1405ndash1408 1987

[22] A ArimaM Harvey and K Shimizu ldquoPseudo LS coupling andpseudo SU3 coupling schemesrdquo Physics Letters B vol 30 no 8pp 517ndash522 1969

[23] K T Hecht and A Adler ldquoGeneralized seniority for favored J= 0 pairs in mixed configurationsrdquo Nuclear Physics A vol 137

no 1 pp 129ndash143 1969[24] P R Page T Goldman and J N Ginocchio ldquoRelativistic sym-

metry suppresses quark spin-orbit splittingrdquo Physical ReviewLetters vol 86 no 2 pp 204ndash207 2001

[25] S Weinberg The Quantum Theory of Fields vol 1 CambridgeUniversity Press Cambridge UK 1995

[26] D Boyanovsky C Destri and H J de Vega ldquoApproach tothermalization in the classical 1205934 theory in 1 + 1 dimensionsenergy cascades and universal scalingrdquo Physical Review D vol69 no 4 Article ID 045003 2004

[27] R Khordad ldquoEffects of magnetic field and geometrical size onthe interband light absorption in a quantumpseudodot systemrdquoSolid State Sciences vol 12 no 7 pp 1253ndash1256 2010

[28] R Khordad ldquoSimultaneous effects of temperature and pressureon the donor binding energy in a V-groove quantum wirerdquoSuperlattices and Microstructures vol 47 no 3 pp 422ndash4312010

[29] A Cetin ldquoA quantum pseudodot system with a two-dimensional pseudoharmonic potentialrdquo Physics LettersA vol 372 no 21 pp 3852ndash3856 2008

[30] S M Ikhdair M Hamzavi and R Sever ldquoSpectra of cylindricalquantum dots the effect of electrical and magnetic fieldstogether with AB flux fieldrdquo Physica B vol 407 pp 4523ndash45292012

[31] S M Ikhdair and M Hamzavi ldquoA quantum pseudodot systemwith two-dimensional pseudoharmonic oscillator in externalmagnetic and Aharonov-Bohm fieldsrdquo Physica B vol 407 pp4198ndash4207 2012

[32] S M Ikhdair and M Hamzavi ldquoEffects of extermal fieldson a two-dimensional Klein-Gordon particle under pseudo-harmonic oscillator interactionrdquo Chinese Physics B vol 21 no11 pp 110302ndash110306 2012

[33] M Aygun O Bayrak I Boztosun and Y Sahin ldquoThe energyeigenvalues of the Kratzer potential in the presence of amagnetic fieldrdquo European Physical Journal D vol 66 no 2Article ID 20319 2012

[34] S M Ikhdair andM Hamzavi ldquoSpectral properties of quantumdots influenced by a confining potential modelrdquo Physica B vol407 pp 4797ndash4803 2012

Advances in High Energy Physics 11

[35] M Hamzavi and A A Rajabi ldquoSolution of Dirac equation withKillingbeck potential by using wave function ansatz methodunder spin symmetry limitrdquo Communications in TheoreticalPhysics vol 55 no 1 pp 35ndash37 2011

[36] M Hamzavi S M Ikhdair and K E Thylwe ldquoPseudospinsymmetry in the relativistic Killingbeck potential Quasi-exactsolutionrdquo Zeitschrift fur Naturforschung A vol 67 pp 567ndash5712012

[37] R Kumar and F Chand ldquoSeries solution to the N-dimensionalradial Schrodinger equation for the quark-antiquark interac-tionrdquo Physica Scripta vol 85 no 5 pp 055008ndash055004 2012

[38] R Kumar and F Chand ldquoReply to comments on seriessolution to the N-dimensionalradial Schrodinger equation forthe quark-antiquark interactionrdquo Physica Scripta vol 86 no 2p 027001 2012

[39] S M Ikhdair ldquoA scalar charged particle in presence of magneticandAharonov-Bohmfield plus scalar-vectorKillingbeck poten-tialsrdquo Few-Body Systems vol 54 no 11 pp 1987ndash1995 2013

[40] W Greiner Relativistic Quantum Mechanics Wave EquationsSpringer Berlin Germany 2000

[41] A D Alhaidari H Bahlouli and A Al-Hasan ldquoDirac andKlein-Gordon equations with equal scalar and vector poten-tialsrdquo Physics Letters A vol 349 no 1ndash4 pp 87ndash97 2006

[42] G Plante and A F Antippa ldquoAnalytic solution of theSchrodinger equation for the Coulomb-plus-linear potentialrdquoJournal of Mathematical Physics vol 46 no 6 pp 1ndash20 2005

[43] J Killingbeck ldquoPerturbation theory without wavefunctionsrdquoPhysics Letters A vol 65 no 2 pp 87ndash88 1978

[44] Y Xu S He and C-S Jia ldquoApproximate analytical solutionsof the Klein-Gordon equation with the Poschl-Teller potentialincluding the centrifugal termrdquo Physica Scripta vol 81 no 4Article ID 045001 2010

[45] R L Liboff Introductory Quantum Mechanics PearsonEducation-Addison Wesley San Francisco Calif USA 2003

[46] GChen Z-DChen andZ-M Lou ldquoBound states of theKlein-Gordon and Dirac equation for scalar and vector pseudohar-monic oscillator potentialsrdquo Chinese Physics vol 13 no 3 pp279ndash282 2004

[47] S H Dong Z Ma and G Espozito ldquoExact solutions of theSchrodinger equation with inversepowerpotentialrdquo Founda-tions of Physics Letters vol 12 no 5 pp 465ndash474 1999

[48] A Arda and R Sever ldquoEffective-mass Klein-Gordon equationfor non-PTnon-Hermitian generalizedMorse potentialrdquo Phys-ica Scripta vol 82 no 6 Article ID 065007 2010

[49] S M Ikhdair and J Abu-Hasna ldquoQuantization rule solutionto the Hulthen potential in arbitrary dimension with a newapproximate scheme for the centrifugal termrdquo Physica Scriptavol 83 no 2 Article ID 025002 2011

[50] S M Ikhdair and R Sever ldquoExact quantization rule tothe Kratzer-type potentials An application to the diatomicmoleculesrdquo Journal of Mathematical Chemistry vol 45 no 4pp 1137ndash1152 2009

Submit your manuscripts athttpwwwhindawicom

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 9

Table 1 The KG energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120596119871values and 120585 = 0 Here ℎ = 119888 =

119898119890= 119896 = 1

119898 119899

119864119899119898

120596119871= 0 120596

119871= 1 120596

119871= 2 120596

119871= 3 120596

119871= 4 120596

119871= 5 120596

119871= 6 120596

119871= 7 120596

119871= 8

0

0 183929 204353 240325 275615 308137 338066 365815 391752 4161661 309625 34289 403986 465113 521808 574094 622602 667941 7106092 412383 451488 526359 603138 675146 74193 804087 862301 9171583 503104 545806 630283 718744 802542 880667 953605 102205 108663

1

0 250976 316597 389307 455265 514478 568284 617802 663858 707061 362919 427262 512444 593706 668087 736315 799437 858337 9137072 458916 522885 616824 709549 795634 87516 949036 101815 1083233 545354 609216 709891 811893 90768 996689 107966 115745 123081

Table 2 The KG energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120585 values and 120596119871= 0

119898 119899

119864119899119898

120585 = 0 120585 = 1 120585 = 2 120585 = 3 120585 = 4 120585 = 5 120585 = 6 120585 = 7 120585 = 8

0

0 183929 250976 309625 362919 412383 458916 503104 545354 5859661 309625 362919 412383 458916 503104 545354 585966 625166 663132 412383 458916 503104 545354 585966 625166 6631370 70 7358923 503104 545354 585966 625166 66313 70 735892 770901 805108

1

0 250976 309625 362919 412383 458916 503104 545354 585966 6251661 362919 412383 458916 503104 545354 585966 625166 66313 702 458916 503104 545354 585966 625166 66313 70 735892 7709013 545354 585966 625166 66313 70 735892 770901 805108 838582

Table 3 The nonrelativistic energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120596119871values and 120585 = 0

119898 119899

119864119899119898

120596119871= 0 120596

119871= 1 120596

119871= 2 120596

119871= 3 120596

119871= 4 120596

119871= 5 120596

119871= 6 120596

119871= 7 120596

119871= 8

0

0 10 141421 223607 316228 412311 509902 608276 707107 8062261 30 424264 67082 948683 123693 152971 182483 212132 2418682 50 707107 111803 158114 206155 254951 304138 353553 4031133 70 989949 156525 221359 288617 356931 425793 494972 564358

1

0 20 382843 647214 932456 122462 15198 181655 211421 2412451 40 665685 109443 156491 204924 253961 303311 352843 402492 60 948528 154164 219737 287386 355941 424966 494264 5637353 80 123137 198885 282982 369848 457922 546621 635685 724981

Table 4 The nonrelativistic energy eigenvalues (119864119899119898

in atomic units) of the harmonic oscillator potential for various 120585 values and 120596119871= 0

119898 119899

119864119899119898

120585 = 0 120585 = 1 120585 = 2 120585 = 3 120585 = 4 120585 = 5 120585 = 6 120585 = 7 120585 = 8

0

0 10 20 30 40 50 60 70 80 901 30 40 50 60 70 80 90 100 1102 50 60 70 80 90 100 110 120 1303 70 80 90 100 110 120 130 140 150

1

0 20 30 40 50 60 70 80 90 1001 40 50 60 70 80 90 100 110 1202 60 70 80 90 100 110 120 130 1403 80 90 100 110 120 130 140 150 160

10 Advances in High Energy Physics

Acknowledgments

One of the authors (Sameer M Ikhdair) would like to thankthe president of An-Najah National University (ANU) Pro-fessor Rami Alhamdallah and the acting president ProfessorMaher Natsheh for their continuous support during hispresent work at ANU The authors acknowledge the partialsupport of the Scientific and Technological Research Councilof Turkey

References

[1] WGreiner andBMullerQuantumMechanics An IntroductionSpringer Berlin Germany 1994

[2] S Ozcelik and M Simsek ldquoExact solutions of the radialSchrodinger equation for inverse-power potentialsrdquo PhysicsLetters A vol 152 no 3-4 pp 145ndash150 1991

[3] S-H Dong ldquoSchrodinger equation with the potential 119881(119903) =119860119903minus4+ 119861119903minus3+ 119862119903minus2+ 119863119903minus1rdquo Physica Scripta vol 64 no 4 pp

273ndash276 2001[4] S-H Dong and Z-Q Ma ldquoExact solutions to the Schrodinger

equation for the potential 119881(119903) = 1198861199032+ 119887119903minus4

+ 119888119903minus6 in two

dimensionsrdquo Journal of Physics A vol 31 no 49 pp 9855ndash98591998

[5] M S Child S-H Dong and X-GWang ldquoQuantum states of asextic potential hidden symmetry and quantum monodromyrdquoJournal of Physics A vol 33 no 32 pp 5653ndash5661 2000

[6] S-H Dong ldquoExact solutions of the two-dimensional Schrodin-ger equation with certain central potentialsrdquo InternationalJournal of Theoretical Physics vol 39 no 4 pp 1119ndash1128 2000

[7] S-H Dong ldquoA new approach to the relativistic schrodingerequation with central potential Ansatz methodrdquo InternationalJournal of Theoretical Physics vol 40 no 2 pp 559ndash567 2001

[8] S-H Dong ldquoOn the solutions of the Schrodinger equation withsome anharmonic potentials wave function ansatzrdquo PhysicaScripta vol 65 no 4 pp 289ndash295 2002

[9] S Flugge Practical Quantum Mechanics 1 Springer BerlinGermany 1994

[10] X-Q Zhao C-S Jia and Q-B Yang ldquoBound states of rel-ativistic particles in the generalized symmetrical double-wellpotentialrdquo Physics Letters A vol 337 no 3 pp 189ndash196 2005

[11] S M Ikhdair ldquoExact solution of Dirac equation with chargedharmonic oscillator in electric field bound statesrdquo Journal ofModern Physics vol 3 no 2 pp 170ndash179 2012

[12] W C Qiang ldquoBound states of the Klein-Gordon equation forring-shaped Kratzer-type potentialrdquo Chinese Physics vol 13 no5 pp 575ndash578 2004

[13] S Ikhdair and R Sever ldquoExact polynomial eigensolutions ofthe Schrodinger equation for the pseudoharmonic potentialrdquoJournal of Molecular Structure vol 806 no 1ndash3 pp 155ndash1582007

[14] F-L Lu C-Y Chen and D-S Sun ldquoBound states of Klein-Gordon equation for double ring-shaped oscillator scalar andvector potentialsrdquo Chinese Physics vol 14 no 3 pp 463ndash4672005

[15] S M Ikhdair and R Sever ldquoExact solutions of the D-dimensional Schrodinger equation for a ring-shaped pseudo-harmonic potentialrdquo Central European Journal of Physics vol 6no 3 pp 685ndash696 2008

[16] S M Ikhdair and R Sever ldquoExact bound states of the D-dimensional Klein-Gordon equation with equal scalar and

vector ring-shaped pseudoharmonic potentialrdquo InternationalJournal of Modern Physics C vol 19 no 9 pp 1425ndash1442 2008

[17] S M Ikhdair and R Sever ldquoExactly solvable effective massd-dimensional schrodinger equation for pseudoharmonic andmodified kratzer problemsrdquo International Journal of ModernPhysics C vol 20 no 3 pp 361ndash372 2009

[18] R-CWang andC-YWong ldquoFinite-size effect in the Schwingerparticle-productionmechanismrdquo Physical ReviewD vol 38 no1 pp 348ndash359 1988

[19] J N Ginocchio ldquoA relativistic symmetry in nucleirdquo PhysicsReport vol 315 no 1ndash4 pp 231ndash240 1999

[20] A Bohr I Hamarnoto and B R Motelson ldquoPseudospln inrotating nuclear potentialsrdquo Physica Scripta vol 26 no 4 pp267ndash272 1982

[21] J Dudek W Nazarewicz Z Szymanski and G A LeanderldquoAbundance and systematics of nuclear superdeformed statesrelation to the pseudospin and pseudo-SU(3) symmetriesrdquoPhysical Review Letters vol 59 no 13 pp 1405ndash1408 1987

[22] A ArimaM Harvey and K Shimizu ldquoPseudo LS coupling andpseudo SU3 coupling schemesrdquo Physics Letters B vol 30 no 8pp 517ndash522 1969

[23] K T Hecht and A Adler ldquoGeneralized seniority for favored J= 0 pairs in mixed configurationsrdquo Nuclear Physics A vol 137

no 1 pp 129ndash143 1969[24] P R Page T Goldman and J N Ginocchio ldquoRelativistic sym-

metry suppresses quark spin-orbit splittingrdquo Physical ReviewLetters vol 86 no 2 pp 204ndash207 2001

[25] S Weinberg The Quantum Theory of Fields vol 1 CambridgeUniversity Press Cambridge UK 1995

[26] D Boyanovsky C Destri and H J de Vega ldquoApproach tothermalization in the classical 1205934 theory in 1 + 1 dimensionsenergy cascades and universal scalingrdquo Physical Review D vol69 no 4 Article ID 045003 2004

[27] R Khordad ldquoEffects of magnetic field and geometrical size onthe interband light absorption in a quantumpseudodot systemrdquoSolid State Sciences vol 12 no 7 pp 1253ndash1256 2010

[28] R Khordad ldquoSimultaneous effects of temperature and pressureon the donor binding energy in a V-groove quantum wirerdquoSuperlattices and Microstructures vol 47 no 3 pp 422ndash4312010

[29] A Cetin ldquoA quantum pseudodot system with a two-dimensional pseudoharmonic potentialrdquo Physics LettersA vol 372 no 21 pp 3852ndash3856 2008

[30] S M Ikhdair M Hamzavi and R Sever ldquoSpectra of cylindricalquantum dots the effect of electrical and magnetic fieldstogether with AB flux fieldrdquo Physica B vol 407 pp 4523ndash45292012

[31] S M Ikhdair and M Hamzavi ldquoA quantum pseudodot systemwith two-dimensional pseudoharmonic oscillator in externalmagnetic and Aharonov-Bohm fieldsrdquo Physica B vol 407 pp4198ndash4207 2012

[32] S M Ikhdair and M Hamzavi ldquoEffects of extermal fieldson a two-dimensional Klein-Gordon particle under pseudo-harmonic oscillator interactionrdquo Chinese Physics B vol 21 no11 pp 110302ndash110306 2012

[33] M Aygun O Bayrak I Boztosun and Y Sahin ldquoThe energyeigenvalues of the Kratzer potential in the presence of amagnetic fieldrdquo European Physical Journal D vol 66 no 2Article ID 20319 2012

[34] S M Ikhdair andM Hamzavi ldquoSpectral properties of quantumdots influenced by a confining potential modelrdquo Physica B vol407 pp 4797ndash4803 2012

Advances in High Energy Physics 11

[35] M Hamzavi and A A Rajabi ldquoSolution of Dirac equation withKillingbeck potential by using wave function ansatz methodunder spin symmetry limitrdquo Communications in TheoreticalPhysics vol 55 no 1 pp 35ndash37 2011

[36] M Hamzavi S M Ikhdair and K E Thylwe ldquoPseudospinsymmetry in the relativistic Killingbeck potential Quasi-exactsolutionrdquo Zeitschrift fur Naturforschung A vol 67 pp 567ndash5712012

[37] R Kumar and F Chand ldquoSeries solution to the N-dimensionalradial Schrodinger equation for the quark-antiquark interac-tionrdquo Physica Scripta vol 85 no 5 pp 055008ndash055004 2012

[38] R Kumar and F Chand ldquoReply to comments on seriessolution to the N-dimensionalradial Schrodinger equation forthe quark-antiquark interactionrdquo Physica Scripta vol 86 no 2p 027001 2012

[39] S M Ikhdair ldquoA scalar charged particle in presence of magneticandAharonov-Bohmfield plus scalar-vectorKillingbeck poten-tialsrdquo Few-Body Systems vol 54 no 11 pp 1987ndash1995 2013

[40] W Greiner Relativistic Quantum Mechanics Wave EquationsSpringer Berlin Germany 2000

[41] A D Alhaidari H Bahlouli and A Al-Hasan ldquoDirac andKlein-Gordon equations with equal scalar and vector poten-tialsrdquo Physics Letters A vol 349 no 1ndash4 pp 87ndash97 2006

[42] G Plante and A F Antippa ldquoAnalytic solution of theSchrodinger equation for the Coulomb-plus-linear potentialrdquoJournal of Mathematical Physics vol 46 no 6 pp 1ndash20 2005

[43] J Killingbeck ldquoPerturbation theory without wavefunctionsrdquoPhysics Letters A vol 65 no 2 pp 87ndash88 1978

[44] Y Xu S He and C-S Jia ldquoApproximate analytical solutionsof the Klein-Gordon equation with the Poschl-Teller potentialincluding the centrifugal termrdquo Physica Scripta vol 81 no 4Article ID 045001 2010

[45] R L Liboff Introductory Quantum Mechanics PearsonEducation-Addison Wesley San Francisco Calif USA 2003

[46] GChen Z-DChen andZ-M Lou ldquoBound states of theKlein-Gordon and Dirac equation for scalar and vector pseudohar-monic oscillator potentialsrdquo Chinese Physics vol 13 no 3 pp279ndash282 2004

[47] S H Dong Z Ma and G Espozito ldquoExact solutions of theSchrodinger equation with inversepowerpotentialrdquo Founda-tions of Physics Letters vol 12 no 5 pp 465ndash474 1999

[48] A Arda and R Sever ldquoEffective-mass Klein-Gordon equationfor non-PTnon-Hermitian generalizedMorse potentialrdquo Phys-ica Scripta vol 82 no 6 Article ID 065007 2010

[49] S M Ikhdair and J Abu-Hasna ldquoQuantization rule solutionto the Hulthen potential in arbitrary dimension with a newapproximate scheme for the centrifugal termrdquo Physica Scriptavol 83 no 2 Article ID 025002 2011

[50] S M Ikhdair and R Sever ldquoExact quantization rule tothe Kratzer-type potentials An application to the diatomicmoleculesrdquo Journal of Mathematical Chemistry vol 45 no 4pp 1137ndash1152 2009

Submit your manuscripts athttpwwwhindawicom

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

10 Advances in High Energy Physics

Acknowledgments

One of the authors (Sameer M Ikhdair) would like to thankthe president of An-Najah National University (ANU) Pro-fessor Rami Alhamdallah and the acting president ProfessorMaher Natsheh for their continuous support during hispresent work at ANU The authors acknowledge the partialsupport of the Scientific and Technological Research Councilof Turkey

References

[1] WGreiner andBMullerQuantumMechanics An IntroductionSpringer Berlin Germany 1994

[2] S Ozcelik and M Simsek ldquoExact solutions of the radialSchrodinger equation for inverse-power potentialsrdquo PhysicsLetters A vol 152 no 3-4 pp 145ndash150 1991

[3] S-H Dong ldquoSchrodinger equation with the potential 119881(119903) =119860119903minus4+ 119861119903minus3+ 119862119903minus2+ 119863119903minus1rdquo Physica Scripta vol 64 no 4 pp

273ndash276 2001[4] S-H Dong and Z-Q Ma ldquoExact solutions to the Schrodinger

equation for the potential 119881(119903) = 1198861199032+ 119887119903minus4

+ 119888119903minus6 in two

dimensionsrdquo Journal of Physics A vol 31 no 49 pp 9855ndash98591998

[5] M S Child S-H Dong and X-GWang ldquoQuantum states of asextic potential hidden symmetry and quantum monodromyrdquoJournal of Physics A vol 33 no 32 pp 5653ndash5661 2000

[6] S-H Dong ldquoExact solutions of the two-dimensional Schrodin-ger equation with certain central potentialsrdquo InternationalJournal of Theoretical Physics vol 39 no 4 pp 1119ndash1128 2000

[7] S-H Dong ldquoA new approach to the relativistic schrodingerequation with central potential Ansatz methodrdquo InternationalJournal of Theoretical Physics vol 40 no 2 pp 559ndash567 2001

[8] S-H Dong ldquoOn the solutions of the Schrodinger equation withsome anharmonic potentials wave function ansatzrdquo PhysicaScripta vol 65 no 4 pp 289ndash295 2002

[9] S Flugge Practical Quantum Mechanics 1 Springer BerlinGermany 1994

[10] X-Q Zhao C-S Jia and Q-B Yang ldquoBound states of rel-ativistic particles in the generalized symmetrical double-wellpotentialrdquo Physics Letters A vol 337 no 3 pp 189ndash196 2005

[11] S M Ikhdair ldquoExact solution of Dirac equation with chargedharmonic oscillator in electric field bound statesrdquo Journal ofModern Physics vol 3 no 2 pp 170ndash179 2012

[12] W C Qiang ldquoBound states of the Klein-Gordon equation forring-shaped Kratzer-type potentialrdquo Chinese Physics vol 13 no5 pp 575ndash578 2004

[13] S Ikhdair and R Sever ldquoExact polynomial eigensolutions ofthe Schrodinger equation for the pseudoharmonic potentialrdquoJournal of Molecular Structure vol 806 no 1ndash3 pp 155ndash1582007

[14] F-L Lu C-Y Chen and D-S Sun ldquoBound states of Klein-Gordon equation for double ring-shaped oscillator scalar andvector potentialsrdquo Chinese Physics vol 14 no 3 pp 463ndash4672005

[15] S M Ikhdair and R Sever ldquoExact solutions of the D-dimensional Schrodinger equation for a ring-shaped pseudo-harmonic potentialrdquo Central European Journal of Physics vol 6no 3 pp 685ndash696 2008

[16] S M Ikhdair and R Sever ldquoExact bound states of the D-dimensional Klein-Gordon equation with equal scalar and

vector ring-shaped pseudoharmonic potentialrdquo InternationalJournal of Modern Physics C vol 19 no 9 pp 1425ndash1442 2008

[17] S M Ikhdair and R Sever ldquoExactly solvable effective massd-dimensional schrodinger equation for pseudoharmonic andmodified kratzer problemsrdquo International Journal of ModernPhysics C vol 20 no 3 pp 361ndash372 2009

[18] R-CWang andC-YWong ldquoFinite-size effect in the Schwingerparticle-productionmechanismrdquo Physical ReviewD vol 38 no1 pp 348ndash359 1988

[19] J N Ginocchio ldquoA relativistic symmetry in nucleirdquo PhysicsReport vol 315 no 1ndash4 pp 231ndash240 1999

[20] A Bohr I Hamarnoto and B R Motelson ldquoPseudospln inrotating nuclear potentialsrdquo Physica Scripta vol 26 no 4 pp267ndash272 1982

[21] J Dudek W Nazarewicz Z Szymanski and G A LeanderldquoAbundance and systematics of nuclear superdeformed statesrelation to the pseudospin and pseudo-SU(3) symmetriesrdquoPhysical Review Letters vol 59 no 13 pp 1405ndash1408 1987

[22] A ArimaM Harvey and K Shimizu ldquoPseudo LS coupling andpseudo SU3 coupling schemesrdquo Physics Letters B vol 30 no 8pp 517ndash522 1969

[23] K T Hecht and A Adler ldquoGeneralized seniority for favored J= 0 pairs in mixed configurationsrdquo Nuclear Physics A vol 137

no 1 pp 129ndash143 1969[24] P R Page T Goldman and J N Ginocchio ldquoRelativistic sym-

metry suppresses quark spin-orbit splittingrdquo Physical ReviewLetters vol 86 no 2 pp 204ndash207 2001

[25] S Weinberg The Quantum Theory of Fields vol 1 CambridgeUniversity Press Cambridge UK 1995

[26] D Boyanovsky C Destri and H J de Vega ldquoApproach tothermalization in the classical 1205934 theory in 1 + 1 dimensionsenergy cascades and universal scalingrdquo Physical Review D vol69 no 4 Article ID 045003 2004

[27] R Khordad ldquoEffects of magnetic field and geometrical size onthe interband light absorption in a quantumpseudodot systemrdquoSolid State Sciences vol 12 no 7 pp 1253ndash1256 2010

[28] R Khordad ldquoSimultaneous effects of temperature and pressureon the donor binding energy in a V-groove quantum wirerdquoSuperlattices and Microstructures vol 47 no 3 pp 422ndash4312010

[29] A Cetin ldquoA quantum pseudodot system with a two-dimensional pseudoharmonic potentialrdquo Physics LettersA vol 372 no 21 pp 3852ndash3856 2008

[30] S M Ikhdair M Hamzavi and R Sever ldquoSpectra of cylindricalquantum dots the effect of electrical and magnetic fieldstogether with AB flux fieldrdquo Physica B vol 407 pp 4523ndash45292012

[31] S M Ikhdair and M Hamzavi ldquoA quantum pseudodot systemwith two-dimensional pseudoharmonic oscillator in externalmagnetic and Aharonov-Bohm fieldsrdquo Physica B vol 407 pp4198ndash4207 2012

[32] S M Ikhdair and M Hamzavi ldquoEffects of extermal fieldson a two-dimensional Klein-Gordon particle under pseudo-harmonic oscillator interactionrdquo Chinese Physics B vol 21 no11 pp 110302ndash110306 2012

[33] M Aygun O Bayrak I Boztosun and Y Sahin ldquoThe energyeigenvalues of the Kratzer potential in the presence of amagnetic fieldrdquo European Physical Journal D vol 66 no 2Article ID 20319 2012

[34] S M Ikhdair andM Hamzavi ldquoSpectral properties of quantumdots influenced by a confining potential modelrdquo Physica B vol407 pp 4797ndash4803 2012

Advances in High Energy Physics 11

[35] M Hamzavi and A A Rajabi ldquoSolution of Dirac equation withKillingbeck potential by using wave function ansatz methodunder spin symmetry limitrdquo Communications in TheoreticalPhysics vol 55 no 1 pp 35ndash37 2011

[36] M Hamzavi S M Ikhdair and K E Thylwe ldquoPseudospinsymmetry in the relativistic Killingbeck potential Quasi-exactsolutionrdquo Zeitschrift fur Naturforschung A vol 67 pp 567ndash5712012

[37] R Kumar and F Chand ldquoSeries solution to the N-dimensionalradial Schrodinger equation for the quark-antiquark interac-tionrdquo Physica Scripta vol 85 no 5 pp 055008ndash055004 2012

[38] R Kumar and F Chand ldquoReply to comments on seriessolution to the N-dimensionalradial Schrodinger equation forthe quark-antiquark interactionrdquo Physica Scripta vol 86 no 2p 027001 2012

[39] S M Ikhdair ldquoA scalar charged particle in presence of magneticandAharonov-Bohmfield plus scalar-vectorKillingbeck poten-tialsrdquo Few-Body Systems vol 54 no 11 pp 1987ndash1995 2013

[40] W Greiner Relativistic Quantum Mechanics Wave EquationsSpringer Berlin Germany 2000

[41] A D Alhaidari H Bahlouli and A Al-Hasan ldquoDirac andKlein-Gordon equations with equal scalar and vector poten-tialsrdquo Physics Letters A vol 349 no 1ndash4 pp 87ndash97 2006

[42] G Plante and A F Antippa ldquoAnalytic solution of theSchrodinger equation for the Coulomb-plus-linear potentialrdquoJournal of Mathematical Physics vol 46 no 6 pp 1ndash20 2005

[43] J Killingbeck ldquoPerturbation theory without wavefunctionsrdquoPhysics Letters A vol 65 no 2 pp 87ndash88 1978

[44] Y Xu S He and C-S Jia ldquoApproximate analytical solutionsof the Klein-Gordon equation with the Poschl-Teller potentialincluding the centrifugal termrdquo Physica Scripta vol 81 no 4Article ID 045001 2010

[45] R L Liboff Introductory Quantum Mechanics PearsonEducation-Addison Wesley San Francisco Calif USA 2003

[46] GChen Z-DChen andZ-M Lou ldquoBound states of theKlein-Gordon and Dirac equation for scalar and vector pseudohar-monic oscillator potentialsrdquo Chinese Physics vol 13 no 3 pp279ndash282 2004

[47] S H Dong Z Ma and G Espozito ldquoExact solutions of theSchrodinger equation with inversepowerpotentialrdquo Founda-tions of Physics Letters vol 12 no 5 pp 465ndash474 1999

[48] A Arda and R Sever ldquoEffective-mass Klein-Gordon equationfor non-PTnon-Hermitian generalizedMorse potentialrdquo Phys-ica Scripta vol 82 no 6 Article ID 065007 2010

[49] S M Ikhdair and J Abu-Hasna ldquoQuantization rule solutionto the Hulthen potential in arbitrary dimension with a newapproximate scheme for the centrifugal termrdquo Physica Scriptavol 83 no 2 Article ID 025002 2011

[50] S M Ikhdair and R Sever ldquoExact quantization rule tothe Kratzer-type potentials An application to the diatomicmoleculesrdquo Journal of Mathematical Chemistry vol 45 no 4pp 1137ndash1152 2009

Submit your manuscripts athttpwwwhindawicom

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 11

[35] M Hamzavi and A A Rajabi ldquoSolution of Dirac equation withKillingbeck potential by using wave function ansatz methodunder spin symmetry limitrdquo Communications in TheoreticalPhysics vol 55 no 1 pp 35ndash37 2011

[36] M Hamzavi S M Ikhdair and K E Thylwe ldquoPseudospinsymmetry in the relativistic Killingbeck potential Quasi-exactsolutionrdquo Zeitschrift fur Naturforschung A vol 67 pp 567ndash5712012

[37] R Kumar and F Chand ldquoSeries solution to the N-dimensionalradial Schrodinger equation for the quark-antiquark interac-tionrdquo Physica Scripta vol 85 no 5 pp 055008ndash055004 2012

[38] R Kumar and F Chand ldquoReply to comments on seriessolution to the N-dimensionalradial Schrodinger equation forthe quark-antiquark interactionrdquo Physica Scripta vol 86 no 2p 027001 2012

[39] S M Ikhdair ldquoA scalar charged particle in presence of magneticandAharonov-Bohmfield plus scalar-vectorKillingbeck poten-tialsrdquo Few-Body Systems vol 54 no 11 pp 1987ndash1995 2013

[40] W Greiner Relativistic Quantum Mechanics Wave EquationsSpringer Berlin Germany 2000

[41] A D Alhaidari H Bahlouli and A Al-Hasan ldquoDirac andKlein-Gordon equations with equal scalar and vector poten-tialsrdquo Physics Letters A vol 349 no 1ndash4 pp 87ndash97 2006

[42] G Plante and A F Antippa ldquoAnalytic solution of theSchrodinger equation for the Coulomb-plus-linear potentialrdquoJournal of Mathematical Physics vol 46 no 6 pp 1ndash20 2005

[43] J Killingbeck ldquoPerturbation theory without wavefunctionsrdquoPhysics Letters A vol 65 no 2 pp 87ndash88 1978

[44] Y Xu S He and C-S Jia ldquoApproximate analytical solutionsof the Klein-Gordon equation with the Poschl-Teller potentialincluding the centrifugal termrdquo Physica Scripta vol 81 no 4Article ID 045001 2010

[45] R L Liboff Introductory Quantum Mechanics PearsonEducation-Addison Wesley San Francisco Calif USA 2003

[46] GChen Z-DChen andZ-M Lou ldquoBound states of theKlein-Gordon and Dirac equation for scalar and vector pseudohar-monic oscillator potentialsrdquo Chinese Physics vol 13 no 3 pp279ndash282 2004

[47] S H Dong Z Ma and G Espozito ldquoExact solutions of theSchrodinger equation with inversepowerpotentialrdquo Founda-tions of Physics Letters vol 12 no 5 pp 465ndash474 1999

[48] A Arda and R Sever ldquoEffective-mass Klein-Gordon equationfor non-PTnon-Hermitian generalizedMorse potentialrdquo Phys-ica Scripta vol 82 no 6 Article ID 065007 2010

[49] S M Ikhdair and J Abu-Hasna ldquoQuantization rule solutionto the Hulthen potential in arbitrary dimension with a newapproximate scheme for the centrifugal termrdquo Physica Scriptavol 83 no 2 Article ID 025002 2011

[50] S M Ikhdair and R Sever ldquoExact quantization rule tothe Kratzer-type potentials An application to the diatomicmoleculesrdquo Journal of Mathematical Chemistry vol 45 no 4pp 1137ndash1152 2009

Submit your manuscripts athttpwwwhindawicom

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 athttpwwwhindawicom

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