Research Article Dirac Equation under Scalar and Vector ...downloads.hindawi.com/journals/ahep/2014/831938.pdf · Research Article Dirac Equation under Scalar and Vector Generalized

Research ArticleDirac Equation under Scalar and Vector Generalized IsotonicOscillators and Cornell Tensor Interaction

H Hassanabadi1 E Maghsoodi2 Akpan N Ikot3 and S Zarrinkamar4

1 Physics Department Shahrood University Shahrood 36199-95161-316 Iran2Department of Basic Sciences Islamic Azad University Shahrood Branch Shahrood Iran3Theoretical Physics Group Department of Physics University of Port Harcourt PMB 5323 Choba Port Harcourt Nigeria4Department of Basic Sciences Islamic Azad University Garmsar Branch Garmsar Iran

Correspondence should be addressed to H Hassanabadi hhasanabadishahroodutacir

Received 6 September 2014 Accepted 13 November 2014 Published 30 November 2014

Academic Editor Shi-Hai Dong

Copyright copy 2014 H Hassanabadi et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

Spin and pseudospin symmetries of Dirac equation are solved under scalar and vector generalized isotonic oscillators and Cornellpotential as a tensor interaction for arbitrary quantum number via the analytical ansatz approach The spectrum of the system isnumerically reported for typical values of the potential parameters

1 Introduction

The solution of the fundamental dynamical equations isan interesting phenomenon in many fields of physics Therelativistic Dirac equationwhich describes themotion of spin12 particle [1] Within the framework of Dirac equationpseudospin and spin symmetries are used to study featuresof deformed nuclei superdeformation and effective shellmodel [2] The concept of SUSYQM provides theoreticalphysicists with a powerful tool to deal with nonrelativisticSchrodinger equation The pseudospin symmetry is referredto as a quasidegeneracy of single nucleon doublets withnonrelativistic quantum number (119899 119897 119895 = 119897 + 12) and(119899 minus 1 119897 + 2 119895 = 119897 + 32) where 119899 119897 and 119895 are singlenucleon radial orbital and total angular quantum numbers[3] and it was shown that the exact pseudospin symmetryoccurs in the Dirac equation when 119889Σ(119903)119889119903 = 0 that isΣ(119903) = 119881(119903) + 119878(119903) = 119888ps = const where 119881(119903) and 119878(119903)are repulsive vector potential and attractive scalar potentialrespectively Details of recent review of spin and pseudospinsymmetries are given in [4] These symmetries under var-ious phenomenological potentials have been investigatedusing various methods such as Nikiforov-Uvarov (NU)supersymmetric quantummechanics (SUSSYQM) and shape

invariance (SI) ansatz approach and asymptotic iteration(AIM) [5ndash15] In recent years the Dirac equation withdifferent potentials in relativistic quantum mechanics withspin andpseudospin symmetry has been investigated [16ndash25]The main aim of the present paper is to obtain approximatesolutions of the Dirac equation with scalar and vector gen-eralized isotonic oscillators and Cornell tensor interactionunder the above mentioned symmetry limits The isotonicoscillator potential consists of the harmonic oscillator pluscentrifugal barrier which is of great interest in the theoryof coherent states [26] and quantum optics [27] and in theanalysis of the isochronous oscillator [28] This potentialis important because of its relation with supersymmetricquantummechanics [29]The two-dimensional version of theisotonic potential is superintegrable and usually is referredto as the Smorodinsky-Winternitz potential [30] Here wemake use of the ansatz approach to deal with this complicatedequation A survey on the application of this technique toother wave equations including Dirac Schrodinger Klein-Gordon spinless-Salpeter and DKP equations can be foundin [31ndash39] The paper is organized as follows In Section 2we give a brief introduction of the supersymmetry quantummechanics (SUSYQM) In Section 3 the Dirac equation iswritten for spin and pseudospin including the Cornell tensor

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014 Article ID 831938 7 pageshttpdxdoiorg1011552014831938

2 Advances in High Energy Physics

interaction term We next propose a physical ansatz solutionto the equation and we solve the Dirac equation under thesesymmetries in Section 4 Finally conclusion is presented inSection 5

2 Dirac Equation including Tensor Coupling

In spherical coordinates the Dirac equation with both scalarpotential 119878(119903) and vector potential 119881(119903) can be expressed as[1 2]

[ sdot + 120573 (119872 + 119878 (119903)) minus 119894120573 sdot 119903119880 (119903)] 120595 ( 119903)

= [119864 minus 119881 (119903)] 120595 ( 119903)

(1)

where119864 is the relativistic energy of the system 120572 and120573 are the4 times 4 Dirac matrices p is the momentum operator = minus119894nablaFor a particle in a spherical field the total angularmomentumoperator 119869 and spin-orbit matrix operator 119870 = ( sdot + 1)

commute with the Dirac Hamiltonian where 120590 and 119871 arethe Paulimatrix and orbital angularmomentum respectivelyThe eigenvalues of119870 are 120581 = minus(119895 + 12) for aligned spin (119904

12

11990132

etc) and 120581 = (119895+12) for unaligned spin (11990112

11988932

etc)The complete set of the conservative quantities can be takenas (1198672 119870 1198692 119869

119911) As shown in [15] by taking the spherically

symmetric Dirac spinor wave functions as

120595119899119896( 119903) = (

119891119899119896( 119903)

119892119899119896( 119903)) = (

119865119899119896( 119903)

119903119884ℓ

119895119898(120579 120593)

119894119866119899119896( 119903)

119903119884ℓ

119895119898(120579 120593)

) (2)

where 119865119899119896( 119903) and 119866

119899119896( 119903) are the radial wave functions of

the upper and lower components respectively 119884ℓ119895119898(120579 120593) and

119884ℓ

119895119898(120579 120593) respectively stand for spin and pseudospin spher-

ical harmonics that are coupled to the angular momentum 119895

and 119898 is the projection of the angular momentum on the 119911-axisThe orbital angularmomentumquantumnumbers ℓ andℓ refer to the upper and lower components respectively Thequasidegenerate doublet structure can be expressed in termsof pseudospin angularmomentum 119904 = 12 and pseudoorbitalangularmomentum ℓ which is defined as ℓ = ℓ+1 for alignedspin 119895 = ℓ minus 12 and ℓ = ℓ minus 1 for unaligned spin 119895 = ℓ + 12As shown in [1 2] substituting (2) into (1) yields two coupleddifferential equations as follows

(119889

119889119903+120581

119903minus 119880 (119903))119865

119899119896(119903)

= (119872 + 119864119899119896minus 119881 (119903) + 119878 (119903)) 119866

119899119896(119903)

(3)

(119889

119889119903minus120581

119903+ 119880 (119903))119866

119899119896(119903)

= (119872 minus 119864119899119896+ 119881 (119903) + 119878 (119903)) 119865

119899119896(119903)

(4)

We consider the difference potentialΔ(119903) and sum poten-tial Σ(119903) as Δ(119903) = 119881(119903) minus 119878(119903) and Σ(119903) = 119881(119903) + 119878(119903)respectively

The spin-orbit quantum number 120581 is related to the orbitalangular momentum quantum number ℓ By eliminating119866119899119896( 119903) in (3) and 119865

119899119896( 119903) in (4) we obtain the following two

second-order differential equations for the upper and lowercomponents

1198892

1198891199032minus120581 (120581 + 1)

1199032

+2120581

119903119880 (119903) minus

119889119880 (119903)

119889119903minus 1198802(119903)

+119889Δ (119903) 119889119903

119872 + 119864119899120581minus Δ (119903)

(119889

119889119903+120581

119903minus 119880 (119903))119865

119899120581(119903)

= (119872 + 119864119899120581minus Δ (119903)) (119872 minus 119864

119899120581+ Σ (119903)) 119865

119899120581(119903)

(5)

1198892

1198891199032minus120581 (120581 minus 1)

1199032

+2120581

119903119880 (119903) +

119889119880 (119903)

119889119903minus 1198802(119903)

+119889Σ (119903) 119889119903

119872 minus 119864119899120581+ Σ (119903)

(119889

119889119903minus120581

119903+ 119880 (119903))119866

119899120581(119903)

= (119872 + 119864119899120581minus Δ (119903)) (119872 minus 119864

119899120581+ Σ (119903)) 119866

119899120581(119903)

(6)

We have 120581(120581 minus 1) = ℓ(ℓ + 1) and 120581(120581 + 1) = ℓ(ℓ + 1)

3 Pseudospin and Spin Symmetric Solutions

31 Pseudospin Symmetry Limit The pseudospin symmetryoccurs when 119889Σ(119903)119889119903 = 0 or equivalently Σ(119903) = 119862ps =

Const [1 2] Here we work on

Δ (119903) = 1198601199032minus 2119892

21199032minus 1

(21199032+ 1)2 (7)

For the tensor term we consider the Cornell potential

119880 (119903) = 119886ps119903 +119887ps

119903 (8)

Substitution of these two terms into (6) gives

1198892

1198891199032+1

1199032(minus120581 (120581 minus 1) + 2120581119887ps minus 119887ps minus 119887

2

ps)

minus 119860ps (119864ps119899120581minus119872 minus 119862ps) 119903

2minus 1198862

ps1199032

minus 2119892ps21199032minus 1

(21199032+ 1)2(119864

ps119899120581minus119872 minus 119862ps)

+ (119864ps119899120581+119872) (119864

ps119899120581minus119872 minus 119862ps) minus 2120581119886ps

+ 119886ps minus 2119886ps119887ps119866ps119899120581(119903) = 0

(9)

where 120581 = minusℓ and 120581 = ℓ + 1 for 120581 lt 0 and 120581 gt 0 respectively

Advances in High Energy Physics 3

32 Spin Symmetry Limit In the spin symmetry limit 119889Δ(119903)119889119903 = 0 or Δ(119903) = 119862

119904= Const [15] we consider

Σ (119903) = 1198601199032minus 2119892

21199032minus 1

(21199032+ 1)2 (10a)

119880 (119903) = 119886119904119903 +

119887119904

119903 (10b)

Substitution of the later into (5) gives

1198892

1198891199032+1

1199032(minus120581 (120581 + 1) + 2120581119887

119904+ 119887119904minus 1198872

119904)

minus 119860119904(119872 + 119864

119904

119899120581minus 119862119904) 1199032minus 1198862

1199041199032

+ 2119892119904

21199032minus 1

(21199032+ 1)2(119872 + 119864

119904

119899120581minus 119862119904)

+ (119864119904

119899120581minus119872) (119872 + 119864

119904

119899120581minus 119862119904) + 2120581119886

119904

minus 119886119904minus 2119886119904119887119904119865119904

119899120581(119903) = 0

(11)

where 120581 = ℓ and 120581 = minusℓ minus 1 for 120581 lt 0 and 120581 gt 0 respectively

4 The Ansatz Solution

41 Solution of the Pseudospin Symmetry Limit In the previ-ous section we obtained a Schrodinger-like equation of theform1198892119866ps119899120581(119903)

1198891199032

+ [120576ps+120582ps

1199032+ 120594

ps1199032+ 120575

ps 21199032minus 1

(21199032+ 1)2]119866

ps119899120581(119903) = 0

(12)

where120576ps= (119864

ps119899120581+119872) (119864

ps119899120581minus119872 minus 119862ps) minus 2120581119886ps + 119886ps minus 2119886ps119887ps

120582ps= minus 120581 (120581 minus 1) + 2120581119887ps minus 119887ps minus 119887

2

ps

120594ps= minus 119860ps (119864

ps119899120581minus119872 minus 119862ps) minus 119886

2

ps

120575ps= minus 2119892ps (119864

ps119899120581minus119872 minus 119862ps)

(13)

The latter fails to admit exact analytical solutions There-fore we follow the ansatz approach with the starting square

119866ps119899120581(119903) = 119891

ps119899(119903) exp (119892ps

120581(119903)) (14)

where

119891ps119899(119903) =

1 if 119899 = 0119899

prod

119894=1

(119903 minus 120572119899

119894) if 119899 ge 1 (15)

119892ps120581(119903) = minus120572

ps1199032+ 120573

ps ln 119903 + 120574ps ln (21199032 + 1) 120572psgt 0

(16)

By substitution of 119891119899(119903) and 119892

120581(119903) into (14) and then

taking the second-order derivative of the obtained equationwe can get

119866ps119899120581

10158401015840

(119903) = [119892ps120581

10158401015840

(119903) + 119892ps120581

10158402

(119903)

+119891ps119899

10158401015840(119903) + 2119892

ps120581

1015840(119903) 119891

ps119899

1015840(119903)

119891ps119899 (119903)

]119866ps119899120581(119903)

(17)

By considering the case 119899 = 0 from (14)ndash(16) we find

119866ps0120581

10158401015840

(119903) = 1

1199032[minus120573

ps+ 120573

ps2] +

1

(21199032+ 1)2

times [16120574ps21199032+ 16120572

ps120574ps1199032+ 16120573

ps120574ps1199032

minus 8120574ps1199032+ 4120574

ps+ 8120572

ps120574 + 8120573

ps120574ps]

+ 4120572ps21199032minus 2120572

ps

minus 4120572ps120573psminus 8120572

ps120574ps119866

ps0120581(119903)

(18)

By comparing the corresponding powers of (12) and (18) wehave

minus120582ps= 120573

ps(120573

psminus 1)

minus120594ps= 4120572

ps2

minus2120575ps= 16120574

ps2+ 16120572

ps120574ps+ 16120573

ps120574psminus 8120574

ps

120575ps= 4120574

ps+ 8120572

ps120574 + 8120573

ps120574ps

minus120576ps= minus 2120572

psminus 4120572

ps120573psminus 8120572

ps120574ps

(19)

Equation (19) gives

120573ps=1

2(1 + radic1 minus 4120582

ps)

120572ps=1

2radicminus120594

ps

120574ps= minus 2 (120572

ps+ 120573

ps)

(20)

Actually to havewell-behaved solutions of the radial wavefunction at boundaries namely the origin and the infinity weneed to take 120575ps from (19) as

120575ps= 4120574

ps+ 8120572

ps120574 + 8120573

ps120574ps (21)

Form (13) (19) and (20) the ground-state energy satisfies

120576psminus radicminus120594

psminus radicminus120594

ps(1 + radic1 minus 4120582

ps)

+ 4radicminus120594ps(radicminus120594

ps+ (1 + radic1 minus 4120582

ps)) = 0

(22)


Table 1 Bound state for the pseudospin symmetry 119886ps = minus002 119887ps = 001 119872 = 05 fmminus1 and 119862ps = minus1 fmminus1

ℓ 120581 (ℓ 119895)119864ps0120581(fmminus1)

119860 = minus005 119860 = minus 05

1 minus1 011987812

minus0492577461 minus04992578522 minus2 0119875

32minus0492679105 minus0499267983

3 minus3 011988952


72minus0492762185 minus0499276263

Table 2 Bound state for the spin symmetry 119886119904= 002 119887

119904= 001 119872 = 05 fmminus1 and 119862

119904= 1 fmminus1

ℓ 120581 (ℓ 119895)119864119904

0120581(fmminus1)

119860 = minus005 119860 = minus05

1 minus2 011990132

0506363937 05006362632 minus3 0119889

520506651467 0500665063

3 minus4 011989172

0506779923 05006779314 minus5 0119892

92050685245 0500685197

Table 3 Energies in the pseudospin symmetry limit for 119860ps = minus05 119886ps = minus002 119887ps = 001 and119872 = 05 fmminus1

119862ps119864ps0120581(fmminus1)

011987812

011987532

011988952

011989172

minus5 minus4499263699 minus449927201 minus4499276204 minus449927873minus4 minus3499262219 minus3499270997 minus3499275435 minus3499278111minus3 minus2499260752 minus2499269988 minus2499274669 minus2499277494minus2 minus1499259296 minus1499268983 minus1499273904 minus1499276877minus1 minus0499257852 minus0499267983 minus0499273142 minus04992762630 050074358 0500733013 0500727617 0500724351

From (14) (16) and (20) we simply have the upper andlower components of the wave function as

119892ps120581(119903) = minus

1

2radicminus120594

ps1199032+1

2(1 + radic1 minus 4120582

ps)

times ln 119903 minus 2 (120572ps + 120573ps) ln (21199032 + 1) (23a)

119866ps0120581(119903) = 119873

0120581119903(12)(1+radic1minus4120582

ps)(21199032+ 1)minus2(120572

ps+120573

ps)

times exp(minus12radicminus120594

ps1199032)

(23b)

119865ps0120581(119903) =

1

119872 minus 119864ps0120581+ 119862ps

(119889

119889119903minus120581

119903+ 119880 (119903))119866

ps0120581(119903) (23c)

42 Solution of the Spin Symmetry Limit In this case ourordinary differential equation is

1198892119865119904

119899120581(119903)

1198891199032

+ [120576119904+120582119904

1199032+ 1205941199041199032+ 120575119904 21199032minus 1

(21199032+ 1)2]119865119904

119899120581(119903) = 0

(24)

where

120576119904= (119864119904

119899120581minus119872) (119872 + 119864

119904

119899120581minus 119862119904) + 2120581119886

119904minus 119886119904minus 2119886119904119887119904

120582119904= minus 120581 (120581 + 1) + 2120581119887

119904+ 119887119904minus 1198872

119904

120594119904= minus 119860

119904(119872 + 119864

119904

119899120581minus 119862119904) minus 1198862

119904

120575119904= 2119892119904(119872 + 119864

119904

119899120581minus 119862119904)

(25)

In this case we introduce the ansatz

119865119904

119899120581(119903) = 119891

119904

119899(119903) exp (119892119904

120581(119903)) (26)

where

119891119904

119899(119903) =

1 if 119899 = 0119899

prod

119894=1

(119903 minus 120572119899

119894) if 119899 ge 1 (27)

119892119904

120581(119903) = minus120572

1199041199032+ 120573119904 ln 119903 + 120574119904 ln (21199032 + 1) 120572

119904gt 0 (28)

substitution of the proposed ansatz gives

11986511990410158401015840

119899120581(119903)

= [119892119904

120581

10158401015840

(119903) + 119892119904

120581

10158402

(119903) +119891119904

119899

10158401015840(119903) + 2119892

119904

120581

1015840(119903) 119891119904

119899

1015840(119903)

119891119904

119899(119903)

] 119865119904

119899120581(119903)

(29)


Table 4 Energies in the spin symmetry limit for 119860119904= minus05 119886

119904= 002 119887

119904= 001 and119872 = 05 fmminus1

119862119904

119864119904

0120581(fmminus1)

011990132

011988952

011989172

011989292

0 minus0499365805 minus0499336214 minus0499322986 minus04993155171 0500636263 0500665063 0500677931 05006851972 1500638332 1500666338 1500678847 15006859093 2500640401 2500667612 2500679761 25006866214 350064247 3500668885 3500680675 35006873325 4500644539 4500670157 4500681588 4500688043

Table 5 Energies in the pseudospin symmetry limit for 119860ps = minus05 119886ps = minus002 119887ps = 001 and 119862ps = minus1 fmminus1

119872(fmminus1) 119864ps0120581(fmminus1)

011987812

011987532

011988952

011989172

0 minus0999259296 minus0999268983 minus0999273904 minus099927687702 minus0799258717 minus0799268582 minus0799273599 minus079927663104 minus059925814 minus0599268183 minus0599273295 minus059927638506 minus0399257564 minus0399267783 minus039927299 minus03992761408 minus0199256991 minus0199267385 minus0199272686 minus0199275894


119904= 002 119887

119904= 001 and 119862

119904= 1 fmminus1

119872(fmminus1) 119864119904+

119899120581(fmminus1)

011990132

011988952

011989172

011989292

0 1000638332 1000666338 1000678847 100068590902 0800637505 0800665828 080067848 080068562404 0600636677 0600665318 0600678114 060068533906 040063585 0400664807 0400677748 040068505408 0200635022 0200664297 0200677381 0200684769

0 05 1 15 2 25

r

0s320p320d52

0

001

002

003

004

005

G0120581(r)

PSS

(a)

0 05 1 15 2 25

r

0p32

0f72

0d52

0

01

02

03

04

05

06

F0120581(r)

SS

(b)

Figure 1 PSS wavefunction for pseudospin symmetry limit for 119860ps = minus05 119886ps = minus002 119887ps = 001 119872 = 05 fmminus1 and 119862ps = minus1 fmminus1 SS

wavefunction for spin symmetry limit for 119860119904= minus05 119886

119904= 002 119887

119904= 001119872 = 05 fmminus1 and 119862

119904= 1 fmminus1


For 119899 = 0 we have

11986511990410158401015840

0120581(119903)

= 1

1199032[minus120573119904+ 1205731199042] +

1

(21199032+ 1)2

times [1612057411990421199032+ 16120572

1199041205741199041199032+ 16120573

1199041205741199041199032

minus 81205741199041199032+ 4120574119904+ 8120572119904120574119904+ 8120573119904120574119904]

+ 412057211990421199032minus 2120572119904minus 4120572119904120573119904minus 8120572119904120574119904119865119904

0120581(119903)

(30)

By comparing the corresponding powers of (24) and (30)we have

120573119904=1

2(1 + radic1 minus 4120582

119904)

120572119904=1

2radicminus120594119904

120574119904= minus2 (120572

119904+ 120573119904)

120576119904minus 2120572119904minus 4120572119904120573119904minus 8120572119904120574119904= 0

(31)

To have physically acceptable solutions we pick up thevalue by considering the above equation the first nodeeigenvalue satisfies

120576119904minus radicminus120594


119904(1 + radic1 minus 4120582

119904)

+ 4radicminus120594119904(radicminus120594

119904+ (1 + radic1 minus 4120582

119904)) = 0

(32)

From (26) (28) and (31) the upper component of the wavefunction is

119892119904

120581(119903) = minus

1

2radicminus1205941199041199032

+1

2(1 + radic1 minus 4120582

119904) ln 119903 minus 2 (120572119904 + 120573119904) ln (21199032 + 1)

(33a)

119865119904

0120581(119903) = 119873

0120581119903(12)(1+radic1minus4120582

119904)

times (21199032+ 1)minus2(120572119904+120573119904)

exp(minus12radicminus1205941199041199032)

(33b)

and for the lower component of the wave function we have

119866119904

0120581(119903) =

1

119872 + 119864119904

0120581minus 119862119904

(119889

119889119903+120581

119903minus 119880 (119903))119865

119904

0120581(119903) (34)

Wehave given somenumerical values of the energy eigen-values in Tables 1ndash6 for various states We have investigatedthe dependence of the bound-state energy levels 119899 = 0 onpotential parameter 119860 The results in Tables 1 and 2 havefound that case 119860 = minus005 is contrary to case 119860 = minus05

under the condition of the pseudospin and spin symmetries

respectively Tables 3 and 4 present the dependence of thebound-state energy levels on parameters119862ps and119862119904 in view ofthe pseudospin and spin symmetry limits It is seen in Tables3 and 4 that although bound states obtained in view of spinsymmetry become more bounded with increasing 119862

119904 they

become less bounded in the pseudospin symmetry limit withincreasing 119862ps We show the effects of the 119872-parameter onthe bound states under the conditions of the pseudospin andspin symmetry limits The results are given in Tables 5 and 6It is seen that if the119872-parameter increases the bound statesbecome less bounded for both the pseudospin and the spinsymmetry limits In Figure 1 the wave functions are plottedfor spin and pseudospin symmetry limits

5 Conclusion

In this paper we have obtained the approximate solutionsof the Dirac equation for the isotonic oscillator potentialincluding the Cornell tensor interaction within the frame-work of pseudospin and spin symmetry limits using theansatz approach which stands as a strong tool of mathemat-ical physics We have obtained the energy eigenvalues andcorresponding lower and upper wave functions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors wish to give their sincere gratitude to the refereesfor a technical comment on the paper

References

[1] J NGinocchio ldquoRelativistic symmetries in nuclei and hadronsrdquoPhysics Reports vol 414 no 4-5 pp 165ndash261 2005

[2] D Troltenier C Bahri and J P Draayer ldquoGeneralized pseudo-SU(3) model and pairingrdquo Nuclear Physics A vol 586 no 1 pp53ndash72 1995

[3] J N Ginocchio ldquoPseudospin as a relativistic symmetryrdquo Physi-cal Review Letters vol 78 no 3 pp 436ndash439 1997

[4] J N Ginocchio A Leviatan J Meng and S-G Zhou ldquoTest ofpseudospin symmetry in deformed nucleirdquo Physical Review Cvol 69 no 3 Article ID 034303 2004

[5] S-H Dong and M Lozada-Cassou ldquoGeneralized hypervirialand recurrence relations for radial matrix elements in arbitrarydimensionsrdquo Modern Physics Letters A Particles and FieldsGravitation Cosmology Nuclear Physics vol 20 no 20 pp1533ndash1540 2005

[6] G-F Wei C-Y Long and S-H Dong ldquoThe scattering of theManning-Rosen potential with centrifugal termrdquoPhysics LettersA vol 372 no 15 pp 2592ndash2596 2008

[7] W C Qiang and S H Dong ldquo The ManningndashRosen potentialstudied by a new approximate scheme to the centrifugal termrdquoPhysica Scripta vol 79 Article ID 045004 2009

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


[9] G FWei and S H Dong ldquoAlgebraic approach to energy spectraof the Scarf type and generalized PoschlndashTeller potentialsrdquoCanadian Journal of Physics vol 89 no 12 pp 1225ndash1231 2011

[10] G-F Wei and S-H Dong ldquoApproximately analytical solutionsof the Manning-Rosen potential with the spin-orbit couplingterm and spin symmetryrdquo Physics Letters A vol 373 no 1 pp49ndash53 2008

[11] G F Wei and S H Dong ldquoPseudospin symmetry in therelativistic Manning-Rosen potential including a Pekeris-typeapproximation to the pseudo-centrifugal termrdquo Physics LettersB vol 686 no 4-5 pp 288ndash292 2010

[12] J Meng K Sugawara-Tanabe S Yamaji and A Arima ldquoPseu-dospin symmetry in Zr and Sn isotopes from the proton dripline to the neutron drip linerdquo Physical Review C vol 59 p 1541999

[13] F S StephenM A Deleplanque J E Draper et al ldquoPseudospinsymmetry and quantized alignment in nucleirdquo Physical ReviewLetters vol 65 no 3 pp 301ndash304 1990

[14] A E Stuchbery ldquoMagnetic properties of rotational states in thepseudo-Nilsson modelrdquo Nuclear Physics A vol 700 no 1-2 pp83ndash116 2002

[15] E Maghsoodi H Hassanabadi S Zarrinkamar and H Rahi-mov ldquoRelativistic symmetries of the Dirac equation under thenuclear WoodsSaxon potentialrdquo Physica Scripta vol 85 no 5Article ID 055007 2012

[16] C-S Jia P Guo and X-L Peng ldquoExact solution of the Dirac-Eckart problemwith spin and pseudospin symmetryrdquo Journal ofPhysics A Mathematical and General vol 39 no 24 pp 7737ndash7744 2006

[17] C-S Jia and A de Souza Dutra ldquoPosition-dependent effec-tive mass Dirac equations with PT-symmetric and non-PT-symmetric potentialsrdquo Journal of Physics A Mathematical andGeneral vol 39 no 38 pp 11877ndash11887 2006

[18] H Hassanabadi E Maghsoodi S Zarrinkamar and H Rahi-mov ldquoDirac equation for generalized Poschl-Teller scalarand vector potentials and a Coulomb tensor interaction byNikiforov-Uvarov methodrdquo Journal of Mathematical Physicsvol 53 no 2 Article ID 022104 2012

[19] H Akcay ldquoDirac equation with scalar and vector quadraticpotentials and Coulomb-like tensor potentialrdquo Physics LettersA vol 373 no 6 pp 616ndash620 2009

[20] C-S Jia T Chen and L-G Cui ldquoApproximate analyticalsolutions of the Dirac equation with the generalized Poschl-Teller potential including the pseudo-centrifugal termrdquo PhysicsLetters A vol 373 no 18-19 pp 1621ndash1626 2009

[21] L H Zhang X P Li and C S Jia ldquoApproximate analyticalsolutions of the Dirac equation with the generalized Morsepotential model in the presence of the spin symmetry andpseudo-spin symmetryrdquo Physica Scripta vol 80 no 3 ArticleID 035003 2009

[22] H Akcay ldquoThe Dirac oscillator with a Coulomb-like tensorpotentialrdquo Journal of Physics A Mathematical and Theoreticalvol 40 no 24 pp 6427ndash6432 2007

[23] O Aydogdu and R Sever ldquoExact pseudospin symmetric solu-tion of the dirac equation for pseudoharmonic potential in thepresence of tensor potentialrdquo Few-Body Systems vol 47 pp 193ndash200 2010

[24] S-H Dong Factorization Method in Quantum MechanicsSpringer Berlin Germany 2007

[25] J Meng K Sugawara-Tanabe S Yamaji P Ring and AArima ldquoPseudospin symmetry in relativisticmean field theoryrdquoPhysical Review C vol 58 no 2 article R628 1998

[26] K Thirulogasanther and N Saad ldquoCoherent states associatedwith the wavefunctions and the spectrum of the isotonicoscillatorrdquo Journal of Physics A Mathematical and General vol37 no 16 p 4567 2004

[27] J-S Wang T-K Liu and M-S Zhan ldquoNonclassical propertiesof even and odd generalized coherent states for an isotonicoscillatorrdquo Journal of Optics B Quantum and SemiclassicalOptics vol 2 no 6 pp 758ndash763 2000

[28] O A Chalykh and A P Veselov ldquoA remark on rationalisochronous potentialsrdquo Journal of Nonlinear MathematicalPhysics vol 12 no 1 pp 179ndash183 2005

[29] J Casahorran ldquoOn a novel supersymmetric connectionsbetween harmonic and isotonic oscillatorsrdquo Physica A vol 217no 3-4 pp 429ndash439 1995

[30] C Grosche G S Pogosyan and A N Sissakian ldquoPath integraldiscussion for Smorodinsky-Winternitz potentials II The two-and three-dimensional sphererdquo Fortschritte der PhysikProgressof Physics vol 43 no 6 pp 523ndash563 1995

[31] N Akpan E Ikot S Maghsoodi H Zarrinkamar and ZHassanabadi ldquoShape-invariant approach to study relativisticsymmetries of the dirac equation with a new hyperbolicalpotential combinationrdquoZeitschrift fur NaturforschungA vol 68pp 499ndash509 2013

[32] H Hassanabadi E Maghsoodi N Salehi A N Ikot and SZarrinkamar ldquoScattering states of the dirac equation underasymmetric Hulthen potentialrdquo The European Physical JournalPlus vol 128 no 10 pp 1ndash9 2013

[33] S-H Dong ldquoExact solutions of the two-dimensional Schrding-er equation with certain central potentialsrdquo International Jour-nal of Theoretical Physics vol 39 no 4 pp 1119ndash1128 2000

[34] S H Dong ldquoCorrelations of spin states for icosahedral doublegrouprdquo International Journal of Theoretical Physics vol 40 no2 pp 569ndash581 2001

[35] H Hassanabadi and A A Rajabi ldquoEnergy levels of a sphericalquantumdot in a confining potentialrdquoPhysics Letters A vol 373no 6 pp 679ndash681 2009

[36] H Hassanabadi and A A Rajabi ldquoRelativistic versus nonrel-ativistic solution of the N-fermion problem in a hyperradius-confining potentialrdquo Few-Body Systems vol 41 no 3-4 pp 201ndash210 2007

[37] S-H Dong Z-Q Ma and G Esposito ldquoExact solutionsof the Schrodinger equation with inverse-power potentialrdquoFoundations of Physics Letters vol 12 no 5 pp 465ndash474 1999

[38] H Hassanabadi H Rahimov and S Zarrinkamar ldquoCornell andCoulomb interactions for the D-dimensional Klein-Gordonequationrdquo Annalen der Physik vol 523 no 7 pp 566ndash575 2011

[39] D Agboola and Y-Z Zhang ldquoUnified derivation of exactsolutions for a class of quasi-exactly solvable modelsrdquo Journal ofMathematical Physics vol 53 no 4 Article ID 042101 13 pages2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


interaction term We next propose a physical ansatz solutionto the equation and we solve the Dirac equation under thesesymmetries in Section 4 Finally conclusion is presented inSection 5

2 Dirac Equation including Tensor Coupling

In spherical coordinates the Dirac equation with both scalarpotential 119878(119903) and vector potential 119881(119903) can be expressed as[1 2]

[ sdot + 120573 (119872 + 119878 (119903)) minus 119894120573 sdot 119903119880 (119903)] 120595 ( 119903)

= [119864 minus 119881 (119903)] 120595 ( 119903)

(1)

where119864 is the relativistic energy of the system 120572 and120573 are the4 times 4 Dirac matrices p is the momentum operator = minus119894nablaFor a particle in a spherical field the total angularmomentumoperator 119869 and spin-orbit matrix operator 119870 = ( sdot + 1)

commute with the Dirac Hamiltonian where 120590 and 119871 arethe Paulimatrix and orbital angularmomentum respectivelyThe eigenvalues of119870 are 120581 = minus(119895 + 12) for aligned spin (119904

12

11990132

etc) and 120581 = (119895+12) for unaligned spin (11990112

11988932

etc)The complete set of the conservative quantities can be takenas (1198672 119870 1198692 119869

119911) As shown in [15] by taking the spherically

symmetric Dirac spinor wave functions as

120595119899119896( 119903) = (

119891119899119896( 119903)

119892119899119896( 119903)) = (

119865119899119896( 119903)

119903119884ℓ

119895119898(120579 120593)

119894119866119899119896( 119903)

119903119884ℓ

119895119898(120579 120593)

) (2)

where 119865119899119896( 119903) and 119866

119899119896( 119903) are the radial wave functions of

the upper and lower components respectively 119884ℓ119895119898(120579 120593) and

119884ℓ

119895119898(120579 120593) respectively stand for spin and pseudospin spher-

ical harmonics that are coupled to the angular momentum 119895

and 119898 is the projection of the angular momentum on the 119911-axisThe orbital angularmomentumquantumnumbers ℓ andℓ refer to the upper and lower components respectively Thequasidegenerate doublet structure can be expressed in termsof pseudospin angularmomentum 119904 = 12 and pseudoorbitalangularmomentum ℓ which is defined as ℓ = ℓ+1 for alignedspin 119895 = ℓ minus 12 and ℓ = ℓ minus 1 for unaligned spin 119895 = ℓ + 12As shown in [1 2] substituting (2) into (1) yields two coupleddifferential equations as follows

(119889

119889119903+120581

119903minus 119880 (119903))119865

119899119896(119903)

= (119872 + 119864119899119896minus 119881 (119903) + 119878 (119903)) 119866

119899119896(119903)

(3)

(119889

119889119903minus120581

119903+ 119880 (119903))119866

119899119896(119903)

= (119872 minus 119864119899119896+ 119881 (119903) + 119878 (119903)) 119865

119899119896(119903)

(4)

We consider the difference potentialΔ(119903) and sum poten-tial Σ(119903) as Δ(119903) = 119881(119903) minus 119878(119903) and Σ(119903) = 119881(119903) + 119878(119903)respectively

The spin-orbit quantum number 120581 is related to the orbitalangular momentum quantum number ℓ By eliminating119866119899119896( 119903) in (3) and 119865

119899119896( 119903) in (4) we obtain the following two

second-order differential equations for the upper and lowercomponents

1198892

1198891199032minus120581 (120581 + 1)

1199032

+2120581

119903119880 (119903) minus

119889119880 (119903)

119889119903minus 1198802(119903)

+119889Δ (119903) 119889119903

119872 + 119864119899120581minus Δ (119903)

(119889

119889119903+120581

119903minus 119880 (119903))119865

119899120581(119903)

= (119872 + 119864119899120581minus Δ (119903)) (119872 minus 119864

119899120581+ Σ (119903)) 119865

119899120581(119903)

(5)

1198892

1198891199032minus120581 (120581 minus 1)

1199032

+2120581

119903119880 (119903) +

119889119880 (119903)

119889119903minus 1198802(119903)

+119889Σ (119903) 119889119903

119872 minus 119864119899120581+ Σ (119903)

(119889

119889119903minus120581

119903+ 119880 (119903))119866

119899120581(119903)

= (119872 + 119864119899120581minus Δ (119903)) (119872 minus 119864

119899120581+ Σ (119903)) 119866

119899120581(119903)

(6)

We have 120581(120581 minus 1) = ℓ(ℓ + 1) and 120581(120581 + 1) = ℓ(ℓ + 1)

3 Pseudospin and Spin Symmetric Solutions

31 Pseudospin Symmetry Limit The pseudospin symmetryoccurs when 119889Σ(119903)119889119903 = 0 or equivalently Σ(119903) = 119862ps =

Const [1 2] Here we work on

Δ (119903) = 1198601199032minus 2119892

21199032minus 1

(21199032+ 1)2 (7)

For the tensor term we consider the Cornell potential

119880 (119903) = 119886ps119903 +119887ps

119903 (8)

Substitution of these two terms into (6) gives

1198892

1198891199032+1

1199032(minus120581 (120581 minus 1) + 2120581119887ps minus 119887ps minus 119887

2

ps)

minus 119860ps (119864ps119899120581minus119872 minus 119862ps) 119903

2minus 1198862

ps1199032

minus 2119892ps21199032minus 1

(21199032+ 1)2(119864

ps119899120581minus119872 minus 119862ps)

+ (119864ps119899120581+119872) (119864

ps119899120581minus119872 minus 119862ps) minus 2120581119886ps

+ 119886ps minus 2119886ps119887ps119866ps119899120581(119903) = 0

(9)

where 120581 = minusℓ and 120581 = ℓ + 1 for 120581 lt 0 and 120581 gt 0 respectively




Σ (119903) = 1198601199032minus 2119892

21199032minus 1

(21199032+ 1)2 (10a)

119880 (119903) = 119886119904119903 +

119887119904

119903 (10b)


1198892

1198891199032+1

1199032(minus120581 (120581 + 1) + 2120581119887

119904+ 119887119904minus 1198872

119904)

minus 119860119904(119872 + 119864

119904

119899120581minus 119862119904) 1199032minus 1198862

1199041199032

+ 2119892119904

21199032minus 1

(21199032+ 1)2(119872 + 119864

119904

119899120581minus 119862119904)

+ (119864119904

119899120581minus119872) (119872 + 119864

119904

119899120581minus 119862119904) + 2120581119886

119904

minus 119886119904minus 2119886119904119887119904119865119904

119899120581(119903) = 0

(11)




1198891199032

+ [120576ps+120582ps

1199032+ 120594

ps1199032+ 120575

ps 21199032minus 1

(21199032+ 1)2]119866

ps119899120581(119903) = 0

(12)

where120576ps= (119864

ps119899120581+119872) (119864



2

ps

120594ps= minus 119860ps (119864


2

ps

120575ps= minus 2119892ps (119864

ps119899120581minus119872 minus 119862ps)

(13)


119866ps119899120581(119903) = 119891

ps119899(119903) exp (119892ps

120581(119903)) (14)

where

119891ps119899(119903) =

1 if 119899 = 0119899

prod

119894=1

(119903 minus 120572119899

119894) if 119899 ge 1 (15)

119892ps120581(119903) = minus120572

ps1199032+ 120573

ps ln 119903 + 120574ps ln (21199032 + 1) 120572psgt 0

(16)


120581(119903) into (14) and then


119866ps119899120581

10158401015840

(119903) = [119892ps120581

10158401015840

(119903) + 119892ps120581

10158402

(119903)

+119891ps119899

10158401015840(119903) + 2119892

ps120581

1015840(119903) 119891

ps119899

1015840(119903)

119891ps119899 (119903)

]119866ps119899120581(119903)

(17)


119866ps0120581

10158401015840

(119903) = 1

1199032[minus120573

ps+ 120573

ps2] +

1

(21199032+ 1)2

times [16120574ps21199032+ 16120572

ps120574ps1199032+ 16120573

ps120574ps1199032

minus 8120574ps1199032+ 4120574

ps+ 8120572

ps120574 + 8120573

ps120574ps]

+ 4120572ps21199032minus 2120572

ps

minus 4120572ps120573psminus 8120572

ps120574ps119866

ps0120581(119903)

(18)


minus120582ps= 120573

ps(120573

psminus 1)

minus120594ps= 4120572

ps2

minus2120575ps= 16120574

ps2+ 16120572

ps120574ps+ 16120573

ps120574psminus 8120574

ps

120575ps= 4120574

ps+ 8120572

ps120574 + 8120573

ps120574ps


psminus 4120572

ps120573psminus 8120572

ps120574ps

(19)

Equation (19) gives

120573ps=1

2(1 + radic1 minus 4120582

ps)

120572ps=1

2radicminus120594

ps

120574ps= minus 2 (120572

ps+ 120573

ps)

(20)


120575ps= 4120574

ps+ 8120572

ps120574 + 8120573

ps120574ps (21)





ps)



ps)) = 0

(22)



ℓ 120581 (ℓ 119895)119864ps0120581(fmminus1)

119860 = minus005 119860 = minus 05

1 minus1 011987812


32minus0492679105 minus0499267983

3 minus3 011988952


72minus0492762185 minus0499276263


119904= 001 119872 = 05 fmminus1 and 119862

119904= 1 fmminus1

ℓ 120581 (ℓ 119895)119864119904

0120581(fmminus1)

119860 = minus005 119860 = minus05

1 minus2 011990132

0506363937 05006362632 minus3 0119889

520506651467 0500665063

3 minus4 011989172

0506779923 05006779314 minus5 0119892

92050685245 0500685197


119862ps119864ps0120581(fmminus1)

011987812

011987532

011988952

011989172



119892ps120581(119903) = minus

1

2radicminus120594

ps1199032+1

2(1 + radic1 minus 4120582

ps)


119866ps0120581(119903) = 119873

0120581119903(12)(1+radic1minus4120582

ps)(21199032+ 1)minus2(120572

ps+120573

ps)


ps1199032)

(23b)

119865ps0120581(119903) =

1

119872 minus 119864ps0120581+ 119862ps

(119889

119889119903minus120581

119903+ 119880 (119903))119866

ps0120581(119903) (23c)


1198892119865119904

119899120581(119903)

1198891199032

+ [120576119904+120582119904

1199032+ 1205941199041199032+ 120575119904 21199032minus 1

(21199032+ 1)2]119865119904

119899120581(119903) = 0

(24)

where

120576119904= (119864119904

119899120581minus119872) (119872 + 119864

119904

119899120581minus 119862119904) + 2120581119886

119904minus 119886119904minus 2119886119904119887119904

120582119904= minus 120581 (120581 + 1) + 2120581119887

119904+ 119887119904minus 1198872

119904

120594119904= minus 119860

119904(119872 + 119864

119904

119899120581minus 119862119904) minus 1198862

119904

120575119904= 2119892119904(119872 + 119864

119904

119899120581minus 119862119904)

(25)


119865119904

119899120581(119903) = 119891

119904

119899(119903) exp (119892119904

120581(119903)) (26)

where

119891119904

119899(119903) =

1 if 119899 = 0119899

prod

119894=1

(119903 minus 120572119899

119894) if 119899 ge 1 (27)

119892119904

120581(119903) = minus120572

1199041199032+ 120573119904 ln 119903 + 120574119904 ln (21199032 + 1) 120572

119904gt 0 (28)


11986511990410158401015840

119899120581(119903)

= [119892119904

120581

10158401015840

(119903) + 119892119904

120581

10158402

(119903) +119891119904

119899

10158401015840(119903) + 2119892

119904

120581

1015840(119903) 119891119904

119899

1015840(119903)

119891119904

119899(119903)

] 119865119904

119899120581(119903)

(29)



119904= 002 119887

119904= 001 and119872 = 05 fmminus1

119862119904

119864119904

0120581(fmminus1)

011990132

011988952

011989172

011989292




011987812

011987532

011988952

011989172



119904= 002 119887

119904= 001 and 119862

119904= 1 fmminus1

119872(fmminus1) 119864119904+

119899120581(fmminus1)

011990132

011988952

011989172

011989292

0 1000638332 1000666338 1000678847 100068590902 0800637505 0800665828 080067848 080068562404 0600636677 0600665318 0600678114 060068533906 040063585 0400664807 0400677748 040068505408 0200635022 0200664297 0200677381 0200684769

0 05 1 15 2 25

r

0s320p320d52

0

001

002

003

004

005

G0120581(r)

PSS

(a)

0 05 1 15 2 25

r

0p32

0f72

0d52

0

01

02

03

04

05

06

F0120581(r)

SS

(b)



119904= 002 119887

119904= 001119872 = 05 fmminus1 and 119862

119904= 1 fmminus1



11986511990410158401015840

0120581(119903)

= 1

1199032[minus120573119904+ 1205731199042] +

1

(21199032+ 1)2

times [1612057411990421199032+ 16120572

1199041205741199041199032+ 16120573

1199041205741199041199032

minus 81205741199041199032+ 4120574119904+ 8120572119904120574119904+ 8120573119904120574119904]


0120581(119903)

(30)


120573119904=1

2(1 + radic1 minus 4120582

119904)

120572119904=1


120574119904= minus2 (120572

119904+ 120573119904)


(31)




119904(1 + radic1 minus 4120582

119904)


119904+ (1 + radic1 minus 4120582

119904)) = 0

(32)


119892119904

120581(119903) = minus

1

2radicminus1205941199041199032

+1

2(1 + radic1 minus 4120582

119904) ln 119903 minus 2 (120572119904 + 120573119904) ln (21199032 + 1)

(33a)

119865119904

0120581(119903) = 119873

0120581119903(12)(1+radic1minus4120582

119904)

times (21199032+ 1)minus2(120572119904+120573119904)


(33b)


119866119904

0120581(119903) =

1

119872 + 119864119904

0120581minus 119862119904

(119889

119889119903+120581

119903minus 119880 (119903))119865

119904

0120581(119903) (34)




119904 they


5 Conclusion




Acknowledgment


References














































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






Σ (119903) = 1198601199032minus 2119892

21199032minus 1

(21199032+ 1)2 (10a)

119880 (119903) = 119886119904119903 +

119887119904

119903 (10b)


1198892

1198891199032+1

1199032(minus120581 (120581 + 1) + 2120581119887

119904+ 119887119904minus 1198872

119904)

minus 119860119904(119872 + 119864

119904

119899120581minus 119862119904) 1199032minus 1198862

1199041199032

+ 2119892119904

21199032minus 1

(21199032+ 1)2(119872 + 119864

119904

119899120581minus 119862119904)

+ (119864119904

119899120581minus119872) (119872 + 119864

119904

119899120581minus 119862119904) + 2120581119886

119904

minus 119886119904minus 2119886119904119887119904119865119904

119899120581(119903) = 0

(11)




1198891199032

+ [120576ps+120582ps

1199032+ 120594

ps1199032+ 120575

ps 21199032minus 1

(21199032+ 1)2]119866

ps119899120581(119903) = 0

(12)

where120576ps= (119864

ps119899120581+119872) (119864



2

ps

120594ps= minus 119860ps (119864


2

ps

120575ps= minus 2119892ps (119864

ps119899120581minus119872 minus 119862ps)

(13)


119866ps119899120581(119903) = 119891

ps119899(119903) exp (119892ps

120581(119903)) (14)

where

119891ps119899(119903) =

1 if 119899 = 0119899

prod

119894=1

(119903 minus 120572119899

119894) if 119899 ge 1 (15)

119892ps120581(119903) = minus120572

ps1199032+ 120573

ps ln 119903 + 120574ps ln (21199032 + 1) 120572psgt 0

(16)


120581(119903) into (14) and then


119866ps119899120581

10158401015840

(119903) = [119892ps120581

10158401015840

(119903) + 119892ps120581

10158402

(119903)

+119891ps119899

10158401015840(119903) + 2119892

ps120581

1015840(119903) 119891

ps119899

1015840(119903)

119891ps119899 (119903)

]119866ps119899120581(119903)

(17)


119866ps0120581

10158401015840

(119903) = 1

1199032[minus120573

ps+ 120573

ps2] +

1

(21199032+ 1)2

times [16120574ps21199032+ 16120572

ps120574ps1199032+ 16120573

ps120574ps1199032

minus 8120574ps1199032+ 4120574

ps+ 8120572

ps120574 + 8120573

ps120574ps]

+ 4120572ps21199032minus 2120572

ps

minus 4120572ps120573psminus 8120572

ps120574ps119866

ps0120581(119903)

(18)


minus120582ps= 120573

ps(120573

psminus 1)

minus120594ps= 4120572

ps2

minus2120575ps= 16120574

ps2+ 16120572

ps120574ps+ 16120573

ps120574psminus 8120574

ps

120575ps= 4120574

ps+ 8120572

ps120574 + 8120573

ps120574ps


psminus 4120572

ps120573psminus 8120572

ps120574ps

(19)

Equation (19) gives

120573ps=1

2(1 + radic1 minus 4120582

ps)

120572ps=1

2radicminus120594

ps

120574ps= minus 2 (120572

ps+ 120573

ps)

(20)


120575ps= 4120574

ps+ 8120572

ps120574 + 8120573

ps120574ps (21)





ps)



ps)) = 0

(22)



ℓ 120581 (ℓ 119895)119864ps0120581(fmminus1)

119860 = minus005 119860 = minus 05

1 minus1 011987812


32minus0492679105 minus0499267983

3 minus3 011988952


72minus0492762185 minus0499276263


119904= 001 119872 = 05 fmminus1 and 119862

119904= 1 fmminus1

ℓ 120581 (ℓ 119895)119864119904

0120581(fmminus1)

119860 = minus005 119860 = minus05

1 minus2 011990132

0506363937 05006362632 minus3 0119889

520506651467 0500665063

3 minus4 011989172

0506779923 05006779314 minus5 0119892

92050685245 0500685197


119862ps119864ps0120581(fmminus1)

011987812

011987532

011988952

011989172



119892ps120581(119903) = minus

1

2radicminus120594

ps1199032+1

2(1 + radic1 minus 4120582

ps)


119866ps0120581(119903) = 119873

0120581119903(12)(1+radic1minus4120582

ps)(21199032+ 1)minus2(120572

ps+120573

ps)


ps1199032)

(23b)

119865ps0120581(119903) =

1

119872 minus 119864ps0120581+ 119862ps

(119889

119889119903minus120581

119903+ 119880 (119903))119866

ps0120581(119903) (23c)


1198892119865119904

119899120581(119903)

1198891199032

+ [120576119904+120582119904

1199032+ 1205941199041199032+ 120575119904 21199032minus 1

(21199032+ 1)2]119865119904

119899120581(119903) = 0

(24)

where

120576119904= (119864119904

119899120581minus119872) (119872 + 119864

119904

119899120581minus 119862119904) + 2120581119886

119904minus 119886119904minus 2119886119904119887119904

120582119904= minus 120581 (120581 + 1) + 2120581119887

119904+ 119887119904minus 1198872

119904

120594119904= minus 119860

119904(119872 + 119864

119904

119899120581minus 119862119904) minus 1198862

119904

120575119904= 2119892119904(119872 + 119864

119904

119899120581minus 119862119904)

(25)


119865119904

119899120581(119903) = 119891

119904

119899(119903) exp (119892119904

120581(119903)) (26)

where

119891119904

119899(119903) =

1 if 119899 = 0119899

prod

119894=1

(119903 minus 120572119899

119894) if 119899 ge 1 (27)

119892119904

120581(119903) = minus120572

1199041199032+ 120573119904 ln 119903 + 120574119904 ln (21199032 + 1) 120572

119904gt 0 (28)


11986511990410158401015840

119899120581(119903)

= [119892119904

120581

10158401015840

(119903) + 119892119904

120581

10158402

(119903) +119891119904

119899

10158401015840(119903) + 2119892

119904

120581

1015840(119903) 119891119904

119899

1015840(119903)

119891119904

119899(119903)

] 119865119904

119899120581(119903)

(29)



119904= 002 119887

119904= 001 and119872 = 05 fmminus1

119862119904

119864119904

0120581(fmminus1)

011990132

011988952

011989172

011989292




011987812

011987532

011988952

011989172



119904= 002 119887

119904= 001 and 119862

119904= 1 fmminus1

119872(fmminus1) 119864119904+

119899120581(fmminus1)

011990132

011988952

011989172

011989292

0 1000638332 1000666338 1000678847 100068590902 0800637505 0800665828 080067848 080068562404 0600636677 0600665318 0600678114 060068533906 040063585 0400664807 0400677748 040068505408 0200635022 0200664297 0200677381 0200684769

0 05 1 15 2 25

r

0s320p320d52

0

001

002

003

004

005

G0120581(r)

PSS

(a)

0 05 1 15 2 25

r

0p32

0f72

0d52

0

01

02

03

04

05

06

F0120581(r)

SS

(b)



119904= 002 119887

119904= 001119872 = 05 fmminus1 and 119862

119904= 1 fmminus1



11986511990410158401015840

0120581(119903)

= 1

1199032[minus120573119904+ 1205731199042] +

1

(21199032+ 1)2

times [1612057411990421199032+ 16120572

1199041205741199041199032+ 16120573

1199041205741199041199032

minus 81205741199041199032+ 4120574119904+ 8120572119904120574119904+ 8120573119904120574119904]


0120581(119903)

(30)


120573119904=1

2(1 + radic1 minus 4120582

119904)

120572119904=1


120574119904= minus2 (120572

119904+ 120573119904)


(31)




119904(1 + radic1 minus 4120582

119904)


119904+ (1 + radic1 minus 4120582

119904)) = 0

(32)


119892119904

120581(119903) = minus

1

2radicminus1205941199041199032

+1

2(1 + radic1 minus 4120582

119904) ln 119903 minus 2 (120572119904 + 120573119904) ln (21199032 + 1)

(33a)

119865119904

0120581(119903) = 119873

0120581119903(12)(1+radic1minus4120582

119904)

times (21199032+ 1)minus2(120572119904+120573119904)


(33b)


119866119904

0120581(119903) =

1

119872 + 119864119904

0120581minus 119862119904

(119889

119889119903+120581

119903minus 119880 (119903))119865

119904

0120581(119903) (34)




119904 they


5 Conclusion




Acknowledgment


References














































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





ℓ 120581 (ℓ 119895)119864ps0120581(fmminus1)

119860 = minus005 119860 = minus 05

1 minus1 011987812


32minus0492679105 minus0499267983

3 minus3 011988952


72minus0492762185 minus0499276263


119904= 001 119872 = 05 fmminus1 and 119862

119904= 1 fmminus1

ℓ 120581 (ℓ 119895)119864119904

0120581(fmminus1)

119860 = minus005 119860 = minus05

1 minus2 011990132

0506363937 05006362632 minus3 0119889

520506651467 0500665063

3 minus4 011989172

0506779923 05006779314 minus5 0119892

92050685245 0500685197


119862ps119864ps0120581(fmminus1)

011987812

011987532

011988952

011989172



119892ps120581(119903) = minus

1

2radicminus120594

ps1199032+1

2(1 + radic1 minus 4120582

ps)


119866ps0120581(119903) = 119873

0120581119903(12)(1+radic1minus4120582

ps)(21199032+ 1)minus2(120572

ps+120573

ps)


ps1199032)

(23b)

119865ps0120581(119903) =

1

119872 minus 119864ps0120581+ 119862ps

(119889

119889119903minus120581

119903+ 119880 (119903))119866

ps0120581(119903) (23c)


1198892119865119904

119899120581(119903)

1198891199032

+ [120576119904+120582119904

1199032+ 1205941199041199032+ 120575119904 21199032minus 1

(21199032+ 1)2]119865119904

119899120581(119903) = 0

(24)

where

120576119904= (119864119904

119899120581minus119872) (119872 + 119864

119904

119899120581minus 119862119904) + 2120581119886

119904minus 119886119904minus 2119886119904119887119904

120582119904= minus 120581 (120581 + 1) + 2120581119887

119904+ 119887119904minus 1198872

119904

120594119904= minus 119860

119904(119872 + 119864

119904

119899120581minus 119862119904) minus 1198862

119904

120575119904= 2119892119904(119872 + 119864

119904

119899120581minus 119862119904)

(25)


119865119904

119899120581(119903) = 119891

119904

119899(119903) exp (119892119904

120581(119903)) (26)

where

119891119904

119899(119903) =

1 if 119899 = 0119899

prod

119894=1

(119903 minus 120572119899

119894) if 119899 ge 1 (27)

119892119904

120581(119903) = minus120572

1199041199032+ 120573119904 ln 119903 + 120574119904 ln (21199032 + 1) 120572

119904gt 0 (28)


11986511990410158401015840

119899120581(119903)

= [119892119904

120581

10158401015840

(119903) + 119892119904

120581

10158402

(119903) +119891119904

119899

10158401015840(119903) + 2119892

119904

120581

1015840(119903) 119891119904

119899

1015840(119903)

119891119904

119899(119903)

] 119865119904

119899120581(119903)

(29)



119904= 002 119887

119904= 001 and119872 = 05 fmminus1

119862119904

119864119904

0120581(fmminus1)

011990132

011988952

011989172

011989292




011987812

011987532

011988952

011989172



119904= 002 119887

119904= 001 and 119862

119904= 1 fmminus1

119872(fmminus1) 119864119904+

119899120581(fmminus1)

011990132

011988952

011989172

011989292

0 1000638332 1000666338 1000678847 100068590902 0800637505 0800665828 080067848 080068562404 0600636677 0600665318 0600678114 060068533906 040063585 0400664807 0400677748 040068505408 0200635022 0200664297 0200677381 0200684769

0 05 1 15 2 25

r

0s320p320d52

0

001

002

003

004

005

G0120581(r)

PSS

(a)

0 05 1 15 2 25

r

0p32

0f72

0d52

0

01

02

03

04

05

06

F0120581(r)

SS

(b)



119904= 002 119887

119904= 001119872 = 05 fmminus1 and 119862

119904= 1 fmminus1



11986511990410158401015840

0120581(119903)

= 1

1199032[minus120573119904+ 1205731199042] +

1

(21199032+ 1)2

times [1612057411990421199032+ 16120572

1199041205741199041199032+ 16120573

1199041205741199041199032

minus 81205741199041199032+ 4120574119904+ 8120572119904120574119904+ 8120573119904120574119904]


0120581(119903)

(30)


120573119904=1

2(1 + radic1 minus 4120582

119904)

120572119904=1


120574119904= minus2 (120572

119904+ 120573119904)


(31)




119904(1 + radic1 minus 4120582

119904)


119904+ (1 + radic1 minus 4120582

119904)) = 0

(32)


119892119904

120581(119903) = minus

1

2radicminus1205941199041199032

+1

2(1 + radic1 minus 4120582

119904) ln 119903 minus 2 (120572119904 + 120573119904) ln (21199032 + 1)

(33a)

119865119904

0120581(119903) = 119873

0120581119903(12)(1+radic1minus4120582

119904)

times (21199032+ 1)minus2(120572119904+120573119904)


(33b)


119866119904

0120581(119903) =

1

119872 + 119864119904

0120581minus 119862119904

(119889

119889119903+120581

119903minus 119880 (119903))119865

119904

0120581(119903) (34)




119904 they


5 Conclusion




Acknowledgment


References














































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119904= 002 119887

119904= 001 and119872 = 05 fmminus1

119862119904

119864119904

0120581(fmminus1)

011990132

011988952

011989172

011989292




011987812

011987532

011988952

011989172



119904= 002 119887

119904= 001 and 119862

119904= 1 fmminus1

119872(fmminus1) 119864119904+

119899120581(fmminus1)

011990132

011988952

011989172

011989292

0 1000638332 1000666338 1000678847 100068590902 0800637505 0800665828 080067848 080068562404 0600636677 0600665318 0600678114 060068533906 040063585 0400664807 0400677748 040068505408 0200635022 0200664297 0200677381 0200684769

0 05 1 15 2 25

r

0s320p320d52

0

001

002

003

004

005

G0120581(r)

PSS

(a)

0 05 1 15 2 25

r

0p32

0f72

0d52

0

01

02

03

04

05

06

F0120581(r)

SS

(b)



119904= 002 119887

119904= 001119872 = 05 fmminus1 and 119862

119904= 1 fmminus1



11986511990410158401015840

0120581(119903)

= 1

1199032[minus120573119904+ 1205731199042] +

1

(21199032+ 1)2

times [1612057411990421199032+ 16120572

1199041205741199041199032+ 16120573

1199041205741199041199032

minus 81205741199041199032+ 4120574119904+ 8120572119904120574119904+ 8120573119904120574119904]


0120581(119903)

(30)


120573119904=1

2(1 + radic1 minus 4120582

119904)

120572119904=1


120574119904= minus2 (120572

119904+ 120573119904)


(31)




119904(1 + radic1 minus 4120582

119904)


119904+ (1 + radic1 minus 4120582

119904)) = 0

(32)


119892119904

120581(119903) = minus

1

2radicminus1205941199041199032

+1

2(1 + radic1 minus 4120582

119904) ln 119903 minus 2 (120572119904 + 120573119904) ln (21199032 + 1)

(33a)

119865119904

0120581(119903) = 119873

0120581119903(12)(1+radic1minus4120582

119904)

times (21199032+ 1)minus2(120572119904+120573119904)


(33b)


119866119904

0120581(119903) =

1

119872 + 119864119904

0120581minus 119862119904

(119889

119889119903+120581

119903minus 119880 (119903))119865

119904

0120581(119903) (34)




119904 they


5 Conclusion




Acknowledgment


References














































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





11986511990410158401015840

0120581(119903)

= 1

1199032[minus120573119904+ 1205731199042] +

1

(21199032+ 1)2

times [1612057411990421199032+ 16120572

1199041205741199041199032+ 16120573

1199041205741199041199032

minus 81205741199041199032+ 4120574119904+ 8120572119904120574119904+ 8120573119904120574119904]


0120581(119903)

(30)


120573119904=1

2(1 + radic1 minus 4120582

119904)

120572119904=1


120574119904= minus2 (120572

119904+ 120573119904)


(31)




119904(1 + radic1 minus 4120582

119904)


119904+ (1 + radic1 minus 4120582

119904)) = 0

(32)


119892119904

120581(119903) = minus

1

2radicminus1205941199041199032

+1

2(1 + radic1 minus 4120582

119904) ln 119903 minus 2 (120572119904 + 120573119904) ln (21199032 + 1)

(33a)

119865119904

0120581(119903) = 119873

0120581119903(12)(1+radic1minus4120582

119904)

times (21199032+ 1)minus2(120572119904+120573119904)


(33b)


119866119904

0120581(119903) =

1

119872 + 119864119904

0120581minus 119862119904

(119889

119889119903+120581

119903minus 119880 (119903))119865

119904

0120581(119903) (34)




119904 they


5 Conclusion




Acknowledgment


References














































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Documents

Research Article Dirac Equation under Scalar and Vector ...downloads.hindawi.com/journals/ahep/2014/831938.pdf · Research Article Dirac Equation under Scalar and Vector Generalized