Exact solutions of the Manning–Rosen potential plus a ring-shaped like potential for the Dirac

equation: spin and pseudospin symmetry

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Phys. Scr. 87 (2013) 025703 (10pp) doi:10.1088/0031-8949/87/02/025703

Exact solutions of the Manning–Rosenpotential plus a ring-shaped like potentialfor the Dirac equation: spin andpseudospin symmetryS Asgarifar and H Goudarzi

Department of Physics, Faculty of Science, Urmia University, PO Box 165, Urmia, Iran

E-mail: h.goudarzi@urmia.ac.ir and goudarzia@phys.msu.ru

Received 14 August 2012Accepted for publication 19 December 2012Published 17 January 2013Online at stacks.iop.org/PhysScr/87/025703

AbstractThe analytical bound state solutions of the Dirac equation for the Manning–Rosen potentialplus a ring-shaped like potential in spin and pseudospin symmetries are investigated by aproper approximation to the centrifugal term. Using the Nikiforov–Uvarov method, an explicitform of the energy eigenvalue and the corresponding energy eigenstates, expressed in terms ofthe Jacobi polynomials, are derived. The normalization coefficients of wave functions for spinand pseudospin symmetry solutions are found explicitly.

PACS numbers: 03.65.Ge, 03.65.Pm, 02.30.Gp

1. Introduction

Forty-three years ago, to clarify a quasi-degeneracybetween single-nucleon states in heavy nuclei, pseudospinsymmetry was put forward in nuclear physics [1, 2].These degenerative single-nucleon states (n, l, j = l + 1

2 )

and (n − 1, l + 2, j = l + 32 ) (where n, l and j are the

radial, orbital and total angular quantum numbers of thesingle nucleon, respectively) are considered as a doubletstructure with (n = n − 1, l = l + 1, j = l ±

12 ), where l and

s =12 are the pseudo-orbital angular momentum and the

pseudospin quantum numbers, respectively. Deformation,superdeformation, identical bands and magnetic momentin the nuclear structures have been successfully explainedby using this doublet structure [3–8]. More than 15 yearsago, the relativistic feature of the symmetry was noted byGinocchio [9]. Ginocchio showed that the pseudo-orbitalangular momentum l is nothing but the usual orbital angularmomentum l of the lower component of the Dirac spinor.

One of the inherent characteristics of the relativisticmean field theory is that an attractive scalar potential S(Er)and a repulsive vector potential V (Er) are nearly equal inmagnitude but different in sign [10]. Ginocchio indicatedthat this near equality V (Er)+ S(Er)∼ 0 leads to pseudospinsymmetry in nuclei (see [10] and references therein).

Meng et al [11] have proved that exact pseudospin symmetryoccurs in the Dirac equation when d

d r6(Er)= 0, where∑(Er)= V (Er)+ S(Er)= constant, following that the exact spin

symmetry occurs in the Dirac equation when dd r1(Er)= 0,

where 1(Er)= V (Er)− S(Er)= constant.After the pioneering work of Ginocchio, pseudospin and

spin symmetry solutions have been investigated by solvingthe Dirac equation in terms of different methods for exactlysolvable potentials such as a new oscillatory ring-shapednoncentral, Woods–Saxon, Morse, Eckart, Poschl–Tellerpotentials [12] and so on, since solutions of the Dirac equationwith pseudospin and spin symmetries are very important todescribe the nuclear shell structure.

Actually, the Manning–Rosen potential is one of themost useful and convenient models for studying the energyeigenvalues of diatomic molecules [13]. As an empiricalpotential, the Manning–Rosen potential gives an excellentdescription of the interaction between the two atoms ina diatomic molecule; in addition it is very reasonableto describe the interactions close to the surface. On theother hand, ring-shaped potentials, which have applicationin ring-shaped cyclic polyene and benzene-type organicmolecules, have been considered to solve the Schrodingerand Dirac equations [14–16]. Hence, it is worth investigatingthe solution of the Dirac equation for the Manning–Rosen

0031-8949/13/025703+10$33.00 Printed in the UK & the USA 1 © 2013 The Royal Swedish Academy of Sciences

Phys. Scr. 87 (2013) 025703 S Asgarifar and H Goudarzi

potential plus a ring-shaped like potential under conditions ofspin and pseudospin symmetries. In the spherical coordinatessuch a potential can be given as

V (r, θ)=h2

2µb2

[α (α− 1) e−2 r

b(1 − q e−

rb)2 −

A e−rb(

1 − q e−rb)

+ρ1

r2sin2θ+ρ2 cos θ

r2sin2θ

], (1)

where q is the deformation parameter and for simplicity incalculation we set q = 1. ρ1 and ρ2 are the positive constants.A and α are two-dimensionless parameters [17, 18]. Screeningparameter b has dimension of length and corresponds to thepotential ring [19, 20] and µ=

m1m2m1+m2

is the reduced mass.For specific cases, ρ1 = 0 and ρ2 = 0, the potential reduces toManning–Rosen potential [21]. The Hulthen potential can begiven as an example for the Manning–Rosen potential plus aring-shaped like potential by setting α = 0 or 1, q = 1, ρ1 = 0and ρ2 = 0 [22–24]. However, in this work, we consider thetwo last terms in equation (1) in addition to the well-knownManning–Rosen potential.

Many authors have used different methods to study thepartially exactly solvable and exactly solvable Schrodinger,Klein–Gordon and Dirac equation. These methods includethe supersymmetric and shape invariant method [25], thevariational method [26], the path integral approach [27],the standard methods [28], the asymptotic iterationmethod [29], the exact quantization rule [30–32], thehypervirial perturbation [33], the shifted 1/N expansion [34]and the modified shifted 1/N expansion [35], the seriesmethod [36], smooth transformation [37], the algebraicapproach [38], the perturbative treatment [39, 40], theNikiforov–Uvarov (NU) method [41] and others. In thiswork, we intend to use the algebraic technique NU to solvethe Dirac equation for scalar and vector Manning–Rosenpotential plus a ring-shaped like potential with spin andpseudospin symmetries.

This work is organized as follows. In section 2, wegive a brief introduction to the Dirac formalism and obtainthe second-order Schrodinger-like differential equations forthe radial and angular functions in the pseudospin andspin symmetry limits. In section 3, we briefly reviewthe NU method. In section 4, we derive l 6= 0 boundstate eigensolutions of the Manning–Rosen potential plus aring-shaped like potential via the NU method. The concludingremarks are given in section 5.

2. Spin and pseudospin symmetry limits of the Diracequation

In the relativistic mean field theory, the Dirac equation for asingle nucleon with mass M moving in a repulsive vector V (Er)and an attractive scalar potential S(Er) is given by (h = c = 1)[

α · p +β (M + S (Er))+ V (Er)]ψnk (Er)= Enkψnk (Er) , (2)

where Enk is the relativistic bound energy of the systemand p = −i h∇ is the three-dimensional momentum operator.α and β are the 4 × 4 usual Dirac matrices. The total

angular momentum operator J and spin–orbit couplingoperator K = −β(σ · L + 1), where L is the orbital angularmomentum, of the spherical nucleons commute with theDirac Hamiltonian. The eigenvalues of spin–orbit couplingoperator are k = ( j + 1

2 ) > 0 and k = −( j + 12 ) < 0 for the

unaligned spin (p1/2, d3/2, . . .), j = l −12 and the aligned

spin (s1/2, p3/2, . . .), j = l + 12 , respectively (the set of

(H, K , J 2, Jz) can be taken as the complete set of theconservative quantities). Hence, the Dirac spinors can bewritten according to radial quantum number n and spin–orbitcoupling quantum number k as follows:

ψnk (Er)=1

r

(Fnk (r) Y l

jm (θ, ϕ)

i Gnk (r) Y ljm (θ, ϕ)

)=

(fnk (Er)gnk (Er)

). (3)

Substituting equation (3) into equation (2), we obtain thefollowing coupled differential equations:

fnk(Er)=(σ · p)[

Enk − M −6(Er)]gnk(Er), (4)

gnk(Er)=(σ · p)[

Enk + M −1(Er)] fnk(Er), (5)

where 6(Er)= V (Er)+ S(Er) and 1(Er)= V (Er)− S(Er).The spin symmetry occurs in the Dirac equation when

1(Er)= S(Er)− V (Er)= Cs = Const. By eliminating gnk(Er)from equation (4), the following uncoupled equation for theupper component can be obtained:{

p2−[Enk − M −6(Er)

][Enk + M − Cs]

}fnk(Er)= 0, (6)

where we consider 6(Er)= W (r)+ W (θ)

r2 . The function fnk(Er)can be separated as

fnk (Er)=1

rF(r)H (θ)8 (φ) 3ξ (7)

and the second-order differential equations are found asfollows:

d28(φ)

dφ2+ m28(φ)= 0, (8)

d2 H (θ)

dθ2+

cos θ

sin θ

dH (θ)

dθ+

[l (l + 1)−

m2

sin2θ− γW (θ)

]H(θ)= 0, (9)

d2 F(r)

dr2+

[β2

−l (l + 1)

r2− γW (r)

]F(r)= 0 (10)

With

γ = Enk + M − Cs and β2= (Enk − M) (Enk + M − Cs) ,

(11)

where m and l are separation constants and k(k + 1)= l(l + 1).As mentioned in the previous section, the pseudospin

symmetry occurs in equation (2) when 6(Er)= S(Er)+ V (Er)=

Cps = Const and pseudo-orbital angular momentum is thenormal orbital angular momentum of the lower component of

2

Phys. Scr. 87 (2013) 025703 S Asgarifar and H Goudarzi

the Dirac spinor. Thus, we deal with the lower component ofthe Dirac spinor to investigate the pseudospin symmetry in theDirac phenomenology. By eliminating fnk(Er) in equation (5),the following uncoupled equation for the lower componentcan be obtained:{

p2−[Enk − M − Cps

] [Enk + M −1(Er)

]}gnk (Er)= 0.

(12)

We take 1(Er)= W (r)+ W (θ)

r2 . In the spherical coordinates thelower component of the Dirac wave functions can be writtenas follows:

gnk (Er)=1

rG (r) T (θ)8 (φ) 3ξ , (13)

where ξ = ±12 is the projection of pseudo-angular momentum

on the z-axis and 3ξ is two component spinors, i.e.(

10

)or(

01

). However, pseudo-orbital angular momentum l and

pseudospin s can be separately seen in the lower spinorcomponent. Inserting equation (13) into equation (12), weobtain three second-order differential equations for theangular and the radial wave functions as

d28(φ)

dφ2+ m28(φ)= 0, (14)

d2T (θ)

dθ2+

cos θ

sin θ

dT (θ)

dθ+

[l(l+1)−

m2

sin2θ−γW (θ)

]T (θ)= 0,

(15)

d2G(r)

dr2+

[β2

−l(l + 1)

r2− γW (r)

]G(r)= 0 (16)

with

γ = En′k − M − Cps and β2= (En′k + M)

(En′k − M − Cps

),

(17)

and m and l are the separation constants andk(k − 1)= l(l + 1).

3. The Nikiforov–Uvarov method

The NU method is briefly outlined here and the detailscan be found in [41]. This method was proposed tosolve the second-order differential wave equation of thehypergeometric type. In this method, after employingan appropriate coordinate transformation, the second-orderdifferential equation can be given in the following form:

d2ψ(s)

ds2+τ (s)

σ (s)

dψ(s)

ds+σ (s)

σ 2(s)ψ(s)= 0, (18)

where σ(s) and σ (s) are at most second-degree polynomialsand τ (s) is a first-degree polynomial. In the NU method,the second-order differential equation can be reducedto hypergeometric type by using ψn(s)= ϕn(s)yn(s) and

choosing an appropriate function as ϕn(s) that must satisfya logarithmic derivative

ϕ′(s)−

(π(s)

σ (s)

)ϕ(s)= 0, (19)

consequently equation (18) can be reduced into thehypergeometric type

d2 yn(s)

ds2+τ(s)

σ (s)

dyn(s)

ds+

λ

σ(s)yn(s)= 0, (20)

where yn(s) is the hypergeometric-type function whosepolynomial solutions are given by the Rodrigues relation

yn(s)=Bn

ρ(s)

dn

d sn

[σ n(s)ρ (s)

], (21)

where Bn is the normalization constant and the weightfunction ρ (s) must satisfy the condition

[σ(s)ρ (s)]′ = τ(s)ρ (s). (22)

The function π(s) and the parameter λ required for thismethod are defined as

π(s)=σ ′(s)− τ (s)

2±

√(σ ′(s)− τ (s)

2

)2

− σ (s)+ kσ(s),

(23)

λ= k +π ′(s). (24)

The expression under the square root in equation (23) must bethe square of a polynomial of first degree [41] because of thatπ(s) is the first-degree polynomial. Thus, discriminant of thesquare root has to be zero. Using this condition, an equationfor k is found. Then, π(s) can be easily obtained from thecorresponding k values. So we have a new eigenvalue equation

λ= λn = −nτ ′(s)−n (n − 1)

2σ ′′(s), n = 0, 1, 2, . . . , (25)

where the derivation of the function τ(s), which is defined as

τ(s)= τ (s)+ 2π(s) (26)

should be negative and by comparing (24) and (25), we canobtain the energy eigenvalues.

4. Bound state solution of the Manning–Rosenpotential plus a ring-shaped like potential

For the Manning–Rosen potential plus a ring-shaped likepotential, the relations of W(θ) and W (r) are defined

W (r)=1

2µb2

[α (α− 1) e−2 r

b(1 − q e−

rb)2 −

A e−rb(

1 − q e−rb)] ,

W (θ)=1

2µb2

[ρ1 + ρ2 cos θ

sin2θ

].

(27)

Now, we try to obtain the normalized wave function andcorresponding energy eigenvalue for such potential via the

3

Phys. Scr. 87 (2013) 025703 S Asgarifar and H Goudarzi

NU method in the spin and pseudospin symmetry cases.Taking into account the boundary condition 8(φ + 2π)=

8(φ), solution of the φ-dependent differential equation canbe obtained immediately as

8m(φ)=1

√2π

ei mφ; m = 0,±1,±2, . . . . (28)

Note that, H(θ) for θ = 0, π must have a finite value. In orderto obtain the solution of equation (9), we introduce a newvariable S = cos θ and set

l (l + 1)= χ,γρ2

2µb2= %, χ − m2

−γρ1

2µb2= η. (29)

Then, equation (9) becomes

d2 H(S)

dS2−

2S

(1 − S2)

dH(S)

dS+

1

(1 − S2)2

[−χ S2

− %S + η]

H(S)= 0. (30)

Comparing equation (30) with equation (18), we obtain thefollowing relations:

τ (s)= −2S, σ (s)=(1 − S2

), σ (s)= −χ S2

− %S + η.

(31)

Substituting these expressions into equation (24), we obtainthat

π (s)= ±

√(χ − k) S2 + %S + (k − η). (32)

As mentioned, discriminant of the square root has to be zero.Then, the function π(s) is obtained in the following fourpossible relations:

π (s)=

√χ −

(χ+η)+√(χ−η)2−%

2 S +

√(χ+η)+

√(χ−η)2−%

2 − η,

k− =(χ+η)+

√(χ−η)2−%

2 ,

−

√χ −

(χ+η)+√(χ−η)2−%

2 S −

√(χ+η)+

√(χ−η)2−%

2 − η,

k− =(χ+η)+

√(χ−η)2−%

2 ,√χ −

(χ+η)−√(χ−η)2−%

2 S +

√(χ+η)−

√(χ−η)2−%

2 − η,

k+ =(χ+η)−

√(χ−η)2−%

2 ,

−

√χ −

(χ+η)−√(χ−η)2−%

2 S −

√(χ+η)−

√(χ−η)2−%

2 − η,

k+ =(χ+η)−

√(χ−η)2−%

2 .

From equation (20) one can obtain for the function τ(s)as

τ (s)=

2

[(√χ−

(χ+η)+√(χ−η)2−%

2 −1

)S+

√(χ+η)+

√(χ−η)2−%

2 − η

],

k− =(χ+η)+

√(χ−η)2−%

2 ,

−2

[(√χ−

(χ+η)+√(χ−η)2−%

2 +1

)S−

√(χ+η)+

√(χ−η)2−%

2 −η

],

k− =(χ+η)+

√(χ−η)2−%

2 ,

2

[(√χ−

(χ+η)−√(χ−η)2−%

2 −1

)S+

√(χ+η)+

√(χ−η)2−%

2 −η

],

k+ =(χ+η)−

√(χ−η)2−%

2 ,

−2

[(√χ−

(χ+η)−√(χ−η)2−%

2 +1

)S+

√(χ+η)+

√(χ−η)2−%

2 −η

],

k+ =(χ+η)−

√(χ−η)2−%

2 .

(33)

In the NU method, τ ′(s) < 0 must be satisfied in order toobtain a physical solution.

Thus, one can choose the suitable for k through abovevalues to satisfy this condition

k+ =

(χ + η)−√(χ − η)2 − %

2, (34)

which leads to

π(s)= −

√√√√χ −

(χ + η)−√(χ − η)2 − %

2S

−

√√√√ (χ + η)−√(χ − η)2 − %

2− η (35)

and function τ(s) is calculated as

τ(s)= − 2

1 +

√√√√χ −

(χ + η)−√(χ − η)2 − %

2

S

− 2

√√√√ (χ + η)−√(χ − η)2 − %

2− η. (36)

Using equations (22) and (25), we obtain

λn = 2n

√√√√χ −

(χ + η)−√(χ − η)2 − %

2

+ n (n − 1) ; n = 0, 1, 2, . . . , (37)

4

Phys. Scr. 87 (2013) 025703 S Asgarifar and H Goudarzi

λ=

(χ + η)−√(χ − η)2 − %

2

−

√√√√χ −

(χ + η)−√(χ − η)2 − %

2. (38)

In order to find an expression, which is relating to Enk , theright-hand sides of equations (37) and (38) must be comparedwith each other. In this case, the result obtained will dependon the ring-shaped like potential constants as well as the usualquantum numbers

l =−1

2±

√(2n + 1) L1 + L2

1 + n (n − 1)+1

4, (39)

where

L1 =

(m2 +

ρ1

2µb2(Enk + M − Cs)

)

+

√(m2 +

ρ1

2µb2(Enk + M − Cs)

)2

−ρ2

2µb2(Enk + M − Cs).

(40)

The separation constant l in equation (39) contains thecontributions that come from the θ -dependent part of thering-shaped like potential. To obtain the θ -dependent angularwave function, we should first calculate ϕ(s) and y(s). ϕ(s)can be found by using equation (26) as

ϕ(s)= (1 − S)12 (41+61)(1 + S)

12 (41−61) . (41)

Using equations (23), (31) and (36), we obtain the followingweight function:

ρ(S)= 2(1 − S)12 (41+61)(1 + S)

12 (41−61). (42)

In the above equations 41 and 61 are defined as

41 =

√√√√χ −

(χ + η)−√(χ − η)2 − %

2,

∑1=

√√√√ (χ + η)−√(χ − η)2 − %

2− η.

(43)

After calculating the weight function ρ (S), we can obtain thesolution of hypergeometric type equation (19)

ynk (S) = Dnk (1 − S)−

(41+61

2

)(1 + S)

−

(41−61

2

)dn

d Sn

×

[(1 + S)

n+(41−61

2

)(1 − S)

n+(41+61

2

)], (44)

where Dnk is the normalization constant. The ynk(S) can beexpressed in terms of the Jacobi polynomial, yielding

ynk(S)= Nnk P

(41+61

2 ,41−61

2

)n (S), (45)

where Nnk =2nn!(−1)n Dnk is a new normalization constant and

can be determined as in the appendix A. Finally, we obtain the

θ -dependent wave function in terms of the Jacobi polynomialby setting S = cos θ as follows:

Hnk (θ) = Nnk (1 − cos θ)12 (41+61) (1 + cos θ)

12 (41−61)

× P

(41+61

2 ,41−61

2

)n (cos θ). (46)

Now, we study the radial equation given in equation (10)taking into account the boundary condition F(0)= F(∞)

= 0. Since the radial equation with the Manning–Rosen plus aring-shaped like potential has no analytical solution for l = 0states, an approximation to the centrifugal term has to bemade. The good approximation for 1

r2 in the centrifugal barrieris taken as [19, 20, 42]

1

r2≈

1

b2

[e−

rb(

1 − e−rb)2

](47)

in a short potential range. Considering above approximationand introducing a new variable S = e−

rb , equation (10)

becomes

d2 F(S)

d S2+(1 − S)

S (1 − S)

dF(S)

d S+

1

[S (1 − S)]2

×[C S2

− DS − B]

F(r)= 0, (48)

where

B = − b2β2,C = B +γ

2µ[A −α (α− 1)] ,

D = 2B − l (l + 1)+γ

2µA. (49)

Comparing equation (48) with equation (18) and trackingsimilar calculations used to find out the solution of theangle-dependent second-order differential equation, we obtainthe following polynomials:

τ (S) = (1 − S) , σ (S)=S (1−S) , σ (S)=C S2− DS−B,

π (S) = −

(ϒ +

1

2

)S −

√B,

τ (S) = − 2 (ϒ + 1) S +(

1 − 2√

B),

λn = n2 + n (2ϒ + 1) ,

λ= −

(ϒ2 +ϒ + C +

1

4

), (50)

where

ϒ =

√√√√1

4− C + (2B + D)+

√4B

(B + D − C +

1

4

). (51)

Recalling B,C, D and ϒ from equations (49) and (51) andcomparing equation (50), we obtain the following energyeigenvalue equation:

− 4b2β2− l (l + 1)+ n2 + n +

h2γ

2µA +

1

2

+

√−4b2β2

(−2b2β2−l (l + 1)+

h2γ

2µα (α− 1)+

1

4

)+(2n + 1)

5

Phys. Scr. 87 (2013) 025703 S Asgarifar and H Goudarzi

×

√√√√√√−3b2β2

− l (l + 1)+ h2γ

2µ α (α− 1)+ 14

+

√−4b2β2

(−2b2β2 − l (l + 1)+ h2γ

2µ α (α− 1)+ 14

)

= 0. (52)

By setting l from equation (39), one can obtain theexact energy spectrum. Let us now find the correspondingeigenfunctions for the radial part. Using the same procedureused to find the θ -dependent angular wave function, we obtainϕ(S) and ρ(S) for radial equation as

ϕ (S)= (S)−√

B (1 − S)(

12 +ϒ+

√B), (53)

ρ (S)= (S)−2√

B (1 − S)2(ϒ+

√B). (54)

We can obtain the solution of hypergeometric typeequation (19)

χnk (S) =tnk

(S)−2√

B (1 − S)2(ϒ+

√B) dn

d Sn

×

[(S)n−2

√B (1 − S)

2(ϒ+

√B)

+n], (55)

where tnk is the normalization constant. The χnk(S) can beexpressed in terms of the Jacobi polynomial, yielding

χnk (S)= Qnk P

(−2

√B,2

(ϒ+

√B))

n (1 − 2S) , (56)

where Qnk =n!

(−1)n2n tnk is a new normalization constant anddetermined in appendix B. Finally, we obtain the radial wavefunction in terms of the Jacobi polynomial by setting S = e−

rb

as follows:

Fnk(r)= Qnk er√

Bb

(1 − e−

rb

)( 12 +ϒ+

√B)

P

(−2

√B,2

(ϒ+

√B))

n

×

(1 − 2 e−

rb

), (57)

where B and ϒ are defined from equations (49) and (51).In the case of pseudospin symmetry, by substituting

relations for W (θ) and W (r) equations (14)–(16), we obtain

8m(φ)=1

√2π

ei mφ; m = 0,±1,±2, . . . , (58)

d2T (θ)

dθ2+

cos θ

sin θ

dT (θ)

dθ+

[l(l + 1)−

m2

sin2θ−γ

b2

×

[ρ1 + ρ2 cos θ

sin2θ

]]T (θ)= 0, (59)

d2G(r)

dr2+

β2−

l(

l + 1)

r2−

γ

2µb2

×

[α (α− 1) e−2 r

b(1 − e−

rb)2 −

A e−rb(

1 − e−rb)]]G(r)= 0. (60)

To avoid repetition in procedure obtaining solution for caseof pseudospin symmetry, by using the below parameter map

we can obtain solution of the θ -dependent equation for thecase of pseudospin symmetry, directly from solution of spinsymmetry

m → m, l → l, H (θ)→ T (θ) , γ → γ=En′k − M − Cps.

(61)

Following the previous results with the above transformation,we finally obtain the separation constant l for the θ -dependentequation with pseudospin symmetry:

l =−1

2±

√(2n′ + 1) L2 + L2

2 + n′(n′ − 1)+1

4, (62)

where

L2 =

(m2 +

ρ1

2µb2

(En′k − M − Cps

))

+

√(m2+

ρ1

2µb2

(En′k−M−Cps

))2

−ρ2

2µb2

(En′k − M − Cps

)(63)

and the wave function in terms of the Jacobi polynomial canbe obtained as

Tn′k (θ) = Hn′k (1 − cos θ)12 (42+62) (1 + cos θ)

12 (42−62)

× P

(42+62

2 ,42−62

2

)n (cos θ) , (64)

where 42 and 62 are defined as

42 =

√χ −

(χ+η)−√(χ−η)

2−%

2 ,∑2

=

√(χ+η)−

√(χ−η)

2−%

2 − η,

(65)

where

l(l + 1)= χ ,γ ρ2

2µb2= %, χ − m2

−γ ρ1

2µb2= η (66)

and using the same procedure in appendix A, we obtainnormalization constant Hn′k =

1√σ(n′)

σ(n′)= 2

(−

1

2

)n′

×

(0(n′ +

(42+62

2

)+ 1))20(n′ +

(42−62

2

)+ 1)0(n′ + 1)

n′!0(n′ +42 + 1)

×

n′∑p,r=0

(−1)p+r

(2)00(r + 1)0(n′ − r + 1)

×0(n′ +42 + r + 1)[

0(r +

(42+62

2

)+ 1)0(p + 1)0

(n′ +

(42+62

2

)− p + 1

)0(n′

− p + 1)0(

p +(42−62

2

)+ 1) ]

×

[−

1

262 + n′ − 2p + r + 1+ 2

∞∑x=0

(−1)x

×(262 + n′

− 2p + r)!(42 −62 + p −

12 + x

)!

(262 + n′ − 2p + r + x + 3)!(42 −62 + p −

32

)!

].

6

Phys. Scr. 87 (2013) 025703 S Asgarifar and H Goudarzi

The boundary condition G(0)= G(∞)= 0 is applied to G(r)by using approximation assumed in the previous sectionand perform similar procedure and also by using the belowparameter map we can obtain solution of the radial equation:

m → m, l → l, F (r)→ G (r) ,

γ → γ = En′k − M − Cps, β2→ β2

= (En′k + M)(En′k − M − Cps

). (67)

The energy eigenvalue equation for the radial equation isfound as

− 4b2β2− l(l + 1)+ n′2 + n′ +

γ

2µA +

1

2

+

√−4b2β2

(−2b2β2 − l(l + 1)+

γ

2µα (α− 1)+

1

4

)+(2n′ + 1

)×

√√√√√−3b2β2

−l(l + 1)+ γ

2µα (α−1)+ 14

+

√−4b2β2

(−2b2β2−l (l + 1)+ γ

2µα (α− 1)+ 14

)= 0.

Finally, we obtain the radial wave function in terms of theJacobi polynomial as follows:

Gn′k (r) = Wn′k er√�

b (1 − e−rb )(

12 +1+

√�)

× P(−2

√�,2(1+

√�))

n (1 − 2 e−rb ), (68)

where � and 1 are defined as

�=−b2β2,1=

√√√√1

4−2+ (2�+ω)+

√4�

(�+ω−2+

1

4

)(69)

that we have

2=�+γ

2µ[A −α (α− 1)] , ω = 2�− l

(l + 1

)+γ

2µA,

(70)

and Wn′k is the normalization constant.

5. Conclusions

In this paper, we have proposed a new exactly solvablepotential which consists of the Manning–Rosen potential plusa ring-shaped like potential and introduced a quasi-analyticsolutions to the Dirac equation with pseudospin and spinsymmetries via the NU method. The radial and angularwave functions are obtained in terms of special orthogonalfunctions in the pseudospin and spin symmetry limits.Calculating the radial wave functions, we have applied aproper approximation to the too singular orbital centrifugalterm ∼ r−2, that this means the quasi-analytical solution.As a remarkable point, the normalization constant of wavefunctions is exactly obtained for such complicated potential.This potential model may have some applications in different

fields, such as for describing interactions between deformedpair of nuclei in the nuclear physics.

Appendix A

Equation (50) satisfies the requirements; Hnk(S)= 0 as S = 1(θ → 0) and −1 (θ → π). Therefore, the wave functionsHnk(S) is valid physically in the closed interval S ∈

[−1, 1] or θ ∈ [0, π]. Further, the wave functions satisfy thenormalization condition∫ π

0|Hnk (θ)|

2 d θ = 2∫ 1

0

(1 − S2

)− 12 |Hnk (S)|

2 d S = 1,

(A.1)where

Hnk (S) = Nnk (1 − S)12 (41+61) (1 + S)

12 (41−61)

× P

(41+61

2 ,41−61

2

)n (S).

Nnk can be determined by

2N 2nk

∫ 1

0(1 − S)(41+61−

12 ) (1 + S)(41−61−

12 )

×

∣∣∣∣∣ P

(41+61

2 ,41−61

2

)n (S)

∣∣∣∣∣2

d S = 1. (A.2)

The Jacobi polynomials P (ω,υ)n (S) can be explicitly written in

two different ways [43, 44]:

P (ω,υ)n (ξ) = 2−n

n∑p=0

(−1)n−p

(n +ω

p

)(n + υn − p

)(1 − ξ)n−p (1 + ξ)p , (A.3)

P (ω,υ)n (ξ) =

0 (n +ω + 1)

n!0 (n +ω + υ + 1)

n∑r=0

(nr

)

×0 (n +ω + υ + r + 1)

0 (r +ω + 1)

(ξ − 1

2

)r

, (A.4)

where ( nr )=

n!r !(n−r)! =

0(n+1)0(r+1)0(n−r+1) . Thus, the expressions

for P(41+61

2 ,41−61

2 )n (S) become

P

(41+61

2 ,41−61

2

)n (S)=

(−

1

2

)n n∑p=0

(−1)p

×0(n +

(41+61

2

)+ 1)

0 (p + 1) 0(n +

(41+61

2

)− p + 1

)×

0(n +

(41−61

2

)+ 1)

0 (n − p + 1) 0(

p +(41−61

2

)+ 1) (1 − S)n−p (1 + S)p,

P

(41+61

2 ,41−61

2

)n (S)=

0(n +

(41+61

2

)+ 1)

n!0 (n +41 + 1)

×

n∑r=0

0 (n + 1) 0 (n +41 + r + 1)

0 (r + 1) 0 (n − r + 1) 0(r +

(41+61

2

)+ 1) ( S − 1

2

)r

.

7

Phys. Scr. 87 (2013) 025703 S Asgarifar and H Goudarzi

Substituting above relations in equation (A.2), one obtainseasily

2N2

nk

(−

1

2

)n

×

(0(n +

(41+61

2

)+ 1))20(n +

(41−61

2

)+ 1)0 (n + 1)

n!0 (n +42 + 1)

×

n∑p,r=0

(−1)p+r

(2)r 0 (r + 1) 0 (n − r + 1),

where

Ink (p, r)=∫ 1

0(1 − S)(41+61+n−p+r−

12 ) (1+S)(41−61+p−

12 ) d S.

(A.5)

If we introduce a =41 +61 + n − p + r −12 and b =41 −

61 + p −12 and c = 261 + n − 2p + r , then equation (A.5)

becomes

Ink (p, r)=

∫ 1

0(1 − S)c

(1 − S2

)bd S. (A.6)

Via integration by parts we obtain

Ink (p, r)= −1

261 + n − 2p + r + 1+ 2

∞∑z=0

(−1)z

×(261 + n − 2p + r)!

(41 −61 + p −

12 + z

)!

(261 + n − 2p + r + z + 3)!(41 −61 + p −

32

)!, (A.7)

and normalization constant Nnk is determine by Nnk =1

√I (n)

where I (n) is

I(n)= 2

(−

1

2

)n

×

(0(n +

(41+61

2

)+ 1))20(n +

(41−61

2

)+ 1)0(n + 1)

n!0(n′ +41 + 1)

×

n′∑p,r=0

(−1)p+r

(2)00(r + 1)0(n − r + 1)

×0(n +41 + r + 1)[

0(r +

(41+61

2

)+ 1)0(p + 1)0

(n +

(41+61

2

)− p + 1

)0(n − p + 1)0

(p +

(41−61

2

)+ 1) ]

×

[−

1

261 + n − 2p + r + 1+ 2

∞∑z=0

(−1)z

×(261 + n − 2p + r)!

(41 −61 + p −

12 + z

)!

(261 + n − 2p + r + z + 3)!(41 −61 + p −

32

)!

].

Appendix B

The radial equation satisfies the requirements; Fnk(S)= 0 asS = 1 (r → 0) and = −1 (r → ∞). Consequently, the wave

function, Fnk(S), is valid physically in the closed intervalS ∈ [−1, 1] or r ∈ [0,.∞). Further, the wave functions satisfythe normalization condition∫

∞

0|Fnk (r)|

2 d r = b∫ 1

0S−1

|Fnk (S)|2 d S = 1, (B.1)

where

Fnk (S) = Qnk S−√

B(1 − S)(12 +ϒ+

√B)P (−2

√B,2(ϒ+

√B))

n

× (1 − 2S), (B.2)

where Qnk can be determined by

bQ2nk

∫ 1

0(S)−2

√B−1 (1 − S)2(

12 +ϒ+

√B)∣∣∣P (−2

√B,2(ϒ+

√B))

n

× (1 − 2S)|2 d S = 1. (B.3)

Using equations (A.3) and (A.4), we obtain explicitexpressions for P (−2

√B,2(ϒ+

√B))

n (1 − 2S):

P (−2√

B,2(ϒ+√

B))n (1 − 2S)= (−1)n

n∑p=0

(−1)p

×0(n − 2

√B + 1)

0 (p + 1) 0(n − 2√

B − p + 1)

0(n + 2ϒ + 2√

B + 1)

0 (n − p + 1) 0(p + 2ϒ + 2√

B + 1)

(S)n−p (1 − S)p P (−2√

B,2(ϒ+√

B))n

× (1 − 2S)=0(n − 2

√B + 1)

n!0 (n + 2ϒ + 1)

×

n∑r=0

(−1)r 0 (n + 1) 0 (n + 2ϒ + r + 1)

0 (r + 1) 0 (n − r + 1) 0(r − 2√

B + 1)Sr .

Inserting above into equation (B.3), one obtains

bQ2nk (−1)n

×

(0(n − 2

√B + 1)

)20(n + 2ϒ + 2

√B + 1)0 (n + 1)

n!0 (n + 2ϒ + 1)

×

n∑p,r=0

(−1)p+r 0 (n + 2ϒ + r + 1)[r !p!0 (n − r + 1) 0 (n − p + 1)0(n − 2

√B − p + 1)0(r − 2

√B + 1)

]

×1

0 (n − r + 1) 0(p + 2ϒ + 2√

B + 1)Vnk (p, r)= 1,

(B.4)

where

Vnk (p, r)=

∫ 1

0(S)(n−p−2

√B+r−1) (1 − S)(2ϒ−2

√B+p+1) d S.

(B.5)

8

Phys. Scr. 87 (2013) 025703 S Asgarifar and H Goudarzi

Using the following integral representation of thehypergeometric function [44, 45]

2 F1 (α0, β0, γ0; 1)=0 (γ0)

0 (α0) 0 (γ0 −α0)

×

∫ 1

0Sα0−1 (1 − S)γ0−α0−1 (1 − S)−β0 d S,

× Re (γ0) > Re (α0) > 0. (B.6)

When we take α0 = n − p − 2√

B + r , β0 = −2Y + 2√

B −

p − 1 and γ0 = α0 + 1 the Vnk(p, r) is calculated as follows:

Vnk (p, r)= 2 F1(α0, β0, γ0; 1)

α0

=0(n − p − 2

√B + r + 1)0(2Y − 2

√B + p + 2)

(n − p − 2√

B + r)0(n + 2Y − 4√

B + r + 2). (B.7)

Finally, we find the Qnk in equation (B.4) as

Qnk =1

√l (n)

, (B.8)

which we obtain for l(n) the following relation:

l (n)= b (−1)n

×(0(n − 2

√B + 1))20(n + 2ϒ + 2

√B + 1)0 (n + 1)

n!0 (n + 2ϒ + 1)

×

n∑p,r=0

(−1)p+r 0 (n + 2ϒ + r + 1)[r !p! [0 (n − r + 1)]2 0 (n − p + 1)0(n − 2

√B − p + 1)0(r − 2

√B + 1)

]×

1

0(p + 2ϒ + 2√

B + 1)

×0(n − p − 2

√B + r + 1)0(2ϒ − 2

√B + p + 2)

(n − p − 2√

B + r)0(n + 2ϒ − 4√

B + r + 2)(B.9)

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