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: [email protected] and [email protected]
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)
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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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