# Relativistic Symmetries of Hulthén Potential Incorporated with Generalized Tensor Interactions

Authors: A. N. Ikot; H. Hassanabadi; E. Maghsoodi; S. Zarrinkamar
Journal: Advances in High Energy Physics
 (2013)

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## Abstract

Spin and pseudospin symmetries of the Dirac equation for a Hulthén potential with a novel tensor interaction, that is, a combination of the Coulomb and Yukawa potentials, are investigated using the Nikiforov-Uvarov method. The bound-state energy spectra and the radial wave functions are approximately obtained in the case of spin and pseudospin symmetries. The tensor interactions and the degeneracy-removing role are presented in details.


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## Body

## 1. Introduction

The exact solutions of wave equations are still an interesting problem in fundamental quantum mechanics. Unfortunately, there are only a few potentials for which the Schrödinger, Dirac, Klein-Gordon, and Duffin-Kemmer-Petiau (DKP) equations can be exactly solved. Several potential models have been introduced to explore the relativistic and nonrelativistic energy spectra and the corresponding wave functions [1–5]. Jia et al. [6] have derived the bound-state solution of the Klein-Gordon equation under unequal scalar and vector kink-like potentials. By using the series expansion method, the authors in [7] have obtained the analytical solutions of the two-dimensional Schrödinger equation with the Morse potential. Pseudospin symmetry solution of Dirac equation for the modified Rosen-Morse potential is investigated in [8]. The DKP equation under a scalar Coulomb interaction is solved in [9] where the authors have used the ansatz approach. Ginocchio et al. [10–14] established the relationship between the pseudospin and the Dirac equation by recognizing that pseudo-orbital angular momentum l~=l+1 is nothing but the usual orbital angular momentum l of the lower component of the Dirac spinor. They went further to show that, within the framework of the Dirac theory, the spin symmetry occurs when the difference of the potential between the repulsive Lorentz vector potential V(r) and the attractive Lorentz scalar potential S(r) is a constant; that is, Δ(r)=V(r)-S(r)=const. On the other hand, the pseudospin symmetry arises when the sum of the potentials is a constant; that is, Σ(r)=V(r)+S(r)=const. Many researchers in the field have investigated the symmetries in the presence of various interactions. The list includes Manning-Rosen [15], Eckart [16], Hyllerass [17], Deng-Fan [18], Mobious square [19], Tietz [20], and Hyperbolical potentials [21]. The analysis of the symmetries has been done by using various methods such as Nikiforov-Uvarov (NU) technique [22], supersymmetric quantum mechanics (SUSYQM) [23], and many others [24, and references therein]. Due to the mathematical structure of the problem, the tensor interaction is often considered as a Coulomb or Cornell interaction [25]. Hassanabadi et al. were the first who introduced the Yukawa tensor interaction [26] besides the Coulomb-like term.Here, we intend to report the solution of Dirac equation for the Hulthén potential under the generalized Coulomb-like and Yukawa-like potentials as tensor interactions. The Hulthén potential is defined as [27, 28]
(1)V(r)=-V0e-2αr(1-e-2αr),
where V0, α are two positive real parameters and represent the strength and the screening range of the potential, respectively.


## 2. Dirac Equation with a Tensor Interaction

The Dirac equation under an attractive scalar potentialS(r), a repulsive vector potential V(r), and a tensor potential U(r) in the relativistic unit (ℏ=c=1) is [13, 14]
(2)[α→·p→+β(M+S(r))-iβα→·r^U(r)]ψ(r)=[E-V(r)]ψ(r),
where E is the relativistic energy of the system, p→=-i∇→ denotes the three-dimensional momentum operator, and M stands for the mass of the fermionic particle. α→, β are the 4×4 Dirac matrices given as
(3)α→=(0σ→iσ→i0),β=(I00-I),
where I is 2×2 unitary matrix and σ→i are the Pauli three-vector matrices as follows:
(4)σ1=(0110),σ2=(0-ii0),σ3=(100-1).
The eigenvalues of the spin-orbit coupling operator are κ=(j+(1/2))>0, κ=-(j+(1/2))<0 for unaligned j=l-(1/2) and the aligned spinj=l+(1/2), respectively. The set (H,K,J2,Jz) forms a complete set of conserved quantities. Thus, we can write the spinors as
(5)ψnκ(r)=1r(Fnκ(r)Yjml(θ,φ)iGnκ(r)Yjml~(θ,φ)),
where Fnκ(r), Gnκ(r) represent the upper and lower components of the Dirac spinors. Yjml(θ,φ), Yjml~(θ,φ) are the spin and pseudospin spherical harmonics and m is the projection on the z-axis. With other known identities,
(6)(σ→·A→)(σ→·B→)=A→·B→+iσ→·(A→×B→),σ→·p→=σ→·r^(r^·p→+iσ→·L→r)
as well as
(7)(σ→·L→)Yjml~(θ,φ)=(κ-1)Yjml~(θ,φ),(σ→·L→)Yjml(θ,φ)=-(κ+1)Yjml(θ,φ),(σ→·r^)Yjml(θ,φ)=-Yjml~(θ,φ),(σ→·r^)Yjml~(θ,φ)=-Yjml(θ,φ),
we find the following two coupled first-order Dirac equations:
(8)(ddr+κr-U(r))Fnκ(r)=(M+Enκ-Δ(r))Gnκ(r),(ddr-κr+U(r))Gnκ(r)=(M-Enκ+Σ(r))Fnκ(r),
where
(9)Δ(r)=V(r)-S(r),Σ(r)=V(r)+S(r).
Decoupling the components, we obtain the second-order Schrödinger-like equation as
(10){d2dr2-κ(κ+1)r2+2κU(r)r-dU(r)dr-U2(r)-(M+Enκ-Δ(r))(M-Enκ+Σ(r))+(dΔ(r)dr(ddr+κr-U(r)))(M+Enκ-Δ(r))-1}×Fnκ(r)=0,(11){{(dΣ(r)dr(ddr-κr+U(r)))}d2dr2-κ(κ-1)r2+2κU(r)r+dU(r)dr-U2(r)-(M+Enκ-Δ(r))(M-Enκ+Σ(r))-(dΣ(r)dr(ddr-κr+U(r)))(M-Enκ+Σ(r))-1}×Gnκ(r)=0,
where κ(κ-1)=l~(l~+1), κ(κ+1)=l(l+1).


## 3. Pseudospin Symmetry Limit

The pseudospin symmetry limit occurs in Dirac equation whendΣ(r)/dr=0 or Σ(r)=Cps=const. In this limit, we take Δ(r) as the Hulthén potential:
(12)Δ(r)=-V0(e-2αr1-e-2αr).
In addition, we propose a novel generalized tensor interaction of the form
(13)U(r)=UC(r)+UY(r),
where UC(r) and UY(r) are the Coulomb-like and Yukawa-like potentials defined as
(14)UC(r)=-Hr,UY(r)=-V1e-αrr
with
(15)H=zazbe24πε0,
where Re is the Coulomb radius and za and zb denote the charges of the projectile a and the target nuclei b, respectively. Also, V1 is the depth of the potential. Substituting  (14) into (13), we can write the tensor potential as
(16)U(r)=-1r(H+V1e-αr).
Substitution of (16) and (29) into (9) gives
(17){d2dr2-κ(κ-1)r2-2κ(H+V1e-αr)r2+Hr2+αV1e-αrr+V1e-αrr2-H2r2-2HV1e-αrr2-V12e-2αrr2-(M+Enκps)×(M-Enκps+Cps)-(M-Enκps+Cps)V0e-2αr(1-e-2αr)}×Gnκps(r)=0.
It is well known that the above equation cannot be exactly solved due to the centrifugal term r-2. In order to get rid of the centrifugal term, we make use of the following approximation [27, 29] (see Figure 1):
(18)1r2≈4α2e-2αr(1-e-2αr)2,1r2≈4α2e-αr(1-e-2αr)2,
where κ=-ℓ~ and κ=ℓ~+1 for κ<0 and κ>0, respectively. Using a new variable of the form s=e-2αr and introducing
(19)f1ps=εps24α2-βps4α2+V1(V1+12),f2ps=2εps24α2-βps4α2-V1(2H+2κ-32)-ηκ(ηκ-1),f3ps=εps24α2,
where
(20)εps2=(M+Enκps)(M-Enκps+Cps),(21)βps=(M-Enκps+Cps)V0,(22)κ(κ-1)+2κH-H+H2=(κ+H)(κ+H-1)=ηκ(ηκ-1)⟶ηκ=(κ+H),
(17) is transformed to
(23)d2Gnκpsds2+1sdGnκpsds+-f1pss2+f2pss-f3pss2(1-s)2Gnκps=0.
Comparing (23) with (A.1), we find
(24)α1=α2=α3=1,ξ1ps=f1ps,ξ2ps=f2ps,ξ3ps=f3ps.
From (A.4) one can determine the rest of coefficients as
(25)α4=0,α5=-12,α6=14+f1ps,α7=-f2ps,α8=f3ps,α9=14+f1ps+f3ps-f2ps,α10=1+2f3ps,α11=2+2(14+f1ps+f3ps-f2ps+f3ps),α12=f3ps,α13=-12-(14+f1ps+f3ps-f2ps+f3ps).
Substitution of the values of the parameters given by (25) into (A.2) and (A.3) gives us the following relations for the eigenfunctions and energy eigenvalues:
(26)Gnκps(r)=e-2αf3psr(1-e-2αr)(1/2)+(1/4)+f1ps+f3ps-f2ps×Pn(2f3ps,2(1/4)+f1ps+f3ps-f2ps)(1-2e-2αr),ρp+λpf3ps+βp4α2=0,
where
(27)ρp=n2+n+12+(2n+1)×14+(κ+V1+H)(κ+V1+H-1)+ηκ(ηκ-1)+V1(2H+2κ-32),λp=2n+1+214+(κ+V1+H)(κ+V1+H-1),
On the other hand, the upper component can be found by using the following relation:
(28)Fn,κps(r)=1M-Enκps+Cps(ddr-κr+U(r))Gnκps(r).Figure 1
1
/
r
2 and its approximations for α=0.05 fm.


## 4. Spin Symmetry Limit

In this section, we consider the spin symmetry limit wheredΔ(r)/dr=0 or Δ(r)=Cps=const. As in the previous section, we consider
(29)Σ(r)=-V0(e-2αr1-e-2αr),U(r)=-1r(H+V1e-αr).
Substitution of (16) and (22) into (10) gives
(30){d2dr2-κ(κ+1)r2-2κHr2-H2r2-Hr2-2κV1e-αrr2-αV1e-αrr-V1e-αrr2-2HV1e-αrr2-V12e-2αrr2-εs2+βse-2αr(1-e-2αr)}Fnκs(r)=0,
with εs2=(Enκs+M-Cs)(M-Enκs), βs=(Enκs+M-Cs)V0, and k=ℓ and k=-ℓ-1 for k<0 and k>0, respectively. Substituting (18) into (30), we have
(31){ddr2-4α2Λκ(Λκ-1)e-2αr(1-e-2αr)2-8α2κV1e-2αr(1-e-2αr)2-2α2V1e-2αr(1-e-2αr)-4α2V1e-2αr(1-e-2αr)2-8α2HV1e-2αr(1-e-2αr)2-4α2V12e-4αr(1-e-2αr)2-εs2+βse-2αr(1-e-2αr)}Fnκs(r)=0,
where
(32)κ(κ+1)+2κH+H+H2=(κ+H+1)(κ+H)=Λκ(Λκ-1)⟶Λκ=(κ+H+1).
Introducing a new transformation of the form y=e-2αr, we arrive at the Schrödinger-like equation:
(33){d2dy2+1yddy+-f1sy2+f2sy-f3sy2(1-y)2}Fnκs(y)=0,
where
(34)f1s=εs24α2+βs4α2+V1(V1-12),f2s=2εs24α2+βs4α2-Λκ(Λκ-1)-(2κ+2H+32)V1,f3s=εs24α2.
Using the same approach we used to solve (23), the energy equation is obtained as
(35)ρs+λsf3s-βs4α2=0,
where
(36)ρs=n2+n+12+(2n+1)×14+(κ+V1+H+1)(κ+V1+H)+Λκ(Λκ-1)+V1(2H+2κ+32),λs=2n+1+214+(κ+V1+H+1)(κ+V1+H),
and the corresponding upper and lower radial wave functions are
(37)Fn,ks(r)=e-2αf3sr(1-e-2αr)(1/2)+(1/4)+f1s+f3s-f2s×Pn(2f3s,2(1/4)+f1s+f3s-f2s)(1-2e-2αr),(38)Gn,κs(r)=1M+Enκs-Cs(ddr+κr-U(r))Fnκs(r).


## 5. Discussion and Numerical Results

We obtained the energy eigenvalues in the absence(H=0,V1=0) and the presence (H=0.5,V1=0.5) of the Coulomb-like plus Yukawa tensor interaction for various values of the quantum numbers n and κ. The results are reported in Tables 1 and 2 under the condition of the pseudospin and spin symmetries, respectively, where we can see the way the tensor interaction affects the degeneracy of the system. If we set H=V1=0, the potential reduces into the Hulthén potential and our result is consistent with that of Aydoğdua et al. [27]. In Tables 1 and 2, we have compared our results with [27]. We represent the effects of the α-parameter on the bound states for H=V1=0.5 in Figure 2. It is seen that if the α-parameter increases, the bound states become more bounded in both for symmetry limits. Figures 3 and 4 present the effect of the tensor interaction on the bound states. We have plotted the energy versus M in Figure 5. Figure 6 shows the effects of Cps and Cs on the bound-states. In Figures 7 and 8, the wave functions are plotted for pseudospin and spin symmetry limits with and without a tensor interaction, respectively. It is seen in Figures 7 and 8 that the tensor interaction affects only the shape of the wave functions and does not change the node structure of the radial upper and lower components of the Dirac spinors.Table 1
Bound state for the pseudospin symmetryα=0.05fm, M=5fm-1, V0=4fm-1, and Cps=-5fm-1.
ℓ
~
n
,
κ
<
0
(
ℓ
,
j
)
E
n
κ
ps
(
f
m
-
1
)
= Ref. [27]
(V1=0,H=0)
E
n
κ
ps
(
f
m
-
1
)
(V1=0.5,H=0.5)
n
-
1
,
κ
>
0
(
ℓ
+
2
,
j
+
1
)
E
n
κ
ps
(
f
m
-
1
)
= Ref. [27]
(V1=0,H=0)
E
n
κ
ps
(
f
m
-
1
)
(V1=0.5,H=0.5)
1
1, −1
1
S
1
/
2
−0.00862755, −0.05738712
−0.00255933, −0.02961889
0, 2
0
d
3
/
2
−0.00862755, −0.05738712
−0.01389734, −0.10389112
2
1, −2
1
P
3
/
2
−0.01538462, −0.10000000
−0.00721029, −0.06124936
0, 3
0
f
5
/
2
−0.01538462, −0.10000000
−0.02260173, −0.15622180
3
1, −3
1
d
5
/
2
−0.02413384, −0.15233675
−0.01389734, −0.10389112
0, 4
0
g
7
/
2
−0.02413384, −0.15233675
−0.03335506, −0.21664494
4
1, −4
1
f
7
/
2
−0.03492388, −0.21278254
−0.02260173, −0.15622180
0, 5
0
h
9
/
2
−0.03492388, −0.21278254
−0.04621304, −0.28340834
1
2, −1
2
S
1
/
2
−0.01538462, −0.10000000
−0.00721029, −0.06124936
1, 2
1
d
3
/
2
−0.01538462, −0.10000000
−0.02260173, −0.15622180
2
2, −2
2
P
3
/
2
−0.02413384, −0.15233675
−0.01389734, −0.10389112
1, 3
1
f
5
/
2
−0.02413384, −0.15233675
−0.03335506, −0.21664494
3
2, −3
2
d
5
/
2
−0.03492388, −0.21278254
−0.02260173, −0.15622180
1, 4
1
g
7
/
2
−0.03492388, −0.21278254
−0.04621304, −0.28340834
4
2, −4
2
f
7
/
2
−0.04781723, −0.27957698
−0.03335506, −0.21664494
1, 5
1
h
9
/
2
−0.04781723, −0.27957698
−0.06125073, −0.35469754Table 2
Bound state for the spin symmetryα=0.05fm, M=5fm-1, V0=4fm-1, and Cs=5fm-1.
ℓ
n
,
κ
<
0
(
ℓ
,
j
)
E
n
κ
s
(
f
m
-
1
)
= Ref. [27]
(V1=0,H=0)
E
n
κ
s
(
f
m
-
1
)
(V1=0.5,H=0.5)
n
,
κ
>
0
(
ℓ
,
j
)
E
n
κ
s
(
f
m
-
1
)
= Ref. [27]
(V1=0,H=0)
E
n
κ
s
(
f
m
-
1
)
(V1=0.5,H=0.5)
1
0, −2
0
P
3
/
2
0.06784763, 0.00145930
0.01709159, 0.00036476
0, 1
0
P
1
/
2
0.06784763, 0.00145930
0.15074992, 0.00328431
2
0, −3
0
d
5
/
2
0.15074992, 0.00328431
0.06784763, 0.00145929
0, 2
0
d
3
/
2
0.15074992, 0.00328431
0.26338976, 0.00584101
3
0, −4
0
f
7
/
2
0.26338976, 0.00584101
0.15074992, 0.00328431
0, 3
0
f
5
/
2
0.26338976, 0.00584101
0.40263365, 0.00913106
4
0,−5
0
g
9
/
2
0.40263365, 0.00913106
0.26338976, 0.00584101
0, 4
0
g
7
/
2
0.40263365, 0.00913106
0.56482502, 0.01315663
1
1, −2
1
P
3
/
2
0.15074992, 0.00328431
0.06784763, 0.00145929
1, 1
1
P
1
/
2
0.15074992, 0.00328431
0.26338976, 0.00584101
2
1, −3
1
d
5
/
2
0.26338976, 0.00584101
0.15074992, 0.00328431
1, 2
1
d
3
/
2
0.26338976, 0.00584101
0.40263365, 0.00913106
3
1, −4
1
f
7
/
2
0.40263365, 0.00913106
0.26338976, 0.00584101
1, 3
1
f
5
/
2
0.40263365, 0.00913106
0.56482502, 0.01315663
4
1, −5
1
g
9
/
1
0.56482502, 0.01315663
0.40263365, 0.00913106
1, 4
1
g
7
/
2
0.56482502, 0.01315663
0.74599944, 0.01792038Figure 2
PSS: energy versusα for pseudospin symmetry limit for V0=4fm-1, V1=0.5fm-1, H=0.5, M=5fm-1, and Cps=-5fm-1. SS: energy versus α for spin symmetry limit for V0=4fm-1, V1=0.5fm-1, H=0.5, M=5fm-1, and Cs=5fm-1.Figure 3
PSS: energy versusV1 for pseudospin symmetry limit for V0=4fm-1, H=0.5, α=0.05fm, M=5fm-1, and Cps=-5fm-1. SS: energy versus V1 for spin symmetry limit for V0=4fm-1, H=0.5, α=0.05fm, M=5fm-1, and Cs=5fm-1.Figure 4
PSS: energy versusH for pseudospin symmetry limit for V0=4fm-1, V1=0.5, α=0.05fm, M=5fm-1, and Cps=-5fm-1. SS: energy versus H for spin symmetry limit for V0=4fm-1, V1=0.5, α=0.05fm, M=5fm-1, and Cs=5fm-1.Figure 5
PSS: energy versusM for pseudospin symmetry limit for V0=4fm-1, V1=0.5, H=0.5, α=0.05fm, and Cps=-5fm-1. SS: energy versus M for spin symmetry limit for V0=4fm-1, V1=0.5, H=0.5, α=0.05fm, and Cs=5fm-1.Figure 6
PSS: energy versusCps for pseudospin symmetry limit for V0=4fm-1, V1=0.5, H=0.5, α=0.05fm, and M=5fm-1. SS: energy versus Cs for spin symmetry limit for V0=4fm-1, V1=0.5, H=0.5, α=0.05fm, and M=5fm-1.Figure 7
Wavefunction for pseudospin symmetry limit forV0=4fm-1, M=5fm-1, Cps=-5fm-1, and α=0.05fm.Figure 8
Wavefunction for spin symmetry limit forV0=4fm-1, M=5fm-1, Cs=5fm-1, and α=0.05fm.


## 6. Conclusion

In this paper, we obtained the approximate analytical solutions of the Dirac equation for the Hulthén potential with a novel generalized tensor interaction consisting of the Coulomb and Yukawa interactions within the framework of pseudospin and spin symmetry limits using the NU technique. We have obtained the energy levels in a closed form and the corresponding wave functions in terms of the Jacobi polynomials. We also included some numerical results to investigate the way the combination of Coulomb and Yukawa potentials. Finally, the results of our work find many applications in both nuclear and Hadron physics and therefore provide more general solutions compared to other previous works performed in [30, 31].



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