# Spin and Pseudospin Symmetries with Trigonometric Pöschl-Teller Potential including Tensor Coupling

Authors: M. Hamzavi; A. A. Rajabi
Journal: Advances in High Energy Physics
 (2013)

---

## Abstract

We study approximate analytical solutions of the Dirac equation with the trigonometric Pöschl-Teller (tPT) potential and a Coulomb-like tensor potential for arbitrary spin-orbit quantum numberκ under the presence of exact spin and pseudospin (p-spin) symmetries. The bound state energy eigenvalues and the corresponding two-component wave functions of the Dirac particle are obtained using the parametric generalization of the Nikiforov-Uvarov (NU) method. We show that tensor interaction removes degeneracies between spin and pseudospin doublets. The case of nonrelativistic limit is studied too.


---

## Body

## 1. Introduction

The Dirac equation, which describes the motion of a spin-1/2 particle, has been used in solving many problems of nuclear and high-energy physics. The spin and thep-spin symmetries of the Dirac Hamiltonian had been discovered many years ago; however, these symmetries have recently been recognized empirically in nuclear and hadronic spectroscopes [1]. Within the framework of Dirac equation, p-spin symmetry used to feature the deformed nuclei and the super deformation to establish an effective shell-model [2–4], whereas spin symmetry is relevant for mesons [5]. The spin symmetry occurs when the scalar potential S(r) is nearly equal to the vector potential V(r) or equivalently S(r)≈V(r), and p-spin symmetry occurs when S(r)≈-V(r) [6, 7]. The p-spin symmetry refers to a quasidegeneracy of single nucleon doublets with nonrelativistic quantum number (n,l,j=l+1/2) and (n-1,l+2,j=l+3/2), where n, l, and j are single nucleon radial, orbital, and total angular quantum numbers, respectively [8, 9]. The total angular momentum is given by  j=l~+s~, where l~=l+1pseudoangular momentum and s~ is p-spin angular momentum [10, 11]. Liang et al. [12] investigated the symmetries of the Dirac Hamiltonian and their breaking in realistic nuclei in the framework of perturbation theory. Guo [13] used the similarity renormalization group to transform the spherical Dirac operator into a diagonal form, and then the upper (lower) diagonal element became an operator describing Dirac (anti-)particle, which holds the form of the Schrödinger-like operator with the singularity disappearing in every component. Chen and Guo [14] investigated the evolution toward the nonrelativistic limit from the solutions of the Dirac equation by a continuous transformation of the Compton wavelengthλ. Lu et al. [15] recently showed that the p-spin symmetry in single particle resonant states in nuclei is conserved when the attractive scalar and repulsive vector potentials have the same magnitude but opposite sign.Tensor potentials were introduced into the Dirac equation with the substitutionp→→p→-imωβ·r^U(r), and a spin-orbit coupling is added to the Dirac Hamiltonian [16, 17]. Lisboa et al. [18] have studied a generalized relativistic harmonic oscillator for spin-1/2 particles by considering a Dirac Hamiltonian that contains quadratic vector and scalar potentials together with a linear tensor potential, under the conditions of pseudospin and spin symmetry. Alberto et al. [19] studied the contribution of the isoscalar tensor coupling to the realization of pseudospin symmetry in nuclei. Akcay showed that the Dirac equation with scalar and vector quadratic potentials and a Coulomb-like tensor potential can be solved exactly [20]; also, he exactly solved Dirac equation with tensor potential containing a linear and Coulomb-like terms too [21]. Aydoğdu and Sever obtained exact solution of Dirac equation for the pseudoharmonic potential in the presence of linear tensor potential under the pseudospin symmetry and showed that tensor interactions remove all degeneracies between members of pseudospin doublets [22]. Zhou et al. solved Dirac equation approximately for Hulthén potential including Coulomb-like tensor potential with arbitrary spin-orbit coupling number κ under spin and pseudospin symmetry limit [10]. Aydoğdu and Sever solved approximately for the Woods-Saxon potential and a tensor potential with the arbitrary spin-orbit coupling quantum number κ under pseudospin and spin symmetry [23]. Very recently, Hamzavi et al. gave exact solutions of the Dirac equation for Mie-type potential and position-dependent mass Coulomb potential with a Coulomb-like tensor potential [24, 25] and pseudoharmonic potential with linear plus Coulomb-like tensor potential [26].The tPT potential has been proposed for the first time by Pöschl and Teller [27] in 1933 to describe the diatomic molecular vibration. Chen [28] and Zhang and Wang [29] have studied the relativistic bound state solutions for the tPT potential and hyperbolical PT (Second PT) potential, respectively. Liu et al. [30] studied the tPT potential within the framework of the Dirac theory. Recently, Candemir [31] investigated the analytical s~-wave solutions of Dirac equation for tPT potential under the p-spin symmetry condition. Very recently, Hamzavi and Rajabi [32] studied the exact s-wave solution (l=0) of the Schrödinger equation for the vibrational tPT potential. The tPT takes the form:
(1)VtPT(r)=V1sin2(αr)+V2cos2(αr),V1>0,V2>0,
where the parameters V1 and V2 describe the property of the potential well, while the parameter α is related to the range of this potential [30]. We find out that this potential has a minimum value at r0=(1/α)tan-1(V1/V24). The case when V1=V2, the minimum value is at r0=π/4α∈(0,∞) for α>0. The second derivative which determines the force constants at r=r0is given by
(2)d2Vdr2|r=r0=8α2(V2+V1V2)cos2[tan-1(V1/V24)],
for any α value, and thus
(3)V(r0)=V1V2+V2cos2[tan-1(V1/V24)],
which means that V(r)at r=r0 has a relative minimum for α>0. When V1=V2=V then minimum value is V(r0)=4V and d2V/dr2|r=r0=32α2V. In Figures 1(a) and 1(b), we draw the tPT potential (1) for parameter values V1=5.0fm-1,V2=3.0fm-1,α=0.02fm-1,  and α=0.30fm-1. Here the potential has a minimum value at r0=0.27027π/α. The curve is nodeless in αr∈(0,π/2). For example, with α=0.30fm-1,r0=2.8303fmand minimum potential V(r0=2.8303fm)=15.746fm-1. It is worthy to note that in the limiting case when α→0, the tPT potential can be reduced to the Kratzer potential [33, 34] V(r)=De((r-re)/r)2+η, where re is the equilibrium intermolecular separation and De is the dissociation energy between diatomic molecules. In our case, De=V1,η=V2,  and re=1/α. In the case of η=0 it reduces to the molecular potential which is called the modified Kratzer potential proposed by Simons et al. [35] and Molski and Konarski [36]. In the case of η=-De, this potential turns into the Kratzer potential, which includes an attractive Coulomb potential and a repulsive inverse square potential introduced by Kratzer in 1920 [37].(a) A plot of the tPT potential forα=0.2fm-1. (b) A plot of the tPT potential for α=0.8fm-1.
(a)
(b)The aim of the present work is to extend our previous work [32] to the relativistic case and κ≠±1 (rotational case) including a Coulomb-like tensor potential. We introduce a convenient approximation scheme to deal with the strong singular centrifugal term. The ansätz of this approximation possesses the same form of the potential and is singular as the centrifugal term r-2.Since the relativistic solution is indispensable, we need to solve the Dirac equation with flexible parameters tPT potential model. However, the Dirac-tPT problem can no longer be solved in a closed form due to the existence of spin-orbit coupling termκ(κ±1)r-2and it is necessary to resort to approximation methods. Therefore, we use an approximation scheme to deal with this term and solve approximately the Dirac equation with the tPT potential for arbitrary spin-orbit quantum number κ. In the presence of spin and p-spin symmetric limitation, we obtain the approximate relativistic bound state solutions including the energy eigenvalue equations and the corresponding unnormalized upper- and lower-spinor components of the wave functions using the concepts of parametric generalization of the NU method [38] since the relativistic corrections are not neglected.Over the past years, the Nikiforov-Uvarov (NU) method [39] has shown to be a powerful tool in solving the second-order differential equation. It was applied successfully to a large number of potential models [40–46]. This method has also been used to solve the spinless (spin-0) Schrödinger [47–51] and Klein-Gordon (KG) [52–57] equations and also relativistic spin-1/2 Dirac equation [58–63] with different potential models.The structure of the paper is as follows. In Section2, in the context of spin and p-spin symmetry, we briefly introduce the Dirac equation with scalar and vector tPT and also tensor potential potentials for arbitrary spin-orbit quantum number κ. The parametric generalization of the NU method is displayed in  the appendix. In the presence of the spin and p-spin symmetry, the approximate energy eigenvalue equations and corresponding two-component wave functions of the Dirac-tPT problem are obtained, and effect of tensor potential is shown in this section. The nonrelativistic limit of the problem is discussed in this section too. Finally, our final concluding remarks are given in Section 3.


## 2. Dirac Equation including Tensor Coupling

The Dirac equation for fermionic massive spin-1/2 particles moving in an attractive scalar potential S(r), a repulsive vector potential V(r), and a tensor potential U(r) is [ℏ=c=1](4)[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∇→ is the three-dimensional momentum operator, and M is the mass of the fermionic particle [64]. α→ and β are the 4×4 usual Dirac matrices given as
(5)α→=(0σ→σ→0),β=(I00-I),
where I is 2×2 unitary matrix and σ→ are three-vector spin matrices
(6)σ1=(0110),σ2=(0-ii0),σ3=(100-1).
The total angular momentum operator J→ and spin-orbit operator K=(σ→·L→+1), where L→ is orbital angular momentum operator, of the spherical nucleons commute with the Dirac Hamiltonian. The eigenvalues of spin-orbit coupling operator are κ=(j+(1/2))>0 and κ=-(j+(1/2))<0 for unaligned spin j=l-(1/2) and the aligned spin j=l+(1/2), respectively. (H2,K,J2,Jz) can be taken as the complete set of the conservative quantities. Thus, the spinor wave functions can be classified according to their angular momentum j, spin-orbit quantum number κ, and the radial quantum number n and can be written as follows:
(7)ψnκ(r→)=(fnκ(r→)gnκ(r→))=(Fnκ(r)rYjml(θ,φ)iGnκ(r)rYjml~(θ,φ)),
where fnκ(r→) is the upper (large) component and gnκ(r→) is the lower (small) component of the Dirac spinors. Yjml(θ,φ) and Yjml~(θ,φ) are spin and pseudospin spherical harmonics, respectively, and m is the projection of the angular momentum on the z-axis. Substituting (7) into (4) and using the following relations: (8a)(σ→·A→)(σ→·B→)=A→·B→+iσ→·(A→×B→),(8b)(σ→·P→)=σ→·r^(r^·P→+iσ→·L→r),
with the following properties:
(8c)(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),one obtains two coupled differential equations for upper and lower radial wave functions Fnκ(r) and Gnκ(r)as(9a)(ddr+κr-U(r))Fnκ(r)=(M+Enκ-Δ(r))Gnκ(r),(9b)(ddr-κr+U(r))Gnκ(r)=(M-Enκ+Σ(r))Fnκ(r),where (10a)Δ(r)=V(r)-S(r),(10b)Σ(r)=V(r)+S(r).Eliminating Fnκ(r) and Gnκ(r) from (10a) and (10b), we obtain the following two Schrödinger-like differential equations for the upper and lower radial spinor components, respectively:
(11)[d2dr2-κ(κ+1)r2+2κrU(r)-dU(r)dr-U2(r)]Fnκ(r)+dΔ(r)/drM+Enκ-Δ(r)(ddr+κr-U(r))Fnκ(r)=[(M+Enκ-Δ(r))(M-Enκ+Σ(r))]Fnκ(r),(12)[d2dr2-κ(κ-1)r2+2κrU(r)+dU(r)dr-U2(r)]Gnκ(r)+dΣ(r)/drM-Enκ+Σ(r)(ddr-κr+U(r))Gnκ(r)=[(M+Enκ-Δ(r))(M-Enκ+Σ(r))]Gnκ(r),
where κ(κ-1)=l~(l~+1) and κ(κ+1)=l(l+1). The quantum number κ is related to the quantum numbers for spin symmetry l and pseudospin symmetry l~ as
(13)κ={-(l+1)=-(j+12)(s1/2,p3/2,etc.),j=l+12,alignedspin(κ<0),+l=+(j+12)(p1/2,d3/2,etc.),j=l-12,unalignedspin(κ>0),
and the quasidegenerate doublet structure can be expressed in terms of a pseudospin angular momentum s~=1/2 and pseudoorbital angular momentum l~, which is defined as
(14)κ={-l~=-(j+12)(s1/2,p3/2,etc.),j=l~-12,alignedpseudospin(κ<0),+(l~+1)=+(j+12)(d3/2,f5/2,etc.),j=l~+12,unalignedspin(κ>0),
where κ=±1,±2,…. For example, (1s1/2,0d3/2) and (1p3/2,0f5/2) can be considered as pseudospin doublets.


### 2.1. Spin Symmetric Limit

In the spin symmetric limitation,dΔ(r)/dr=0 or Δ(r)=Cs=constant [7, 65–67]; then, (11) withΣ(r)=VtPT(r) and tensor potential as Coulomb potential, U(r)=-H/r becomes(15a)[d2dr2-ηκ(ηκ+1)r2-γ(V1sin2(αr)+V2cos2(αr))-β2]×Fnκ(r)=0,(15b)γ=M+Enκ-Cs,β2=(M-Enκ)(M+Enκ-Cs),where ηκ=κ+H+1 is new spin-orbit centrifugal term and κ=l and κ=-l-1 for κ<0 and κ>0, respectively. The Schrödinger-like equation (15a) that results from the Dirac equation is a second-order differential equation containing a spin-orbit centrifugal term ηκ(ηκ+1)r-2which has a strong singularity at r=0,and needs to be treated very carefully while performing the approximation. In absence of tensor interaction, (15a) has an exact rigorous solution only for the states with κ=-1because of the existence of the centrifugal termκ(κ+1)/r2. However, when this term taken into account, the corresponding radial Dirac equation can no longer be solved in a closed form, and it is necessary to resort to approximate methods. Over the last few decades, several schemes have been used to calculate the energy spectrum. The main idea of these schemes relies on using different approximations of the spin-orbit centrifugal coupling term ηκ(ηκ+1)/r2. So, we need to perform a new approximation for the spin-orbit term as a function of the tPT potential parameters. Therefore, we resort to use an appropriate approximation scheme to deal with the centrifugal potential term as
(16)1r2=limα→0α2(d0+1sin2(αr)),0<αr<π/2,
where d0=1/12 is a dimensionless shifting parameter and αr≪1. The approximation (16) is done on the basis that sin(z)=z-z3/3!+z5/5!-z7/7!+⋯ and in the limit when z→0,sin(z)≈z. To show the validity and accuracy of our choice to the approximation scheme (16), we plot the centrifugal potential term 1/r2and its approximations: α2/sin2(αr) and α2(d0+1/sin2(αr))in Figure 2. As illustrated, the three curves coincide together and show how accurate this replacement is.Figure 2
The centrifugal term1/r2 (red line) and its approximations (16) with d0=0 (green dash line) and d0=1/12 (blue dot line), while α=0.002fm-1.Thus, employing such an approximation scheme, we can then write (15a) as
(17)[d2dr2-ηκ(ηκ+1)α2(d0+1sin2(αr))-γ(V1sin2(αr)+V2cos2(αr))-β2d2dr2-ηκ(ηκ+1)α2(d0+1sin2(αr))]Fnκ(r)=0.
Followed by making a new change of variables s(r)=sin2(αr), this allows us to decompose the spin-symmetric Dirac equation (17) into the Schrödinger-type equation satisfying the upper-spinor component Fn,κ(s),
(18){d2ds2+(1/2)-ss(1-s)dds-1s2(1-s)2×(1/2)-ss(1-s)[-As2+Bs-C]}Fn,κ(s)=0,A=-14α2[ηκ(ηκ-1)α2d0+β2],B=14α2[ηκ(ηκ-1)α2(1-d0)+γ(V1-V2)-β2],C=14α2[ηκ(ηκ-1)α2+γV1],
where Fnκ(r)≡Fn,κ(s)has been used. If the previous equation is compared with (A.2), we can obtain the specific values for constants ci(i=1,2,3) as
(19)c1=12,c2=1,c3=1.
In order to obtain the bound state solutions of (17), it is necessary to calculate the remaining parametric constants, that is, ci(i=4,5,…,13), by means of the relation (A.5). Their specific values are displayed in Table 1 for the relativistic tPT potential model. Further, using these constants along with (A.7), we can readily obtain the energy eigenvalue equation for the Dirac-tPT problem as
(20)(n+12+A-B+C+116+C+116)2=A,
or equivalently
(21)(2n+1+121+4V2(M+Enκ-Cs)α2+12(2ηκ-1)2+4V1(M+Enκ-Cs)α2)2=1α2(Enκ-M)(M+Enκ-Cs)-ηκ(ηκ-1)d0.
To show the procedure of determining the energy eigenvalues from (21), we take a set of physical parameter values,M=10fm-1, V1=5.0fm-1, V2=3.0fm-1, Cs=0fm-1, and α=0.8,0.6,0.4,0.2,0.04, and 0.02 [30].Table 1
The specific values of the parametric constants  for the spin symmetric Dirac-tPT problem.
Constant
Analytic value
c
4
1
4
c
5
-
1
2
c
6
1
4
(
1
+
4
A
)
c
7
-
1
4
(
1
+
4
B
)
c
8
1
16
(
1
+
16
C
)
c
9
A
-
B
+
C
+
1
16
c
10
1
2
1
+
16
C
c
11
2
A
-
B
+
C
+
1
16
c
12
1
4
(
1
+
1
+
16
C
)
c
13
1
4
+
A
-
B
+
C
+
1
16In Tables2 and 3, we present the energy spectrum for the spin symmetric case. Obviously, the pairs (np1/2,np3/2),(nd3/2,nd5/2),(nf5/2,nf7/2),(ng7/2,ng9/2), and so on are degenerate states. Thus, each pair is considered as spin doublet and has positive energy. In Table 2, we can see that tensor interaction removes degeneracies between spin doublets. Further, when potential range parameter α approaches zero, the energy eigenvalues approaches a constant. From (21), we find that this constant is M+V1+V2+2V1V2; that is, limα→0En,κ=M+V1+V2+2V1V2, which can be seen from Table 3 too.Table 2
The bound state energy eigenvalues in units offm-1 of the spin symmetry tPT potential for several values of n and κ with α=0.8.
l
n
,
κ
<
0
(
l
,
j
=
l
+
1
/
2
)
E
n
,
κ
<
0  H≠0
E
n
,
κ
<
0
H
=
0
n
,
κ
>
0
(
l
,
j
=
l
-
1
/
2
)
E
n
,
κ
>
0
H
≠
0
E
n
,
κ
>
0
H
=
0
1
0, −2
0
p
3
/
2
26.82780
26.89327
0, 1
0
p
1
/
2
27.02322
26.89327
2
0, −3
0
d
5
/
2
26.89327
27.02322
0, 2
0
d
3
/
2
27.21577
27.02322
3
0, −4
0
f
7
/
2
27.02322
27.21577
0, 3
0
f
5
/
2
27.46825
27.21577
4
0, −5
0
g
9
/
2
27.21577
27.46825
0, 4
0
g
7
/
2
27.77739
27.46825
1
1, −2
1
p
3
/
2
28.96349
29.02725
1, 1
1
p
1
/
2
29.15386
29.02725
2
1, −3
1
d
5
/
2
29.02725
29.15386
1, 2
1
d
3
/
2
29.34155
29.15386
3
1, −4
1
f
7
/
2
29.15386
29.34155
1, 3
1
f
5
/
2
29.58785
29.34155
4
1, −5
1
g
9
/
2
29.34155
29.58785
1, 4
1
g
7
/
2
29.88968
29.58785Table 3
The spin symmetric bound state energy eigenvalues in units offm-1of the tPT potential for several values of n and κ with M=10.0,V1=5.0,V2=3.0,cs=0, and H=0.
l
n
,
κ
(
l
,
j
=
l
±
1
/
2
)
α
=
0.8
α
=
0.4
α
=
0.2
α
=
0.04
α
=
0.02
1
1, −2,1
1
p
3
/
2
,
1
p
1
/
2
29.02725
27.36375
26.54876
25.90551
25.82567
2
1, −3,2
1
d
5
/
2
,
1
d
3
/
2
29.15386
27.39629
26.55699
25.90584
25.82575
3
1, −4,3
1
f
7
/
2
,
1
f
5
/
2
29.34155
27.44495
26.56933
25.90634
25.82588
4
1, −5,4
1
g
9
/
2
,
1
g
7
/
2
29.58785
27.50956
26.585773
25.90701
25.82605
1
2, −2,1
2
p
3
/
2
,
2
p
1
/
2
31.16696
28.43165
27.08151
26.01179
25.87879
2
2, −3,2
2
d
5
/
2
,
2
d
3
/
2
31.29020
28.46376
27.08969
26.01212
25.87887
3
2, −4,3
2
f
7
/
2
,
2
f
5
/
2
31.47301
28.51179
27.10195
26.01262
25.878100
4
2, −5,4
2
g
9
/
2
,
2
g
7
/
2
31.71308
28.57555
27.11828
26.01328
25.87916In Figure3, we have investigated the effect of the tensor potential on the spin doublet splitting by considering the following pairs of orbital: (1p3/2,1p1/2)and (1f7/2,1f5/2). We observe that in the case of T=0 (no tensor interaction), members of spin doublets have same energy. However, in the presence of the tensor potentialT≠0, these degeneracies are removed. We can also see that spin doublet splitting increases with increasingT. The reason is that term 2κT gives different contributions to each level in the spin doublet because T takes different values for each state in the spin doublet.Figure 3
Effect of tensor potential on spin doublets.On the other hand, in order to establish the upper-spinor component of the wave functionsFn,κ(r), namely, (15a), the relations (A.8), (A.9), (A.10), and (A.11) are used. Firstly, we find the first part of the wave function as
(22)ϕ(s)=s(1/4)(1+1+16C)(1-s)(1/4)+A-B+C+(1/16).
Secondly, we calculate the weight function as
(23)ρ(s)=s(1/2)1+16C(1-s)2A-B+C+(1/16),
which gives the second part of the wave function as
(24)yn(s)=Pn((1/2)1+16C,2A-B+C+(1/16))(1-2s),
where Pn(a,b)(y) are the orthogonal Jacobi polynomials. Finally the upper spinor component for arbitrary κ can be found through the relation (A.11)(25)Fnκ(s)=Nnκs(1/4)(1+1+16C)(1-s)(1/4)+A-B+C+(1/16)×Pn((1/2)1+16C,2A-B+C+(1/16))(1-2s),
or
(26)Fnκ(r)=Nnκ(sin(αr))(1/2)(1+ξκ)(cos(αr))(1/2)(1+δ)×Pn((1/2)ξκ,(1/2)δ)(cos(2αr)),
where
(27)ξκ=1+4α2[ηκ(ηκ-1)α2+γV1],δ=1+4γV2α2,
and Nnκ is the normalization constant. Further, the lower-spinor component of the wave function can be calculated by using
(28)Gnκ(r)=1M+Enκ-Cs(ddr+κr-U(r))Fnκ(r),
where E≠-M+Cs and in the presence of the exact spin symmetry (Cs=0), only positive energy states do exist.


### 2.2.p-Spin Symmetric Limit

Ginocchio showed that there isp-spin symmetry in case when the relationship between the vector potential and the scalar potential is given by V(r)=-S(r) [7]. Further, Meng et al. showed that if d[V(r)+S(r)]/dr=dΣ(r)/dr=0, then Σ(r)=Cps=constant, for which the p-spin symmetry is exact in the Dirac equation [65–67]. Thus, choosing the Δ(r) as tPT potential and U(r) as Coulomb potential, (12) under this symmetry becomes(29a)[d2dr2-λκ(λκ-1)r2-γ~(V1sin2(αr)+V2cos2(αr))-β~2]×Gnκ(r)=0,(29b)γ~=Enκ-M-Cps,β~2=(M+Enκ)(M-Enκ+Cps),whereλκ=κ+H, κ=-l~ and κ=l~+1 for κ<0 and κ>0, respectively. Employing the new approximation for the spin-orbit pseudocentrifugal term,λκ(λκ-1)/r2, that is, (16), the p-spin Dirac equation (29a) can be written as
(30)[d2dr2-λκ(λκ-1)α2(d0+1sin2(αr))-γ~(V1sin2(αr)+V2cos2(αr))-β~2d2dr2-λκ(λκ-1)α2(d0+1sin2(αr))]Gnκ(r)=0.
To avoid repetition of the negative energy solution of (30), the p-spin symmetric case can be readily obtained directly via the spin symmetric solution throughout the following parametric mappings:
(31)Fnκ(r)⟷Gnκ(r),κ→κ-1,V(r)→-V(r)(i.e.,V1→-V1,V2→-V2),Enκ→-Enκ,Cs→-Cps.
Following the previous procedure, one can obtain the p-spin symmetric energy equation as
(32)-(2n+1+121+4V2(Enκ-M-Cps)α2+12(2λκ-1)2+4V1(Enκ-M-Cps)α2)2=1α2(M+Enκ)(M-Enκ+Cps)+λκ(λκ-1)d0.
Furthermore, the lower-spinor component of the wave functions is found as
(33)Gnκ(r)=N~n,κ(sin(αr))(1/2)(1+ξ~κ)×(cos(αr))(1/2)(1+δ~)Pn((1/2)ξ~,(1/2)δ~)(cos(2αr)),
with
(34)ξ~κ=(2λκ-1)2+4V1(Enκ-M-Cps)α2,δ~=1+4V2(Enκ-M-Cps)α2,
where N~nκ is the normalization constant.In Tables4 and 5, we give the numerical results for the p-spin symmetric case. In this case, we take the set of parameter values, M=10fm-1, V1=-5.0fm-1, V2=-3.0fm-1, cps=0fm-1, and α=0.8,0.4,0.2,0.04, and 0.02 [30]. We observe the degeneracy in the following doublets(1s1/2,0d3/2),(1p3/2,0f5/2),(1d5/2,0g7/2), (1f7/2,0h9/2), and so on. Thus, each pair is considered as p-spin doublet and has negative energy. In Table 4, we can see that tensor interaction removes degeneracies between spin doublets.Table 4
The bound state energy eigenvalues in units offm-1 of the p-spin symmetry tPT potential for several values of n and κ with α=0.8.
l
~
n
,
κ
<
0
(
l
,
j
)
E
n
,
κ
<
0 H≠0
E
n
,
κ
<
0 H=0
n
-
1
,
κ
>
0
(
l
+
2
,
j
+
1
)
E
n
-
1
,
κ
>
0
H
≠
0
E
n
-
1
,
κ
>
0
H
=
0
1
1, −1
1
s
1
/
2
−28.96349
−29.02725
0, 2
0
d
3
/
2
−29.15386
−29.02725
2
1, −2
1
p
3
/
2
−29.02725
−29.15386
0, 3
0
f
5
/
2
−29.34155
−29.15386
3
1, −3
1
d
5
/
2
−29.15386
−29.34155
0, 4
0
g
7
/
2
−29.58785
−29.34155
4
1, −4
1
f
7
/
2
−29.34155
−29.58785
0, 5
0
h
9
/
2
−29.88968
−29.58785
1
2, −1
2
s
1
/
2
−31.10492
−31.16696
1, 2
1
d
3
/
2
−31.29020
−31.16696
2
2, −2
2
p
3
/
2
−31.16696
−31.29020
1, 3
1
f
5
/
2
−31.47301
−31.29020
3
2, −3
2
d
5
/
2
−31.29020
−31.47301
1, 4
1
g
7
/
2
−31.71308
−31.47301
4
2, −4
2
f
7
/
2
−31.47301
−31.71308
1, 5
1
h
9
/
2
−32.00754
−31.71308Table 5
The bound state energy eigenvalues in units offm-1 of the p-spin symmetry tPT potential for several values of n and κ with  H=0.
l
~
n
,
κ
(
l
,
j
)
α
=
0.8
α
=
0.4
α
=
0.2
α
=
0.04
α
=
0.02
1
1, −1,2
1
s
1
/
2
,
0
d
3
/
2
−29.02725
−27.36375
−26.54876
−25.90551
−25.82567
2
1, −2,3
1
p
3
/
2
,
0
f
5
/
2
−29.15386
−27.39629
−26.55699
−25.90584
−25.82575
3
1, −3,4
1
d
5
/
2
,
0
g
7
/
2
−29.34155
−27.44495
−26.56933
−25.90634
−25.82588
4
1, −4,5
1
f
7
/
2
,
0
h
9
/
2
−29.58785
−27.50956
−26.58577
−25.90701
−25.82605
1
2, −1,2
2
s
1
/
2
,
1
d
3
/
2
−31.16696
−28.43165
−27.08151
−26.01179
−25.87879
2
2, −2,3
2
p
3
/
2
,
1
f
5
/
2
−31.29020
−28.46376
−27.08969
−26.01212
−25.87887
3
2, −3,4
2
d
5
/
2
,
1
g
7
/
2
−31.47301
−28.51179
−27.10195
−26.01262
−25.87810
4
2, −4,5
2
f
7
/
2
,
1
h
9
/
2
−31.71308
−28.57555
−27.11828
−26.01328
−25.87916In Figure4, we have investigated the effect of the tensor potential on the p-spin doublet splitting by considering the following pairs of orbital: (1d5/2,0g7/2) and  (2f7/2,1h9/2), and one can observe that the results obtained in the p-spin symmetric limit resemble the ones observed in the spin symmetric limit.Figure 4
Effect of tensor potential onp-spin doublets.On the other hand, the upper-spinor component of the Dirac wave function can be calculated by(35)Fnκ(r)=1M-Enκ+Cps(ddr-κr+U(r))Gnκ(r),
where E≠M+Cps, and in the presence of the exact p-spin symmetry (Cps=0), only negative energy states do exist.


### 2.3. The Nonrelativistic Limiting Case

In this section, we study the energy eigenvalue equation (21) and upper-spinor component of wave function (26) of the Dirac-tPT problem under the nonrelativistic limits: Cs=0,κ→l,Enκ-M≃Enl, and M+Enκ≃2m. Thereby, we obtain the energy equation of the Schrödinger equation with any arbitrary orbital state for the tPT potential as
(36)Enl=ℏ2α2l(l+1)d02m+2ℏ2α2m×[n+12+14((2l+1)2+8mV1ℏ2α2+1+8mV2ℏ2α2)]2.
In the limit when α→0, the vibration-rotation energy formula (36) reduces into a constant value:
(37)limα→0Enl=(V1+V2)2.
Further, there is no loss of generality if d0=0; then, (36) becomes
(38)Enl=2ℏ2α2m×[n+12+14((2l+1)2+8mV1ℏ2α2+1+8mV2ℏ2α2)]2,
where n=0,1,2,… and l=0,1,2,… are the vibration and rotation quantum numbers, respectively. To obtain numerical energy eigenvalues for the present potential model, we take the following set of parameter values; namely, M=10fm-1, V1=5.0fm-1, V2=3.0fm-1, and α=1.2,0.8,0.4,0.2,0.02, and 0.002 [30]. As seen from Table 6, in the limit when potential range parameter α approaches zero, the energy eigenvalues approach a constant value given by (37).Table 6
The bound state energy levelsEnl of the Schrödinger equation for the tPT potential.
E
n
l
State(n,l)
M
=
10.0
f
m
-
1
,
V
1
=
5
f
m
-
1
,
V
2
=
3
f
m
-
1 [30]
α
=
1.2
α
=
0.8
α
=
0.4
α
=
0.2
α
=
0.02
α
=
0.002
1
s [32]
22.87051710
20.32991862
17.95616357
16.83082621
15.85264289
15.75661628
2
s [32]
28.29143398
23.68420415
19.50420742
17.57271070
15.92394680
15.76371786
2
p
28.64395419
23.82847894
19.53712286
17.58054181
15.92402153
15.76371860
3
s [32]
34.28835086
27.2944896
21.11625126
18.33059518
15.99541071
15.77082105
3
p
34.67512504
27.44896381
21.15044543
18.33858626
15.99548560
15.77082179
3
d
35.43921159
27.75631556
21.21875330
18.35456399
15.99563534
15.77082328
4
s [32]
40.86126774
31.16077522
22.79229510
19.10447967
16.06703463
15.77792584
4
p
41.28229584
31.32544868
22.82776800
19.11263070
16.06710967
15.77792658
4
d
42.11348590
31.65300783
22.89862721
19.12892817
16.06725974
15.77792806
4
f
43.33519178
32.14003977
23.00470171
19.15336297
16.06748485
15.77793030Also, we can get the radial wave functions of the Schrödinger equation with tPT potential as(39)Rn,l(s)=s(1/4)[1+(2l+1)2+(8mV1/ℏ2α2)]×(1-s)(1/4)(1+1+(8mV2/ℏ2α2))×Pn((1/2)(2l+1)2+(8mV1/ℏ2α2),(1/2)1+(8mV2/ℏ2α2))×(1-2s).
Inserting s=sin2(αr) in the previous equation, we can obtain (40a)Rn,l(r)=Nnl(sin(αr))(1+ηl)/2×(cos(αr))(1+δ)/2Pn(ηl/2,δ/2)(cos(2αr)),(40b)ηl=(2l+1)2+8mV1ℏ2α2,δ=1+8mV2ℏ2α2,where Nnl is a normalization factor to be calculated from the normalization conditions.


## 2.1. Spin Symmetric Limit

In the spin symmetric limitation,dΔ(r)/dr=0 or Δ(r)=Cs=constant [7, 65–67]; then, (11) withΣ(r)=VtPT(r) and tensor potential as Coulomb potential, U(r)=-H/r becomes(15a)[d2dr2-ηκ(ηκ+1)r2-γ(V1sin2(αr)+V2cos2(αr))-β2]×Fnκ(r)=0,(15b)γ=M+Enκ-Cs,β2=(M-Enκ)(M+Enκ-Cs),where ηκ=κ+H+1 is new spin-orbit centrifugal term and κ=l and κ=-l-1 for κ<0 and κ>0, respectively. The Schrödinger-like equation (15a) that results from the Dirac equation is a second-order differential equation containing a spin-orbit centrifugal term ηκ(ηκ+1)r-2which has a strong singularity at r=0,and needs to be treated very carefully while performing the approximation. In absence of tensor interaction, (15a) has an exact rigorous solution only for the states with κ=-1because of the existence of the centrifugal termκ(κ+1)/r2. However, when this term taken into account, the corresponding radial Dirac equation can no longer be solved in a closed form, and it is necessary to resort to approximate methods. Over the last few decades, several schemes have been used to calculate the energy spectrum. The main idea of these schemes relies on using different approximations of the spin-orbit centrifugal coupling term ηκ(ηκ+1)/r2. So, we need to perform a new approximation for the spin-orbit term as a function of the tPT potential parameters. Therefore, we resort to use an appropriate approximation scheme to deal with the centrifugal potential term as
(16)1r2=limα→0α2(d0+1sin2(αr)),0<αr<π/2,
where d0=1/12 is a dimensionless shifting parameter and αr≪1. The approximation (16) is done on the basis that sin(z)=z-z3/3!+z5/5!-z7/7!+⋯ and in the limit when z→0,sin(z)≈z. To show the validity and accuracy of our choice to the approximation scheme (16), we plot the centrifugal potential term 1/r2and its approximations: α2/sin2(αr) and α2(d0+1/sin2(αr))in Figure 2. As illustrated, the three curves coincide together and show how accurate this replacement is.Figure 2
The centrifugal term1/r2 (red line) and its approximations (16) with d0=0 (green dash line) and d0=1/12 (blue dot line), while α=0.002fm-1.Thus, employing such an approximation scheme, we can then write (15a) as
(17)[d2dr2-ηκ(ηκ+1)α2(d0+1sin2(αr))-γ(V1sin2(αr)+V2cos2(αr))-β2d2dr2-ηκ(ηκ+1)α2(d0+1sin2(αr))]Fnκ(r)=0.
Followed by making a new change of variables s(r)=sin2(αr), this allows us to decompose the spin-symmetric Dirac equation (17) into the Schrödinger-type equation satisfying the upper-spinor component Fn,κ(s),
(18){d2ds2+(1/2)-ss(1-s)dds-1s2(1-s)2×(1/2)-ss(1-s)[-As2+Bs-C]}Fn,κ(s)=0,A=-14α2[ηκ(ηκ-1)α2d0+β2],B=14α2[ηκ(ηκ-1)α2(1-d0)+γ(V1-V2)-β2],C=14α2[ηκ(ηκ-1)α2+γV1],
where Fnκ(r)≡Fn,κ(s)has been used. If the previous equation is compared with (A.2), we can obtain the specific values for constants ci(i=1,2,3) as
(19)c1=12,c2=1,c3=1.
In order to obtain the bound state solutions of (17), it is necessary to calculate the remaining parametric constants, that is, ci(i=4,5,…,13), by means of the relation (A.5). Their specific values are displayed in Table 1 for the relativistic tPT potential model. Further, using these constants along with (A.7), we can readily obtain the energy eigenvalue equation for the Dirac-tPT problem as
(20)(n+12+A-B+C+116+C+116)2=A,
or equivalently
(21)(2n+1+121+4V2(M+Enκ-Cs)α2+12(2ηκ-1)2+4V1(M+Enκ-Cs)α2)2=1α2(Enκ-M)(M+Enκ-Cs)-ηκ(ηκ-1)d0.
To show the procedure of determining the energy eigenvalues from (21), we take a set of physical parameter values,M=10fm-1, V1=5.0fm-1, V2=3.0fm-1, Cs=0fm-1, and α=0.8,0.6,0.4,0.2,0.04, and 0.02 [30].Table 1
The specific values of the parametric constants  for the spin symmetric Dirac-tPT problem.
Constant
Analytic value
c
4
1
4
c
5
-
1
2
c
6
1
4
(
1
+
4
A
)
c
7
-
1
4
(
1
+
4
B
)
c
8
1
16
(
1
+
16
C
)
c
9
A
-
B
+
C
+
1
16
c
10
1
2
1
+
16
C
c
11
2
A
-
B
+
C
+
1
16
c
12
1
4
(
1
+
1
+
16
C
)
c
13
1
4
+
A
-
B
+
C
+
1
16In Tables2 and 3, we present the energy spectrum for the spin symmetric case. Obviously, the pairs (np1/2,np3/2),(nd3/2,nd5/2),(nf5/2,nf7/2),(ng7/2,ng9/2), and so on are degenerate states. Thus, each pair is considered as spin doublet and has positive energy. In Table 2, we can see that tensor interaction removes degeneracies between spin doublets. Further, when potential range parameter α approaches zero, the energy eigenvalues approaches a constant. From (21), we find that this constant is M+V1+V2+2V1V2; that is, limα→0En,κ=M+V1+V2+2V1V2, which can be seen from Table 3 too.Table 2
The bound state energy eigenvalues in units offm-1 of the spin symmetry tPT potential for several values of n and κ with α=0.8.
l
n
,
κ
<
0
(
l
,
j
=
l
+
1
/
2
)
E
n
,
κ
<
0  H≠0
E
n
,
κ
<
0
H
=
0
n
,
κ
>
0
(
l
,
j
=
l
-
1
/
2
)
E
n
,
κ
>
0
H
≠
0
E
n
,
κ
>
0
H
=
0
1
0, −2
0
p
3
/
2
26.82780
26.89327
0, 1
0
p
1
/
2
27.02322
26.89327
2
0, −3
0
d
5
/
2
26.89327
27.02322
0, 2
0
d
3
/
2
27.21577
27.02322
3
0, −4
0
f
7
/
2
27.02322
27.21577
0, 3
0
f
5
/
2
27.46825
27.21577
4
0, −5
0
g
9
/
2
27.21577
27.46825
0, 4
0
g
7
/
2
27.77739
27.46825
1
1, −2
1
p
3
/
2
28.96349
29.02725
1, 1
1
p
1
/
2
29.15386
29.02725
2
1, −3
1
d
5
/
2
29.02725
29.15386
1, 2
1
d
3
/
2
29.34155
29.15386
3
1, −4
1
f
7
/
2
29.15386
29.34155
1, 3
1
f
5
/
2
29.58785
29.34155
4
1, −5
1
g
9
/
2
29.34155
29.58785
1, 4
1
g
7
/
2
29.88968
29.58785Table 3
The spin symmetric bound state energy eigenvalues in units offm-1of the tPT potential for several values of n and κ with M=10.0,V1=5.0,V2=3.0,cs=0, and H=0.
l
n
,
κ
(
l
,
j
=
l
±
1
/
2
)
α
=
0.8
α
=
0.4
α
=
0.2
α
=
0.04
α
=
0.02
1
1, −2,1
1
p
3
/
2
,
1
p
1
/
2
29.02725
27.36375
26.54876
25.90551
25.82567
2
1, −3,2
1
d
5
/
2
,
1
d
3
/
2
29.15386
27.39629
26.55699
25.90584
25.82575
3
1, −4,3
1
f
7
/
2
,
1
f
5
/
2
29.34155
27.44495
26.56933
25.90634
25.82588
4
1, −5,4
1
g
9
/
2
,
1
g
7
/
2
29.58785
27.50956
26.585773
25.90701
25.82605
1
2, −2,1
2
p
3
/
2
,
2
p
1
/
2
31.16696
28.43165
27.08151
26.01179
25.87879
2
2, −3,2
2
d
5
/
2
,
2
d
3
/
2
31.29020
28.46376
27.08969
26.01212
25.87887
3
2, −4,3
2
f
7
/
2
,
2
f
5
/
2
31.47301
28.51179
27.10195
26.01262
25.878100
4
2, −5,4
2
g
9
/
2
,
2
g
7
/
2
31.71308
28.57555
27.11828
26.01328
25.87916In Figure3, we have investigated the effect of the tensor potential on the spin doublet splitting by considering the following pairs of orbital: (1p3/2,1p1/2)and (1f7/2,1f5/2). We observe that in the case of T=0 (no tensor interaction), members of spin doublets have same energy. However, in the presence of the tensor potentialT≠0, these degeneracies are removed. We can also see that spin doublet splitting increases with increasingT. The reason is that term 2κT gives different contributions to each level in the spin doublet because T takes different values for each state in the spin doublet.Figure 3
Effect of tensor potential on spin doublets.On the other hand, in order to establish the upper-spinor component of the wave functionsFn,κ(r), namely, (15a), the relations (A.8), (A.9), (A.10), and (A.11) are used. Firstly, we find the first part of the wave function as
(22)ϕ(s)=s(1/4)(1+1+16C)(1-s)(1/4)+A-B+C+(1/16).
Secondly, we calculate the weight function as
(23)ρ(s)=s(1/2)1+16C(1-s)2A-B+C+(1/16),
which gives the second part of the wave function as
(24)yn(s)=Pn((1/2)1+16C,2A-B+C+(1/16))(1-2s),
where Pn(a,b)(y) are the orthogonal Jacobi polynomials. Finally the upper spinor component for arbitrary κ can be found through the relation (A.11)(25)Fnκ(s)=Nnκs(1/4)(1+1+16C)(1-s)(1/4)+A-B+C+(1/16)×Pn((1/2)1+16C,2A-B+C+(1/16))(1-2s),
or
(26)Fnκ(r)=Nnκ(sin(αr))(1/2)(1+ξκ)(cos(αr))(1/2)(1+δ)×Pn((1/2)ξκ,(1/2)δ)(cos(2αr)),
where
(27)ξκ=1+4α2[ηκ(ηκ-1)α2+γV1],δ=1+4γV2α2,
and Nnκ is the normalization constant. Further, the lower-spinor component of the wave function can be calculated by using
(28)Gnκ(r)=1M+Enκ-Cs(ddr+κr-U(r))Fnκ(r),
where E≠-M+Cs and in the presence of the exact spin symmetry (Cs=0), only positive energy states do exist.


## 2.2.p-Spin Symmetric Limit

Ginocchio showed that there isp-spin symmetry in case when the relationship between the vector potential and the scalar potential is given by V(r)=-S(r) [7]. Further, Meng et al. showed that if d[V(r)+S(r)]/dr=dΣ(r)/dr=0, then Σ(r)=Cps=constant, for which the p-spin symmetry is exact in the Dirac equation [65–67]. Thus, choosing the Δ(r) as tPT potential and U(r) as Coulomb potential, (12) under this symmetry becomes(29a)[d2dr2-λκ(λκ-1)r2-γ~(V1sin2(αr)+V2cos2(αr))-β~2]×Gnκ(r)=0,(29b)γ~=Enκ-M-Cps,β~2=(M+Enκ)(M-Enκ+Cps),whereλκ=κ+H, κ=-l~ and κ=l~+1 for κ<0 and κ>0, respectively. Employing the new approximation for the spin-orbit pseudocentrifugal term,λκ(λκ-1)/r2, that is, (16), the p-spin Dirac equation (29a) can be written as
(30)[d2dr2-λκ(λκ-1)α2(d0+1sin2(αr))-γ~(V1sin2(αr)+V2cos2(αr))-β~2d2dr2-λκ(λκ-1)α2(d0+1sin2(αr))]Gnκ(r)=0.
To avoid repetition of the negative energy solution of (30), the p-spin symmetric case can be readily obtained directly via the spin symmetric solution throughout the following parametric mappings:
(31)Fnκ(r)⟷Gnκ(r),κ→κ-1,V(r)→-V(r)(i.e.,V1→-V1,V2→-V2),Enκ→-Enκ,Cs→-Cps.
Following the previous procedure, one can obtain the p-spin symmetric energy equation as
(32)-(2n+1+121+4V2(Enκ-M-Cps)α2+12(2λκ-1)2+4V1(Enκ-M-Cps)α2)2=1α2(M+Enκ)(M-Enκ+Cps)+λκ(λκ-1)d0.
Furthermore, the lower-spinor component of the wave functions is found as
(33)Gnκ(r)=N~n,κ(sin(αr))(1/2)(1+ξ~κ)×(cos(αr))(1/2)(1+δ~)Pn((1/2)ξ~,(1/2)δ~)(cos(2αr)),
with
(34)ξ~κ=(2λκ-1)2+4V1(Enκ-M-Cps)α2,δ~=1+4V2(Enκ-M-Cps)α2,
where N~nκ is the normalization constant.In Tables4 and 5, we give the numerical results for the p-spin symmetric case. In this case, we take the set of parameter values, M=10fm-1, V1=-5.0fm-1, V2=-3.0fm-1, cps=0fm-1, and α=0.8,0.4,0.2,0.04, and 0.02 [30]. We observe the degeneracy in the following doublets(1s1/2,0d3/2),(1p3/2,0f5/2),(1d5/2,0g7/2), (1f7/2,0h9/2), and so on. Thus, each pair is considered as p-spin doublet and has negative energy. In Table 4, we can see that tensor interaction removes degeneracies between spin doublets.Table 4
The bound state energy eigenvalues in units offm-1 of the p-spin symmetry tPT potential for several values of n and κ with α=0.8.
l
~
n
,
κ
<
0
(
l
,
j
)
E
n
,
κ
<
0 H≠0
E
n
,
κ
<
0 H=0
n
-
1
,
κ
>
0
(
l
+
2
,
j
+
1
)
E
n
-
1
,
κ
>
0
H
≠
0
E
n
-
1
,
κ
>
0
H
=
0
1
1, −1
1
s
1
/
2
−28.96349
−29.02725
0, 2
0
d
3
/
2
−29.15386
−29.02725
2
1, −2
1
p
3
/
2
−29.02725
−29.15386
0, 3
0
f
5
/
2
−29.34155
−29.15386
3
1, −3
1
d
5
/
2
−29.15386
−29.34155
0, 4
0
g
7
/
2
−29.58785
−29.34155
4
1, −4
1
f
7
/
2
−29.34155
−29.58785
0, 5
0
h
9
/
2
−29.88968
−29.58785
1
2, −1
2
s
1
/
2
−31.10492
−31.16696
1, 2
1
d
3
/
2
−31.29020
−31.16696
2
2, −2
2
p
3
/
2
−31.16696
−31.29020
1, 3
1
f
5
/
2
−31.47301
−31.29020
3
2, −3
2
d
5
/
2
−31.29020
−31.47301
1, 4
1
g
7
/
2
−31.71308
−31.47301
4
2, −4
2
f
7
/
2
−31.47301
−31.71308
1, 5
1
h
9
/
2
−32.00754
−31.71308Table 5
The bound state energy eigenvalues in units offm-1 of the p-spin symmetry tPT potential for several values of n and κ with  H=0.
l
~
n
,
κ
(
l
,
j
)
α
=
0.8
α
=
0.4
α
=
0.2
α
=
0.04
α
=
0.02
1
1, −1,2
1
s
1
/
2
,
0
d
3
/
2
−29.02725
−27.36375
−26.54876
−25.90551
−25.82567
2
1, −2,3
1
p
3
/
2
,
0
f
5
/
2
−29.15386
−27.39629
−26.55699
−25.90584
−25.82575
3
1, −3,4
1
d
5
/
2
,
0
g
7
/
2
−29.34155
−27.44495
−26.56933
−25.90634
−25.82588
4
1, −4,5
1
f
7
/
2
,
0
h
9
/
2
−29.58785
−27.50956
−26.58577
−25.90701
−25.82605
1
2, −1,2
2
s
1
/
2
,
1
d
3
/
2
−31.16696
−28.43165
−27.08151
−26.01179
−25.87879
2
2, −2,3
2
p
3
/
2
,
1
f
5
/
2
−31.29020
−28.46376
−27.08969
−26.01212
−25.87887
3
2, −3,4
2
d
5
/
2
,
1
g
7
/
2
−31.47301
−28.51179
−27.10195
−26.01262
−25.87810
4
2, −4,5
2
f
7
/
2
,
1
h
9
/
2
−31.71308
−28.57555
−27.11828
−26.01328
−25.87916In Figure4, we have investigated the effect of the tensor potential on the p-spin doublet splitting by considering the following pairs of orbital: (1d5/2,0g7/2) and  (2f7/2,1h9/2), and one can observe that the results obtained in the p-spin symmetric limit resemble the ones observed in the spin symmetric limit.Figure 4
Effect of tensor potential onp-spin doublets.On the other hand, the upper-spinor component of the Dirac wave function can be calculated by(35)Fnκ(r)=1M-Enκ+Cps(ddr-κr+U(r))Gnκ(r),
where E≠M+Cps, and in the presence of the exact p-spin symmetry (Cps=0), only negative energy states do exist.


## 2.3. The Nonrelativistic Limiting Case

In this section, we study the energy eigenvalue equation (21) and upper-spinor component of wave function (26) of the Dirac-tPT problem under the nonrelativistic limits: Cs=0,κ→l,Enκ-M≃Enl, and M+Enκ≃2m. Thereby, we obtain the energy equation of the Schrödinger equation with any arbitrary orbital state for the tPT potential as
(36)Enl=ℏ2α2l(l+1)d02m+2ℏ2α2m×[n+12+14((2l+1)2+8mV1ℏ2α2+1+8mV2ℏ2α2)]2.
In the limit when α→0, the vibration-rotation energy formula (36) reduces into a constant value:
(37)limα→0Enl=(V1+V2)2.
Further, there is no loss of generality if d0=0; then, (36) becomes
(38)Enl=2ℏ2α2m×[n+12+14((2l+1)2+8mV1ℏ2α2+1+8mV2ℏ2α2)]2,
where n=0,1,2,… and l=0,1,2,… are the vibration and rotation quantum numbers, respectively. To obtain numerical energy eigenvalues for the present potential model, we take the following set of parameter values; namely, M=10fm-1, V1=5.0fm-1, V2=3.0fm-1, and α=1.2,0.8,0.4,0.2,0.02, and 0.002 [30]. As seen from Table 6, in the limit when potential range parameter α approaches zero, the energy eigenvalues approach a constant value given by (37).Table 6
The bound state energy levelsEnl of the Schrödinger equation for the tPT potential.
E
n
l
State(n,l)
M
=
10.0
f
m
-
1
,
V
1
=
5
f
m
-
1
,
V
2
=
3
f
m
-
1 [30]
α
=
1.2
α
=
0.8
α
=
0.4
α
=
0.2
α
=
0.02
α
=
0.002
1
s [32]
22.87051710
20.32991862
17.95616357
16.83082621
15.85264289
15.75661628
2
s [32]
28.29143398
23.68420415
19.50420742
17.57271070
15.92394680
15.76371786
2
p
28.64395419
23.82847894
19.53712286
17.58054181
15.92402153
15.76371860
3
s [32]
34.28835086
27.2944896
21.11625126
18.33059518
15.99541071
15.77082105
3
p
34.67512504
27.44896381
21.15044543
18.33858626
15.99548560
15.77082179
3
d
35.43921159
27.75631556
21.21875330
18.35456399
15.99563534
15.77082328
4
s [32]
40.86126774
31.16077522
22.79229510
19.10447967
16.06703463
15.77792584
4
p
41.28229584
31.32544868
22.82776800
19.11263070
16.06710967
15.77792658
4
d
42.11348590
31.65300783
22.89862721
19.12892817
16.06725974
15.77792806
4
f
43.33519178
32.14003977
23.00470171
19.15336297
16.06748485
15.77793030Also, we can get the radial wave functions of the Schrödinger equation with tPT potential as(39)Rn,l(s)=s(1/4)[1+(2l+1)2+(8mV1/ℏ2α2)]×(1-s)(1/4)(1+1+(8mV2/ℏ2α2))×Pn((1/2)(2l+1)2+(8mV1/ℏ2α2),(1/2)1+(8mV2/ℏ2α2))×(1-2s).
Inserting s=sin2(αr) in the previous equation, we can obtain (40a)Rn,l(r)=Nnl(sin(αr))(1+ηl)/2×(cos(αr))(1+δ)/2Pn(ηl/2,δ/2)(cos(2αr)),(40b)ηl=(2l+1)2+8mV1ℏ2α2,δ=1+8mV2ℏ2α2,where Nnl is a normalization factor to be calculated from the normalization conditions.


## 3. Concluding Remarks

In this work, we have studied the bound state solutions of the Dirac equation with trigonometric Pöschl-Teller and Coulomb-like tensor potentials for any spin-orbit quantum numberκ. By making an appropriate approximation to deal with the spin-orbit centrifugal (pseudo-centrifugal) coupling term, we have obtained the approximate energy eigenvalue equation and the unnormalized two components of the radial wave functions expressed in terms of the Jacobi polynomials using the NU method. It is found that tensor interaction removes degeneracies between each pair of pseudospin or spin doublets.



---

*Source: 196986-2013-03-14.xml*