We solve approximately Dirac equation for Eckart plus Hulthen potentials with Coulomb-like and Yukawa-like tensor interaction in the presence of spin and pseudospin symmetry for k≠0. The formula method is used to obtain the energy eigenvalues and wave functions. We also discuss the energy eigenvalues and the Dirac spinors for Eckart plus Hulthen potentials with formula method. To show the accuracy of the present model, some numerical results are shown in both pseudospin and spin symmetry limits.
1. Introduction
One of the interesting problems in nuclear and high energy physics is to obtain analytical solution of the Klein-Gordon, Duffin-Kemmer-Petiau, and Dirac equations for mixed vector and scalar potentials [1]. The study of relativistic effect is always useful in some quantum mechanical systems [2, 3]. Therefore, the Dirac equation has become the most appealing relativistic wave equation for spin-1/2 particles. For example, in the relativistic treatment of nuclear phenomena, the Dirac equation is used to describe the behavior of the nuclei in nucleus and also to solve many problems of high energy physics and chemistry. For this reason, it has been used extensively to study the relativistic heavy ion collisions and heavy ion spectroscopy and more recently in laser-matter interaction (for a review, see [4] and references therein) and condensed matter physics [5, 6]. The idea about spin symmetry and pseudospin symmetry with the nuclear shell model has been introduced in 1969 by Arima et al. (1969) and Hecht and Adler (1969) [7, 8]. Spin and pseudospin symmetries are SU(2) symmetries of a Dirac Hamiltonian with vector and scalar potentials. They are realized when the difference, Δ(r)=V(r)-S(r), or the sum, ∑(r)=V(r)+S(r), is constant. The near realization of these symmetries may explain degeneracy in some heavy meson spectra (spin symmetry) or in single-particle energy levels in nuclei (pseudospin symmetry), when these physical systems are described by relativistic mean-field theories (RMF) with scalar and vector potentials [9]. Recently, some authors have studied various types of potential with a tensor potential, under the conditions of pseudospin and spin symmetry. They have found out that the tensor interaction removes the degeneracy between two states in the pseudospin and spin doublet [10–13]. The pseudospin and spin symmetry appearing in nuclear physics refers to a quasidegeneracy of the single-nucleon doublets and can be characterized with the nonrelativistic quantum numbers (n,l,j=l+1/2) and (n,l+2,j=l+3/2), where n, l, and j are the single-nucleon radial, orbital, and total angular momentum quantum numbers for a single particle, respectively [7, 8]. These kinds of various methods have been used for the analytical solutions of the Klein-Gordon equation and Dirac equation such as the super symmetric quantum mechanics [14–16], asymptotic iteration method (AIM) [17, 18], factorization method [19, 20], Laplace transform approach [21], GPS method [22, 23] and the path integral method [24–26], and Nikiforov-Uvarov method [27–29]. The Klein-Gordon and Dirac wave equations are frequently used to describe the particle dynamics in relativistic quantum mechanics with some typical kinds of potential by using different methods [30]. For example, Kratzer potential [31, 32], Woods-Saxon potential [33, 34], Scarf potential [35, 36], Hartmann potential [37, 38], Rosen Morse potential, [39, 40], Hulthen potential [41], and Eckart potential [42, 43].
In this paper, we attempt to solve approximately Dirac wave equation for k≠0 for Eckart plus Hulthen potentials for the spin and pseudospin symmetry with a tensor potential by using the formula method. The organization of this paper is as follows: in Section 2, the formula method is reviewed [44]. In Section 3 we review basic Dirac equations briefly. In Sections 3.1 and 3.2, solutions of Dirac wave equation for the spin and pseudospin symmetry of these potentials in the presence of Coulomb-like tensor interaction are presented, respectively. In Sections 3.3 and 3.4, solutions of Dirac wave equation for the spin and pseudospin symmetry of these potentials in the presence of Coulomb-like plus Yukawa-like tensor interaction are presented, respectively. In Section 4, we provide results and discussion. The conclusion is given in Section 5.
2. Review of Formula Method
The formula method has been used to solve the Schrodinger, Dirac, and Klein-Gordon wave equations for a certain kind of potential. In this method, the differential equations can be written as follows [44]:(1)Ψn′′s+k1-k2ss1-k3sΨn′s+As2+Bs+Cs21-k3s2Ψns=0.For a given Schrödinger-like equation in the presence of any potential model which can be written in the form of (1), the energy eigenvalues and the corresponding wave function can be obtained by using the following formulas, respectively [44]:(2)k42-k52-1-2n/2-1/2k3k2-k3-k22-4A221-2n/2-1/2k3k2-k3-k22-4A2-k52=0,k3≠0,(3)Ψns=Nnsk41-k3sk5F12-n,n+2k4+k5+k2k3-1;2k4+k1,k3s,where (4)k4=1-k1+1-k12-4C2,k5=12+k12-k22k3+12+k12-k22k32-Ak32+Bk3+C.And Nn is the normalization constant. In special case where k3→0, the energy eigenvalues and the corresponding wave function can be obtained as [44](5)B-k4k2-nk22k4+k1+2n2-k52=0,Ψns=Nnsk4exp-k5sF11-n;2k4+k1;2k5+k2s.The solution provides a valuable means for checking and improving models and numerical methods introduced for solving complicated quantum systems.
3. Basic Dirac Equations
In the relativistic description, the Dirac equation of a single nucleon with the mass moving in an attractive scalar potential S(r) and a repulsive vector potential V(r) can be written as [45](6)-iħcα^·∇^+β^Mc2+SrΨnr,k=E-VrΨnr,k,where E is the relativistic energy, M is the mass of a single particle, and α and β are the 4 × 4 Dirac matrices. For a particle in a central field, the total angular momentum J and K^=-β^(α^·L^+ħ) commute with the Dirac Hamiltonian, where L is the orbital angular momentum. For a given total angular momentum j, the eigenvalues of the K^ are k=±(j+1/2), where negative sign is for aligned spin and positive sign is for unaligned spin. The wave functions can be classified according to their angular momentum j and spin-orbit quantum number k as follows:(7)Ψnr,kr,θ,ϕ=1rFnr,krYjmlθ,ϕiGnr,krYjml~θ,ϕ,where Fnr,k(r) and Gnr,k(r) are upper and lower components and Yjml(θ,ϕ) and Yjml~(θ,ϕ) are the spherical harmonic functions. nr is the radial quantum number and m is the projection of the angular momentum on the z-axis. The orbital angular momentum quantum numbers l and l~ represent the spin and pseudospin quantum numbers. Substituting (7) into (6), we obtain couple equations for the radial part of the Dirac equation as follows by ħ=c=1:(8)ddr+kr-UrFnr,kr=M+En,k-ΔrGnr,kr,ddr-kr+UrGnr,kr=M-En,k+∑rFnr,kr,where Δ(r)=V(r)-S(r) and ∑(r)=V(r)+S(r) are the difference and the sum of the potentials V(r) and S(r), respectively, and U(r) is a tensor potential. We obtain the second-order Schrodinger-like equation as(9)d2dr2-kk+1r2+2kUrr-dUrr-U2r-M+En,k-ΔrM-En,k+∑r+dΔr/drd/dr+k/r-UrM+En,k-ΔrFnr,kr=0,d2dr2-kk-1r2+2kUrr+dUrr-U2r-M+En,k-ΔrM-E+∑r+d∑r/drd/dr-k/r+UrM-En,k+∑rGnr,kr=0.We consider bound state solutions that demand the radial components satisfying Fnr,k(0)=Gnr,k(0)=0, and Fnr,k(∞)=Gnr,k(∞)=0 [45].
3.1. Solution Spin Symmetric with Coulomb-Like Tensor Interaction
Under the condition of the spin symmetry—that is, dΔ(r)/dr=0 or Δ(r)=Cs=const—the upper component Dirac equation can be written as(10)d2dr2-kk+1r2+2kUrr-dUrr-U2r-M+En,k-CsM-En,k+∑rFnr,kr=0.The potential ∑(r) is taken as the Eckart [42, 43] plus Hulthen potentials [41]: (11)∑r=4q1e-2αr1-e-2αr2-q21+e-2αr1-e-2αr+v01-e-2αr-v11-e-2αr2,where the parameters q1, q2, v0, and v1 are real parameters; these parameters describe the depth of the potential well, and the parameter α is related to the range of the potential.
For the tensor term, we consider the Coulomb-like potential [46],(12)Ur=-Hr,H=ZfZge24πε0,r>Rc,where Rc is the Coulomb radius and Zf and Zg stand for the charges of the projectile particle f and the target nucleus g, respectively.
By substituting (11) and (12) into (10), we obtain the upper radial equation of Dirac equation as(13)d2dr2-τkτk+1r2-γ-β4q1e-2αr1-e-2αr2-q21+e-2αr1-e-2αr+v01-e-2αr-v11-e-2αr2Fnr,kr=0,where τk=k+H, γ=(M+En,k-Cs)(M-En,k) and β=(M+En,k-Cs).
Equation (13) is exactly solvable only for the case of k=0. In order to obtain the analytical solutions of (13), we employ the improved approximation scheme suggested by Greene and Aldrich [47, 48] and replace the spin-orbit coupling term with the expression that is valid for α≤1 [49]:(14)1r2≈4α21-e-2αr2.The behavior of the improved approximation is plotted in Figure 1. We can see the good agreement for small α values.
The behavior approximation for α=0.07 fm^{−1}.
By applying the transformation s=exp(-2αr), (13) is brought into the form(15)Fn,k′′s+1-ss1-sFn,k′s+1s21-s2As2+Bs+CFn,ks=0,where the parameters A, B, and C are considered as follows:(16)A=-14α2γ+βq2,B=14α22γ-4βq1+βv0,C=-14α24α2τkτk-1+γ-βq2+βv0-βv1.Now by comparing (15) with (1), we can easily obtain the coefficients ki (i=1,2,3) as follows:(17)k1=k2=k3=1.The values of the coefficients ki (i=4,5) are also found from (4) as below:(18)k4=-C,k5=12+14-A+B+C.Thus, by the use of energy equation (2) for energy eigenvalues, we find(19)-C-1/2+1/4-A+B+C2-1-2n/2-1/21--4A221-2n/2-1/21--4A2-12+14-A+B+C2=0.In Tables 1–3, we give the numerical results for the spin symmetric energy eigenvalues (in units of fm^{−1}).
Energies of the spin symmetry limit in the presence and absence of Coulomb-like tensor interaction by parameters M=10 fm^{−1}, c=1, h=1, α=0.05 fm^{−1}, V1 = 0.4 fm^{−1}, V0 = 0.3 fm^{−1}, q1 = 0.1 fm^{−1}, q2 = 0.2 fm^{−1}, and Cs = 5 fm^{−1}.
l
n,k>0
State (l,j)
En,ks
(H=0)
En,ks (H=0.65)
n,k<0
State (l,j+1)
En,ks (H=0)
En,ks (H=0.65)
1
1,1
1p_{1/2}
3.854830541
5.230932502
1,-2
1p_{3/2}
3.854830541
2.022333483
2
1,2
1d_{3/2}
5.816938791
6.683341758
1,-3
1d_{5/2}
5.816938791
4.646742373
3
1,3
1f_{5/2}
7.054878462
7.612226007
1,-4
1f_{7/2}
7.054878462
6.315225843
4
1,4
1g_{7/2}
7.855626422
8.227670602
1,-5
1g_{9/2}
7.855626422
7.373922639
1
2,1
2p_{1/2}
5.770989098
6.634606185
2,-2
2p_{3/2}
5.770989098
4.607350329
2
2,2
2d_{3/2}
7.005594455
7.562996503
2,-3
2d_{5/2}
7.005594455
6.267433123
3
2,3
2f_{5/2}
7.806806149
8.180021794
2,-4
2f_{7/2}
7.806806149
7.324529686
4
2,4
2g_{7/2}
8.346470004
8.605903663
2,-5
2g_{9/2}
8.346470004
8.019210395
The energy eigenvalues (in units of fm^{−1}) for the spin symmetry limit with parameters M=10 fm^{−1}, c=1, h=1, V1 = 4 fm^{−1}, V0 = 3 fm^{−1}, q1 = 1 fm^{−1}, q2 = 2 fm^{−1}, and Cs = 5 fm^{−1}.
α (fm^{−1})
En,k (H=0)
En,k (H=0.65)
1d_{3/2}
1d_{5/2}
1d_{3/2}
1d_{5/2}
0.3
2.367211785
2.367211785
3.624581074
0.93609384
0.35
3.53989514
3.53989514
4.771398349
2.070547382
0.4
4.521194951
4.521194951
5.690996586
3.069929104
0.5
5.998968975
5.998968975
7.009500454
4.671333928
0.6
6.995852039
6.995852039
7.853318047
5.827299953
0.7
7.670572507
7.670572507
8.400594789
6.652859598
The energy eigenvalues (in units of fm^{−1}) for the spin symmetry limit with parameters c=1, h=1, α=0.05 fm^{−1}, V1 = 4 fm^{−1}, V0 = 3 fm^{−1}, q1 = 1 fm^{−1}, q2 = 2 fm^{−1}, and Cs = 5 fm^{−1}.
M (fm^{−1})
En,k (H=0)
En,k (H=0.65)
1d_{3/2}
1d_{5/2}
1d_{3/2}
1d_{5/2}
5.5
3.335677895
3.335677895
3.893525562
2.662575645
6
3.479942785
3.479942785
4.105505756
2.72071499
6.5
3.618211096
3.618211096
4.311706034
2.7725715
7
3.752572889
3.752572889
4.514086551
2.820419493
8
4.013951033
4.013951033
4.911575769
2.908709541
9
4.269356779
4.269356779
5.303088298
2.991067928
10
4.521194951
4.521194951
5.690996586
3.069929104
Let us find the corresponding wave functions. In reference to (3) and (18), we can obtain the upper wave function as(20)Fn,ksr=Ne-2αr-c1-e-2αr1/2+1/4+A+B+CF12-n,n+2-c+12+14+A+B+C;2-c+1,e-2αr,where N is the normalization constant; on the other hand, the lower component of the Dirac spinor can be calculated from(21)Gn,ksr=1M+En,ks-Csddr+kr-UrFn,ksr.We have obtained the energy eigenvalues and the wave function of the radial Dirac equation for Eckart plus Hulthen potentials with Coulomb-like tensor interaction in the presence of the spin symmetry for k≠0.
3.2. Solution Pseudospin Symmetry with Coulomb-Like Tensor Interaction
For pseudospin symmetry—that is, d∑(r)/dr=0 or ∑(r)=Cps=const—the lower component Dirac equation can be written as(22)d2dr2-kk-1r2+2kUrr+dUrr-U2r-M+En,k-ΔrM-E+∑rGnr,kr=0.We consider the scalar, vector, and tensor potentials as the following [41]: (23)Δr=4q1e-2αr1-e-2αr2-q21+e-2αr1-e-2αr+v01-e-2αr-v11-e-2αr2,Ur=-Hr,H=ZfZge24πε0,r>Rc.Substituting (23) into (22), we obtain the lower radial equation of Dirac equation as(24)d2dr2-λkλk-1r2-γ′+β′4q1e-2αr1-e-2αr2-q21+e-2αr1-e-2αr+v01-e-2αr-v11-e-2αr2Gnr,kr=0,where λk=k+H, γ′=(M+En,k)(M-En,k+Cps) and β′=(M-En,k+Cps).
By using the transformation s=exp(-2αr) and employing the improved approximation, (24) is brought into the form(25)Gn,k′′s+1-ss1-sGn,k′s+1s21-s2A′s2+B′s+C′Gn,ks=0,where the parameters A′, B′, and C′ are considered as follows:(26)A′=-14α2γ′-β′q2,B′=14α22γ′+4β′q1-β′v0,C′=-14α24α2λkλk-1+γ′+β′q2-β′v0+β′v1.We can easily obtain the coefficients ki (i=1,2,3) by comparing (25) with (1) as follows:(27)k1′=k2′=k3′=1.The values of the coefficients ki′ (i=4,5) are also found from (4) as below:(28)k4′=-C′,k5′=12+14-A′+B′+C′.We have (29) for energy eigenvalues by the use of (2):(29)-C′-1/2+1/4-A′+B′+C′2-1-2n/2-1/21--4A′221-2n/2-1/21--4A′2-12+14-A′+B′+C′2=0.In Tables 4–6, we give the numerical results for the pseudospin symmetric energy eigenvalues (in units of fm^{−1}).
The energy eigenvalues (in units of fm^{−1}) for the pseudospin symmetry limit in the presence and absence of Coulomb-like tensor interaction: M=10 fm^{−1}, c=1, ħ=1, α=0.05 fm^{−1}, V1 = 0.4 fm^{−1}, V0 = 0.3 fm^{−1}, q1 = 0.1 fm^{−1}, q2 = −0.2 fm^{−1}, and Cps = −5 fm^{−1}.
l
n,k<0
State (l,j)
En,kps (H=0)
En,kps (H=0.65)
n,k>0
State (l+2,j+1)
En,kps (H=0)
En,kps (H=0.65)
1
1,-1
1s_{1/2}
-7.065764205
-5.701645852
1,2
1d_{3/2}
-7.065764205
-7.916615587
2
1,-2
1p_{3/2}
-8.242801196
-7.570912051
1,3
1f_{5/2}
-8.242801196
-8.691210574
3
1,-3
1d_{5/2}
-8.872276828
-8.505344072
1,4
1g_{7/2}
-8.872276828
-9.132415328
4
1,-4
1f_{7/2}
-9.241962415
-9.022813407
1,5
1h_{9/2}
-9.241962415
-9.404940153
1
2,-1
2s_{1/2}
-8.20892624
-7.540097339
2,2
2d_{3/2}
-8.20892624
-8.656719162
2
2,-2
2p_{3/2}
-8.837973611
-8.470939351
2,3
2f_{5/2}
-8.837973611
-9.098960172
3
2,-3
2d_{5/2}
-9.209115205
-8.988907697
2,4
2g_{7/2}
-9.209115205
-9.37334647
4
2,-4
2f_{7/2}
-9.44496483
-9.303214658
2,5
2h_{9/2}
-9.44496483
-9.554705712
The energy eigenvalues (in units of fm^{−1}) for the pseudospin symmetry limit with parameters M=10 fm^{−1}, c=1, ħ=1, V1 = 4 fm^{−1}, V0 = 3 fm^{−1}, q1 = 1 fm^{−1}, q2 = −2 fm^{−1}, and Cps = −5 fm^{−1}.
α(fm^{−1})
En,kps (H=0)
En,kps (H=0.65)
1s_{1/2}
1d_{3/2}
1s_{1/2}
1d_{3/2}
0.3
-4.313211747
-4.313211747
-2.304589779
-5.851610879
0.35
-5.441940815
-5.441940815
-3.496832686
-6.846892358
0.4
-6.330481794
-6.330481794
-4.501756572
-7.592836004
0.5
-7.58011028
-7.58011028
-6.029844453
-8.587003959
0.6
-8.366177606
-8.366177606
-7.07440217
-9.176918108
0.7
-8.871669319
-8.871669319
-7.792319679
-9.537028876
The energy eigenvalues (in units of fm^{−1}) for the pseudospin symmetry limit with parameters c=1, ħ=1, α=0.4 fm^{−1}, V1=4 fm^{−1}, V0=3 fm^{−1}, q1=1 fm^{−1}, q2=-2 fm^{−1}, and Cps=-5 fm^{−1}.
M (fm^{−1})
En,kps (H=0)
En,kps (H=0.65)
1s_{1/2}
1d_{3/2}
1s_{1/2}
1d_{3/2}
5.5
-4.282887209
-4.282887209
-3.435345049
-4.891201159
6
-4.519466496
-4.519466496
-3.562129097
-5.200846041
6.5
-4.751863076
-4.751863076
-3.685040453
-5.506165891
7
-4.981434942
-4.981434942
-3.80536706
-5.808539775
8
-5.435100161
-5.435100161
-4.04106051
-6.407528709
9
-5.884189891
-5.884189891
-4.27265172
-7.001673282
10
-6.330481794
-6.330481794
-4.501756572
-7.592836004
By using (3) and (28), we can finally obtain the lower component of the Dirac spinor as below:(30)Gn,kpsr=Ne-2αr-c′1-e-2αr1/2+1/4+A′+B′+C′F21-n,n+2-c′+12+14+A′+B′+C′;2-c′+1,e-2αr,where N′ is the normalization constant; on the other hand, the upper component of the Dirac spinor can be calculated from (30) as(31)Fn,kpsr=1M-En,kps+Cpsddr-kr+UrGn,kpsr.We have obtained the energy eigenvalues and the spinors of the radial Dirac equation for Eckart plus Hulthen potentials with Coulomb-like tensor interaction in the presence of the pseudospin symmetry for k≠0. We show in Figure 2 behavior energy for various H in the spin and pseudospin symmetry.
Energy spectra in the (a) spin and (b) pseudospin symmetries at various H for Coulomb-tensor interaction with parameters M=10 fm^{−1}, c=1, ħ=1, α=0.4 fm^{−1}, V1=4 fm^{−1}, V0=3 fm^{−1}, q1=1 fm^{−1}, q2ps=-2 fm^{−1}, q2s=2 fm^{−1}, Cs=5 fm^{−1}, and Cps=-5 fm^{−1}.
3.3. Solution Spin Symmetry with Coulomb Plus Yukawa-Like Tensor Interaction
In this section, for the spin symmetry, we consider Δ(r)=Cs, ∑(r), and U(r) as the following:(32)∑r=4q1e-2αr1-e-2αr2-q21+e-2αr1-e-2αr+v01-e-2αr-v11-e-2αr2,Ur=-Hr-Aexp-2αrr,where H and A are the real parameters; substitution of (32) into (10) yields(33)Fn,k′′s+1-ss1-sFn,k′s+1s21-s2χ2s2+χ1s+χ0Fn,ks=0,where the parameters χ2, χ1, and χ0 are considered as follows:(34)χ2=-14α2γ+βq2+AA-1,χ1=14α22γ-4βq1+βv0-2Aτk,χ0=-14α24α2τkτk-1+γ-βq2+βv0-βv1.By comparing (33) with (1), we can easily obtain the coefficients ki (i=1,2,3) as follows:(35)k1=k2=k3=1.The values of the coefficients ki (i=4,5) are also found from (4) as below:(36)k4=-χ0,k5=12+14-χ2+χ1+χ0.By the use of energy equation (2) for energy eigenvalues, we have(37)-χ0-1/2+1/4-χ2+χ1+χ02-1-2n/2-1/21--4χ2221-2n/2-1/21--4χ22-12+14-χ2+χ1+χ02=0.In reference to (3) and using (36), we can obtain the upper wave function:(38)Fn,ksr=Ne-2αr-χ01-e-2αr1/2+1/4+χ2+χ1+χ0F21-n,n+2-χ0+12+14+χ2+χ1+χ0;2-χ0+1,e-2αr,where N is the normalization constant; on the other hand, the upper component of the Dirac spinor can be calculated from (38) as(39)Gn,ksr=1M+En,ks-Csddr+kr-UrFn,ksr.The effects of the Yukawa-like tensor interactions on the upper and lower components for the spin symmetry are shown in Figure 3.
(a) Upper and (b) lower components of 1f7/2 in the spin symmetry in the presence and absence of Yukawa tensor interaction for M=10 fm^{−1}, c=1, ħ=1, α=0.4 fm^{−1}, V1=4 fm^{−1}, V0=3 fm^{−1}, q1=1 fm^{−1}, q2=2 fm^{−1}, and Cs=5 fm^{−1}.
We have obtained the energy eigenvalues and the spinors of the radial Dirac equation for Eckart plus Hulthen potentials with the spin symmetry for k≠0 in the presence and absence of Yukawa tensor.
3.4. Solution Pseudospin Symmetry with Coulomb Plus Yukawa-Like Tensor Interaction
In this section, for the pseudospin symmetry, we consider ∑(r)=Cs, Δ(r), and U(r) as the following: (40)Δr=4q1e-2αr1-e-2αr2-q21+e-2αr1-e-2αr+v01-e-2αr-v11-e-2αr2,Ur=-Hr-Aexp-2αrr,where H and A are the real parameters. Substitution of (40) into (22) yields(41)Gn,k′′s+1-ss1-sGn,k′s+1s21-s2χ2′s2+χ1′s+χ0′Gn,ks=0,where the parameters χ2′, χ1′, and χ0′ are considered as follows:(42)χ2′=-14α2γ-βq2+AA+1,χ1′=14α22γ+4βq1-βv0-2Aλk-1,χ0′=-14α24α2λkλk-1+γ+βq2-βv0+βv1.By comparing (41) with (1), we can easily obtain the coefficients ki′ (i=1,2,3) as follows:(43)k1′=k2′=k3′=1.The values of the coefficients ki′ (i=4,5) are also found from (4) as below:(44)k4′=-χ0′,k5′=12+14-χ2′+χ1′+χ0′.By using energy equation (2) for energy eigenvalues, we have(45)-χ0′-1/2+1/4-χ2′+χ1′+χ0′2-1-2n/2-1/21--4χ2′221-2n/2-1/21--4χ2′2-12+14-χ2′+χ1′+χ0′2=0.By the use of (3) and (44), we can finally obtain the lower component of the Dirac spinor as below: (46)Gn,kpsr=Ne-2αr-χ0′1-e-2αr1/2+1/4+χ2′+χ1′+χ0′F21-n,n+2-χ0′+12+14+χ2′+χ1′+χ0′;2-χ0′+1,e-2αr,where N′ is the normalization constant; on the other hand, the upper component of the Dirac spinor can be calculated from (46) as(47)Fn,kpsr=1M-En,kps+Cpsddr-kr+UrGn,kpsr.The effects of the Yukawa-like tensor interactions on the upper and lower components for the pseudospin symmetry are shown in Figure 4.
(a) Upper and (b) lower components of 1d5/2 in the pseudospin symmetry in the presence and absence of Yukawa tensor interaction for M=10 fm^{−1}, c=1, ħ=1, α=0.4 fm^{−1}, V1=4 fm^{−1}, V0=3 fm^{−1}, q1=1 fm^{−1}, q2=-2 fm^{−1}, and Cps=-5 fm^{−1}.
We have obtained the energy eigenvalues and the spinors of the radial Dirac equation for Eckart plus Hulthen potentials with the pseudospin symmetry for k≠0 in the presence and absence of Yukawa tensor interaction. In Figures 5 and 6, we show the effect of Coulomb-like tensor interaction and Yukawa plus Coulomb-like tensor interaction in the remove of degeneracy.
Energy spectra in the (a) spin and (b) pseudospin symmetries versus A for Yukawa-like tensor interaction with parameters M=10 fm^{−1}, c=1, ħ=1, α=0.4 fm^{−1}, V1=4 fm^{−1}, V0=3 fm^{−1}, q1=1 fm^{−1}, q2ps=-2 fm^{−1}, q2s=2 fm^{−1}, Cs=5 fm^{−1}, and Cps=-5 fm^{−1}.
Energy spectra in the (a) spin and (b) pseudospin symmetries versus A=H for Yukawa plus Coulomb-like tensor interaction with parameters M=10 fm^{−1}, c=1, ħ=1, α=0.4 fm^{−1}, V1=4 fm^{−1}, V0=3 fm^{−1}, q1=1 fm^{−1}, q2ps=-2 fm^{−1}, q2s=2 fm^{−1}, Cs=5 fm^{−1}, and Cps=-5 fm^{−1}.
In Figure 7, we show the comparison between Coulomb and Yukawa-like tensor interaction for spin and pseudospin symmetries.
Energy spectra in the (a) spin and (b) pseudospin symmetries at various H and A for comparison between Coulomb and Yukawa-like tensor interaction with parameters M=10 fm^{−1}, c=1, ħ=1, α=0.4 fm^{−1}, V1=4 fm^{−1}, V0=3 fm^{−1}, q1=1 fm^{−1}, q2ps=-2 fm^{−1}, q2s=2 fm^{−1}, Cs=5 fm^{−1}, and Cps=-5 fm^{−1}.
In Figures 5, 6, and 7, we showed that degeneracy is removed by tensor interaction. Furthermore, the amount of the energy difference between the two states in the doublets increases with increasing H and A.
4. Results and Discussion
We obtained the energy eigenvalues in the absence and the presence of the Coulomb-like tensor potential for various values of the quantum numbers n and k. In Tables 1 and 4, in the absence of the tensor interaction (H=0), the degeneracy between spin doublets and pseudospin doublets is observed. For example, we observe the degeneracy in (1p1/2, 1p3/2), (1d3/2, 1d5/2), and so forth in the spin symmetry, and we observe the degeneracy in (1s1/2, 1d3/2), (1p3/2, 1f5/2), and so forth in the pseudospin symmetry. When we consider the tensor interaction, for example, by parameter H=0.65, the degeneracy is removed. In Tables 2 and 3 for the spin symmetry and also in Tables 5 and 6 for the pseudospin symmetry, we show that degeneracy exists between spin doublets for several of the parameters α and M, and we show that degeneracy is removed in the presence of tensor interaction. In Figure 2, the degeneracy is removed by tensor interaction effect in spin symmetry and pseudospin symmetry; also the amount of the energy difference between the two states in the doublets increases with increasing parameter H. The effects of the Yukawa-like tensor interactions on the upper and lower components of radial Dirac equation for the symmetries are shown in Figures 3 and 4. The sensitiveness of the pseudospin doublets (1p3/2, 1f5/2) and (1d5/2, 1g7/2) and spin doublets (1f5/2, 1f7/2) and (1d3/2, 1d5/2) for the effects of the Yukawa-like tensor interactions and for Yukawa plus Coulomb-like tensor interaction is given in Figures 5 and 6. In Figure 7 we show that the Coulomb-tensor interaction is stronger than Yukawa-like tensor interaction to remove degeneracy.
5. Conclusions
In this paper, we have discussed approximately the solutions of the Dirac equation for Eckart plus Hulthen potentials with spin symmetry and pseudospin symmetry for k≠0. We obtained the energy eigenvalues and the wave function in terms of the generalized polynomials functions via the formula method. To show the accuracy of the present model, some numerical values of the energy levels are shown in Figures 3, 4, 5, and 6. We have showed that the energy degeneracy in pseudospin and spin doublets is removed by the tensor interaction effect.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
OyewumiK. J.AkoshileC. O.Bound-state solutions of the Dirac-Rosen-Morse potential with spin and pseudospin symmetryWangI. C.WongC. Y.Finite-size effect in the Schwinger particle-production mechanismAlbertoP.LisboaR.MalheiroM.de CastroA. S.Tensor coupling and pseudospin symmetry in nucleiSalaminY. I.HuS. X.HatsagortsyanK. Z.KeitelC. H.Relativistic high-power laser-matter interactionsKatsnelsonM. I.NovoselovK. S.GeimA. K.Chiral tunnelling and the Klein paradox in grapheneChengY.-F.DaiT.-Q.Exact solutions of the Klein-Gordon equation with a ring-shaped modified Kratzer potentialArimaA.HarveyM.ShimizuK.Pseudo LS coupling and pseudo SU3 coupling schemesHechtK. T.AdlerA.Generalized seniority for favored J≠0 pairs in mixed configurationsGinocchioJ. N.Relativistic symmetries in nuclei and hadronsLisboaR.MalheiroM.de CastroA. S.AlbertoP.FiolhaisM.Pseudospin symmetry and the relativistic harmonic oscillatorEshghiM.Makarov potential in relativistic equation via Laplace transformation approachAydogduO.SeverR.Pseudospin and spin symmetry in the Dirac equation with Woods-Saxon potential and tensor potentialAkcayH.TezcanC.Exact solutions of the Dirac equation with harmonic oscillator potential including a Coulomb-like tensor potentialJiaC.-S.GuoP.PengX.-L.Exact solution of the Dirac-Eckart problem with spin and pseudospin symmetryContreras-AstorgaA.FernándezD. J.NegroJ.Solutions of the Dirac equation in a magnetic field and intertwining operatorsFeiziH.RajabiA. A.ShojaeiM. R.Supersymmetric solution of the Schrödinger equation for Woods-Saxon potential by using the Pekeris approximationCiftciH.HallR. L.SaadN.Asymptotic iteration method for eigenvalue problemsÖzerO.LévaiG.Asymptotic iteration method applied to bound-state problems with unbroken and broken supersymmetryDongS.-H.InfeldL.HullT. E.The factorization methodArdaA.SeverR.Exact solutions of the Morse-like potential, step-up and step-down operators via Laplace transform approachRoyA. K.Calculation of the spiked harmonic oscillators through a generalized pseudospectral methodRoyA. K.Studies on some exponential-screened Coulomb potentialsCaiJ. M.CaiP. Y.InomataA.Path-integral treatment of the Hulthén potentialDiafA.ChouchaouiA.LombardR. J.Feynman integral treatment of the Bargmann potentialDiafA.ChouchaouiA.l-states of the Manning-Rosen potential with an improved approximate scheme and Feynman path integral formalismShojaeiM. R.RajabiA. A.FarrokhM.Zoghi-FoumaniN.Energy levels of spin-1/2 particles with Yukawa interactionNikiforovA. F.UvarovV. B.ShojaeiM. R.MousaviM.Solutions of the Klein-Gordon equation for 1≠0 with positiondependent mass for modified Eckart potential plus Hulthen potentialIkotA. N.UdoimukA. B.AkpabioL. E.Bound states solution of Klein-Gordon equation with type-I equal vector and scalar Poschl-Teller potential for arbitray l-stateQiangW. C.Bound states of the Klein-Gordon equation for ring-shaped Kratzer-type potentialQiangW.-C.Bound states of the Klein-Gordon and Dirac equations for potential Vr=Ar-2-Br-1BerkdemirC.BerkdemirA.SeverR.Systematical approach to the exact solution of the Dirac equation for a deformed form of the Woods-Saxon potentialGuoJ.-Y.ShengZ.-Q.Solution of the Dirac equation for the Woods–Saxon potential with spin and pseudospin symmetryZhangX. C.LiuQ. W.JiaC. S.WangL. Z.Bound states of the Dirac equation with vector and scalar Scarf-type potentialsScarfF.New soluble energy band problemChenC.-Y.Exact solutions of the Dirac equation with scalar and vector Hartmann potentialsde Souza DutraA.HottM.Dirac equation exact solutions for generalized asymmetrical Hartmann potentialsIkhdairS. M.FalayeB. J.Bound states of spatially dependent mass Dirac equation with the Eckart potential including Coulomb tensor interactionAlhaidariA. D.Relativistic extension of shape-invariant potentialsFarrokhM.ShojaeiaM. R.RajabiA. A.Klein-Gordon equation with Hulth'en potential and position-dependent massEckartC.The penetration of a potential barrier by electronsFalayeB. J.Any l-state solutions of the Eckart potential via asymptotic iteration methodFalayeB. J.IkhdairS. M.HamzaviM.Formula method for bound state problemsGreinerW.SatchlerG. R.HillE. L.The theory of vector spherical harmonicsGreeneR. L.AldrichC.Variational wave functions for a screened Coulomb potentialJiaC.-S.ChenT.CuiL.-G.Approximate analytical solutions of the Dirac equation with the generalized Pöschl-Teller potential including the pseudo-centrifugal term