# Exact Solutions of a Spatially Dependent Mass Dirac Equation for Coulomb Field plus Tensor Interaction via Laplace Transformation Method

Authors: M. Eshghi; M. Hamzavi; S. M. Ikhdair
Journal: Advances in High Energy Physics
 (2012)

---

## Abstract

The spatially dependent mass Dirac equation is solved exactly for attractive scalar and repulsive vector Coulomb potentials including a tensor interaction potential under the spin and pseudospin (p-spin) symmetric limits by using the Laplace transformation method (LTM). Closed forms of the energy eigenvalue equation and wave functions are obtained for arbitrary spin-orbit quantum numberκ. Some numerical results are given too. The effect of the tensor interaction on the bound states is presented. It is shown that the tensor interaction removes the degeneracy between two states in the spin doublets. We also investigate the effects of the spatially-dependent mass on the bound states under the conditions of the spin symmetric limit and in the absence of tensor interaction (T=0).


---

## Body

## 1. Introduction

Dirac equation has become one of the most appealing relativistic wave equations for spin-1/2 particles. However, solving such a wave equation is still a very challenging problem even if it has been derived more than 80 years ago and has been utilized profusely. It is always useful to investigate the relativistic effects [1–4]. 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 in solving many problems of high-energy physics and chemistry. For this reason, it has been used extensively to study the relativistic heavy ion collisions, heavy ion spectroscopy, and more recently in laser-matter interaction (for a review, see [5] and references therein) and condensed matter physics [6].On the other hand, systems with position-dependent mass (PDM) have been found to be very useful in studying the physical properties of various microstructures [7–14]. Recently, there has been increased interest in searching for analytical solutions of the Dirac equation with PDM and with constant mass under the spin and p-spin symmetries [15–34].Here, we shall attempt to solve the Dirac equation by using the Laplace transform method (LTM). The LTM is an integral transform and is comprehensively useful in physics and engineering [35] and recently used by many authors to solve the Schrödinger equation for different potential forms [36–40]. This method could be a nearly new formalism in the literature and serve as a powerful algebraic treatment for solving the second-order differential equations. As a result, the LTM describes a simple way for solving of radial and one-dimensional differential equations. The other advantage of this method is that a second-order equation can be converted into more simpler form whose solutions may be obtained easily [36]. In this paper, we obtain solution of the Dirac equation both PDM and tensor interaction for attractive scalar and repulsive vector Coulomb potential under the spin and p-spin symmetry limits. We give some numerical results.


## 2. Review to Dirac Equation including Tensor Coupling

The Dirac equation which describes a nucleon in repulsive vectorV(r) and attractive scalar S(r) and a tensor U(r) potentials is written as
(2.1)[α→·p→+β(M(r)c2+S(r))-iβα→·r^U(r)]ψ(r→)=[E-V(r)]ψ(r→),
where M(r) is the effective mass of the fermionic particle, E is the relativistic energy of the system, p→=-iℏ∇→ is the three-dimensional momentum operator. α→ and β are the 4×4 Dirac matrices give as
(2.2)α→=(0σ→σ→0),β=(I00-I),
where I is 2×2 unitary matrix and σ→ are three-vector spin matrices
(2.3)σ1=(0110),σ2=(0-ii0),σ3=(100-1).
The total angular momentum operator J→ and spin-orbit K=(σ→·L→+1), where L→ is orbital angular momentum, 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 can be written as follows:
(2.4)ψnκ(r→)=(fnκ(r→)gnκ(r→))=1r(Fnκ(r)Yjml(θ,φ)iGnκ(r)Yjml~(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 p-spin spherical harmonics, respectively, and m is the projection of the angular momentum on the z-axis. Substituting (2.4) into (2.1) and using the following relations [41] as(2.5a)(σ→·A→)(σ→·B→)=A→·B→+iσ→·(A→×B→),(2.5b)(σ→·P→)=σ→·r^(r^·P→+iσ→·L→r),
and with the following properties:
(2.5c)σ→·L→{Yjml~(θ,ϕ)Yjml(θ,ϕ)={(κ-1)Yjml~(θ,ϕ),-(κ-1)Yjml(θ,ϕ),(2.5d)σ→·r^{Yjml~(θ,ϕ)Yjml(θ,ϕ)={-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
(2.5e)(ddr+κr-U(r))Fnκ(r)=(M(r)c2+Enκ-Δ(r))Gnκ(r),(2.5f)(ddr-κr+U(r))Gnκ(r)=(M(r)c2-Enκ+Σ(r))Fnκ(r),
where
(2.5g)Δ(r)=V(r)-S(r),(2.5h)Σ(r)=V(r)+S(r).
Eliminating Fnκ(r) and Gnκ(r) from (2.5e) and (2.5f), we finally obtain the following two Schrödinger-like differential equations for the upper and lower radial spinor components, respectively:
(2.5i)[d2dr2-κ(κ+1)r2+2κrU(r)-dU(r)dr-U2(r)]Fnκ(r)-dM(r)/dr-dΔ(r)/drM(r)+Enκ-Δ(r)(ddr+κr-U(r))Fnκ(r)=1(ℏc)2[(M(r)c2+Enκ-Δ(r))(M(r)c2-Enκ+Σ(r))]Fnκ(r),(2.5j)[d2dr2-κ(κ-1)r2+2κrU(r)+dU(r)dr-U2(r)]Gnκ(r)-dM(r)/dr+dΣ(r)/drM(r)-Enκ+Σ(r)(ddr-κr+U(r))Gnκ(r)=1(ℏc)2[(M(r)c2+Enκ-Δ(r))(M(r)c2-Enκ+Σ(r))]Gnκ(r),where κ(κ-1)=l~(l~+1) and κ(κ+1)=l(l+1). These radial wave functions are required to satisfy the necessary boundary conditions. The spin-orbit quantum number κ is related to the quantum numbers for spin symmetry l and p-spin symmetry l~ as
(2.6)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 quasi-degenerate doublet structure can be expressed in terms of a p-spin angular momentum s~=1/2 and pseudo-orbital angular momentum l~, which is defined as
(2.7)κ={-l~=-(j+12);(s1/2,p3/2,etc.)j=l~-12,alignedp-spin(κ<0),l~+1=j+12;(d3/2,f5/2,etc.)j=l~+12,unalignedp-spin(κ>0),
where κ=±1,±2,…. For example, (1s1/2,0d3/2) and (1p3/2,0f5/2) can be considered as p-spin doublets. In the next section, we will consider the spin and p-spin symmetry cases.


## 3. Relativistic Bound State Solutions

### 3.1.  Spin Symmetry Case

Equation (2.5i) cannot be solved analytically because of the last term ((dM(r)/dr-dΔ(r)/dr)/(M(r)+Enκ-Δ(r)))(d/dr+κ/r-U(r)). Therefore, in solving the mathematical relation dM(r)c2/dr=dΔ(r)/dr=dV(r)/dr [42, 43], we can then exactly solve (2.5i). In this stage, we take the vector potential in the form of an attractive Coulomb-like field [18] as
(3.1)Σ(r)=V(r)=-ℏcqνr,qν=q,r≠0,
where qν is being a vector dimensionless real parameter coupling constant and ℏc is being a constant with J·fm dimension. Also, it is convenient to take the mass function [18] as
(3.2)M(r)=m0+m1r,m1=m0λ0b,λ0=ℏm0c,
where m0 and m1 stand for the rest mass of the fermionic particle and the perturbed mass, respectively. Further, b is the dimensionless real constant to be set to zero for the constant mass case and λ0 is the Compton-like wavelength in fm units. Further, the tensor interaction takes the simple form:
(3.3)U(r)=-Tr,T=ZaZbe24πε0,r≥Rc,
where Rc is the coulomb radius, Za and Zb stand for the charges of the projectile particle a and the target nucleus b, respectively.Substituting (3.1)–(3.3) into (2.5i) considering the spin symmetry case where Δ(r)=Cs=const., that is, (dΔ(r)/dr=0) [44, 45]. Thus, the equation obtained for the upper component of the Dirac spinor Fnk(r) becomes
(3.4){d2dr2-(κ+T)(κ+T+1)+b(b-q)r2+1ℏc[q(m0c2+Enκ-Cs)+b(Cs-2m0c2)]r-1(ℏc)2(m0c2-Enκ)(m0c2+Enκ-Cs)}Fnκ(r)=0.
Further, defining the new parameters
(3.5)λ2=(κ+T)(κ+T+1)+b(b-q),δ2=-1ℏc[q(m0c2+Enκ-Cs)+b(Cs-2m0c2)],ε2=1(ℏc)2(m0c2-Enκ)(m0c2+Enκ-Cs),
and introducing Fnκ(r)=r1/2φ(r), then (3.4) turns into the form(3.6a)[r2d2dr2+rddr-r2(ν2r2+δ2r+ε2)]φ(r)=0,(3.6b)ν2=(κ+T+12)2+b(b-q).Setting φ(r)=rαf(r) with α is a constant and then inserting into (3.6a), we have
(3.7)[r2d2dr2+(2α+1)rddr-(ν2+δ2r+ε2r2-α2)]f(r)=0.
Now, to obtain a finite wave function when r→∞, if we take α=-ν in (3.7) then it becomes
(3.8)[rd2dr2-(2ν-1)ddr-(δ2+ε2r)]f(r)=0.
The LTM [46, 47]
(3.9)L{g(ρ)}=f(t)=∫0∞e-tρg(ρ)dρ,
leads to an equation
(3.10)(t2-ε2)df(t)dt+[(2ν+1)t+δ2]f(t)=0.
Equation (3.10) is a first-order differential equation and therefore we may directly make use of the integral to get the expression
(3.11)f(t)=N(t+ε)-(2ν+1)(t-εt+ε)-[δ2+(2ν+1)ε]/2ε,
where N is a constant. Noting that ((t-ε)/(t+ε))-(δ2+(2ν+1)ε)/2ε is a multi-valued function and the wave functions are required to be single-valued, we must take
(3.12)-δ22ε-2ν+12=n,n=0,1,2,3,…
which gives single-values wave functions. Using this requirement and further expanding (3.11) into series, we obtain
(3.13)f(t)=N'∑k=0n(-2ε)kn!(n-k)!k!(t+ε)-(2ν+1+k),
where N′ is a constant. In terms of a simple extension of the inverse Laplace transformation [46, 47], we can immediately obtain
(3.14)f(r)=N′′r2νe-εr∑k=0n(-1)kn!(n-k)!k!Γ(2ν+1)Γ(2ν+1+k)(2εr)k,
and from φ(r)=rαf(r), we then obtain
(3.15)φ(r)=N′′′rνe-εr∑k=0n(-1)kn!(n-k)!k!Γ(2ν+1)Γ(2ν+1+k)(2εr)k,
where N′′′ is a constant. On the other hand, the confluent hyper-geometric functions is defined as a series expansion [48]
(3.16)F11(-n,γ,s)=∑j=0n(-1)jn!(n-j)!j!Γ(γ)Γ(γ+j)sj.
So, we find the upper-spinor component of wave function as
(3.17)Fnκ(r)=Ne-εrrν+1/2Ln2ν(2εr)=NΓ(n+2ν+1)n!Γ(2ν+1)e-εrrν+1/2F11(-n,2ν+1,2εr),
where N is normalization constant. By using the normalization condition given as ∫0∞|Fnκ(r)|2dr=1, and the relation between the Laguerre polynomials and confluent hyper-geometric functions as Lnp(x)=(Γ(n+p+1)/n!Γ(p+1))F11(-n,p+1,x) [47], the normalization constant in (3.17) is written as
(3.18)N=(2ε)ν+1n!Γ(n+2ν+1)(2n+2ν+1),
where we have used [47]
(3.19)∫0∞xq+1e-xLnq(x)Ln′q(x)dx=(2n+q+1)Γ(n+q+1)n!δnn′.
Inserting the parameters in (3.5) and (3.6b) into (3.12), one obtain the transcendental energy eigenvalue equation as follows
(3.20)(m0c2-Enκ)(m0c2+Enκ-Cs)=14[q(m0c2+Enκ-Cs)+b(Cs-2m0c2)n+1/2+(κ+T+1/2)2+b(b-q)]2,n=0,1,2,…
which is identical to [18] when 2q→q and the tensor interaction is removed, that is, T=0. For the constant mass case, we have
(3.21)(m0c2-Enκ)(m0c2+Enκ-Cs)=14[q(m0c2+Enκ-Cs)n+κ+T+1]2,n=0,1,2,….
In case when Cs=0,  T=0,  κ=l,  Σ(r)=V(r),  m0c2-Enκ→-Enl and m0c2+Enκ→m0, the above equation reduces to the non-relativistic energy limit (in units of ℏ=1):
(3.22)Enl=-2m0q2(n+l+1)2.
The wave function (3.17) is finite in the entire range and at the boundaries, that is, Fnκ(r=0)=0 and Fnκ(r→∞)→0.The lower spinor wave function can be obtained via(3.23)Gnκ(r)=1(m0c2+ℏcb/r+Enκ-Cs)×{(-ε+ν+1/2r+κ+Tr)Fnκ(r)-2Nεe-εrrν+1/2Ln-12ν+1(2εr)},
where we have used
(3.24)ddxLnα(x)=-Ln-1(α+1)(x)=1x[nLnα(x)-(n+α)Ln-1α(x)].
In Tables 1, 2, and 3 with ℏ=c=1, we give some numerical results for the energy eigenvalues from energy formula (3.20).Table 1
The bound state energy eigenvalues of the Coulomb potential under the pseudo-spin symmetry limit for several values ofn and κ for m0=5.0fm-1,ℏ=c=1, q=1,T=(0,1,2,5), and Cps=-1.
l
~
n
,
κ
<
0
(
l
,
j
=
l
+
1
/
2
)
m
1
E
n
,
κ
<
0
T
=
0
T
=
1
T
=
2
T
=
5
1
1, −2
1
p
3
/
2
0.0
−4.86154
−4.75676
−4.47059
−4.86154
0.2
−4.80508
−4.66644
−4.36211
−4.80508
0.4
−4.74207
−4.57172
−4.26086
−4.74207
0.5
−4.70859
−4.52376
−4.21286
−4.70859
2
1, −3
1
d
5
/
2
0.0
−4.91089
−4.86154
−4.75676
−4.75676
0.2
−4.87335
−4.80508
−4.66644
−4.66644
0.4
−4.83043
−4.74207
−4.57172
−4.57172
0.5
−4.80716
−4.70859
−4.52376
−4.52376
3
1, −4
1
f
7
/
2
0.0
−4.93793
−4.91089
−4.86154
−4.47059
0.2
−4.91138
−4.87335
−4.80508
−4.36211
0.4
−4.88067
−4.83043
−4.74207
−4.26086
0.5
−4.86385
−4.80716
−4.70859
−4.21286
4
1, −5
1
g
9
/
2
0.0
−4.95431
−4.93793
−4.91089
−4.47059
0.2
−4.93461
−4.91138
−4.87335
−4.36211
0.4
−4.91166
−4.88067
−4.83043
−4.26086
0.5
−4.89903
−4.86385
−4.80716
−4.21286
1
2, −2
2
p
3
/
2
0.0
−4.91089
−4.86154
−4.75676
−4.91089
0.2
−4.87402
−4.80794
−4.69083
−4.87402
0.4
−4.83243
−4.75015
−4.62526
−4.83243
0.5
−4.81014
−4.72024
−4.59287
−4.81014
2
2, −3
2
d
5
/
2
0.0
−4.93793
−4.91089
−4.86154
−4.86154
0.2
−4.9116
−4.87402
−4.80794
−4.80794
0.4
−4.88134
−4.83243
−4.75015
−4.75015
0.5
−4.86487
−4.81014
−4.72024
−4.72024
3
2, −4
2
f
7
/
2
0.0
−4.95431
−4.93793
−4.91089
−4.75676
0.2
−4.93469
−4.9116
−4.87402
−4.69083
0.4
−4.91194
−4.88134
−4.83243
−4.62526
0.5
−4.89945
−4.86487
−4.81014
−4.59287
4
2, −5
2
g
9
/
2
0.0
−4.96498
−4.95431
−4.93793
−4.75676
0.2
−4.94984
−4.93469
−4.91160
−4.69083
0.4
−4.93218
−4.91194
−4.88134
−4.62526
0.5
−4.92244
−4.89945
−4.86487
−4.59287Table 2
The bound state energy eigenvalues of the Coulomb potential under the pseudo-spin symmetry limit for several values ofn and κ for m0=5.0fm-1, q=1,ℏ=c=1,T=(0,1,2,5), and Cps=-1.
l
~
n
-
1
,
κ
>
0
(
l
,
j
=
l
-
1
/
2
)
m
1
E
n
-
1
,
κ
>
0
T
=
0
T
=
1
T
=
2
T
=
5
2
0, 2
0
d
3
/
2
0.0
−4.75676
−4.86154
−4.91089
−4.96498
0.2
−4.66644
−4.80508
−4.87335
−4.94979
0.4
−4.57172
−4.74207
−4.83043
−4.93205
0.5
−4.52376
−4.70859
−4.80716
−4.92225
3
0, 3
0
f
5
/
2
0.0
−4.86154
−4.91089
−4.93793
−4.97231
0.2
−4.80508
−4.87335
−4.91138
−4.96026
0.4
−4.74207
−4.83043
−4.88067
−4.94614
0.5
−4.70859
−4.80716
−4.86385
−4.93833
4
0, 4
0
g
7
/
2
0.0
−4.91089
−4.93793
−4.95431
−4.97756
0.2
−4.87335
−4.91138
−4.93461
−4.96777
0.4
−4.83043
−4.88067
−4.91166
−4.95628
0.5
−4.80716
−4.86385
−4.89903
−4.94991
5
0, 5
0
h
9
/
2
0.0
−4.93793
−4.95431
−4.96498
−4.98144
0.2
−4.91138
−4.93461
−4.94979
−4.97334
0.4
−4.88067
−4.91166
−4.93205
−4.96381
0.5
−4.86385
−4.89903
−4.92225
−4.95852
2
1, 2
1
d
3
/
2
0.0
−4.86154
−4.91089
−4.93793
−4.97231
0.2
−4.80794
−4.87402
−4.91160
−4.96028
0.4
−4.75015
−4.83243
−4.88134
−4.94621
0.5
−4.72024
−4.81014
−4.86487
−4.93843
3
1, 3
1
f
5
/
2
0.0
−4.91089
−4.93793
−4.95431
−4.97756
0.2
−4.87402
−4.91160
−4.93469
−4.96778
0.4
−4.83243
−4.88134
−4.91194
−4.95632
0.5
−4.81014
−4.86487
−4.89945
−4.94997
4
1, 4
1
g
7
/
2
0.0
−4.93793
−4.95431
−4.96498
−4.98144
0.2
−4.91160
−4.93469
−4.94984
−4.97757
0.4
−4.88134
−4.91194
−4.93218
−4.96383
0.5
−4.86487
−4.89945
−4.92244
−4.95856
5
1, 5
1
h
9
/
2
0.0
−4.95431
−4.96498
−4.97231
−4.98440
0.2
−4.93469
−4.94984
−4.96028
−4.97758
0.4
−4.91194
−4.93218
−4.94621
−4.96957
0.5
−4.89945
−4.92244
−4.93843
−4.96512Table 3
The bound state energy eigenvalues of the Coulomb potential for several states under the pseudo-spin symmetry limit with values ofCps with m0=5.0fm-1, q=1,ℏ=c=1,m1=(0,0.5), and T=(0,5).
l
~
n
,
κ
(
l
,
j
=
l
+
1
/
2
)
C
p
s
E
n
,
κ
<
0
T
=
0
T
=
0
T
=
5
T
=
5
m
1
=
0
m
1
=
0.5
m
1
=
0
m
1
=
0.5
1
1
,
2
1
d
3
/
2
−1.00
−4.75676
−4.52376
−4.96498
−4.92225
−1.25
−4.76351
−4.53699
−4.96595
−4.92441
−1.50
−4.77027
−4.55022
−4.96693
−4.92657
−1.75
−4.77703
−4.56345
−4.96790
−4.92873
−2.00
−4.78378
−4.57668
−4.96887
−4.93089
−2.25
−4.79054
−4.58991
−4.96984
−4.93305
−2.50
−4.79730
−4.60313
−4.97082
−4.93521
−2.75
−4.80405
−4.61636
−4.97179
−4.93736
3
0
,
-
4
0
f
7
/
2
−1.00
−4.91089
−4.80543
−3.20000
−2.90681
−1.25
−4.91337
−4.81083
−3.25000
−2.96496
−1.50
−4.91584
−4.81624
−3.30000
−3.0231
−1.75
−4.91832
−4.82164
−3.35000
−3.08124
−2.00
−4.92079
−4.82705
−3.40000
−3.13939
−2.25
−4.92327
−4.83245
−3.45000
−3.19753
−2.50
−4.92574
−4.83786
−3.50000
−3.25568
−2.75
−4.92822
−4.84326
−3.55000
−3.31382
1
1
,
-
2
1
p
3
/
2
−1.00
−4.86154
−4.70859
−4.86154
−4.70859
−1.25
−4.86538
−4.71669
−4.86538
−4.71669
−1.50
−4.86923
−4.72478
−4.86923
−4.72478
−1.75
−4.87308
−4.73288
−4.87308
−4.73288
−2.00
−4.87692
−4.74097
−4.87692
−4.74097
−2.25
−4.88077
−4.74907
−4.88077
−4.74907
−2.50
−4.88462
−4.75716
−4.88462
−4.75716
−2.75
−4.88846
−4.76526
−4.88846
−4.76526
3
1
,
-
4
1
f
7
/
2
−1.00
−4.93793
−4.86385
−4.47059
−4.21286
−1.25
−4.93966
−4.86764
−4.48529
−4.23472
−1.50
−4.94138
−4.87142
−4.50000
−4.25659
−1.75
−4.9431
−4.8752
−4.51471
−4.27845
−2.00
−4.94483
−4.87898
−4.52941
−4.30032
−2.25
−4.94655
−4.88276
−4.54412
−4.32218
−2.50
−4.94828
−4.88654
−4.55882
−4.34405
−2.75
−4.95000
−4.89033
−4.57353
−4.36591


### 3.2. p-Spin Symmetry Case

To avoid repetition in the solution of (2.5j), the negative energy solution for p-spin symmetry can be obtained directly from those of the above positive energy solution for spin symmetry by using the parameter mapping [18]:
(3.25)Fnκ(r)⟷Gnκ(r);V(r)→-V(r)(q→-q);κ+1→κ;Enκ→-Enκ;Cs⟷-Cps.
Following the previous results with the above transformations, we finally arrive at the transcendental energy equation as
(3.26)(m0c2+Enκ)(m0c2-Enκ+Cps)=14[q(m0c2-Enκ+Cps)+b(Cps+2m0c2)n+(κ+T-1/2)2+b(b+q)+1/2]2,n=0,1,2,…
which is identical to [18] when 2q→q and the tensor interaction is removed, that is, T=0. For the constant mass case, we have
(3.27)(m0c2+Enκ)(m0c2-Enκ+Cps)=14[q(m0c2-Enκ+Cps)n+κ+T]2,n=0,1,2,…,
the lower-spinor component of wave function as
(3.28)Gnκ(r)=N~e-ε~rrν+1/2Ln2ν(2ε~r)=N~Γ(n+2ν~+1)n!Γ(2ν~+1)e-ε~rrν~+1/2F11(-n,2ν~+1,2ε~r),
where ε~2=(1/(ℏc)2)(m0c2+Enκ)(m0c2-Enκ+Cps), ν~2=(κ+T-1/2)2+b(b+q) and N~ is normalization constant
(3.29)N~=(2ε~)ν~+1n!Γ(n+2ν~+1)(2n+2ν~+1).
The upper-spinor wave function can be obtained via
(3.30)Fnκ(r)=1(m0c2+ℏcb/r-Enκ+Cps)×{(-ε~+ν~+1/2r-κ+Tr)Gnκ(r)-2N~ε~e-ε~rrν~+1/2Ln-12ν~+1(2ε~r)}.


## 3.1.  Spin Symmetry Case

Equation (2.5i) cannot be solved analytically because of the last term ((dM(r)/dr-dΔ(r)/dr)/(M(r)+Enκ-Δ(r)))(d/dr+κ/r-U(r)). Therefore, in solving the mathematical relation dM(r)c2/dr=dΔ(r)/dr=dV(r)/dr [42, 43], we can then exactly solve (2.5i). In this stage, we take the vector potential in the form of an attractive Coulomb-like field [18] as
(3.1)Σ(r)=V(r)=-ℏcqνr,qν=q,r≠0,
where qν is being a vector dimensionless real parameter coupling constant and ℏc is being a constant with J·fm dimension. Also, it is convenient to take the mass function [18] as
(3.2)M(r)=m0+m1r,m1=m0λ0b,λ0=ℏm0c,
where m0 and m1 stand for the rest mass of the fermionic particle and the perturbed mass, respectively. Further, b is the dimensionless real constant to be set to zero for the constant mass case and λ0 is the Compton-like wavelength in fm units. Further, the tensor interaction takes the simple form:
(3.3)U(r)=-Tr,T=ZaZbe24πε0,r≥Rc,
where Rc is the coulomb radius, Za and Zb stand for the charges of the projectile particle a and the target nucleus b, respectively.Substituting (3.1)–(3.3) into (2.5i) considering the spin symmetry case where Δ(r)=Cs=const., that is, (dΔ(r)/dr=0) [44, 45]. Thus, the equation obtained for the upper component of the Dirac spinor Fnk(r) becomes
(3.4){d2dr2-(κ+T)(κ+T+1)+b(b-q)r2+1ℏc[q(m0c2+Enκ-Cs)+b(Cs-2m0c2)]r-1(ℏc)2(m0c2-Enκ)(m0c2+Enκ-Cs)}Fnκ(r)=0.
Further, defining the new parameters
(3.5)λ2=(κ+T)(κ+T+1)+b(b-q),δ2=-1ℏc[q(m0c2+Enκ-Cs)+b(Cs-2m0c2)],ε2=1(ℏc)2(m0c2-Enκ)(m0c2+Enκ-Cs),
and introducing Fnκ(r)=r1/2φ(r), then (3.4) turns into the form(3.6a)[r2d2dr2+rddr-r2(ν2r2+δ2r+ε2)]φ(r)=0,(3.6b)ν2=(κ+T+12)2+b(b-q).Setting φ(r)=rαf(r) with α is a constant and then inserting into (3.6a), we have
(3.7)[r2d2dr2+(2α+1)rddr-(ν2+δ2r+ε2r2-α2)]f(r)=0.
Now, to obtain a finite wave function when r→∞, if we take α=-ν in (3.7) then it becomes
(3.8)[rd2dr2-(2ν-1)ddr-(δ2+ε2r)]f(r)=0.
The LTM [46, 47]
(3.9)L{g(ρ)}=f(t)=∫0∞e-tρg(ρ)dρ,
leads to an equation
(3.10)(t2-ε2)df(t)dt+[(2ν+1)t+δ2]f(t)=0.
Equation (3.10) is a first-order differential equation and therefore we may directly make use of the integral to get the expression
(3.11)f(t)=N(t+ε)-(2ν+1)(t-εt+ε)-[δ2+(2ν+1)ε]/2ε,
where N is a constant. Noting that ((t-ε)/(t+ε))-(δ2+(2ν+1)ε)/2ε is a multi-valued function and the wave functions are required to be single-valued, we must take
(3.12)-δ22ε-2ν+12=n,n=0,1,2,3,…
which gives single-values wave functions. Using this requirement and further expanding (3.11) into series, we obtain
(3.13)f(t)=N'∑k=0n(-2ε)kn!(n-k)!k!(t+ε)-(2ν+1+k),
where N′ is a constant. In terms of a simple extension of the inverse Laplace transformation [46, 47], we can immediately obtain
(3.14)f(r)=N′′r2νe-εr∑k=0n(-1)kn!(n-k)!k!Γ(2ν+1)Γ(2ν+1+k)(2εr)k,
and from φ(r)=rαf(r), we then obtain
(3.15)φ(r)=N′′′rνe-εr∑k=0n(-1)kn!(n-k)!k!Γ(2ν+1)Γ(2ν+1+k)(2εr)k,
where N′′′ is a constant. On the other hand, the confluent hyper-geometric functions is defined as a series expansion [48]
(3.16)F11(-n,γ,s)=∑j=0n(-1)jn!(n-j)!j!Γ(γ)Γ(γ+j)sj.
So, we find the upper-spinor component of wave function as
(3.17)Fnκ(r)=Ne-εrrν+1/2Ln2ν(2εr)=NΓ(n+2ν+1)n!Γ(2ν+1)e-εrrν+1/2F11(-n,2ν+1,2εr),
where N is normalization constant. By using the normalization condition given as ∫0∞|Fnκ(r)|2dr=1, and the relation between the Laguerre polynomials and confluent hyper-geometric functions as Lnp(x)=(Γ(n+p+1)/n!Γ(p+1))F11(-n,p+1,x) [47], the normalization constant in (3.17) is written as
(3.18)N=(2ε)ν+1n!Γ(n+2ν+1)(2n+2ν+1),
where we have used [47]
(3.19)∫0∞xq+1e-xLnq(x)Ln′q(x)dx=(2n+q+1)Γ(n+q+1)n!δnn′.
Inserting the parameters in (3.5) and (3.6b) into (3.12), one obtain the transcendental energy eigenvalue equation as follows
(3.20)(m0c2-Enκ)(m0c2+Enκ-Cs)=14[q(m0c2+Enκ-Cs)+b(Cs-2m0c2)n+1/2+(κ+T+1/2)2+b(b-q)]2,n=0,1,2,…
which is identical to [18] when 2q→q and the tensor interaction is removed, that is, T=0. For the constant mass case, we have
(3.21)(m0c2-Enκ)(m0c2+Enκ-Cs)=14[q(m0c2+Enκ-Cs)n+κ+T+1]2,n=0,1,2,….
In case when Cs=0,  T=0,  κ=l,  Σ(r)=V(r),  m0c2-Enκ→-Enl and m0c2+Enκ→m0, the above equation reduces to the non-relativistic energy limit (in units of ℏ=1):
(3.22)Enl=-2m0q2(n+l+1)2.
The wave function (3.17) is finite in the entire range and at the boundaries, that is, Fnκ(r=0)=0 and Fnκ(r→∞)→0.The lower spinor wave function can be obtained via(3.23)Gnκ(r)=1(m0c2+ℏcb/r+Enκ-Cs)×{(-ε+ν+1/2r+κ+Tr)Fnκ(r)-2Nεe-εrrν+1/2Ln-12ν+1(2εr)},
where we have used
(3.24)ddxLnα(x)=-Ln-1(α+1)(x)=1x[nLnα(x)-(n+α)Ln-1α(x)].
In Tables 1, 2, and 3 with ℏ=c=1, we give some numerical results for the energy eigenvalues from energy formula (3.20).Table 1
The bound state energy eigenvalues of the Coulomb potential under the pseudo-spin symmetry limit for several values ofn and κ for m0=5.0fm-1,ℏ=c=1, q=1,T=(0,1,2,5), and Cps=-1.
l
~
n
,
κ
<
0
(
l
,
j
=
l
+
1
/
2
)
m
1
E
n
,
κ
<
0
T
=
0
T
=
1
T
=
2
T
=
5
1
1, −2
1
p
3
/
2
0.0
−4.86154
−4.75676
−4.47059
−4.86154
0.2
−4.80508
−4.66644
−4.36211
−4.80508
0.4
−4.74207
−4.57172
−4.26086
−4.74207
0.5
−4.70859
−4.52376
−4.21286
−4.70859
2
1, −3
1
d
5
/
2
0.0
−4.91089
−4.86154
−4.75676
−4.75676
0.2
−4.87335
−4.80508
−4.66644
−4.66644
0.4
−4.83043
−4.74207
−4.57172
−4.57172
0.5
−4.80716
−4.70859
−4.52376
−4.52376
3
1, −4
1
f
7
/
2
0.0
−4.93793
−4.91089
−4.86154
−4.47059
0.2
−4.91138
−4.87335
−4.80508
−4.36211
0.4
−4.88067
−4.83043
−4.74207
−4.26086
0.5
−4.86385
−4.80716
−4.70859
−4.21286
4
1, −5
1
g
9
/
2
0.0
−4.95431
−4.93793
−4.91089
−4.47059
0.2
−4.93461
−4.91138
−4.87335
−4.36211
0.4
−4.91166
−4.88067
−4.83043
−4.26086
0.5
−4.89903
−4.86385
−4.80716
−4.21286
1
2, −2
2
p
3
/
2
0.0
−4.91089
−4.86154
−4.75676
−4.91089
0.2
−4.87402
−4.80794
−4.69083
−4.87402
0.4
−4.83243
−4.75015
−4.62526
−4.83243
0.5
−4.81014
−4.72024
−4.59287
−4.81014
2
2, −3
2
d
5
/
2
0.0
−4.93793
−4.91089
−4.86154
−4.86154
0.2
−4.9116
−4.87402
−4.80794
−4.80794
0.4
−4.88134
−4.83243
−4.75015
−4.75015
0.5
−4.86487
−4.81014
−4.72024
−4.72024
3
2, −4
2
f
7
/
2
0.0
−4.95431
−4.93793
−4.91089
−4.75676
0.2
−4.93469
−4.9116
−4.87402
−4.69083
0.4
−4.91194
−4.88134
−4.83243
−4.62526
0.5
−4.89945
−4.86487
−4.81014
−4.59287
4
2, −5
2
g
9
/
2
0.0
−4.96498
−4.95431
−4.93793
−4.75676
0.2
−4.94984
−4.93469
−4.91160
−4.69083
0.4
−4.93218
−4.91194
−4.88134
−4.62526
0.5
−4.92244
−4.89945
−4.86487
−4.59287Table 2
The bound state energy eigenvalues of the Coulomb potential under the pseudo-spin symmetry limit for several values ofn and κ for m0=5.0fm-1, q=1,ℏ=c=1,T=(0,1,2,5), and Cps=-1.
l
~
n
-
1
,
κ
>
0
(
l
,
j
=
l
-
1
/
2
)
m
1
E
n
-
1
,
κ
>
0
T
=
0
T
=
1
T
=
2
T
=
5
2
0, 2
0
d
3
/
2
0.0
−4.75676
−4.86154
−4.91089
−4.96498
0.2
−4.66644
−4.80508
−4.87335
−4.94979
0.4
−4.57172
−4.74207
−4.83043
−4.93205
0.5
−4.52376
−4.70859
−4.80716
−4.92225
3
0, 3
0
f
5
/
2
0.0
−4.86154
−4.91089
−4.93793
−4.97231
0.2
−4.80508
−4.87335
−4.91138
−4.96026
0.4
−4.74207
−4.83043
−4.88067
−4.94614
0.5
−4.70859
−4.80716
−4.86385
−4.93833
4
0, 4
0
g
7
/
2
0.0
−4.91089
−4.93793
−4.95431
−4.97756
0.2
−4.87335
−4.91138
−4.93461
−4.96777
0.4
−4.83043
−4.88067
−4.91166
−4.95628
0.5
−4.80716
−4.86385
−4.89903
−4.94991
5
0, 5
0
h
9
/
2
0.0
−4.93793
−4.95431
−4.96498
−4.98144
0.2
−4.91138
−4.93461
−4.94979
−4.97334
0.4
−4.88067
−4.91166
−4.93205
−4.96381
0.5
−4.86385
−4.89903
−4.92225
−4.95852
2
1, 2
1
d
3
/
2
0.0
−4.86154
−4.91089
−4.93793
−4.97231
0.2
−4.80794
−4.87402
−4.91160
−4.96028
0.4
−4.75015
−4.83243
−4.88134
−4.94621
0.5
−4.72024
−4.81014
−4.86487
−4.93843
3
1, 3
1
f
5
/
2
0.0
−4.91089
−4.93793
−4.95431
−4.97756
0.2
−4.87402
−4.91160
−4.93469
−4.96778
0.4
−4.83243
−4.88134
−4.91194
−4.95632
0.5
−4.81014
−4.86487
−4.89945
−4.94997
4
1, 4
1
g
7
/
2
0.0
−4.93793
−4.95431
−4.96498
−4.98144
0.2
−4.91160
−4.93469
−4.94984
−4.97757
0.4
−4.88134
−4.91194
−4.93218
−4.96383
0.5
−4.86487
−4.89945
−4.92244
−4.95856
5
1, 5
1
h
9
/
2
0.0
−4.95431
−4.96498
−4.97231
−4.98440
0.2
−4.93469
−4.94984
−4.96028
−4.97758
0.4
−4.91194
−4.93218
−4.94621
−4.96957
0.5
−4.89945
−4.92244
−4.93843
−4.96512Table 3
The bound state energy eigenvalues of the Coulomb potential for several states under the pseudo-spin symmetry limit with values ofCps with m0=5.0fm-1, q=1,ℏ=c=1,m1=(0,0.5), and T=(0,5).
l
~
n
,
κ
(
l
,
j
=
l
+
1
/
2
)
C
p
s
E
n
,
κ
<
0
T
=
0
T
=
0
T
=
5
T
=
5
m
1
=
0
m
1
=
0.5
m
1
=
0
m
1
=
0.5
1
1
,
2
1
d
3
/
2
−1.00
−4.75676
−4.52376
−4.96498
−4.92225
−1.25
−4.76351
−4.53699
−4.96595
−4.92441
−1.50
−4.77027
−4.55022
−4.96693
−4.92657
−1.75
−4.77703
−4.56345
−4.96790
−4.92873
−2.00
−4.78378
−4.57668
−4.96887
−4.93089
−2.25
−4.79054
−4.58991
−4.96984
−4.93305
−2.50
−4.79730
−4.60313
−4.97082
−4.93521
−2.75
−4.80405
−4.61636
−4.97179
−4.93736
3
0
,
-
4
0
f
7
/
2
−1.00
−4.91089
−4.80543
−3.20000
−2.90681
−1.25
−4.91337
−4.81083
−3.25000
−2.96496
−1.50
−4.91584
−4.81624
−3.30000
−3.0231
−1.75
−4.91832
−4.82164
−3.35000
−3.08124
−2.00
−4.92079
−4.82705
−3.40000
−3.13939
−2.25
−4.92327
−4.83245
−3.45000
−3.19753
−2.50
−4.92574
−4.83786
−3.50000
−3.25568
−2.75
−4.92822
−4.84326
−3.55000
−3.31382
1
1
,
-
2
1
p
3
/
2
−1.00
−4.86154
−4.70859
−4.86154
−4.70859
−1.25
−4.86538
−4.71669
−4.86538
−4.71669
−1.50
−4.86923
−4.72478
−4.86923
−4.72478
−1.75
−4.87308
−4.73288
−4.87308
−4.73288
−2.00
−4.87692
−4.74097
−4.87692
−4.74097
−2.25
−4.88077
−4.74907
−4.88077
−4.74907
−2.50
−4.88462
−4.75716
−4.88462
−4.75716
−2.75
−4.88846
−4.76526
−4.88846
−4.76526
3
1
,
-
4
1
f
7
/
2
−1.00
−4.93793
−4.86385
−4.47059
−4.21286
−1.25
−4.93966
−4.86764
−4.48529
−4.23472
−1.50
−4.94138
−4.87142
−4.50000
−4.25659
−1.75
−4.9431
−4.8752
−4.51471
−4.27845
−2.00
−4.94483
−4.87898
−4.52941
−4.30032
−2.25
−4.94655
−4.88276
−4.54412
−4.32218
−2.50
−4.94828
−4.88654
−4.55882
−4.34405
−2.75
−4.95000
−4.89033
−4.57353
−4.36591


## 3.2. p-Spin Symmetry Case

To avoid repetition in the solution of (2.5j), the negative energy solution for p-spin symmetry can be obtained directly from those of the above positive energy solution for spin symmetry by using the parameter mapping [18]:
(3.25)Fnκ(r)⟷Gnκ(r);V(r)→-V(r)(q→-q);κ+1→κ;Enκ→-Enκ;Cs⟷-Cps.
Following the previous results with the above transformations, we finally arrive at the transcendental energy equation as
(3.26)(m0c2+Enκ)(m0c2-Enκ+Cps)=14[q(m0c2-Enκ+Cps)+b(Cps+2m0c2)n+(κ+T-1/2)2+b(b+q)+1/2]2,n=0,1,2,…
which is identical to [18] when 2q→q and the tensor interaction is removed, that is, T=0. For the constant mass case, we have
(3.27)(m0c2+Enκ)(m0c2-Enκ+Cps)=14[q(m0c2-Enκ+Cps)n+κ+T]2,n=0,1,2,…,
the lower-spinor component of wave function as
(3.28)Gnκ(r)=N~e-ε~rrν+1/2Ln2ν(2ε~r)=N~Γ(n+2ν~+1)n!Γ(2ν~+1)e-ε~rrν~+1/2F11(-n,2ν~+1,2ε~r),
where ε~2=(1/(ℏc)2)(m0c2+Enκ)(m0c2-Enκ+Cps), ν~2=(κ+T-1/2)2+b(b+q) and N~ is normalization constant
(3.29)N~=(2ε~)ν~+1n!Γ(n+2ν~+1)(2n+2ν~+1).
The upper-spinor wave function can be obtained via
(3.30)Fnκ(r)=1(m0c2+ℏcb/r-Enκ+Cps)×{(-ε~+ν~+1/2r-κ+Tr)Gnκ(r)-2N~ε~e-ε~rrν~+1/2Ln-12ν~+1(2ε~r)}.


## 4. Numerical Results

In Tables1 to 3, we see that energies of bound states such as: (np1/2,np3/2), (nd3/2,nd5/2), (nf5/2,nf7/2), (ng7/2,ng9/2),… (where each pair is considered as a spin doublet) in the absence of the tensor potential are degenerate but in the presence of the tensor potential, the degeneracies are removed. Also, we investigate the effects of the m1 and Cs parameters on the bound states under the conditions of the spin symmetry limits for T=0. The results are given in Tables 1 to 3. It is readily seen that if m1 and Cs parameters increases, the value of the bound state energy eigenvalues of this potential increases for several states. It is shown that the energy eigenvalues decrease with decreasing m1 and when increase with increasing the tensor strength T for κ>0 and κ<0. In Table 3, for constant values of Cs and T, the energy increases when m1 increasing. The decrease in the energy values is large without tensor interaction while small in presence of tensor interaction and then being large. Further, when Cs increases, the energy increasing.Finally, we plot the relativistic energy eigenvalues under spin and p-spin symmetry limitations in Figures1 to 4. In Figure 2, we plot the energy eigenvalues of spin symmetry limit versus the perturbated mass m1. It is seen that when m1 increases, the energy increases too. In Figure 2, we have shown the variation of the energy as a function of T. We can see the degeneracy removes between spin doublets and also they become far from each other, when the parameter T increases. In Figures 3 to 4, we plot the energy states of the pseudospin symmetry limit for different levels as functions of parameters m1 and T, respectively. The variation of energy can also be seen from these figures.Figure 1
The variation of the energy levels as a functionm1 in the presence of spin symmetry.Figure 2
The variation of the energy levels as a functionT in the presence of spin symmetry.Figure 3
The variation of the energy levels as a functionm1 in the presence of pseudospin symmetry.Figure 4
The variation of the energy levels as a functionT in the presence of pseudospin symmetry.


## 5. Conclusion

In this paper, the relativistic equation for particles with spin 1/2 was solved exactly with both spatially-dependent mass and tensor interaction for attractive scalar and repulsive vector Coulomb potentials under the spin symmetry limit via the Laplace transformation method. Some numerical results are given for specific values of the model parameters. Effects of the tensor interaction on the bound states were presented that tensor interaction removes degeneracy between two states in spin doublets. We also investigated the effects of the spatially-dependent mass on the bound states under the conditions of the spin symmetry limits forT=0.



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*Source: 873619-2012-12-04.xml*