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Pseudospin symmetry solution of the Dirac equation with position dependent mass for
the Hulthén potential within the Yukawa-like tensor interaction
Akpan N Ikot1*, Elham Maghsoodi2, Hillary P.Obong 1,Cecilia N Isonguyo1, Saber
Zarrinkamar3 and Hassan Hassanabadi2
1
Theoretical Physics Group, Department of Physics, University of Port Harcourt-Nigeria.
Department of Basic Sciences, Shahrood Branch, Islamic Azad University, Shahrood, Iran
2
3
Department of Basic Sciences, Garmsar Branch, Islamic Azad University, Garmsar, Iran
*email:[email protected]
Abstract
We present approximate analytical solutions of the Dirac equation for the Hulthen potential
with position dependent mass within the framework of pseudospin symmetry limit using
the Nikiforov Uvarov method. We obtained the relativistic energy spectrum and the
corresponding unnormalized wave function expressed in terms of the Jacobi polynomials.
Keywords: Dirac equation, Hulthén potential, pseudospin symmetry, Yukawa tensor interaction.
PACS numbers: 03.65Ge, 03.65Pm, 03.65Ca.
e-mail: "akpan ndem ikot" <[email protected]>
1. Introduction
The Dirac equation with a position dependent mass (PDM) has attracted a lot of interest in
recent times because of its application in particle, nuclear semiconductor and condensed
matter physics [1]. However, many quantum mechanical systems with a position –
dependent have been found to be very useful for studying physical properties of different
microstructures [2], such as quantum liquids [3], quantum dots [4], quantum wells [5], and
semiconductor hetero structures [6]. Different authors have investigated the PDM for nonrelativistic quantum mechanics using various techniques [7-8]. The Dirac equation with the
PDM on the other hand has been investigated in quantum mechanical systems such as
heavy atoms and heavy ion doping [9]. In the Dirac equation, the pseudospin symmetry
occurs when the magnitude of the attractive Lorentz scalar potential S (r ) and repulsive
vector potential V ( r ) are nearly equal but opposite in sign, i.e., S (r ) V (r ) where the
sum of the potential is (r )  S (r )  V (r )  C ps  const [10-12]. Tensor interactiopotentials
ˆ (r ) with
were introduced into the Dirac equation by the transformation p  p  im .rU
spin-orbit coupling term added to the Dirac Hamiltonian [13]. In most of studies, due to the
mathematical structure of the problem, the tensor interaction is considered as Coulomb-like
[14-15] or Cornell interaction. Hassanabadi et al. were the first who introduces the Yukawa
tensor interaction in the Dirac theory [16]. The Hulthén potential has been used extensively
to describe the interactions between the nucleon and heavy nucleus and is define as [17]
e 2 r
V (r )  V 0
, V  2 z
(1)
1  e 2 r  0
Where V0 ,  and z are three real parameters and represent the strength, the screening
range of the potential and the atomic number respectively.
In this paper, we consider the Hulthén potential with position dependent mass and Yukawa
tensor potential for attractive scalar and repulsive vector Hulthén potential under
pseudospin symmetry.
2. Parametric Nikiforov-Uvarov method
Within this section, we will introduce the simple but powerful NU technique which has
solved many important problems in quantum mechanics [18, 19]. According to the NU
method , a second-order differential equation of the form
d2
c1  c 2s d
1
 2
1s 2  2s  3
 2
2
s (1  c 3s ) ds s (1  c 3s )
 ds

Posses the below solutions [20, 21]

 (s )  0.

(2)
c3n  2nc5  (2n  1)  c9  c3 c8   n  n  1 c3  c7  2c3c8  2 c8c9  0, (3)
 ( s)  s c 1  c3 s  Pn c
c13
12
10 , c11

(1  2c3 s) ,
(4)
where Pn( ,  ) ( s ) are the Jacobi polynomials and
1
1
(1  c1 ), c5  (c2  2c3 ), c6  c52  1 ,
2
2
c7  c4 c5   2 , c8  c42  3 , c9  c3c7  c32c8  c6 ,
c4 

c .
c10  c1  2c4  2 c8 , c11  c2  2c5  2
c12  c4  c8 , c13  c5 

c9  c3

c9  c3 c8 ,
(5)
8
3. Dirac Equation with a Tensor Coupling
Dirac equation with a tensor potential U (r ) in the relativistic unit (  c  1) is written as
[22]
ˆ (r ) (r )   E  V (r ) (r ) ,
. p   (M  S (r)  i.rU
(6)
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.  ,  are the 4  4 Dirac
matrices given as
i 
0
I 0 
,   
,
0
 0 I 
 
 i
(7)
where I is a 2  2 unit matrix and  i are the Pauli three-vector matrices defined as
0 1
 0 i 
1 0 
 , 2  
 , 2  
.
1 0
i 0 
 0 1
1  
(8)
1
1
The eigenvalues of the spin-orbit coupling operator are   ( j  )  0,   ( j  )  0
2
2
for unaligned
H , K , J
2
,Jz

spinors as [23]
j l
1
2
and aligned spin
j l
1
2
cases, respectively. The set
forms a complete set of conserved quantities. Thus, we can write the
l
1  Fn (r )Y jm ( ,  ) 

 n (r )  
r  iG n (r )Y jml ( ,  ) 


(9)
where Fn (r ) and Gn (r ) represent the upper and lower components of the Dirac spinors.
l
l
Yjm
( ,  ), Yjm
( ,  ) are the spin and pseudospin spherical harmonics and m is the projection
on the z-axis. With other known identities [24]
 .A  .B   A.B  i . A  B  ,
(10)

 .L 
 . p   .rˆ  rˆ. p  i

r 

as well as
 .L  Y
 .L  Y
l
jm
l
( ,  )  (  1)Y jm
( ,  )
l
jm
l
( ,  )  (  1)Y jm
( ,  )
 .rˆ  Y jml ( ,  )  Y jml ( ,  )
 .rˆ  Y jml ( ,  )  Y jml ( ,  )
(11)
we obtain the coupled equations [24],
d 

   U (r )  Fn (r )   M (r )  En  (r )  Gn (r ) ,
 dr r

(12)
d 

   U (r )  Gn (r )   M (r )  En  (r )  Fn (r ) ,
 dr r

(13)
where,
(r )  V (r )  S (r ),
(r )  V (r )  S (r ).
(14)
(15)
Eliminating Fn (r ) and Gn (r ) in favor of each other in Eqs. (12) and (13), we obtain the
second-order Schrödinger-like equations
 d 2  (  1) 2U (r ) dU (r )



 U 2 (r )    (r )  En  (r )   ( r )  En  ( r )  
 2
2
r
dr
r
 dr


  d  (r ) d (r )  d 
 Fn (r )  0,
  U (r ) 
  dr  dr 

 dr r



  (r )  En  (r ) 


(16)
 d 2  (  1) 2U (r ) dU (r )



 U 2 (r )    (r )  En  (r )   (r )  En  (r )  
 2
2
r
dr
r
 dr


  d  (r ) d (r )  d 
 Gn (r )  0, (17)
  U (r ) 
  dr  dr 

 dr r



  (r )  En  (r ) 


with  (  1)  l (l  1),  (  1)  l (l  1) .The mathematical relation
d  (r )
d ( r ) d  ( r )


is the necessary relation to obtain exact or approximate
dr
dr
dr
solutions for mass dependent function problems[25].
3.1 The Pseudospin Symmetry Limit
d ( r )
 0 or (r )  C ps  const [10-12]. Here, we
In the pseudospin symmetry limit,
dr
intent to study the Hulthén potential [7, 17]
(r )  
2 z e 2 r
,
1  e 2 r


(18)
2 z e 2 r
 ( r )  0 
1  e 2 r


besides the Yukawa tensor interaction [26]
 e r 
U (r )  V1 
,
 r 
(19)
Where z , 0 , V1 and  are Coulomb charge, the rest reduced mass of the Dirac theory ,
potential depth and range of the nucleon force, respectively [27]. The corresponding
equation is not exactly solvable. Consequently, to provide an analytical solution, we have
to proceed on an approximate basis. Therefore, we introduce the approximations [28]
1
4 2e2 r

,
r 2 (1  e2 r )2
(20)
1
4 2e r
, (21)

r 2 (1  e2 r )2
which are plotted in Fig. (1). The combination of recent equations as well as a change of
variable of the form s  e 2 r , yields
d 2G nps
(1  s ) dG nps
1
Q1ps s 2  Q 2ps s  Q3ps  G nps (s )  0,

 2
2
ds
s (1  s ) ds
s (1  s )2 
(22)
with
 2
1  

Q1ps   2  V1  V1    2  ,
2  4 

 4
 2 2
3
 

ps
Q2   2   (  1)   2   V1  2  ,
2
4 

 4
(23)
2
,
4 2
 2   0  Enps  C ps  0  Enps  ,   2 z  0  Enps  .
Q3ps 
Comparing Eqs. (22) and (2), we obtain the required parameters as
c1  1, 1  Q1Ps ,
c2  1,  2  Q2Ps ,
(24)
c3  1,  3  Q3Ps .
Eq. (5) determines the rest of the coefficients as
1
1
c 4  0, c 5   , c 6   Q1ps ,
2
4
1
c 7  Q 2ps , c 8  Q 3ps , c 9  Q1ps  Q 3ps  Q 2ps  , c10  1  2 Q 3ps ,
4
 1

1  1
c11  2  2 
 Q1ps  Q 3ps  Q 2ps  Q 3ps  , c12  Q 3ps , c13    
 Q1ps  Q 3ps  Q 2ps  Q 3ps
4
2
4



By using Eqs. (24) and (25), we can easily obtain the energy relation
n
 2n  1 
2

 2n  1 

Q1ps  Q3ps  Q2ps 

1
 Q3ps   n  n  1  Q2ps  2Q3ps
4

1

2 Q3ps  Q1ps  Q3ps  Q2ps    0,
4

and lower component of the wave function is
(26)

 . (25)


Gnps  r   N n e 2 r
 
Q3ps
 2 F1 (n, n  2 Q3ps  2
1  e 2 r

1 1
  Q ps  Q3ps Q2ps
2 4 1
1
 Q1ps  Q3ps  Q2ps  1; 2 Q3ps  1; e 2 r ).
4
(27)
The corresponding upper component can be simply obtained from Eq.(12), i.e.
Fnps (r ) 

1
M  Enps  C ps

d 
 e r


V

1
 r
 dr r
  ps
  Gn (r ).

(28) ,
where Enps  M  C ps .
4 Discussions and Numerical Results
Approximate solutions of the Dirac equation for the Hulthén potential is obtained in the
absence and the presence of the Yukawa tensor potential for various values of the quantum
numbers n and κ. The bound states are reported in tables (1) under the condition of the
pseudospin symmetry and we can clearly see that there is the degeneracy between the
bound states and in the presence of the Yukawa tensor interaction, these degeneracies are
removed or changed. In Figs. (2) , we show the effects of the α-parameter on the bound
states in the presence of the tensor potential (V1 = 0.5) and It is seen that if the α-parameter
increases, the bound states become more bounded both for the conditions of the pseudospin
limit. In Fig. (3), we have obtained behavior of energy vs. Cps for Pseudospin symmetry
limit and we can see that bound states become less bounded in the pseudospin symmetry
limit with increasing Cps. Fig. (4) Present the dependence of the bound-state energy levels
on potential parameter z for Pseudospin symmetry limit. It is seen in Fig. (4) that bound
states obtained in view of pseudospin symmetry become more bounded with increasing z.
In Figs. (5) and (6), we obtain the effects of the tensor interaction parameter V 1 and 0 on
the bound states in view of the pseudospin symmetry limit respectively. Fig. (5) shows that
the magnitude of the energy difference between the degenerate states increases as H
increases and in Fig. (6), we can see that bound states become less bounded in the
pseudospin symmetry limit with increasing 0 . In Fig. (7), the wavefunctions for
Pseudospin symmetry limit plotted for without Tensor Interaction and for condition Tensor
Interaction and Fig. (7) shows that tensor interaction affects only the shape of the wave
functions and does not change the node structures of the radial upper and lower
components of the Dirac spinors.
5 Conclusions
In this paper, we have obtained the approximate solutions of the Dirac equation for the
Hulthén potential with position dependent mass including the Yukawa tensor interaction
term within the framework of pseudospin and spin symmetry limit using the NU method.
We have obtained the energy eigenvalues and corresponding lower and upper wave
functions in terms of the Jacobi polynomials. Moreover, the results obtained in this work
have been compared with the previous work of others authors as given in the literature.
Finally, this work can be extended to others potentials model [7, 8, 25] which has many
applications in physics and related fields.
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Table (1). Energies in the Pseudospin Symmetry Limit
for   0.01, 0 =1fm-1 , z = 1, C ps  5.
1
n ,  0
( , j)
1,-1
1S 1
E nps (fm 1 )
E nps (fm 1 )
(V 1  0)
-3.928349615
(V 1  0.5)
n  1,
 0
(  2,
j  1)
-3.893285734
0,2
0d 3
2
2
1,-2
1P3
1,-3
1d 5
-3.963160651
-3.949025469
0,3
1,-4
1f 7
-3.979367223
-3.972408645
0,4
2,-1
2S 1
-3.988044887
-3.984191529
0,5
2,-2
2P3
-3.963160651
-3.949025469
1,2
2,-3
2d 5
-3.979367223
-3.972408645
1,3
2,-4
2f 7
-3.988044887
-3.984191529
1,4
3,-1
3S 1
-3.993092019
-3.990805160
1,5
3,-2
3P3
-3.979367223
-3.972408645
2,2
3,-3
3d 5
-3.988044887
-3.984191529
2,3
3,-4
3f 7
2
-3.979367223
-3.984191529
0h 9
-3.988044887
-3.990805160
1d 3
-3.963160651
-3.972408645
1f 5
-3.979367223
-3.984191529
1g 7
-3.988044887
-3.990805160
1h 9
-3.993092019
-3.994756044
2d 3
-3.979367223
-3.984191529
2f 5
-3.988044887
-3.99080516
-3.993092019
-3.994756044
-3.996160182
-3.997182574
2
-3.993092019
-3.990805160
2,4
2g 7
2
2
4
0g 7
2
2
3
-3.972408645
2
2
2
-3.963160651
2
2
1
0f 5
2
2
4
-3.949025469
2
2
3
-3.928349615
2
2
2
(V 1  0.5)
2
2
1
(V 1  0)
2
2
4
E nps (fm 1 )
2
2
3
E nps (fm 1 )
-3.996160182
-3.994756044
2,5
2h 9
2
Fig. (1)
1
and its approximations for α=0.01
r2
Fig. (2). Energy vs.  for pseudospin Symmetry limit for
0 =1fm-1 , z = 1, C ps  5, V 1  0.5.
Fig. (3). Energy vs. C ps for pseudospin Symmetry limit for
  0.01, 0 =1fm -1 , z = 1, V 1  0.5.
Fig. (4). Energy vs. z for pseudospin Symmetry limit for
  0.01, 0 =1fm -1 , C ps  5, V 1  0.5.
Fig. (5). Energy vs. V 1 for pseudospin Symmetry limit for
  0.01, 0 =1fm -1 , z = 1, C ps  5.
Fig. (6). Energy vs. 0 for pseudospin Symmetry limit for
  0.01, z = 1, C ps  5, V 1  0.5.
Fig. (7). Wavefunction for Pseudospin Symmetry Limit for
for   0.01, 0 =1fm-1 , z = 1, C ps  5, V 1  0.5.