Download Relativistic Problem by Choosing Spatially

World Applied Sciences Journal 16 (3): 415-420, 2012
ISSN 1818-4952
© IDOSI Publications, 2012
Relativistic Problem by Choosing Spatially-Dependent
Mass Coupled with a Tensor Potential
Mehdi Eshghi
Department of Physics, Central Tehran Branch, Islamic Azad University, Tehran, Iran
Abstract: In this research, we have solved the Dirac equation with spin symmetry for Pöschl-Teller potential
including a Coulomb-like tensor interaction by choosing a spatially-dependent mass. The energy eigenvalues
equation and the corresponding unnormalized wave functions have obtained in terms of the Jacobi
polynomials. The Nikiforov-Uvarov method had used in the calculations. Some numerical results are given for
this potential.
Pöschl-Teller potential
Key words:Dirac equation
Nikiforov-Uvarov.
INTRODUCTION
M (=
r ) m0 + 4V0
(1 + e −2
r 2
)
and the Pöschl-Teller potential of the form [26-28] is
V (r ) = −
V0
cosh 2 r
r ≥ Rc
(3)
where Rc = 7.78 fm is the Coulomb radius, Za and Zb
denote the charges of the projectile a and the target nuclei
b, respectively.
Our aim in this paper is study the Dirac equation for
Pöschl-Teller potential including a Coulomb-like tensor
coupling in the case of SDM distribution Eq. (1) under the
spin symmetry. We have obtained the energy eigenvalues
equation and the corresponding spinor wave functions by
using the Nikiforov-Uvarov (NU) method.
a spin-orbit coupling term is added to the Dirac
Hamiltonian. Recently, tensor couplings have been used
widly in the studies of nuclear properties. In this regard,
see [8-16].
In recent years, the solution of Dirac, Klein-Gordon
and Schrödinger equations with a spatially-dependent
mass (SDM) are useful for the investigation of some
physical systems. They are used, for example, in the
determination of the electronic properties of the
semiconductors [17], 3He clusters [18], in quantum liquids
[19], in quantum dotes [20], etc. Some authers have
investigated the exact solutions of the Dirac equation with
position-depedent mass [12, 21-25].
According to the report which was given in the
research [21, 26] the SDM for q = 1 of the form is
r
Spatially-dependent mass
Z aZ be 2
H
U (r ) =
H=
− ,
,
r
4 0
Concepts of spin symmetry, pseudo-spin symmetry
and a tensor potential have been found interesting
applications in the field of nuclear physics [1-5]. Tensor
potentials were introduced into the Dirac equation with


ˆ ( r ) [6, 7]. In this way,
the substitution p → p − im ⋅ rU
e −2
Coulomb-like
Review of the Nikiforov-Uvarov Method: We give a brief
description of the conventional NU method [29]. Recently,
this method has been introduced for solving the
Schrödinger, Klien-Gordon and Dirac equations with the
well known potentials. For example, see [30-37].
The NU method redues the second order differential
equations to the hypergeometric type with an appropriate
coordinate transformation s =s(r) as
n′′ ( s ) +
(1)
(s)
(s)
n′ ( s ) +
 (s)
2
(s)
n (s)
(4)
0
=
where (s) and  ( s ) are polynomials, at the most of the
second degree and  ( s ) is a polynomials, at most of the
first degree. If we take the following factorization
n ( s ) = ( s ) yn ( s ) (4) becomes
(2)
and tensor potential Coulomb-like [8] is
Corresponding Author: Mehdi Eshghi, Department of Physics, Central Tehran Branch, Islamic Azad University, Tehran, Iran.
Tel.: +98-21-73026693, Fax: +98-21-77104938.
415
World Appl. Sci. J., 16 (3): 415-420, 2012
where
( s) yn′′ ( s ) + ( s ) y′n ( s ) + yn ( s ) =
0
(5)
d
(ln ( s ))
ds
′( s ) < 0
( s ) =  ( s ) + 2 ( s ),
(6)
(s) = (s)
total angular momentum j, the eigenvalues of K̂ are
=
− ( j + (1/ 2)) for aligned spin (s1/2, p3/2, etc.) and
= ( j + (1/ 2)) for unaligned spin (p1/2, d3/2, etc.). By using
the radial quantum number n and spin-orbit coupling
quantum number k, the Dirac spinors wave functions can
be classified and had given by
(7)
where the polynomial solutions yn(s) have been given by
the Rodrigues formula
a dn 
yn ( s ) = n
( s ) ds n 
n
(s ) ( s) 

n
(8)
The function (s) and the parameter
equation are defined as follows
=
(s)
and
the spin and pseudo-spin spherical harmonics,
respectively and m is the projection of the total angular
momentum on the z-axis. The orbital angular-momentum
quantum number l and l are the labeles of upper and
(9)
in the above
lower components. For a given spin-orbit quantum
number = ±1,±2,±3,..., the orbital angular momentum and
pseudo-orbital angular momentum are given by
l = + 1/ 2 − 1/ 2 and l = − 1/ 2 − 1/ 2 , respectively.
2
′( s ) −  ( s )
 ′( s ) −  ( s ) 
± 
−  (s) + q (s)

2
2


(10)
= q + ′( s )
Substituting (14) into (13) and using the following
relations [38],
(11)
    
   
( . A)( .B ) =A.B + i .( A × B)


 

.L
( .P )
.rˆ (rˆ.P + i
=
)
r
and properties
The determination of q is the essential point in the
calculation of (s). It is simply defined by setting the
discriminant of the square root wich must be zero. The
eigenvalues equation have calculated from the above
equation
n( n − 1)
′′( s ). n =
=
− n ′( s ) −
0,1, 2,...
n =
(12)
2
Dirac Equation: According to the report which was given
by researcher [8], the spatially-dependent mass Dirac
equation including tensor interaction for spin-1/2 particles
with both the scalar and the vector potential, in units
where = c= 1 , is
 

ˆ (r )]
[ ⋅ P + ( M ( r ) + Vs ( r ) ) − i ⋅ rU


) [ E − Vv ( r )] nk ( r ),
nk ( r=
(14)
where Fnk(r) is upper and G nk(r) is the lower radial wave
functions of the Dirac spinors, Y l ( , ) and Y l ( , ) are
jm
jm
where n is a normalization constant and the weight
function (s) must satisfy the differential equation
 (s) 
′( s ) − 
0, ( s ) =
( s ) ( s ).
 ( s) =
 (s) 
l


 1 Fnk ( r )Y jm ( , ) 
(r ) = 
r iG (r )Y l ( , ) 
jm

 nk
(15)
(16)
  l
l
( .L )Y jm
( , =
) ( − 1)Y jm
( , )
(17)
  l
l
( .L)Y jm
( , )=
−( − 1)Y jm
( , )
(18)

l
l
( .rˆ)Y jm
( , ) = −Y jm
( , )
(19)

l
l
( .rˆ)Y jm
( , ) = −Y jm
( , )
(20)
yields two coupled differential equations as follows
(13)
 d

+ − U ( r )  Fn =
( r )  En + M ( r ) − ∆ ( r )  Gn ( r )

(21)
dr
r


where and the 4×4 matrices, E is the relativistic energy


of the system, P = −i∇ is the three-dimensional
 d

− + U ( r )  Gn=
( r )  M ( r ) − En + Σ ( r )  Fn ( r )

(22)
 dr r

momentum operator. For a particle in a central field, the
total momentum operator J and operator
is the spinK̂
orbit matrix operator and have written in terms of the
orbital angular momentum operator as Kˆ =
− ( ˆ .Lˆ + 1) ,
L̂
which commute with the Dirac Hamiltonian. For a given
) Vv (r ) + Vs (r ) . By
where ∆(r )= Vv (r ) − Vs (r ) and Σ(r =
substituting Gnk(r) from (21) into (22) and Fnk(r) from (22)
into (21), we obtain the following two second-order
differential equations for the upper and lower
components,
416
World Appl. Sci. J., 16 (3): 415-420, 2012
 d2
( + 1) 2
dU ( r )
−
+
− U 2 (r )
U (r ) −

2
2
r
dr
r
 dr
+ ( En + M ( r ) − ∆ (r ))( En − M (r ) − Σ( r ))
where
b1 = ( + H )( + H + 1) b2 =
(23)
En + M 0 − C s
4 2
(28)
V0
V0 =
4 2
b3 =
M 0 − En
dM ( r ) d ∆( r )
−
 d k  
dr
dr
0,
+
+
 F (r ) =
+
−
∆
M
r
E
r
(
)
(
)
(
)  dr r    n
n
By comparing (27) with (4), we determine polynomials as
( s ) = 2s ( s − 1)  ( s ) = 1 − 3s
 (s) =
−b3V0 s 2 + (b1 − b2b3 − b3V0 ) s − b1
2
 d
( − 1) 2
dU ( r )
−
+
− U 2 (r )
U (r ) +

2
2
r
dr
dr
r

+ ( En + M ( r ) − ∆ ( r ))( En − M ( r ) − Σ( r ))
Substituting them into (10), we obtain
(24)
dM ( r ) d Σ (r )
+
 d k  
dr
dr
0.
+
+
 G (r ) =
−
+
Σ
M
r
E
r
(
)
(
)
(
)  dr r    n
n
V0
cosh
2
(25)
≈

×  −b3V0 s 2 + (b1 − b2b3 − b3V0 ) s − b1   Fnk ( s ) =0,

 
)
(
)
(
)
)
= n 4 + 2 b2b3 + 1 + 4b1 + 2n(n − 1)
(26)
(32)
Some numerical results are given in table 1. we use
1
the parameters =
Cs 5,=
M 1 fm−=
,
0.01,=
V0 10 .
By using a transformation of the form s = tanh2 ar, we
rewrite it as follows
 d2
1 − 3s d
1
+
+

2
−
2
s
(1
s
)
ds
[2s (1 − s)]2
 ds
(
(
2
sinh 2 r
(30)
1
− (b1 + b3V0 + b2b3 )
2
1
1
1
−
b2b3 (1 + 4b1) − 2 b2b3 + 1 + 4b1 −
2
2
2
We take the following approximation [42] as
r2
)
2

1 1 − 4V0b3 − 8q s +
2 (8q − 4(b − b V − b b ) − 2) s + 4b + 1
1
3 0
2 3
1
(31)
By using (31) and (12), we obtain the eigenvalue
equation to be

0
 Fn (r ) =
r 
1
(
1− s
2
The constant q is determined in the same way.
Therefore, we get
1 

 2  −2 b2b3 + 1 + 4b1 s − 1 + 4b1  ,


−(b1 + b3V0 + b2b3 ) + b2b3 (1 + 4b1 )
 for q =
1− s 
2
(s)
=
±
2
 1  +2 b b + 1 + 4b s − 1 + 4b  ,
2 3
1
1 
 2 


 for q = −(b1 + b3V0 + b2b3 ) − b2b3 (1 + 4b1 )

2
the equation obtained for the upper component of the
Dirac spinor Gnk(r) becomes
− ( M 0 − En )
(s) =
±
( + 1) = l (l + 1) and
In the above equations


( − 1) = l (l + 1) . Equations (23) and (24) can not be
solved analytically because of the last term in the
equations. It is convenient to solve the mathematical
relation dM (r ) d ∆ (r ) [23, 39]. By using this relation,
=
dr
dr
we can exactly solve Eq. (23).
Substituting (1), (2) and (3) into (23) and considering
spin symmetry and taking r = Cs as the Pöschl-Teller
potential and ∆r= Cs= const.( d ∆ (r ) / dr= 0) , i.e.[40, 41].
 d 2 ( + H )( + H + 1)
−

2
r2
 dr
−( M 0 − En )(M 0 + En − Cs )
(29)
The wave function Fnk(s) is obtained from (6) by
taking (s) and (s),
=
(s)
(27)
b2b3
( k + H +1)
2
s
(1 − s ) 2
and using (11), we have
417
(33)
World Appl. Sci. J., 16 (3): 415-420, 2012
Table 1: The bound state energy eigenvalues En,k in unit of fm
1
of the spin symmetry Pöschl-Teller potential for several values of n and k.
l
n,k<0
l,j
En,k<0(H=0)
En,k<0(H=5)
n-1, k>0
l+2,j+1
En,k>0(H=0)
En,k>0(H=5)
1
1, -1
2s1/2
1.00237
1.01093
0, 2
4d3/2
1.00588
1.02135
2
1, -2
3p3/2
1.00393
1.00821
0, 3
5f5/2
1.00821
1.02557
3
1, -3
4d5/2
1.00588
1.00588
0, 4
6g7/2
1.01093
1.03016
4
1, -4
5f7/2
1.00821
1.00393
0, 5
7h9/2
1.01403
1.03512
1
( s)
=
s
2
(2k + 2 H −1)
bb
2
(1 − s ) 2 3
ACKNOWLEDGMENT
(34)
The author would like to thank the editor of the
journal WASJ Dr. Najafpoor in the present work.
Substitution (34) into (8), we obtain
=
yn ( s ) an s
−
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× n s

ds

b2b3
(35)




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=
Fn ( s ) an s
 (2 + 2 H −1)
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CONCLUSION
In this work, we have obtained analytically the
approximate energy equation and the corresponding
wavefunctions of the Dirac equation for the Pöschl-Teller
potential coupled with a Coulomb-like tensor under the
condition of the spin symmetry by choosing SDM and
using the NU method. Some numerical results are given
for this potential.
418
World Appl. Sci. J., 16 (3): 415-420, 2012
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