International Journal of Physical Sciences Vol. 7(45), pp. 5954-5964, 30 November, 2012 Available online at http://www.academicjournals.org/IJPS DOI: 10.5897/IJPS12.628 ISSN 1992 - 1950 ©2012 Academic Journals

Full Length Research Paper

The Dirac equation with position-dependent mass for the modified Pöschl-Teller potential and its solution

Yasuk, F.1* and Bahar, M. K.1,2

1Deparment of Physics, Erciyes University, 38039, Kayseri, Turkey.

2Department of Physics, Karamanoglu Mehmetbey University, 70100, Karaman, Turkey.

Accepted 5 November, 2012

The Dirac-modified Pöschl-Teller problem with position dependent mass is investigated within the framework of the asymptotic iteration method in N dimensions. Any -state solutions are obtained by using the exponential approximation for the centrifugal term. The energy eigenvalues and two-

component spinor corresponding eigenfunctions are determined for different screening parameters

explicitly. The corresponding eigenfunctions obtained in the form of hypergeometric functions are normalized and plotted for different quantum states. Effects on modified Pöschl-Teller potential, bound

state energy eigenvalues and normalized corresponding eigenfunctions of the screening parameters

are investigated and its results are discussed. Key words: Dirac equation, position-dependent mass (PDM), Pöschl-Teller (PT) potential, asymptotic iteration method, bound state.

INTRODUCTION It is known that the solutions of the wave equations are very important since they contain all necessary information for quantum systems. In recent years, the solutions of the wave equations with a position-dependent mass (PDM) have been investigated more efficiently. Systems with PDM have been found to be very useful in studying the physical and electronic properties of semiconductors, quantum wells and quantum dots, quantum liquids, Helium clusters (Alhaidari, 2002, 2003), graded alloys and semiconductor heterostructures. The analytical solutions of the non-relativistic Schrödinger equation with PDM for solvable potentials have been discussed by a number of methods (Plastino et al., 1999; de Souza and Almeida, 2000; Alhaidari, 2002, 2003; Roy and Roy, 2002; Koç et al., 2002, 2005; Gönül et al., 2002a, 2002b; Bagchi et al., 2004; Quesne and Tkachuk, 2004; Quesne, 2006). Moreover, the investigation of relativistic effects is important in quantum mechanical *Corresponding author. E-mail: yasuk@erciyes.edu.tr.

PACS numbers: 03.65.-w; 03.65.Ge, 03.65.Pm.

systems such as heavy atoms and heavy ion doping. So, for these types of materials, the study of the properties of the Dirac equation with PDM is certainly of great interest (Panella et al., 2010). In the recent literature, considerable studies have been done to investigate of the exact or quasi-exact solutions of the Dirac equation with PDM and constant mass properties of these solutions for different potentials and mass distributions (Panella et al., 2010; Alhaidari, 2007; Alhaidari et al., 2007; Alhaidari, 2004; de Souza and Jia, 2006; Vakarchuk, 2005; Ikhdair and Sever, 2010; Jia et al., 2009; Jia and de Souza, 2008; Dekar et al., 1998; Peng et al., 2006; Chen et al., 2012; Esghi and Mehreban, 2012a; Esghi, 2011; Hamzavi et al., 2012; Esghi and Mehreban, 2012b; Esghi and Hamzavi, 2012; Agboola, 2012).

The asymptotic iteration method (AIM) of this study is to solve the PDM Dirac equation with the modified Pöschl-Teller (PT) potential and suitable mass function by using a different and more practical method, the AIM (Ciftci et al., 2003; Ciftci et al., 2005; Saad et al., 2006) within the frame of exponential approximation for centrifugal term. The PT potential as an important diatomic molecular potential has received much attention and considerable interest because of its wide application

in physics and chemical physics. This potential is a short- range model potential and has been used to describe bending molecular vibrations. In the work, the basic equations of the AIM used in our calculations are given briefly. Then, the Dirac equation in N-dimension is presented. Approximate solutions of the N-dimensional Dirac-modified PT problem with PDM are also given. Finally, a brief conclusion and discussion are presented.

BASIC EQUATIONS OF THE AIM

We briefly outline the AIM here; the details can be found in Ciftci et al. (2003), Ciftci et al. (2005) and Saad et al. (2006). The AIM was proposed to solve second-order differential equations of the form:

'

0 0'' ( ) ( )y x y s x y (1)

Where 0 ( ) 0x and 0 0( ), ( )s x x are in ( , )C a b . The

variables, 0 ( )s x and 0 ( )x , sufficiently differentiable.

The differential Equation 1 has a general solution:

2 1 0

'' '' '' '' '( ) exp exp [ ( ) 2 ( )]

x x x

y x dx C C x x dx dx

(2)

If, for sufficiently large n ,

1

1

n n

n n

s s

, (3)

Where

'

1 1 0 1

'

1 0 1

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ), 1,2,3...

n n n n

n n n

x x s x x x

s x s x s x x n

(4)

The termination condition of the method together with Equation 4 can be also be written as follows:

1 1( ) ( ) ( ) ( ) ( ) 0.n n n nx x s x x s x (5)

For a given potential, the idea is to convert the relativistic

wave equation to the form of Equation 1. Then, 0s and

0 are determined and ns and n parameters are

calculated. The energy eigenvalues are obtained by the termination condition given by Equation 5. However, the exact eigen functions can be derived from the following wave function generator:

'

2( ) exp

x

n ky x C dx

. (6)

Yasuk and Bahar 5955 Where n = 0, 1, 2, . . . and k is the iteration step number and usually greater than n. THE DIRAC EQUATION IN N-DIMENSIONS The N-dimensional Dirac equation with a central potential

( )V r and PDM ( )r can be written in natural units

1c as (Hall, 2010; Greiner, 1981):

( ) ( )rnH r E r where

1

( ) ( )

N

j j

j

H p r V r (7)

Where rnE is the relativistic energy,

j and are

Dirac matrices, which satisfy anticommutation relations

2 2

2

0

1

j k k j jk

j j

j

(8)

and

j j

j

p i ix

1 j N (9)

The orbital angular momentum operators jkL , the spinor

operators jkS and the total angular momentum operators

jkJ can be defined as follows:

, / 2,jk jk j k jk kj j k jk jk jk

k j

L L ix ix S S i J L Sx x

.2 2 2 2 2 2, , ,1

N N N

jk jk jk

j k j k j k

L L S S J J j k N (10)

For a spherically symmetric potential, total angular

momentum operator jkJ and the spin-orbit operator

2 2 2( ( 1) / 2) K J L S N commutate with the

Dirac Hamiltonian. For a given total angular momentum

j , the eigenvalues of K are ( ( 2) / 2) j N for

unaligned spin 1

2j and ( ( 2) / 2) j N for

unaligned spin1

2j . Also, since ( )V r is spherically

5956 Int. J. Phys. Sci. symmetric, the symmetry group of the system is SO(N) group. Thus, we can introduce the hyperspherical coordinates

1 1

1 1

1 2

cos

sin ...sin cos ,

sin ...sin sinN N

x r

x r

x r

2 1 N

Where the volume element of the configuration space is given as:

1

1

N

N

j

j

dx r drd

11

1

(sin )

N

j

j j

j

d d (11)

with 0 ,r 0 ,k

1,2,... 2, k N 0 2 , such that the spinor wave

functions can be classified according to the hyper radial

quantum number rn and the spin-orbit quantum number

and can be written using the Pauli-Dirac representation

1

2( ) ( )

( , )( ) ( )

r

r

r

Nn jm N

n N

n jm N

F r Yfr r

g iG r Y

(12)

Where ( )rnF r and ( )

rnG r are the radial wave function

of the upper and the lower-spinor components,

respectively, ( )jm NY and ( )

jm NY are the

hyperspherical harmonic functions coupled with the total

angular momentum j . The orbital and the pseudo-orbital

angular momentum quantum numbers for spin symmetry

and pseudospin symmetry refer to the upper and lower-component, respectively (Agboola, 2012). Inserting Equation 12 into the first of Equation 7, and separating the variables, we obtain the following coupled radial Dirac equation for the spinor component:

( ) ( ) [ ( ) ( )] ( )r r rn n n

dF r r E V r G r

dr r

(13)

( ) ( ) [ ( ) ( )] ( )r r rn n n

dG r r E V r F r

dr r

(14)

Where (2 1) / 2 N . Using Equation 13 as the

upper component and substituting into Equation 14, we obtain the following second-order differential equations:

2

2 2

( ) ( )( )( )

( 1)( ( ) ( ))( ( ) ( )) ( ) 0

( ( ) ( ))r r r

r

n n n

n

d r dV r d

d dr dr dr rr E V r r E V r F rdr r r E V r

(15)

2

2 2

( ) ( )( )( )

( 1)( ( ) ( ))( ( ) ( )) ( ) 0

( ( ) ( ))r r r

r

n n n

n

d r dV r d

d dr dr dr rr E V r r E V r G rdr r r E V r

(16)

The energy eigenvalues in these equations depend on

the angular momentum quantum number and N-dimension. To solve these equations, we used an approximation for the centrifugal barrier as discussed later in the work. APPROXIMATE SOLUTIONS TO THE N-DIMENSIONAL DIRAC EQUATION WITH POSITION-DEPENDENT MASS FOR THE MODIFIED PT POTENTIAL In this section, with in the framework of the AIM, we construct the relativistic energy spectrum and corresponding eigenfunctions of the PDM Dirac equation

with the modified PT potential in N-dimension by using approximation of the centrifugal term.

Energy eigenvalues of the N-dimensional Dirac equation with PDM for the modified PT potential

The modified PT potential given by Flügge (1999) can be expressed in the form:

2 2

2 2

( 1) AV(r)

2M Cosh ( r) Cosh ( r)

(17)

Where α is the screening parameter and is a

dimensionless parameter for the depth of the potential

Yasuk and Bahar 5957

Figure 1. Comparison of 1/r2 (red line) with 2 2/ ( )Sinh r for different values of

screening parameter.

well. Equation 15 cannot be solved analytically because of the last term in the equation, we use the equality d (r) dV(r)

0dr dr

to eliminate this term. Thus, using

this equality condition, the mass function is obtained as the following:

0 2

Ar

Cosh ( r)

(18)

Where μ0 is the integral constant and corresponds to the rest mass of the Dirac particle. The mass function μ(r) has the same form as the PT potential. The PT potential and mass function is inserted in Equation 15, then we get

20

0 02 2 2

2 ( )( 1)( )( ) ( ) 0

( )

r

r r r

n

n n n

A EdE E F r

dr r Cosh r

. (19)

Equation 19 cannot be solved exactly for any of the

wave case due to the spin-orbit coupling term 21/ r .

Therefore, to solve the Dirac equation for a single particle with PDM in the modified PT potential, we shall use an approximation as discussed in the following. To obtain analytical approximate solutions for the modified PT potential with centrifugal term, we have to use an approximation in the following form (Wei and Dong, 2009; Greene and Aldrich, 1976):

2 2 2

2 2 2 2

1 4

(1 ) ( )

r

r

e

r e Sinh r

(20)

Greene and Aldrich (1976) proposed a considerable approximation for the 1/r

2 term in the Dirac equation. By

using this approximation it is possible to find the approximate bound state solutions of the Dirac equation for coupling to pure 1/r

2 potentials with any values for

higher excitation levels with more accuracy. As shown in

Figure 1, 1/r2 and

2 2/ ( )Sinh r are consistent with

each other.

The

2 2 2

2 2 2 2

1 4

(1 ) ( )

r

r

e

r e Sinh r

exponential

approximation is used for the centrifugal term and

defined2tanh ( )z r transformation, Equation 19

becomes

r

2

0 1 2n2 2 2

d 1 3z dF (z) 0

dz 2z(1 z) dz z 1 z z(1 z) z 1 z

(21)

Where 0

( 1)

4

, r

2 2

0 n

1 2

E

4

and r0 n

2 2

2A( E )

4

.

In order to solve Equation 21 with the aid of AIM, we should transform Equation 21 to the form of Equation 1.

The upper spinor component rnF (r) / r has to satisfy the

boundary conditions, that is, rnF (0) 0 and

rnF ( ) 0 . For the bound states, the regularity

conditions are 1

(1 1 16 ) 004

and 1 .

5958 Int. J. Phys. Sci.

Therefore, the reasonable physical wave function is proposed as follows:

0 1

r r

1(1 1 16 )

4n nF z z 1 z f z

(22)

and if Equation 22 is inserted into Equation 21, the second-order homogeneous linear differential equations in the following form are obtained:

r r

0 0 1'' '

n n

2 1 16 z(4 1 16 4 )f (z) f (z)

4z 1 z

r

0 0 0 1 1 2

n

(1 4 1 16 2(2 1 16 ) 4 4f (z)

2z 1 z

(23)

Where

0 1 1

0

1 (2 1 4 2 ) 2( )

( 1 )

zz

z z

,

0 0 0 1 2

0

1 2 1 4 2(1 1 4 ) 2( )

2 ( 1 )s z

z z

.

By means of Equation 4, we may calculate n (z) and

ns (z) . This gives:

0 101 2

2

0 1 1 2

2 2

0 1 1 2

2 2

2 1 16 ( 4 (5 2 ))1 16( )

16 16 ( 1 )

6 (15 4 20 4 4 )

8( 1 )

( 15 4 4 4 4 ),

8( 1 )

zz

z z z

z

z z

z

z z

(24)

0 1

1 2 2

0 0 1 0 1 1 2

2 2

( 6 ( 1 ) 1 16 4 (3 ))( )

32 ( 1 )

(1 4 1 16 4 2 1 16 4 4.

32 ( 1 )

z zs z

z z

z z

…etc. Combining these results obtained by the AIM with quantization condition given by Equation 5 yields:

0 1 1 0 1,0 0 0 1 0 1 2

10 ( 1 4 1 16 4 2 1 16 4 )

4s s

1 2 2 1 1,1 0 0 1 0 1 2

10 ( 9 4 3 1 16 12 2 1 16 4 )

4s s

2 3 3 2 1,2 0 0 1 0 1 2

10 ( 33 4 5 1 16 20 2 1 16 4 )

4s s (25)

When the above expressions are generalized, 0 1 2, ,

statements containing the energy eigenvalues turns out to be

2

1, 0 2 0 1 0 1

1(4 4 (2 1) 1 16 (8 4) 2 1 16 8 1)

4n n n n .

If the above 0 1 2, , expressions are inserted, the

energy eigenvalues of the PDM Dirac equation for the PT potential in N-dimension is as follows:

2 2

2 20

0

2 2

2 2 2 2

0 0 2

( 1)( )

1( ( 1) 1 4 ( 1)(2 1)

4 4

(4 2) 1 4 ( 1) 8 1 0

rr

r r

nn

n n

EE M n

E En n

(26)

When the dimension of the system is N = 3, N = 5 and N = 7 for different values and quantum states, the

energy eigenvalues are calculated from Equation 25. The results are shown in Table 1.

As shown Table 1, due to increase values of

screening parameter, nE eigenvalues also decrease. To

understand how to change the modified PT potential for different values of screening parameter, we should

also connect to the modified PT potential with

screening parameter. In this case, the modified PT potential is shown in Figure 2. As can be seen in Figure 2, as a result of increasing values of screening

parameter, potential depth increases and more effective the modified PT potential, which will cause more bound states into the modified PT potential well.

Eigenfunctions of the N-dimensional Dirac equation with PDM for the modified PT potential

Also, the corresponding eigenfunctions can be obtained for PDM Dirac equation with the PT potential by using the wave function generator given by Equation 6. The first lowest states are:

0( ) 1f z ,

Yasuk and Bahar 5959

Table 1. The dimension of the system when N = 3, N = 5 and N = 7.

=0.1 Energy

= 0.4 Energy

= 0.8 Energy

, ,n l

, ,n l

, ,n l

N = 3

0,0,1

-0.993509 0,0,1

-0.996452 0,0,1

-0.996743

1,0,1

-0.978634 1,0,1

-0.988204 1,0,1

-0.989157

2,1,2

-0.935159 2,1,2

-0.962698 2,1,2

-0.965506

3,2,3

-0.868729 3,2,3

-0.922395 3,2,3

-0.928001

4,3,4

-0.781247 4,3,4

-0.867434 4,3,4

-0.876702

N = 5

0,0,2

-0.987121 0,0,2

-0.992916 0,0,2

-0.993492

1,0,2

-0.968071 1,0,2

-0.982316 1,0,2

-0.983740

2,1,3

-0.918600 2,1,3

-0.953297 2,1,3

-0.956846

3,2,4

-0.846641 3,2,4

-0.909509 3,2,4

-0.916111

4,3,5

-0.754236 4,3,5

-0.851108 4,3,5

-0.861599

N = 7

0,0,3

-0.978634 0,0,3

-0.988204 0,0,3

-0.989157

1,0,3

-0.955463 1,0,3

-0.975255 1,0,3

-0.977241

2,1,4

-0.900127 2,1,4

-0.942730 2,1,4

-0.947109

3,2,5

-0.822812 3,2,5

-0.895471 3,2,5

-0.903148

4,3,6

-0.725687 4,3,6

-0.833645 4,3,6

-0.845429

The bound state energy levels nE are shown in the case of PDM. 01, 0.001, 1, 1M

.

0 1

0

1 2

0

1 16 4( 1)1 16 2 4( ) (1 )

4 1 16 2

4

f z C z

,

0 0

2 2

0 1 0 1 0 1 2

0 0 0

1 16 2 1 16 6( ) ( )

4 4

2( 1 16 4 6) ( 1 16 4 6)( 1 16 4 12)(1 ),

1 16 2 ( 1 16 2)( 1 16 6)

f z C

z z

5960 Int. J. Phys. Sci.

V (

r)

F2(r

)

Figure 2. The variation of the PT potential according to various values of screening parameter. Plot of

probability density for n = 0 quantum state for different screening parameter.

0 0 0

3 2

0 1 0 1

0 1 2

0 0 0

0 1 0 1 0 1

0 0 0

1 16 2 1 16 6 1 16 10( ) ( )( )

4 4 4

1 16 4 1 16 43( 3)( 4)3( 1 16 4 12) 4 4(1

1 16 2 1 16 2 1 16 2( 1)

4 4

( 1 16 4 12)( 1 16 4 16)( 1 16 4 20)

( 1 16 2)( 1 16 6)( 1 16 10

f z C

z z

3))

z

(27)

etc.

Thus, the wave function nf z can be written as:

0

0 1 0

2 2 1

0

1 16 2( ) 1 16 4 1 16 24( ) ( 1) ( , , ; )

4 41 16 2( )

4

n

n

n

f z C F n n z

(28)

Hence, we can write the total radial wave function as follows:

01

1(1 1 16 )

0 1 0' 2 222 1

1 16 4 1 16 2( ) tanh ( )(1 tanh ( )) ( , ; ; tanh ( ))

4 4rnF r N r r F n n r

(29)

In order to normalize the wave functions, we should

determine ( )rnG z . Therefore, if we insert Equation 17

and 0 2

Ar

Cosh ( r)

into Equation 16, we

obtain that

202 2

02 2 2

2 ( )( 1)[ ( ] ( ) 0

( )

r

r r

n

n n

A EdE G r

dr r Cosh r

. (30)

When 21/ r exponential approximation and defined

2tanh ( )z r transformation are used, Equation 30

becomes

2

3 1 4

2 2 2

1 3[ ] ( ) 0

2 (1 ) (1 ) (1 ) (1 ) rn

d z dG z

dz z z dz z z z z z z

(31)

Where 3

( 1)

4

, 0

4 2

( )

2

rnA E

. If the

Yasuk and Bahar 5961

processes to find ( )rnF z are repeated, we obtain

1

(1 1 16 )32

1 3 1 3' 2 2

2 1

1 16 4 1 16 2( ) ( )(1 tanh ( )) ( , ; ; tanh ( ))

4 4rnG r N tanh r r F n n r

. (32)

The total radial wave functions are obtained in terms of confluent hypergeometric functions. In this part of our

work, we calculate the normalization constant 'N in

Equations 29 and 32. Calculation of the normalization constant is very necessary, which has not been explicitly worked out in most of the studies. How to compute the normalization constant is given in some studies (Saad,

2007). To compute the normalization constant'N , firstly,

we start with the normalization condition.

2 2 1

0

( ) 1Nf g r dr

(Greiner, 1981). According to the

transformation of2tanh ( )z r , 0z for 0r and

1z for r . Therefore, 1 12 2

0 0

( ) ( )1

2 (1 ) 2 (1 )

F z G zdz dz

z z z z

. So, we have

by means of the above normalization condition that is:

0

1

32 11

21 161

2 1 0 1 022 1

0' 2

21 161

3 1 322 1

0

1 16 4 1 16 2(1 ) ( , ; ; )

4 4( ) 2

1 16 4 1 16 2(1 ) ( ,1 ; ; )

4 4

z z F n n z dz

N

z z F n n z dz

(33)

The series representation of the confluent

hypergeometric function 2 1F is

1

1 1

0 1

( ) ...( )( ,.. ; ,.. ; )

( ) ...( ) !

ii p i

p q p q

i i q i

a a sF a a c c s

c c i

. (34)

Being Pochhammer symbols 1( )ia , 1( )ic and a

polynomial of degree n in s, we obtain the normalization constant in the following:

0 1 0

0

1

0 0 0 1

0 1 0 0 0 1

3 2' 2

3 1 3

3

1 16 4 1 16 2( ) ( ) ( ) 1 16 24 2 (2 , )

21 16 2 1 16 4 2( ) !( )

4 4

1 16 4 1 16 2 1 16 2 1 16 4 2( , , ; , ;1)

4 2 4 4( )

1 16 4 1 16 2( ) ( ) ( )

4 2

1 16 2 1( ) !(

4

n i i i

i

i i

i i i

i

n n

B

i

F n n i i

N

n n

i

3

1

0 3 1

3 1 3 3 3 1

3 2

2

1 16 2(2 , )

216 4 2)

4

1 16 4 1 16 2 1 16 2 1 16 4 2( , , ; , ;1)

4 2 4 4

n

i

i

B

F n n i i

(35)

In Figure 3, the normalized wave functions and probability densities for the mass dependent Dirac equation with the PT potential are plotted when the dimension of system is N = 3, for n = 0,1 and 2.

If ( )rnF r and ( )

rnG r are inserted into Equation 12, it

obtained that the spinor wave functions of the Dirac equation with PDM for the modified PT potential in N-dimensions. In case of 0.001, 1M , the negative

bound states be formed for 0 . If further considering

the limit case 0 , it is found from Equation 26 that

5962 Int. J. Phys. Sci.

Figure 3. The dimension is N=3. The position dependent mass. a) Plot of the normalized wavefunction for n=0 quantum state,

b) Plot of probability density for n=0 quantum state ( 00.1, 1, 1, 0.01M ), c) Plot of the normalized

wavefunction for n=1 quantum state, d) Plot of probability density for n=1 quantum state

( 00.1, 1, 1, 0.01M ), e) Plot of the normalized wavefunction for n=2 quantum state, f) Plot of probability

density for n=2 quantum state( 00.1, 1, 1, 0.01M ). g) Plot of the normalized wavefunction for n=3

quantum state, h) Plot of probability density for n=3 quantum state( 00.1, 1, 1, 0.01M ).

there are no energy eigenvalues. Because, in the limit

case 0 , the 1 2, and

4 statements diverge. The

corresponding upper and lower components approach zero, that is,

0lim ( ) 0

rnF r

, 0

lim ( ) 0rnG r

.

Conclusions We have considered approximately analytical bound state solutions of the Dirac equation with the modified PT potential and mass function using the AIM by applying an approximation to the centrifugal like term, which is a different approach. For arbitrary spin-orbit quantum number, state and parameter, we have obtained

the energy eigenvalues and corresponding radial wave functions for the case of PDM and approximation on the spin-orbit coupling term in a systematic way. The corresponding eigenfunctions have been obtained in terms of confluent hypergeometric functions. We have also obtained the normalization constants in the form of hypergeometric series. Regarding the method presented in this study, we should point that it is a systematic one and it is very efficient and practical in solving the within PDM systems for any state.

ACKNOWLEDGEMENTS The authors are grateful to the anonymous referees for their illuminating criticism and suggestions. We also thank Professor Shi-Hai Dong for many helpful discussion and suggested improvements to the paper. REFERENCES Agboola D (2012). Dirac-Hulthen Problem with Position-dependent

Mass in D-dimensions. Afr. Rev. Phys. P. 7. Alhaidari A (2004). Solution of the Dirac equation with position-

dependent mass in the Coulomb field. Phys. Lett. A. 322:72. Alhaidari AD (2002). Solutions of the nonrelativistic wave equation with

position-dependent effective mass. Phys. Rev. A 66: 042116. Alhaidari AD (2003). Nonrelativistic Green’s Function for Systems With

Position-Dependent Mass. Int. J. Theor. Phys. 42:2999. Alhaidari AD (2007). Relativistic extension of the complex scaling

method. Phys. Rev. A. 75:042707. Alhaidari AD, Bahlouli H, Al Hasan A, Abdelmonem MS (2007).

Relativistic scattering with a spatially dependent effective mass in the Dirac equation. Phys. Rev. A. 75:062711.

Arfken GB, Weber HJ (1995). Mathematical Methods for Physicists, Academic Press, San Diego.

Bagchi B, Gorain P, Quesne C, Roychoudhury R (2004). A general scheme for the effective-mass Schrödinger equation and the generation of the associated potentials. Mod. Phys. Lett. A. 19:2765.

Chen CY, Lu FL, You Y (2012). Scattering states of modified P¨oschl Teller potential in D-dimension. Chin. Phys. B. 21(3):030302.

Ciftci H, Hall RL, Saad N (2003). Asymptotic iteration method for eigenvalue problems. J. Phys. A. 36:11807.

Ciftci H, Hall RL, Saad N (2005). Construction of exact solutions to eigenvalue problems by the asymptotic iteration method. J. Phys. A. 38:1147.

de Souza DA, Almeida CAS (2000). Exact solvability of potentials with spatially dependent effective masses. Phys. Lett. A. p. 275.

Yasuk and Bahar 5963 de Souza Dutra A, Jia CS (2006). Classes of exact Klein–Gordon

equations with spatially dependent masses: Regularizing the one-dimensional inversely linear potential. Phys. Lett. 352:484.

Dekar L, Chetouani L, Hammann TF (1998). An exactly soluble Schrödinger equation with smooth position-dependent mass. J. Math. Phys. 39: 2551.

Esghi M (2011). Relativistic Pöschl-Teller Potential with Position-dependent Mass in D-dimensions. Int. J. Sci. Adv. Tech. 1(10).

Esghi M, Hamzavi M (2012). Spin Symmetry in Dirac-Attractive Radial Problem and Tensor Potential. Commun. Theor. Phys. 57(3):355.

Esghi M, Mehreban H (2012a). Eigen Spectra in the Dirac-Hyperbolic Problem with Tensor Coupling. Chin. J. Phys. 50(4):533.

Esghi M, Mehreban H (2012b). Solution of the Dirac Equation with Position-DependentMass for q-Parameter Modified Pöschl–Teller and Coulomb-like Tensor Potential. Few Body Syst. 52:41-47.

Flügge S (1999). Practical Quantum Mechanics. Berlin: Springer Press. Gönül B, Gönül B, Tutcu D, Özer O (2002b). Supersymmetric approach

to exactly solvable systems with position dependent effective masses. 17:2057.

Gönül B, Özer O, Gönül B, Üzgün F (2002a). Exact solutions of effective-mass Schrödinger equations. Mod. Phys. Lett. A. 17:2453.

Greene RL, Aldrich C (1976). Variational wave functions for a screened Coulomb potential. Phys. Rev. A. 14:2363.

Greiner W (1981). Relativistic Quantum Mechanics. Springer, Verlag. Hall RL (2010). Relativistic comparison theorems. Phys. Rev. A.

81:052101. Hamzavi M. Esghi M, Ikhdair SM (2012). Effect of tensor interaction in

the Dirac-attractive radial problem under pseudospin symmetry limit. J. Math. Phys. 53:082101.

Ikhdair SM, Sever R (2010). Approximate bound state solutions of Dirac equation with Hulth´en potential including Coulomb-like tensor potential. Appl. Math. Comput. 216:911.

Jia CS, de Souza Dutra A (2008). Extension of PT-symmetric quantum mechanics to the Dirac theory with position-dependent mass. Ann. Phys. 323:566.

Jia CS. Che T, Cui LG (2009). Approximate analytical solutions of the Dirac equation with the generalized Pöschl–Teller potential including the pseudo-centrifugal term. Phys. Lett. A. 373:1621.

Koç R, Koca M, Körcük E (2002). A new class of quasi-exactly solvable potentials with a position-dependent mass. J. Phys. A. 35:L527.

Koç R, Koca M, Sahinoglu G (2005). Scattering in abrupt heterostructures using a position dependent mass Hamiltonian. Eur. Phys. J. B. 48:583.

Panella O, Biondini S, Arda A (2010). New exact solution of the one-dimensional Dirac equation for theWoods–Saxon potential within the effective mass case. J. Phys. A: Math. Theor. 43:325302.

Peng XL, Liu JY, Jia CS (2006). Approximation solution of the Dirac equation with position-dependent mass for the generalized Hulthén potential. Phys. Lett. A. 352:478.

Plastino AR, Rigo A, Casas M, Garcias F, Plastino A (1999). Supersymmetric approach to quantum systems with position-dependent effective mass. Phys. Rev. A. 60 4318.

Quesne C (2006). First-order intertwining operators and position-dependent mass Schrödinger equations in d dimensions. Ann. Phys. N.Y. 321:1221.

Quesne C, Tkachuk VM (2004). Lorentz-covariant deformed algebra with minimal length and application to the (1 + 1)-dimensional Dirac oscillator. J. Phys. A. 37:4267.

Roy B, Roy P (2002). A Lie algebraic approach to effective mass Schrödinger equations. J. Phys. A. 35:3961.

Saad N (2007). The Klein–Gordon equation with a generalized Hulthén potential in D-dimensions. Phys. Scr. 76:623.

Saad N, Hall RL, Ciftci H (2006). Criterion for polynomial solutions to a class of linear differential equations of second order. J. Phys. A. 39:13445.

Szego G (1939). Orthogonal Polynomials, Am. Math. Soc. New York. P. 397.

Vakarchuk IO (2005). The Kepler problem in Dirac theory for a particle with position-dependent mass. J. Phy. A: Math. Gen. 38:4727.

Wei GF, Dong SH (2009). Algebraic approach to pseudospin symmetry for the Dirac equation with scalar and vector modified Pöschl-Teller potentials. Eur. Phys. Lett. 87:40004.

5964 Int. J. Phys. Sci. APPENDIX To normalize the wave functions, some of the special procedures for the beta function are given in the following form (Arfken and Weber, 1995; Szego, 1939):

i) (1 )

( 1, ) ( , )x y

q q

x q qB x y B x y

x y x y

ii) (1 )

( , ) ( 1, )x y y

q q

q qI x y I x y

x y

2 1(1 ) (1 )

( 2, )( 2) ( 2, ) ( 1) ( 1, )

x y x y

q

q q q qI x y

x B x y x B x y

3 2 1(1 ) (1 ) (1 )

( 3, )( 3) ( 3, ) ( 2) ( 2, ) ( 1) ( 1, )

x y x y x y

q

q q q q q qI x y

x B x y x B x y x B x y

.......

1

( , ) (1 )( ) ( , )

kmx y

q

k

qI x m y q q

x k B x k y

1,2,...m

iii)

1

0

(1 )(1 )( , ) ( )

(1 ) 1

x ykk

q

k k

yq q qB x y

x x q

1

2 1

(1 )(1,1 ;1 ; )

1

x yq q qF y x

x q

for

1( ,0) (0, )

2q

and

1

0

(1 )(1 ) 1( , ) ( , ) ( )

(1 )

x ykk

q

k k

xq q qB x y B x y

y y q

1

2 1

(1 ) 1( , ) (1,1 ;1 ; )

x yq q qB x y F x y

y q

4) ( ) ( ) ( )i j i ja a a i