Indian J Phys DOI 10.1007/s12648-017-1009-z

ORIGINAL PAPER

Approximate bound and scattering solutions of Dirac equation for the modified deformed Hylleraas potential with a Yukawa-type tensor interaction H Hassanabadi1, A N Ikot2*, C P Onyenegecha2 and S Zarrinkamar3 1

Physics Department, Shahrood University of Technology, Shahrood, Iran

2

Theoretical Physics Group, Department of Physics, University of Port Harcourt-Nigeria, Port Harcourt, Nigeria 3

Department of Basic Sciences, Garmsar Branch, Islamic Azad University, Garmsar, Iran Received: 25 August 2016 / Accepted: 07 February 2017

Abstract: Analytical bound and scattering state solutions of Dirac equation are investigated for the modified deformed Hylleraas potential with a Yukawa-type tensor interaction. The energy equation, phase shifts and normalization constants of the pseudospin and spin symmetry limits are represented. Since the modified deformed Hylleraas potential reduces to the Po¨schl–Teller, Hulthe´n and deformed Hylleraas potential, we have obtained energy equation and scattering properties of the Dirac equation for these potentials within a Yukawa-type tensor interaction. We have also reported some numerical results to show the effect of tensor interaction. Keywords: Dirac equation; Modified deformed Hylleraas potential; Yukawa-type potential; Pseudospin symmetry; Spin symmetry; Tensor interaction; Scattering state; Phase shift PACS Nos.: 03.65.Ge; 03.65.Pm; 03.65.Nk

1. Introduction Obtaining the exact or approximate-analytical solutions of wave equations is an interesting problem in fundamental quantum mechanics. Unfortunately, there are only a few potentials for which the Schro¨dinger, Dirac, Klein–Gordon and Duffin–Kemmer–Petiau (DKP) equations can be exactly solved. Several model potentials have been introduced to explore the relativistic and non-relativistic energy spectra and wave function behaviors [1–5]. Jia et al. [6] have derived the bound state solution of Klein–Gordon equation under unequal scalar and vector Kink-like Potentials. By using the series expansion method, the authors in [7] have obtained the analytical solutions of the two-dimensional Schro¨dinger equation with the Morse potential. Pseudospin symmetry solution of Dirac equation

for the modified Rosen-Morse potential is investigated in [8]. The DKP equation under a scalar Coulomb interaction is solved in [9] via the ansatz approach. Here, we report the solution of bound and scattering states of Dirac equation for the modified deformed Hylleraas potential with a Yukawa-type potential as a tensor interaction. The Hylleraas potential is originally considered for diatomic molecules [10] and recent works investigate [11] the relativistic rotational-vibrational energy levels of such molecules [12–15]. The modified deformed Hylleraas potential is defined as [16]    2ar  V0 a  e2ar e VðrÞ ¼  V 1 b 1  e2ar 1  e2ar e2ar þ V2 ð1Þ ð1  e2ar Þ2

*Corresponding author, E-mail: [email protected]

Ó 2017 IACS

H Hassanabadi et al.

where V0 ; b; a; V1 ; a and V2 are constant parameters. The Po¨schl–Teller, Hulthe´n and deformed Hylleraas potential are special cases of the modified deformed Hylleraas potential. For V0 ¼ V1 ¼ 0, the potential reduces to the Po¨schl–Teller potential. When a ¼ V2 ¼ 0, the potential changes into the Hulthe´n potential which is widely used in nuclear, particle physics and atomic physics [17, 18]. For V1 ¼ V2 ¼ 0, the interaction reduces to the deformed Hylleraas potential.

2. Dirac equation and tensor interaction The Dirac equation with an attractive scalar potential SðrÞ, a repulsive vector potential VðrÞ and a tensor potential UðrÞ in the relativistic unit ðh ¼ c ¼ 1Þ is [19, 20] ~  r^UðrÞwðrÞ ¼ ½E  VðrÞwðrÞ; ½~ a ~ p þ bðM þ SðrÞÞ  iba ð2Þ ~ where E is the relativistic energy of the system, ~ p ¼ ir denotes the three dimensional momentum operator and M represents the mass of the fermionic particle. ~; a b are the 4  4 Dirac matrices given as     0 ~ ri I 0 ~ a¼ ;b ¼ ; ð3Þ ~ ri 0 0 I where I is the 2  2 unitary matrix and ~ ri are the Pauli three-vector matrices:       0 1 0 i 1 0 r1 ¼ ; r2 ¼ ; r3 ¼ : 1 0 i 0 0 1 ð4Þ of the spin-orbit coupling operator are j ¼ The eigenvalues   j þ 12  0; j ¼  j þ 12  0 for unaligned j ¼ l  12 and

the aligned spin j ¼ l þ 12, respectively. The set ðH; K; J 2 ; Jz Þ forms a complete set of conserved quantities. Thus, we can write the spinors as, ! l Fnj ðrÞYjm ðh; uÞ 1 wnj ðrÞ ¼ ð5Þ r iGnj ðrÞY l~ ðh; uÞ jm

where Fnj ðrÞ; Gnj ðrÞ represent the upper and lower ~l l components of the Dirac spinors. Yjm ðh; uÞ; Yjm ðh; uÞ are the spin and pseudospin spherical harmonics and m is the projection on the z-axis. Using the well-known identities,      ~ ; ~ ~ ~ r~ A ~ r ~ B ¼~ A~ B þ ir AxB ! ð6Þ ~ r ~ L ~ r ~ p ¼~ r  r^ r^  ~ pþi r as well as   ~l l~ ~ ðh; uÞ ¼ ðj  1ÞYjm ðh; uÞ r ~ L Yjm   l l ~ r ~ L Yjm ðh; uÞ ¼ ðj  1ÞYjm ðh; uÞ ~

l l ðh; uÞ ¼ Yjm ðh; uÞ ð~ r  r^ÞYjm

ð7Þ

~

l l ð~ r  r^ÞYjm ðh; uÞ ¼ Yjm ðh; uÞ

we find the following two coupled equations [19, 20]   d j þ  UðrÞ Fnj ðrÞ ¼ ðM þ Enj  DðrÞÞGnj ðrÞ; ð8Þ dr r   d j  þ UðrÞ Gnj ðrÞ ¼ ðM  Enj þ RðrÞÞFnj ðrÞ ð9Þ dr r where, DðrÞ ¼ VðrÞ  SðrÞ

ð10Þ

RðrÞ ¼ VðrÞ þ SðrÞ

ð11Þ

Eliminating Gnj ðrÞ in favor of Fnj ðrÞ, we obtain the second-order Schro¨dinger-like equation

9 8 2 d jðj þ 1Þ 2jUðrÞ dUðrÞ > > 2 > > >   U ðrÞ  ðM þ Enj  DðrÞÞðM  Enj þ RðrÞÞ > þ = < dr 2  2 r r dr   Fnj ðrÞ ¼ 0; dDðrÞ d j > > > > dr dr þ r  UðrÞ > > ; :þ ðM þ Enj  DðrÞÞ 9 8 2 d jðj  1Þ 2jUðrÞ dUðrÞ > > 2 > > > þ  U ðrÞ  ðM þ Enj  DðrÞÞðM  Enj þ RðrÞÞ > þ = < dr 2  r2 r dr   Gnj ðrÞ ¼ 0; dRðrÞ d > >  j þ UðrÞ > > > > þ dr dr r ; : ðM  Enj þ RðrÞÞ

ð12Þ

ð13Þ

Approximate bound and scattering solutions of Dirac equation

where jðj  1Þ ¼ ~lð~l þ 1Þ; jðj þ 1Þ ¼ lðl þ 1Þ [19, 20].

W ps ¼

2.1. Pseudospin symmetry limit The Pseudospin symmetry limit occurs in Dirac equation when dRðrÞ dr ¼ 0 or RðrÞ ¼ Cps ¼ const. In this limit, we take DðrÞ as modified deformed Hylleraas potential and the Yukawa potential for the tensor interaction    2ar  V0 a  e2ar e DðrÞ ¼  V1 b 1  e2ar 1  e2ar e2ar þ V2 ð14Þ ð1  e2ar Þ2 ear UðrÞ ¼ V00 r

(

  eps V0 a ps  M  E nj 4a2 b

ð19  cÞ

Eq. (19) appears as

) d2 1  x d Lps x2 þ Pps x þ W ps þ þ Gps nj ðxÞ ¼ 0 dx2 xð1  xÞ dx x2 ð1  xÞ2 ð20Þ Comparing Eq. (20) with Eq. (77) gives

a1 ¼ a2 ¼ a3 ¼ 1; ps nps 3 ¼ W

ps nps 1 ¼ L ;

ps nps 2 ¼P ;

ð21Þ

From Eq. (80) one can determine the rest of coefficients as

ð15Þ

where V0 ; V1 ; V2 ; a; b and V00 are constant coefficients. Substitution of Eqs. (14) and (15) in Eq. (13) gives   9 8 2   d e2ar V0 eps e2ar > > 2 2 0 2 0 ps 2 0 ps > >  V þ 4a kðk  1Þ  8a kV þ 4a V þ V e þ 2a V  e > > 2 1 0 0 0 < dr 2 b ð1  e2ar Þ = ps ð1  e2ar Þ2 Gnj ðrÞ ¼ 0; > > e 4ar V0 aeps 1 > > 2 02 ps ps > >  e ðM þ Enj Þ þ ; :  4a V0 b ð1  e2ar Þ ð1  e2ar Þ2 ð16Þ

where   ps eps ¼ M  Enj þ Cps

ð17Þ

and we have used the following approximation 1 4a2 e2ar  ; r 2 ð1  e2ar Þ2

ð18Þ

where j ¼ ‘~ and j ¼ ‘~þ 1 for j\0 and j [ 0, respectively. Using a new variable of the form z ¼ e2ar and introducing   V0 eps V0 ps þ V1  M  Enj Lps ¼ V002  0 þ 2 ð19  aÞ 2 4a b 3V 0 Pps ¼ kðk  1Þ  2kV00 þ 0 2   eps V0 V0 a ps þ 2M þ 2Enj þ 2 V 2  V1   b 4a b ð19  bÞ

1 1 a5 ¼  ; a6 ¼  Lps ; a7 ¼ Pps ; 2 4 1 a8 ¼ W ps ; a9 ¼  Lps  Pps  W ps ; pffiffiffiffiffiffiffiffiffiffiffiffi 4 a10 ¼ 1 þ 2 W ps ; ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 1 ps ps ps ps  L  P  W þ W ; a11 ¼ 2 þ 2 4 pffiffiffiffiffiffiffiffiffiffiffiffi a12 ¼ W ps ; ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 1 1  Lps  Pps  W ps þ W ps a13 ¼   2 4 a4 ¼ 0;

ð22Þ Substituting the values of the parameters of Eq. (22) in Eqs. (78) and (79), the energy equation and the lower component are respectively found as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2a W ps r 2ar 12þ 14Lps Pps W ps Gps ðrÞ ¼ e ð1  e Þ nk  pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð23Þ 2 W ps ;2 14Lps Pps W ps ð1  2e2ar Þ Pn

H Hassanabadi et al.

1 nðn þ 1Þ þ þ ð2n þ 1Þ 2 ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 1 ps ps ps ps  L  P  W þ W 4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1 ps ps ps ps ps ps L P W ¼ 0;  P  2W þ 2 W 4 ð24Þ On the other hand, the upper component can be simply calculated using   1 d j ps Fn;j  þ UðrÞ Gps ðrÞ ¼ ð25Þ ps nj ðrÞ: M  Enj þ Cps dr r 2.2. Spin symmetry limit

3V 0 Ps ¼ kðk þ 1Þ  2kV00  0 2   es V 0 V0 a s þ 2M  2Enj þ 2 V2 þ V1 þ þ b 4a b ð30  bÞ Ws ¼



es V0 a s  M þ Enj  b 4a2

 ð30  cÞ

The energy equation in spin symmetry limit is ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 1 1 nðn þ 1Þ þ þ ð2n þ 1Þ  Ls  Ps  W s þ W s 2 4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi 1 s s  L s  Ps  W s  P  2W þ 2 W s 4 ¼ 0;

In this section, we consider the spin symmetry limit corresponding to dDðrÞ dr ¼ 0 or DðrÞ ¼ Cps ¼ const. As in the previous section, we consider    2ar  V0 a  e2ar e RðrÞ ¼  V1 b 1  e2ar 1  e2ar e2ar þ V2 ð26Þ ð1  e2ar Þ2 UðrÞ ¼ V00

ear r

ð27Þ

ð31Þ and the corresponding upper and lower radial components are pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffi 1 1 s s s s s Fn;k ðrÞ ¼ e2a W r ð1  e2ar Þ2þ 4L P W  pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð32Þ 2 W s ;2 14Ls Ps W s 2ar Pn ð1  2e Þ;   1 d j s þ þ UðrÞ Fnj ðrÞ; ð33Þ Gsn;j ðrÞ ¼ s C M þ Enj r s dr

Substitution of Eqs. (18), (26) and (27) into Eq. (12) gives   9 8 2   d e2ar V0 e s e2ar > > 2 2 0 2 0 s 2 0 s > > > = < dr 2 þ 4a kðk þ 1Þ  8a kV0  4a V0  V2 e ð1  e2ar Þ2 þ 2a V0 þ b þ V1 e ð1  e2ar Þ > s Fnj ðrÞ ¼ 0;  4ar s > > e V ae 1 0 > > 2 02 s s > >  e  ðM  E Þ ; :  4a V0 nj b ð1  e2ar Þ ð1  e2ar Þ2 ð28Þ

 s  where es ¼ Enj þ M  Cs , and k ¼ ‘ and k ¼ ‘  1 for k\0 and k [ 0, respectively. Introducing a new variable of the form y ¼ e2ar , gives us the Schrodinger-like equation: ( ) d2 1  y d Ls y2 þ Ps y þ W s s þ þ ðyÞ ¼ 0 Fnj dy2 yð1  yÞ dy y2 ð1  yÞ2 ð29Þ where   V00 es V0 02 s  V0 þ 2   V1  M þ Enj L ¼ 2 4a b s

ð30  aÞ

In Tables 1 and 2 we have calculated the energy of the Dirac equation in pseudospin and spin symmetry limits, respectively. It is clear that in the absence of tensor interaction we have degenerate states and these degeneracies are removed in the presence of tensor interaction. In Figs. 1 and 2, we have presented the effect of tensor interaction on the upper and lower component of Dirac wave functions for the pseudospin and spin symmetries, respectively. The sensitivity of the pseudospin doublets ð1s1=2 ; 0d3=2 Þ and ð1p3=2 ; 0f5=2 Þ and spin doublets ð0p1=2 ; 0p3=2 Þ and ð0d3=2 ; 0d5=2 Þ to the parameters V00 are given in Figs. 3 and 4.

Approximate bound and scattering solutions of Dirac equation Table 1 The energy of the Pseudo spin symmetry limit for different states and M ¼ 5ðfm1 Þ; a ¼ 0:01ðfm1 Þ; V0 ¼ 0:5ðfm1 Þ; V1 ¼ 0:3ðfm1 Þ; V2 ¼ 0:4ðfm1 Þ; Cps ¼ 5ðfm1 Þ; a ¼ b ¼ 12 ~l

n; k

state

En;kðV0 ¼0Þ

En;kðV0 ¼0:6Þ

n  1; k

state

En1;kðV0 ¼0Þ

En1;kðV0 ¼0:6Þ

1

1, -1

1s12

-5.411820852

-5.411725522

0, 2

0d32

-5.411820852

-5.411907696

2

1, -2

1p32

-5.412023060

-5.411867242

0, 3

0f52

-5.412023060

-5.412170238

3

1, -3

1d52

-5.412325591

-5.412109714

0, 4

0g72

-5.412325591

-5.412532597

4

1, -4

1f72

-5.412727511

-5.412452189

0, 5

0h92

-5.412727511

-5.412993658

1

2, -1

2s12

-5.419044543

-5.418954852

1, 2

1d32

-5.419044543

-5.419126084

2

2, -2

2p32

-5.419234604

-5.419088060

1, 3

1f52

-5.419234604

-5.419372854

3

2, -3

2d52

-5.419518958

-5.419315966

1, 4

1g72

-5.419518958

-5.419713440

4

2, -4

2f72

-5.419896724

-5.419637864

1, 5

1h92

-5.419896724

-5.420146787

3. Scattering state solutions We now investigate the scattering state solutions of Dirac equation under the modified deformed Hylleraas potential. First, we obtain the phase shift and the normalization constant for pseudospin symmetry limit, and then discuss the scattering properties for the spin symmetry. 3.1. Pseudospin symmetry limit By using a change of variable of the form x ¼ 1  e2ar , Eq. (16) appears as

d2 d v v xð1  xÞ 2  x þ 1 þ 2 þ v3 Gps n;j ðxÞ ¼ 0; dx x 1  x dx ð34Þ with V2 eps  V002 ; v1 ¼ jðj  1Þ  2jV00 þ V00 þ 4a2   eps V0 a ps v2 ¼ 2 M  En;j þ ; b 4a  V 0 V0 eps V1 eps eps  02 ps þ V þ M þ E v3 ¼ 0  2  0 n;j ; 2 4a b 4a2 4a2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ 1  4v1 ; 2 pffiffiffiffiffiffiffiffiffiffiffiffi ik b ¼  ; k ¼ 4a2 v2 : 2a c¼

ð38Þ

with pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ik pffiffiffiffiffi 1 1 þ 1  4v1  þ v3 ; 2 2a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ik pffiffiffiffiffi 1 1 þ 1  4v   v3 ; g2 ¼ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2a g3 ¼ 1 þ 1  4v1 ;

g1 ¼

ð39Þ

The solution of Eq. (37) is gps n;j ðxÞ ¼ 2 F1 ðg1 ; g2 ; g3 ; xÞ;

ð40Þ

we can easily find the total radial wave function as   ps ikr 2ar c Gps Þ 2 F1 g1 ; g2 ; g3 ; 1  e2ar ; n;j ðrÞ ¼ Nn;j e ð1  e ð41Þ from Eq. (39) one can obtain

ð35Þ

g3  g1  g2 ¼

ik ¼ ðg1 þ g2  g3 Þ ; a

g3  g1 ¼ g 2 ;

ð42Þ

g3  g2 ¼ g 1 ;

In order to obtain the scattering properties, we need to transform Eq. (34) into the form of hypergeometric equation. Therefore, we use the following transformation

By using the properties of hypergeometric function [21] and Eq. (42), at the infinity limit we obtain

b ps c Gps n;j ðxÞ ¼ x ð1  xÞ gn;j ðxÞ;

2 F1

ð36Þ

which gives

d2 d xð1  xÞ 2 þ ðg3  ð1 þ g1 þ g2 ÞÞ  g1 g2 gps n;j ðxÞ dx dx ¼ 0; ð37Þ where



 g1 ; g2 ; g3 ; 1  e2ar ¼ Cðg3 Þ  

Cðg3  g1  g2 Þ Cðg3  g1  g2 Þ þ e2ikr ; Cðg3  g1 ÞCðg3  g2 Þ Cðg3  g1 ÞCðg3  g2 Þ

ð43Þ with the help of Eq. (43) and    Cðg3  g1  g2 Þ  id Cðg3  g1  g2 Þ  e ps ; ¼ Cðg3  g1 ÞCðg3  g2 Þ Cðg3  g1 ÞCðg3  g2 Þ ð44Þ

H Hassanabadi et al. Table 2 The energy of the spin symmetry limit for different states and M ¼ 5ðfm1 Þ; a ¼ 0:01ðfm1 Þ; V0 ¼ 0:5ðfm1 Þ; V1 ¼ 0:3ðfm1 Þ; V2 ¼ 0:4ðfm1 Þ; Cs ¼ 5ðfm1 Þ; a ¼ b ¼  12 l

n; k

state

s En;kðV 0 ¼0Þ

s En;kðV 0 ¼0:6Þ

n  1; k

state

s En;kðV 0 ¼0Þ

s En;kðV 0 ¼0:6Þ

1 2 3 4 1 2 3 4

0, -2 0, -3 0, -4 0, -5 1, -2 1, -3 1, -4 1, -5

0p32 0d52 0f72 0g92 1p32 1d52 1f72 1g92

3.938108097 3.938886382 3.940049930 3.941594132 4.009133309 4.009840775 4.010898581 4.012302691

3.937839297 3.938384829 3.939317660 3.940634083 4.008889321 4.009385188 4.010233176 4.011430041

0, 0, 0, 0, 1, 1, 1, 1,

0p12 0d32 0f52 0g72 1p12 1d32 1f52 1g72

3.938108097 3.938886382 3.940049930 3.941594132 4.009133309 4.009840775 4.010898581 4.012302691

3.938540508 3.939550403 3.940942967 3.942712707 4.009526701 4.010444766 4.011710901 4.013320294

1 2 3 4 1 2 3 4

Fig. 1 Wave functions of 1s1=2 in the pseudospin symmetry for M ¼ 5ðfm1 Þ; a ¼ 0:1ðfm1 Þ; V0 ¼ 0:5ðfm1 Þ; V1 ¼ 0:3ðfm1 Þ; V2 ¼ 0:4 ðfm1 Þ; Cps ¼ 5ðfm1 Þ; a ¼ b ¼ 12

Equation (41) reduces to     Cðg3  g1  g2 Þ   ps   sin kr þ p þ dps : Gps ðrÞ ¼ 2N Cðg Þ 3  nj nj  Cðg3  g1 ÞCðg3  g2 Þ 2 ð45Þ By comparing Eq. (45) with the boundary condition   ps jp [22, 23], we deduce that Gps nj ð1Þ ! 2 sin kr  2 þ dj . Therefore, the phase shift and the normalization constant are determined as p dps j ¼ ðj þ 1Þ þ dps 2   p Cðg3  g1  g2 Þ ¼ ðj þ 1Þ þ arg ; ð46Þ 2 Cðg3  g1 ÞCðg3  g2 Þ   1 Cðg3  g1 ÞCðg3  g2 Þ ps : ð47Þ Nnj ¼ Cðg3 Þ  Cðg3  g1  g2 Þ 

3.2. Spin symmetry limit Here, we want to study the scattering properties of spin s symmetry 0 case. Using y ¼ 1  e2ar and Fn;j ðyÞ ¼ b s c0 y ð1  yÞ fn;j ðyÞ, Eq. (28) becomes

d d2  0 0 0 0 0 s yð1  yÞ 2 þ g3  ð1 þ g1 þ g2 Þy ðyÞ ¼ 0;  g1 g2 fn;j dy dy ð48Þ where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ 1  4v01 ; 2 qffiffiffiffiffiffiffiffiffiffiffiffi ik0 0 b ¼  ; k0 ¼ 4a2 v02 : 2a c01 ¼

with

ð49Þ

Approximate bound and scattering solutions of Dirac equation

Fig. 2 Wave functions of 0p32 in the spin symmetry for M ¼ 5ðfm1 Þ; a ¼ 0:1ðfm1 Þ; V0 ¼ 0:5ðfm1 Þ; V1 ¼ 0:3ðfm1 Þ; V2 ¼ 0:4 ðfm1 Þ; Cs ¼ 5ðfm1 Þ; a ¼ b ¼  12

Fig. 3 Energy spectra in the pseudospin symmetry versus V00 for M ¼ 5ðfm1 Þ; a ¼ 0:01ðfm1 Þ; V0 ¼ 0:5ðfm1 Þ; V1 ¼ 0:3ðfm1 Þ; V2 ¼ 0:4 ðfm1 Þ; Cps ¼ 5ðfm1 Þ; a ¼ b ¼ 12

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ik0 qffiffiffiffiffi 1 1 þ 1  4v01  þ v03 ; 2 2a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ik0 qffiffiffiffiffi  1 1 þ 1  4v0   v03 ; g02 ¼ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2a

g01 ¼

g03 ¼ 1 þ and

1  4v01 ;

ð50Þ

V2 e s v01 ¼ jðj þ 1Þ  2jV00  V00  2  V002 ;   4a s e V a 0 s v02 ¼ 2 M þ En;j  ; b 4a  V 0 V 0 e s V1 e s es  s v03 ¼ 0 þ 2 þ 2 þ V002 þ 2 M  En;j ; 2 4a b 4a 4a

ð51Þ

H Hassanabadi et al.

Fig. 4 Energy spectra in the spin symmetry versus V00 for M ¼ 5ðfm1 Þ; a ¼ 0:01ðfm1 Þ; V0 ¼ 0:5ðfm1 Þ; V1 ¼ 0:3ðfm1 Þ; V2 ¼ 0:4ðfm1 Þ; Cs ¼ 5ðfm1 Þ; a ¼ b ¼  12

Therefore,  0 0 0  0 0 s s Fn;j ðrÞ ¼ Nn;j eik r ð1  e2ar Þc2 F1 g1 ; g2 ; g3 ; 1  e2ar ; ð52Þ by repeating the procedure of Sect. 3.1, we can obtain p dsj ¼ ðj þ 1Þ þ ds 2   p Cðg03  g01  g02 Þ ¼ ðj þ 1Þ þ arg ; ð53Þ 2 Cðg03  g01 ÞCðg03  g02 Þ   1 Cðg03  g01 ÞCðg03  g02 Þ s : ð54Þ Nnj ¼ Cðg03 Þ  Cðg03  g01  g02 Þ  4. Some special cases

VðrÞ ¼

V20 sinh2 ðarÞ

where V20 ¼ V42 . Therefore, for the pseudospin symmetry limit we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffi 1 1 ps ps ps 2a W ps r Gps ð1  e2ar Þ2þ 4L P W nk ðrÞ ¼ e  pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð56Þ 2 W ps ;2 14Lps Pps W ps ð1  2e2ar Þ Pn 1 nðn þ 1Þ þ þ ð2n þ 1Þ 2 ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 1  Lps  Pps  W ps þ W ps 4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1 ps ps  Lps  Pps  W ps ¼ 0;  P  2W þ 2 W ps 4

As already mentioned, the Po¨schl–Teller, Hulthe´n and deformed Hylleraas Potentials are special cases of modified deformed Hylleraas potential. Here, we are going to study the energy eigenvalues and scattering properties for these potentials under the Yukawa-type tensor interaction.

where

4.1. Po¨schl–Teller potential

Pps ¼ kðk  1Þ  2kV00 þ

If we choose V0 ¼ V1 ¼ 0 the modified deformed Hylleraas reduces to the Po¨schl–Teller potential [24]

ð55Þ

ð57Þ

Lps ¼ V002 

 V00 eps  ps  2 M þ Enj 2 4a

ð58  aÞ

 3V00 eps  ps þ 2 V20 þ 2M þ 2Enj 2 4a ð58  bÞ

Approximate bound and scattering solutions of Dirac equation

W ps ¼ 

 eps  ps M þ Enj 2 4a

ð58  cÞ

and energy eigenvalues and eigenfunctions of the spin symmetry have the following form. ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 1 1 s s s s  L  P  W þ W nðn þ 1Þ þ þ ð2n þ 1Þ 2 4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi 1  Ls  Ps  W s ¼ 0;  Ps  2W s þ 2 W s 4 ð59Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffi 1 1 s s s s s Fn;k ðrÞ ¼ e2a W r ð1  e2ar Þ2þ 4L P W  pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 W s ;2 14Ls Ps W s ð1  2e2ar Þ; Pn

ð60Þ

with Ls ¼

 V00 es  s  V002  2 M  Enj 2 4a

Ps ¼ kðk þ 1Þ  2kV00 

Ws ¼

 es  s Enj  M 2 4a

ð61  aÞ

 3V00 es  s þ 2 V20 þ 2M  2Enj 2 4a ð61  bÞ ð61  cÞ

4.2. Hulthe´n potential

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffi 1 1 ps ps ps 2a W ps r ð1  e2ar Þ2þ 4L P W Gps nk ðrÞ ¼ e  pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 W ps ;2 14Lps Pps W ps Pn ð1  2e2ar Þ

ð65Þ

where Lps ¼ V002 

 V00 eps  ps þ 2 V10  M  Enj 2 4a

3V 0 Pps ¼ kðk  1Þ  2kV00 þ 0 2  eps  0 ps þ 2 V1 þ 2M þ 2Enj 4a  eps  ps W ps ¼  2 M þ Enj 4a

ð66  aÞ

ð66  bÞ ð66  cÞ

For the spin symmetry limit we have ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 1 1  Ls  Ps  W s þ W s nðn þ 1Þ þ þ ð2n þ 1Þ 2 4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi 1  Ls  Ps  W s ¼ 0;  Ps  2W s þ 2 W s 4 ð67Þ pffiffiffiffiffiffiffi 2a W s r

1 2þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 s s s

s 4L P W Fn;k ðrÞ ¼ e ð1  e2ar Þ  pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 W s ;2 14Ls Ps W s ð1  2e2ar Þ; Pn

ð68Þ

where For a ¼ V2 ¼ 0, the modified deformed Hylleraas potential turns into the Hulthe´n potential [25] VðrÞ ¼ 

V10 1

edr

where   V0 þ V1 V10 ¼ b d ¼ 2a

ð62Þ

 V00 es  s  V002 þ 2 V10  M þ Enj ð69  aÞ 2 4a  3V 0 es  s Ps ¼ kðk þ 1Þ  2kV00  0 þ 2 V10 þ 2M  2Enj 2 4a ð69  bÞ

Ls ¼

 es  s Enj  M 2 4a

ð63  aÞ

Ws ¼

ð63  bÞ

which is the result of Ref. [26].

According to Eqs. (23) and (24), the energy eigenvalues and the corresponding eigenfunctions of the pseudospin symmetry can be obtained as 1 nðn þ 1Þ þ þ ð2n þ 1Þ 2 ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 1  Lps  Pps  W ps þ W ps 4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1 ps ps  Lps  Pps  W ps ¼ 0;  P  2W þ 2 W ps 4 ð64Þ

ð69  cÞ

4.3. Deformed Hylleraas potential By considering V1 ¼ V2 ¼ 0, the modified deformed Hylleraas potential changes into the deformed Hylleraas potential   V0 a  e2ar VðrÞ ¼ ð70Þ b 1  e2ar which possesses the pseudospin energy equation and lower component

H Hassanabadi et al.

1 nðn þ 1Þ þ þ ð2n þ 1Þ 2 ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 1 ps ps ps ps  L  P  W þ W  Pps  2W ps 4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1 ps ps ps ps L P W ¼ 0; þ 2 W 4 ð71Þ pffiffiffiffiffiffiffiffiffi 2a W ps r

1 2þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 ps ps ps

4L P W Gps ð1  e2ar Þ nk ðrÞ ¼ e  pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 W ps ;2 14Lps Pps W ps Pn ð1  2e2ar Þ

ð72Þ

with

Appendix

  e V0 ps  M  Enj Lps ¼ V002  þ 2 2 4a b

ð73  aÞ

3V 0 Pps ¼ kðk  1Þ  2kV00 þ 0 2   eps V 0 V0 a ps þ 2M þ 2Enj þ 2   b 4a b   eps V0 a ps  M  Enj W ps ¼ 2 b 4a

ð73  bÞ

V00

bound state energy eigenvalues and scattering state solutions for the modified deformed Hylleraas potential with Yukawa-like tensor interaction. By choosing appropriate values for the potential constants, we obtained the energy eigenvalues and the eigenfunctions of the Dirac equation for the Po¨schl–Teller, Hulthe´n and deformed Hylleraas potentials which are the special cases of the modified deformed Hylleraas potential. We also discussed the effect of tensor interaction on the wave functions and energy of the system via various figures and tables.

ps

ð73  cÞ

and for the spin symmetry we have ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 1 1  Ls  Ps  W s þ W s nðn þ 1Þ þ þ ð2n þ 1Þ 2 4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi 1 s s s s s s  L  P  W ¼ 0;  P  2W þ 2 W 4 pffiffiffiffiffiffiffi 2a W s r

1 2þ

(

) d2 a1  a2 s d 1 2 þ þ ½n1 s þ n2 s  n3  w ds2 sð1  a3 sÞ ds ½sð1  a3 sÞ2 ¼0 ð77Þ

ð74Þ

According to the NU method, the eigenfunctions is 

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 s s s

s 4L P W ðrÞ ¼ e ð1  e2ar Þ Fn;k  pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 W s ;2 14Ls Ps W s Pn ð1  2e2ar Þ;

The NU method solves many linear second order differential equations by reducing them to a generalized equation of hypergeometric type. Here, instead of the original formulation, we use the parametric version which enables us to solve a second-order differential equation of the form [27, 28, 29]

wðsÞ ¼ sa12 ð1  a3 sÞ

a a12  a13 3

Pn



a

a10 1; a11 a10 1 3

ð1  2a3 sÞ

ð75Þ

ð78Þ and the energy of the system satisfies

where   V0 es V0 s L ¼ 0  V002 þ 2   M þ Enj 2 4a b s

Ps ¼ kðk þ 1Þ  2kV00 

ð76  aÞ

  3V00 es V0 V0 a s þ 2M  2Enj þ 2 þ b 2 4a b

ð76  bÞ s

Ws ¼



e V0 a s  M þ Enj  b 4a2

 ð76  cÞ

5. Conclusions In this paper, the Dirac equation was studied for the modified deformed Hylleraas potential with a Yukawatype potential as tensor interaction. We reported the

pffiffiffiffiffi pffiffiffiffiffi a2 n  ð2n þ 1Þa5 þ ð2n þ 1Þð a9 þ a3 a8 Þ pffiffiffiffiffiffiffiffiffi þ nðn  1Þa3 þ a7 þ 2a3 a8 þ 2 a8 a9 ¼ 0;

ð79Þ

where 1 1 a4 ¼ ð1  a1 Þ; a5 ¼ ða2  2a3 Þ; a6 ¼ a25 þ n1 ; 2 2 a7 ¼ 2a4 a5  n2 ; a8 ¼ a24 þ n3 pffiffiffiffiffi a9 ¼ a3 a7 þ a23 a8 þ a6 ; a10 ¼ a1 þ 2a4 þ 2 a8 pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi a11 ¼ a2  2a5 þ 2ð a9 þ a3 a8 Þ; a12 ¼ a4 þ a8 pffiffiffiffiffi pffiffiffiffiffi a13 ¼ a5  ð a9 þ a3 a8 Þ !   n n Cða þ b þ n þ m þ 1Þ x  1 m Cða þ n þ 1Þ X Pða;bÞ ðxÞ ¼ n n!Cða þ b þ n þ 1Þ m¼0 m Cða þ m þ 1Þ 2

ð80Þ And

ða;bÞ Pn

is Jacobi polynomial.

Approximate bound and scattering solutions of Dirac equation

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