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
References [1] Y Xu, S He and C S Jia, Phys. Scr. 81 0453001 (2010) [2] X Y Gu and S H Dong, J. Math. Chem. 49 2053 (2011) [3] M R Pahlavani and S A Alavi, Commun. Theor. Phys. 58 739 (2012) [4] G F Wei and S H Dong Eur. Phys. J. A 43 185 (2010) [5] C Y Chen, S D Sheng and L F Lin, Phys. Scr. 74 405 (2006) [6] C S Jia, X P Li and L H Zhang Few-Body Syst. 52 11 (2012) [7] S H Dong and G H Sun Phys. Lett. A 314 261 (2003) [8] G F Wei and S H Dong Eur. Phys. J. A 46 207 (2010) [9] H Hassanabadi, B H Yazarloo, S Zarrinkamar and A A Rajabi, Phys. Rev. C 84 064003 (2011) [10] Y P Varshni, Rev. Mod. Phys. 29 664 (1957) [11] C S Jia, L H Zhang and C W Wang Chem. Phys. Lett. 667 211 (2017) [12] C S Jia, J W Dai, L H Zhang, J Y Liu and X L Peng Phys. Lett. A 379 137 (2015) [13] C S Jia and Z W Shui Eur. Phys. J. A 51 144 (2015) [14] C S Jia, T Chen and S He Phys. Lett. A 377 682 (2013) [15] C S Jia, J W Dai, L H Zhang, J Y Liu and G D Zhang Chem. Phys. Lett. 619 54 (2015)
[16] H Hassanabadi, E Maghsoodi, S Zarrinkamar and H Rahimov Chin. Phys. B 21 120302 (2012) [17] R Barnan and R Rajkumar J. Phys. A: Math. Gen. 20 3051 (1987) [18] M Jameelt J. Phys. A: Math. Gen. 19 1967 (1986) [19] J N Ginocchio Phys. Rev. Lett.78 (1997) 436 [20] J N Ginocchio Phys. Rep. 414 (2005) 165 [21] B H Yazarloo, L L Lu, G. Liu, S. Zarrinkamar and H. Hassanabadi, Adv. High Energy Phys. 2013 317605 (2013) [22] L D Landau and E M Lifshitz, Quantum Mechanics (Non-Relativistic Theory) 3rd ed. (New York: Pergamon) (1979) [23] G F Wei, C Y Long and S H Dong Phys. Lett. A 372 2592 (2008) [24] H Hassanabadi, B H Yazarloo and S Zarrinkamar Few-Body Syst. 54 2017 (2013) [25] S Zarrinkamar, A A Rajabi, B H Yazarloo and H Hassanabadi Chin. Phys. C 37 023101 (2013) [26] O Aydogdu, E Maghsoodi and H Hassanabadi Chin. Phys. B 22 010302 (2013) [27] A F Nikiforov, V B Uvarov, Special Functions of Mathematical Physics (Springer Basel) (1988) [28] B H Yazarloo, H Hassanabadi and S Zarrinkamar Eur. Phys. J. Plus 127 51 (2012) [29] C Tezcan and R Sever Int. J. Theor. Phys. 48 337 (2009)