Exact pseudospin symmetry solution of the Dirac equation for spatially-dependent mass Coulomb potential including a Coulomb-like tensor interaction via asymptotic iteration method

Physics Letters A 374 (2010) 4303–4307

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Physics Letters A www.elsevier.com/locate/pla

Exact pseudospin symmetry solution of the Dirac equation for spatially-dependent mass Coulomb potential including a Coulomb-like tensor interaction via asymptotic iteration method M. Hamzavi ∗ , A.A. Rajabi, H. Hassanabadi Physics Department, Shahrood University of Technology, Shahrood, Iran

a r t i c l e

i n f o

Article history: Received 9 June 2010 Received in revised form 8 August 2010 Accepted 26 August 2010 Available online 27 August 2010 Communicated by R. Wu

a b s t r a c t In this Letter, the Dirac equation is exactly solved for spatially-dependent mass Coulomb potential including a Coulomb-like tensor potential under pseudospin symmetry limit by using asymptotic iteration method with arbitrary spin–orbit coupling number κ . The energy eigenvalues and corresponding eigenfunctions are obtained and some numerical results are given. © 2010 Elsevier B.V. All rights reserved.

Keywords: Dirac equation Spatially-dependent mass Coulomb potential Tensor potential Pseudospin symmetry Asymptotic iteration method

1. Introduction In the framework of the Dirac equation, the pseudospin symmetry occurs when the magnitude of the attractive Lorentz scalar potential S (r ) and the time-component repulsive vector potential V (r ), are nearly equal but in opposite sign, i.e. S (r )  − V (r ), where the sum of the potential is Σ(r ) = S (r ) + V (r ) = C ps = constant [1–19]. Tensor potentials were introduced into the Dirac  → p − imωβ . rˆ U (r ) and a spin– equation with the substituting p orbit coupling is added to the Dirac Hamiltonian [20–30]. In the relativistic and non-relativistic cases, the solution of Dirac, Klein– Gordon and Schrödinger equations with effective mass is useful for the investigation of some physical systems [31–50]. In this Letter, we consider both spatially-dependent mass and tensor potential for attractive scalar and repulsive vector Coulomb potential under pseudospin symmetry limit. In Section 2, the Dirac equation with tensor potential and spatially-dependent mass is briefly introduced. In Section 3, a brief introduction to asymptotic iteration method (AIM) [51–55] is given. We solve the Dirac equation and give some numerical results in Section 4. Finally, conclusion is given in Section 5.

*

Corresponding author. Tel.: +98 273 3395270; fax: +98 273 3395270. E-mail address: [email protected] (M. Hamzavi).

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2. Dirac equation with spatially-dependent mass and tensor coupling The spatially-dependent mass Dirac equation including tensor interaction for spin- 12 particles with scalar potential S (r ) and vector potential V (r ), in units where h¯ = c = 1, is













α . p + β m(r ) + S (r ) − i β α . rˆ U (r ) ψ(r ) = E − V (r ) ψ(r ) (1)

 is the  = −i ∇ where E is the relativistic energy of the system, p three-dimensional momentum operator and m(r ) is the effective  and β are the 4 × 4 usual Dirac mass of the fermionic particle. α matrices give as



α =

0

σ

σ





,

0

β=

I 0 0 −I



(2)

 are three-vector spin matriwhere I is 2 × 2 unitary matrix and σ ces



σ1 =

0 1 1 0





,

σ2 =

0 −i i 0





,

σ3 =

1 0 0 −1



(3)

The total angular momentum operator J and spin–orbit K = (σ . L + 1) commute with Dirac Hamiltonian, where L is the orbital

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M. Hamzavi et al. / Physics Letters A 374 (2010) 4303–4307

angular momentum. The eigenvalues of spin–orbit coupling operator are κ = ( j + 12 )  0 and κ = −( j + 12 ) ≺ 0 for unaligned spin

j = l − 12 and the aligned spin j = l + 12 , respectively. ( H 2 , K , J 2 , J z ) can be taken as the complete set of the conservative quantities. Thus, the Dirac spinors can be written according to radial quantum number n and spin–orbit coupling number κ as follows

 ψnκ (r ) =

f nκ (r )



 =

gnκ (r )

F nκ (r ) l Y jm (θ, r

i

ϕ)

G nκ (r ) ˜l Y jm (θ, r



(4)

ϕ)

where f nκ (r ) is the upper (large) component and gnκ (r ) is the lower (small) component of the Dirac spinors. Y ljm (θ, ϕ ) and ˜

Y ljm (θ, ϕ ) are spin and pseudospin spherical harmonics, respectively, and m is the projection of the angular momentum on the z-axis. Substituting Eq. (4) into Eq. (1) and using the following relations [52]

 )(σ . B ) = A  . B + i σ . ( A  × B ) (σ . A    σ . L (σ . P ) = σ . rˆ rˆ . P + i

(5a) (5b)

r

L )Y l (θ, φ) jm

= −(κ

˜

˜

(σ . rˆ)Y ljm (θ, φ) = −Y ljm (θ, φ)

(6)

one obtains two coupled differential equations for upper and lower radial wave functions F nκ (r ) and G nκ (r ) as



dr d dr

κ

+

r

κ

−

r

   − U (r ) F nκ (r ) = m(r ) + E nκ − (r ) G nκ (r )    + U (r ) G nκ (r ) = m(r ) − E nκ + Σ(r ) F nκ (r )

(7a) (7b)

(r ) = V (r ) − S (r )

(8a)

Σ(r ) = V (r ) + S (r )

(8b)

Eliminating F nκ (r ) and G nκ (r ) from Eqs. (7), we obtain the following two Schrödinger-like differential equations for the upper and lower radial spinor components, respectively

d2 dr 2

−

κ (κ + 1) r2

+

2κ r

U (r ) −

dU (r ) dr

− U 2 (r )

  r) ( dmdr(r ) − d ( ) d κ dr + − U (r ) F nκ (r ) m(r ) + E nκ − (r ) dr r    = m(r ) + E nκ − (r ) m(r ) − E nκ + Σ(r ) F nκ (r )

d

2

dr 2

− −

U (r ) = −

κ (κ − 1) r2

( dmdr(r )

+

+

2κ r

U (r ) +

dΣ(r ) ) dr

m(r ) − E nκ + Σ(r )



d2

r

,

H=

Z a Z b e2 4πε0

, r  Rc

(12)

Λκ (Λκ − 1) + m1 (m1 + C )

−

r2

γ˜ (m1 + C ) − m1 (m0 + E nκ ) r

 − β˜ 2 G nκ (r ) = 0

(13)

Λκ = κ + H

γ˜ = E nκ − m0 − C ps β˜ 2 = ( E nκ + m0 )(m0 − E nκ + C ps )

dU (r ) dr

(9)

AIM is proposed and applied to second-order differential equations in the form

d2 y (x)

dy (x) dx

+ S 0 (x) y (x)

(15)

ykn+1 (x)

 x = N exp

S n (x )

λn (x )



dx

(16)

where N is the integration constant and

dλn−1 (x) dx dS n−1 (x)

dx for some k  0 S k (x) S k−1 (x)

=

λk−1 (x)

+ S n−1 (x) + λ0 (x)λn−1 (x)

(17)

+ S 0 (x)λn−1 (x)

(18)

= α (x)

(19)

energy eigenvalues are found from termination condition of the method as

r

   = m(r ) + E nκ − (r ) m(r ) − E nκ + Σ(r ) G nκ (r )

= λ0 (x)

where λ0 (x) and S 0 (x) are sufficiently many times continuously differentiable and λ0 (x) is different from zero [51–55]. Eq. (15) has given solution as follows

λk (x)

− U 2 (r )

(14)

3. Asymptotic iteration method (AIM)

S n (x) =

 d κ − + U (r ) G nκ (r )

dr

H

where R c = 7.78 fm is the Coulomb radius, Z a and Z b denote the charges of the projectile a and the target nuclei b, respectively [28]. Therefore, from Eq. (10), one obtains

λn (x) =

−



Coulomb potential, i.e. (r ) = − Cr where C = Z α and α = e 2 is the fine structure constant in units where h¯ = c = 1. The Coulomblike tensor potential is added as

dx2

where



The pseudospin symmetry case occurs in the Dirac equation dΣ(r ) when dr = 0 or Σ(r ) = C ps = constant [9]. (r ) is taken as

where

(σ . rˆ)Y ljm (θ, φ) = −Y ljm (θ, φ)

d

(11)

r

2.1. Pseudospin symmetry limit

+

− 1)Y ljm (θ, φ)

m1

where m0 is the rest mass of the fermionic particle and m1 is the perturbated mass [28].

dr 2

˜ ˜ (σ . L )Y ljm (θ, φ) = (κ − 1)Y ljm (θ, φ)



m(r ) = m0 +



and properties

(σ .

By using this relation, we can exactly solve Eq. (10), but Eq. (9) cannot be solved exactly. Also, the mass function should be taken as

(10)

where κ (κ − 1) = ˜l(˜l + 1) and κ (κ + 1) = l(l + 1). It is convenient r) to solve the mathematical relation dmdr(r ) = − dΣ( = − dVdr(r ) [28]. dr



λk (x) λk−1 (x)

=0

k (x) = S (x) S (x) k

k −1

(20)

where k is the iteration number [51–55]. By using Eqs. (20) and (16), one can obtain energy eigenvalues equations and corresponding wave functions, respectively.

M. Hamzavi et al. / Physics Letters A 374 (2010) 4303–4307

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Table 1 The bound state energy eigenvalues in unit of fm−1 of the pseudospin symmetry spatially-dependent mass Coulomb potential including tensor potential with m0 = 5 fm−1 , C = 0.5 and C ps = 0.

˜l

n, κ ≺ 0

(l, j )

E n ,κ ≺0 H =1 m 1 = 0.5

n − 1,

1

1, −1

1s1/2

2

1, −2

1p 3/2

3

1, −3

1d5/2

4

1, −4

1 f 7/2

1

2, −1

2s1/2

2

2, −2

2p 3/2

3

2, −3

2d5/2

4

2, −4

2 f 7/2

−4.553184694 −4.749331473 −4.851130965 −4.902767354 −4.779321871 −4.855413144 −4.903808321 −4.932163183

κ 0

(l + 2, j + 1)

E n−1,κ 0 H =1 m 1 = 0.5

0, 2

0d3/2

0, 3

0 f 5/2

0, 4

0g 7/2

0, 5

0h9/2

1, 2

1d3/2

1, 3

1d3/2

1, 4

1g 7/2

1, 5

1h9/2

−4.851130965 −4.902767354 −4.931815494 −4.949632811 −4.903808321 −4.932163183 −4.949774758 −4.961375704

Table 2 The bound state energy eigenvalues for Coulomb potential in unit of fm−1 under the pseudospin symmetry in the absence of spatially-dependent mass with m0 = 5 fm−1 , C = 0.5 and C ps = 0.

˜l

n, κ ≺ 0

(l, j )

E n ,κ ≺0 H =1 m1 = 0

1

1, −1

1s1/2

2

1, −2

1p 3/2

3

1, −3

1d5/2

4

1, −4

1 f 7/2

1

2, −1

2s1/2

2

2, −2

2p 3/2

3

2, −3

2d5/2

4

2, −4

2 f 7/2

−4.846153846 −4.931034483 −4.961089494 −4.975062344 −4.931034483 −4.961089494 −4.975062344 −4.982668977

n − 1,

k=1

4. Pseudospin symmetry solution of Dirac equation via AIM ˜

1

By using transformation G nκ (r ) = r ε+ 2 e −β r gnk (r ) [56], we convert Eq. (10) into the form of Eq. (15) as follows

d2 gnκ (r ) dr 2

 =

2β˜ − 2ε − 1



dgnκ (r )

r

dr

 +

2ε β˜ + β˜ − Γ 2 r

 gnκ (r ) (21)

where

ε2 = Λκ (Λκ − 1) + m1 (m1 + C ) +

4

Γ = γ˜ (m1 + C ) − m1 (m0 + C )

(22)

Comparing Eqs. (21) and (15), one obtains

S 0 (r ) =

2β˜ − 2ε − 1 2ε β˜ + β˜ − Γ 2

(23b)

r

Substituting Eqs. (23) into Eqs. (17), (18), we obtain

S 1 (r ) =

r

+

−4.961089494 −4.975062344 −4.982668977 −4.987261146 −4.97506234 −4.982668977 −4.987261146 −4.990243902

0, 2

0d3/2

0, 3

0 f 5/2

0, 4

0g 7/2

0, 5

0h9/2

1, 2

1d3/2

1, 3

1d3/2

1, 4

1g 7/2

1, 5

1h9/2

⇒

1 (r ) = 0 Γ2 1 ε0 = − 2 2β˜

⇒

λ1 (r ) S 0 (r ) − λ0 (r ) S 1 (r ) = 0

⇒

⇒

λ2 (r ) S 1 (r ) − λ1 (r ) S 2 (r ) = 0

2 (r ) = 0 Γ2 3 ε1 = − 2 2β˜

.. .

(25)

εn =

  Γ2 1 − n+ 2 2β˜

(26)

 ( E nκ + m0 )(m0 − E nκ + C ps ) 2n + 1  2 + 4(κ + H )(κ + H − 1) + 4m1 (m1 + C ) + 1  2 = ( E nκ − m0 − C ps )(m1 + C ) − m1 (m0 + E nκ )

(27)

when H = m1 = C ps = 0, the problem of this Letter reduces to the Coulomb potential and the energy eigenvalues are obtained as [54]

4ε 2 + 6ε + 2 r2

˜ 2 4ε β˜ 2 + 2β˜ 2 − 2βΓ +

.. .

6β˜ ε + 3β˜ + Γ 2

⇒

E n−1,κ 0 H =1 m1 = 0

By recalling Eqs. (14) and (22) and substituting in above equation, the energy eigenvalues function is obtained as

(23a)

r

λ1 (r ) = 4β˜ 2 −

k=2

(l + 2, j + 1)

and generally for arbitrary k, we have

1

2

λ0 (r ) =

⇒

κ 0

E nκ (Coulomb) = −m0

r 2Γ + 2ε Γ 2 − 4ε β˜ − 2β˜ − 4ε 2 β˜ 2

(n + κ )2 − C 2 (n + κ )2 − Z 2 α 2 = − m 0 (n + κ )2 + C 2 (n + κ )2 + Z 2 α 2 (28)

r (24)

To find energy eigenvalue, we substitute Eqs. (24) into Eq. (20) and obtain

Some numerical results are given in Tables 1–4. In Tables 3 and 4 when H = 0, we see that energies of bound states such as (1s1/2 , 0d3/2 ), (1p 3/2 , 0 f 5/2 ), (1d5/2 , 0g 7/2 ), (1 f 7/2 , 0h9/2 ), . . . are degenerate, where each pair is considered as pseudospin doublet. In Tables 1–2, in the presence or tensor potential, i.e. when H = 0,

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M. Hamzavi et al. / Physics Letters A 374 (2010) 4303–4307

Table 3 The bound state energy eigenvalues for Coulomb potential in unit of fm−1 under the pseudospin symmetry in the absence of tensor potential with m0 = 5 fm−1 , C = 0.5 and C ps = 0.

˜l

n, κ ≺ 0

(l, j )

E n ,κ ≺0 H =0 m 1 = 0.5

1

1, −1

1s1/2

2

1, −2

1p 3/2

3

1, −3

1d5/2

4

1, −4

1 f 7/2

1

2, −1

2s1/2

2

2, −2

2p 3/2

3

2, −3

2d5/2

4

2, −4

2 f 7/2

−4.749331473 −4.851130965 −4.902767354 −4.931815494 −4.855413144 −4.903808321 −4.932163183 −4.949774758

n − 1,

κ 0

(l + 2, j + 1)

E n−1,κ 0 H =0 m 1 = 0.5

−4.749331473 −4.851130965 −4.902767354 −4.931815494 −4.855413144 −4.903808321 −4.932163183 −4.949774758

0, 2

0d3/2

0, 3

0 f 5/2

0, 4

0g 7/2

0, 5

0h9/2

1, 2

1d3/2

1, 3

1d3/2

1, 4

1g 7/2

1, 5

1h9/2

Table 4 The bound state energy eigenvalues for pure Coulomb potential in unit of fm−1 under the pseudospin symmetry with m0 = 5 fm−1 , C = 0.5 and C ps = 0.

˜l

n, κ ≺ 0

(l, j )

1

1, −1

1s1/2

2

1, −2

1p 3/2

3

1, −3

1d5/2

4

1, −4

1 f 7/2

1

2, −1

2s1/2

2

2, −2

2p 3/2

3

2, −3

2d5/2

4

2, −4

2 f 7/2

E n ,κ ≺0 H = m1 = 0

−4.931034483 −4.961089494 −4.975062344 −4.982668977 −4.961089494 −4.975062344 −4.982668977 −4.987261146

the degeneracy between two states in the pseudospin is removed. For eigenfunctions, we use Eq. (16) and obtain

g 0 (r ) = N



g 1 (r ) = N Γ 2 − 2β˜



 1−



2β˜ 2

.. .

(29)

1 ˜ G nκ (r ) = B n r εn + 2 e −β r (−1)n

 2n−1 



Γ − (k + 1)β˜ 2

× 1 F 1 (−n, 2εn + 1; 2β˜ r )

(30)

where B n is new normalization constant. We can also write hypergeometric function 1 F 1 in terms of Laguerre polynomials, therefore ˜

G nκ (r ) = D nκ r εn + 2 e −β r L n n (2β˜ r ) 2ε

(31)

where D nκ is normalization constant given as [57]

D nκ =

1 n!



˜ εn +1/2 (2β)

(n − 2εn )! n!

(32)

from Eq. (7b), the upper spinor component of the Dirac equation can be calculated as

F nκ (r ) =

1 m(r ) − E nκ + C ps



d dr

−

(l + 2, j + 1)

0, 2

0d3/2

0, 3

0 f 5/2

0, 4

0g 7/2

0, 5

0h9/2

1, 2

1d3/2

1, 3

1d3/2

1, 4

1g 7/2

1, 5

1h9/2

E n−1,κ 0 H = m1 = 0

−4.931034483 −4.961089494 −4.975062344 −4.982668977 −4.961089494 −4.975062344 −4.982668977 −4.987261146

degeneracy between two states in the pseudospin doublets. In Tables 1–4, we have presented some numerical results in the presence and absence of spatially-dependent mass and tensor potential.

κ r

 + U (r ) G nκ (r )

We would like to thank the kind referee for positive suggestions which have improved the present Letter. References



k=n

1

κ 0

Acknowledgement

Γ 2 − 2β˜

which leads to

n − 1,

(33)

5. Conclusion We investigated the Coulomb–Dirac problem including spatiallydependent mass and Coulomb-like tensor potential under pseudospin symmetry limit by using AIM. We have shown that this symmetry has exact solutions and also, tensor interaction removes

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