The spin symmetry for deformed generalized Pöschl–Teller potential

Physics Letters A 373 (2009) 2428–2431

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

The spin symmetry for deformed generalized Pöschl–Teller potential Gao-Feng Wei a,∗ , Shi-Hai Dong b a b

Department of Physics, Xi’an University of Arts and Science, Xi’an 710065, People’s Republic of China Departamento de Física, Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, Edificio 9, Unidad Profesional Adolfo López Mateos, Mexico D. F. 07738, Mexico

a r t i c l e

i n f o

Article history: Received 25 February 2009 Received in revised form 30 April 2009 Accepted 8 May 2009 Available online 13 May 2009 Communicated by R. Wu PACS: 03.65.Ge 34.20.Cf

a b s t r a c t In the case of spin symmetry we solve the Dirac equation with scalar and vector deformed generalized Pöschl–Teller (DGPT) potential and obtain exact energy equation and spinor wave functions for s-wave bound states. We find that there are only positive energy states for bound states in the case of spin symmetry based on the strong regularity restriction condition λ < −η for the wave functions. The energy eigenvalue approaches a constant when the potential parameter α goes to zero. Two special cases such as generalized PT potential and standard PT potential are also briefly discussed. © 2009 Elsevier B.V. All rights reserved.

Keywords: Dirac equation Deformed generalized Pöschl–Teller potential Spin symmetry

1. Introduction The spin and pseudospin symmetry concepts introduced in nuclear theory [1,2] are used to explain the features of deformed nuclei [3] and superdeformation [4], and establish an effective shell-model coupling scheme [5]. Since their introduction some contribution related to this problem has been made both in nuclear theory and in quantum theory [6–20]. For example, within the framework of the relativistic mean field theory, Ginocchio [6,7] found that a Dirac Hamiltonian with scalar and vector harmonic oscillator potentials in the case of V (r ) − S (r ) = 0 possesses not only a spin symmetry but also a U(3) symmetry, while a Dirac Hamiltonian in the case of V (r ) + S (r ) = 0 possesses a pseudospin symmetry and a pseudo-U(3) symmetry. Meng et al. [8] showed that the pseudospin symmetry is exact under the condition of d( V (r ) + S (r ))/dr = 0 and spin symmetry is exact under the condition of d( V (r ) − S (r ))/dr = 0. On the other hand, some typical physical models have been studied such as harmonic oscillator [9,10], Woods–Saxon potential [11], Morse potential [12–14], Eckart potential [15–17], Pöschl–Teller potential [18], Manning–Rosen potential [19], etc. In addition, Alhaidari et al. investigated in some detail physical interpretation on the three-dimensional Dirac equa-

*

Corresponding author. Tel.: +86 29 88272280; fax: +86 29 88272280. E-mail addresses: [email protected] (G.-F. Wei), [email protected] (S.-H. Dong). 0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.05.011

tion in the case of spin symmetry V (r ) = S (r ) and pseudospin symmetry V (r ) = − S (r ) [20]. Recently, the DPT potential has been discussed by path integral method [21]. The nonrelativistic bound and scattering states for GPT potential were studied [22,23]. Also, Jia et al. [24] have studied the pseudospin symmetry properties for GPT potential by SUSYQM and shape invariance methods. Nevertheless, the spin symmetry case for DGPT potential has not been touched to our knowledge, which is the main aim of this Letter. This Letter is organized as follows. In Section 2 we present basic equation of Dirac theory. In Section 3 in the case of spin symmetry, we solve Dirac equation and obtain the s-wave bound state solutions to DGPT potential. In Section 4 we briefly discuss two special cases of DGPT i.e. GPT potential and standard PT potential. We summarize the conclusions in Section 5. 2. Dirac spinors The Dirac equation of a nucleon with mass M moving in an attractive scalar potential S (r ) and a repulsive vector potential V (r ) can be written as (h¯ = c = 1)











α · p + β M + S (r ) Ψ (r ) = E − V (r ) Ψ (r ),

where E is the relativistic energy of the system, and the 4 × 4 Dirac matrices

(1)

α and β are

G.-F. Wei, S.-H. Dong / Physics Letters A 373 (2009) 2428–2431



α=

0

σi

σi

0



σ1 =

0 1 1 0



 β=

,





I 0 0 −I

σ2 =

,

−i

0 i

 , 

0

 ,

σ3 =

1 0 0 −1

 .

(2)

For spherically symmetrical nuclei, the total angular momentum of the nucleon J and spin–orbit matrix K = −β(σ · L + 1) commute with the Dirac Hamiltonian, where L is the orbital angular momentum. The eigenvalues of K are k = ±( j + 1/2), where k = −( j + 1/2) < 0 is for the aligned spin j = l + 1/2 (s1/2 , p 3/2 , etc.) and k = ( j + 1/2) > 0 is for the unaligned spin j = l − 1/2 (p 1/2 , d3/2 , etc.). The complete set of the conservative quantities can be taken as (H , K , J 2 , J z ) and the spinor wave functions can be classified according to their angular momentum j , k and the radial quantum number n. The spherically symmetric Dirac spinor wave functions can be written as

Ψnk =



1



F nk (r )Y ljm (θ, φ)

(3)

,

˜

r

iG nk (r )Y ljm (θ, φ)

By using them E˘grifes et al. studied the bound states of deformed Rosen–Morse potential by NU approach [28]. De Souza Dutra [29] found that the deformed hyperbolic functions can be reduced to the non-deformed ordinary hyperbolic functions by using the coordinate translation transformation. Substituting Eq. (6) into Eq. (5a) allows us to obtain the following Schrödinger-like equation for the upper component,



−



d dr d dr

+ −

k r k











F nk (r ) = M + E nk + S (r ) − V (r ) G nk (r ),



G nk (r ) = M − E nk + S (r ) + V (r ) F nk (r ).

r

(4a)



d2 dr 2

+

d2 dr 2

−

k(k + 1) r2

dΔ(r ) d ( dr dr



  − M + E nk − Δ(r ) M − E nk + Σ(r ) k ) r

−

k(k − 1) r2

(5a)

  − M + E nk − Δ(r ) M − E nk + Σ(r )

 − kr ) − G nk (r ) = 0, M − E nk + Σ(r )

(5b)

(8)

 ˜ V 1 − V˜ 2 coshq (α r ) sinhq2 (α r )

F n,−1 (r ) = 0,

(9)





E˜ n,−1 = E n2,−1 − M 2 + C ( M − E n,−1 ) , V˜ 1 = ( M + E n,−1 − C ) V 1 ,

V˜ 2 = ( M + E n,−1 − C ) V 2 .

(10)

We are now in the position to solve Eq. (9). After introducing a new variable x = q−1/2 coshq (α r ), Eq. (9) can be rearranged as



˜˜  ˜˜ ˜˜ n,−1 + V 1 − V 2 x F n,−1 (x) = 0, − E dx dx2 1 − x2   −1/2   1/ 2 x ∈ 1/2 q +q ,∞ , 1 − x2

 d2

−x

d

(11)

where

˜

E˜ n,−1 =

1

α2

E˜ n,−1 ,

˜

V˜ 1 =

1

α2q

V˜ 1 ,

˜

V˜ 2 =

1

α 2 q 1/ 2

V˜ 2 .

F n,−1 (x) = (1 − x)λ (1 + x)η f n,−1 (x),

 d2 f n,−1 (x) dx2

(12)

(13)

  df n,−1 (x) + 2η − 2λ − (1 + 2η + 2λ)x

  + − E˜˜ n,−1 − (η + λ)2 f n,−1 (x) = 0, where

dx

(14)



 1 ˜ ˜ 1 + σ 1 + 4 V˜ 1 − 4 V˜ 2 , 4   1 η = 1 + τ 1 + 4 V˜˜ 1 + 4 V˜˜ 2 , 4 λ=

Now, let us consider the exact spin symmetry, i.e., dΔ/dr = 0 or Δ = C = constant. Here we take Σ(r ) as DGPT potential V 1 − V 2 coshq (α r ) sinhq2 (α r )

(6)

,

which can be obtained by modifying GPT potential [22–24]. The potential parameters V 1 and V 2 describe the property of potential well, V 1 > V 2 , while α is related to the range of the potential. Notice that this potential is a special case of five-parameter exponential-type potential [25,26]. The deformed hyperbolic functions [27] are denoted by (we assume without loss of generality q > 0 and study real bound states)

2

F nk (r ) = 0,

+ E˜ n,−1 −

1 − x2

3. Spin symmetry

e x − qe −x

sinhq2 (α r )



r2



where Δ(r ) = V (r ) − S (r ) and Σ(r ) = V (r ) + S (r ).

sinhq (x) =

and inserting it into Eq. (11), we have



dΣ(r ) d ( dr dr

V q (r ) =

V 1 − V 2 coshq (α r )

By taking wave function of the form



+ F nk (r ) = 0, M + E nk − Δ(r )

k(k + 1)



where

(4b)

By eliminating G nk (r ) in Eq. (4a) and F nk (r ) in Eq. (4b), one obtains two second-order differential equations for the upper and lower components as follows:

d2 dr 2

˜l

and Y jm (θ, φ) are the spherical square-integral functions, harmonic functions, and m is the projection of the total angular momentum on the third axis, and l(l + 1) = k(k + 1), ˜l(˜l + 1) = k(k − 1). The substitution of Eq. (3) into Eq. (1) yields two coupled differential equations as follows:

+ E˜ nk − [ M + E nk − C ]

2 − M 2 + C ( M − E nk )]. Note that this equation can where E˜ nk = [ E nk be solved exactly only for s-wave (k = −1) case because of the spin–orbit coupling term k(k + 1)/r 2 . In this work we study the spin symmetry only for s-wave case. In this case, Eq. (8) is simplified as



Y ljm (θ, φ)

d2 dr 2

where the upper and lower components F nk (r ) and G nk (r ) are real



2429

,

coshq (x) =

e x + qe −x 2

.

(7)

(15)

where σ = ±1 and τ = ±1. Thus, the choice of parameters λ and η is not unique. However, for bound states in order to satisfy the boundary conditions of wave functions, i.e. F n,−1 (r )/r becomes zero when r is infinite, and F n,−1 (r )/r is finite when r goes to zero. These regularity conditions require λ > 0, η < 0 and λ < −η . Therefore, the parameters λ and η are taken as



 1 λ = 1 + 1 + 4 V˜˜ 1 − 4 V˜˜ 2 , 4   1 η = 1 − 1 + 4 V˜˜ 1 + 4 V˜˜ 2 . 4

(16)

Further inserting such a transformation z = (1 − x)/2 into Eq. (14) leads to the following hypergeometric differential equation [30],

2430

G.-F. Wei, S.-H. Dong / Physics Letters A 373 (2009) 2428–2431

Table 1 The bound state energy levels E n,−1 are shown in the case of spin symmetry. The numerical results show that the energy levels approach a constant when potential range parameter α approaches zero. n

E n,−1 C = 2, V 1 = 5, V 2 = 3, M = 10, q = 0.3

0 1 2 3 4 5

z(1 − z)

α = 1.2

α = 0.8

α = 0.4

α = 0.2

α = 0.02

α = 0.002

α = 0.00002

9.711281 9.935042 9.999259 9.902187 9.632656 9.166280

9.657836 9.843139 9.956666 9.999593 9.971418 9.869882

9.599803 9.710864 9.803498 9.879074 9.934847 9.973961

9.569055 9.629276 9.684802 9.735696 9.782011 9.823793

9.540389 9.546840 9.553242 9.559597 9.565904 9.572163

9.537471 9.538120 9.538769 9.539417 9.540065 9.540713

9.537149 9.537156 9.537162 9.537169 9.537175 9.537182

d2 f n,−1 ( z) dz2

1 df n,−1 ( z) + 2λ + − (1 + 2η + 2λ)z 2

−

dz

  + − E˜˜ n,−1 − (η + λ)2 f n,−1 (z) = 0,

2λ+η n(2η + 2λ + n)α sinhq (α r ) zλ (1 − z)η



(4λ + 1)q1/2 ( M + E n,−1 − C )

 3 × 2 F 1 1 − n; 1 + n + 2(η + λ); 2λ + ; z .

(17)

2

whose solutions are given by ∞  (a)γ (b)γ zγ f n,−1 ( z) = 2 F 1 (a; b; c ; z) = , (c )γ γ !

(18)

γ =0

where

 − E˜˜ n,−1 ,

a=η+λ+

c = 2λ + 1/2,

 b=η+λ−

(z)γ =

According to the properties of hypergeometric functions, know that series f n,−1 ( z) given in Eq. (18) approaches infinite less parameter a is a negative integer. Therefore, considering finiteness of solutions, the general quantum condition is given

we unthe by

a=η+λ+

(19)

n = 0, 1, 2, . . . ,

from which, together with Eqs. (10), (12) and (16), one obtains the energy equation E n2,−1 − M 2 + C ( M − E n,−1 )



= −α

2

n+

 −

1+

1 2

1

+



1+

4

4( M + E n,−1 − C )



α2

4( M + E n,−1 − C )



V1

α2

q

+

V1 q

V2

−



V2 q 1/ 2

 2 (20)

.

q 1/ 2





1 z (1 − z)η 2 F 1 −n; 2(η + λ) + n; 2λ + ; z .

λ+η λ

(21)

2

By using the following recurrence relation between hypergeometric functions d  dz



2 F 1 (a; b ; c ; z)



=

ab



c

2 F 1 (a

+ 1; b + 1; c + 1; z)

(22)

and inserting Eq. (21) into Eq. (4a), we obtain the corresponding lower component G n,−1 (r ) as G n,−1 (r ) =

=

1 M + E n,−1 − C F n,−1 (r ) M + E n,−1 − C



dF n,−1 (r ) dr



1

− + r

−

1 r

F n,−1 (r )

q−1/2 α sinh

αr )

q(

2

−

V 12 − qV 22 /2q is not a physically acceptable value as shown

by numerical calculation below. In order to show how to calculate numerically the energy levels based on Eq. (20), we take a set of parameters C = 2, V 1 = 5, V 2 = 3, M = 10, q = 0.3 to do that. For example, when n = 0 and α = 1.2 we solve Eq. (20) numerically and obtain two energy values E 0,−1 : 9.711281, −7.983582. If choosing the first solution E 0,−1 = 9.711281, we have λ = 6.121 and η = −8.00543 by solving Eq. (16). It is evident to see that they satisfy the regularity restrictions: λ > 0, η < 0 and λ < −η . However, if taking the second value E 0,−1 = −7.983582, one has λ = 0.557236 and η = −0.104426. Obviously, they do not satisfy the regularity condition λ < −η . In a similar way we calculate the energy eigenvalues E n,−1 and illustrate the effect of potential parameter α on the energy levels as shown in Table 1. It is found that the  energy levels approach a constant limα →0 E n,−1 = M − V 1 /2q +

when potential parameter

This is a rather complicated transcendental energy equation and determines the energy levels E n,−1 . The corresponding upper component F n,−1 (r ) can be finally expressed as F n,−1 (r ) = 2

V 12 − qV 22 /2q or limα →0 E n,−1 =

C − M. However, the limit energy limα →0 E n,−1 = C − M is not physically acceptable since it makes the lower component G n,−1 (r ) diverge. Actually, the limit limα →0 E n,−1 = M − V 1 /2q 

( z + γ ) . (z)

 − E˜˜ n,−1 = −n,

It is obvious to see from Eqs. (21) and (23) that the spinors F n,−1 (r ) and G n,−1 (r ) satisfy the regularity boundary conditions for bound states when λ > 0, η < 0, λ < −η and λ, η ∈ R. Moreover, after considering the limit case α → 0, we find from Eq. (20) that the energy eigenvalue approaches a constant, i.e.,  limα →0 E n,−1 = M − V 1 /2q ±

− E˜˜ n,−1 ,

(23)



η 1−z

−

λ z



V 12 − qV 22 /2q

α approaches zero.

4. Discussions Let us briefly study the special case q = 1. In this case, the energy equations and spinor wave functions for GPT potential and standard PT potential can be easily obtained through some parametric mapping, e.g. the results for GPT can be easily realized by considering q = 1, coshq (x) = cosh(x), sinhq (x) = sinh(x) in Eqs. (16), (20), (21) and (23). For more special case Δ(r ) = C = 0, we have S (r ) = V (r ) = 12 Σ(r ), it is obvious that the obtained results above shall reduce to the bound states of the Dirac equation with equally scalar and vector GPT potentials. As far as the standard PT potential [18] V (r ) = − A ( A + α )/ cosh2 (α r ) + B ( B − α )/ sinh2 (αr ), this can also be easily realized by taking q = 1, V 1 = 2( A ( A + α ) + B ( B − α )), V 2 = 2( A ( A + α ) − B ( B − α )), and substituting α → 2α . It is noted that the results obtained here for standard PT potential are consistent with those given in [31]. However, the energy levels approach the constant of limα →0 E n,−1 = M − ( A − B )2 when potential range parameter α approaches zero.

G.-F. Wei, S.-H. Dong / Physics Letters A 373 (2009) 2428–2431

Also, we are able to obtain the bound states of the Dirac equation with equally scalar and vector standard PT potentials by taking Δ(r ) = C = 0. 5. Concluding remarks In this Letter we have presented the exact solutions of s-wave Dirac equation for DGPT potential under the condition of spin symmetry. We have found that there are only positive energy states for bound states in the case of spin symmetry based on the strong regularity restriction condition λ < −η for the wave functions. It is also found from the numerical results that energy levels approach a constant when potential parameter α goes to zero. Finally, two special cases, i.e. the GPT potential and standard PT potential have been discussed briefly. Acknowledgements We would like to thank the kind referees for their positive and invaluable suggestions which have improved the present manuscript greatly. This work was partially supported by Project 20090513-SIP-IPN, COFAA-IPN, Mexico. References [1] A. Arima, M. Harvey, K. Shimizu, Phys. Lett. B 30 (1969) 517. [2] K.T. Hecht, A. Adeler, Nucl. Phys. A 137 (1969) 129. [3] A. Bohr, I. Hamamoto, B.R. Mottelson, Phys. Scr. 26 (1982) 267.

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