JOURNAL OF MATHEMATICAL PHYSICS 52, 013506 (2011)
Solution of Dirac equation with spin and pseudospin symmetry for an anharmonic oscillator H. Goudarzi,a) M. Sohbati, and S. Zarrin Department of Physics, Faculty of Science, Urmia University , Urmia, P.O. Box 165, Iran (Received 26 July 2010; accepted 19 November 2010; published online 7 January 2011)
We present the exact solutions of Dirac equation with anharmonic oscillator potential using the Nikiforov–Uvarov method. Taking into account potentials of vector field V (r ) and scalar field S(r ) in Dirac Hamiltonian, the bound state energy eigenvalues and the corresponding upper and lower two-component spinors of fermion are obtained. These solutions are considered in the framework of the spin and pseudospin C 2011 American Institute of Physics. [doi:10.1063/1.3532930] symmetry concept.
I. INTRODUCTION
The solutions of the Dirac equation in electromagnetic fields play an important role in the relativistic quantum mechanics. One of the application of the relativistic Dirac equation is in the nuclear physics.1–4 The Dirac equation for spin 1/2 particles play a central role in the relativistic description of atoms, nuclei, and hadrons.2 Note that some fundamental concepts as the pseudospin symmetry can be applied to many systems in nuclear physics and related areas.1, 3, 5 It has also been used to explain features of deformed nuclei,6 the super-deformation,7 and to establish an effective nuclear shell model scheme.8–10 The pseudospin symmetry introduced in nuclear theory refers to a quasidegeneracy of the single nucleon doublets and can be characterized with the nonrelativistic quantum numbers (n, l, j = l + 1/2) and (n − 1, l + 2, j = l + 3/2), where n, l, and j are the single nucleon radial, orbital, and total angular momentum quantum numbers for a single particle, respectively.4, 5, 8, 11 The total angular momentum is given as j = l˜ + s˜ , where l˜ = l + 1 is a pseudoangular momentum and s˜ = 1/2 is a pseudospin angular momentum.12 Within the framework of the Dirac equation the spin symmetry arises if the magnitude of the attractive scalar potential S(r ) and repulsive vector potential V (r ) are nearly equal, S(r ) ∼ = V (r ) in nuclei (i.e., when the difference potential (r ) = V (r ) − S(r ) is a constant). However, the pseudospin symmetry accurse when the magnitude of the scalar and vector potentials are nearly ˜ ) = V (r ) + S(r ) equal, but in opposite sign, as V (r ) ∼ = −S(r ) (i.e., when the sum potential (r 1, 3, 13 The Dirac equation under the exact pseudospin and spin symmetry has been is a constant). studied by different types of potentials such as the Hulten potential,14–17 the Morse potential,18–20 the Wood–Saxon potential,21, 22 and the harmonic oscillator potential.11, 23, 24 It is well known that the harmonic oscillator potential is one of the exactly solvable potential in quantum mechanics, and has been widely applied in many fields. In particular, it is a central potential of nuclear shell model, which has provided a good description of nuclear single particle motion and shell structure for the spherical and axially deformed nuclei.25 However, considering the fact that the realistic nuclei often deviate from the spherical and axial harmonic oscillator model, several other oscillator models, such as the nonspherical harmonic oscillator,26 the ring-shaped harmonic oscillator,27, 28 and the ring-shaped nonharmonic oscillator25, 29 have been introduced. So the need for a description of nuclei in which rotational–vibration interaction dominate has led to search for algebraically solvable potentials. It has been recently shown that one such potential
a) Author to whom correspondence should be addressed. Electronic mail:
[email protected]. Tel.: + 98 914 440 6713.
Fax: + 98 441 2776707.
0022-2488/2011/52(1)/013506/7/$30.00
52, 013506-1
C 2011 American Institute of Physics
013506-2
Goudarzi, Sohbati, and Zarrin
J. Math. Phys. 52, 013506 (2011)
is Davidson potential which has algebraic solution both for diatomic molecules and for a liquid drop model of nucleons.30 The Davidson potential is a scalar function of the nuclear quadruple moments and expressed in terms of nucleon coordinates.31 In spherical coordinates the ring-shaped nonspherical harmonic oscillator potential25 is defined as V (r, θ ) =
1 2 α 2 η + Mω2r 2 + , 2 2Mr 2 2Mr 2 sin2 θ
(1)
where M and ω denote the rest mass and frequency of particle, respectively. α and η are the dimensionless parameters. If we confine η = 0, Eq. (1) reduces to the Davidson potential. So the anharmonic oscillator potential with η = 0 is defined as V (r, θ ) =
2 α 1 Mω2r 2 + . 2 2Mr 2
(2)
Many authors have used different methods to study the partially exactly solvable and exactly solvable Schrodinger, Klein–Gordon (KG), and Dirac equation. These methods include the standard method, supersymmetry method,32 the Nikiforov–Uvarov (NU) method,30, 33 and so on. In the recent work, we intend to use the algebraic technique NU to solve the Dirac equation for scalar and vector anharmonic oscillator potential with spin and pseudospin symmetries. The NU method is used to obtain the energy eigenvalue and the corresponding wave function in terms of orthogonal polynomials for a class of noncentral potentials. This paper is organized as follows. In Sec. II we briefly review the Nikiforov–Uvarov method. In Sec. III we briefly discuss the equations for the Dirac spinors including the centrifugal term, and the spin–orbit quantum number κ. Sections IV and V are devoted to study spin and pseudospin symmetries of the Dirac equation for the anharmonic oscillator potential, respectively. Finally, the relevant conclusions are given in Sec. VI. II. NIKIFOROV–UVAROV METHOD
The second-order differential equations whose solutions are the special functions can be solved by using the NU method. This method was proposed to solve the second-order differential equation of hypergeometric type and in this method the differential equations can be written in the following form: σ˜ (z) τ˜ (z) dψ(z) d 2 ψ(z) + 2 ψ(z) = 0, + dz 2 σ (z) dz σ (z)
(3)
where σ (z) and σ˜ (z) are polynomials, at most second degree, and τ˜ (z) is a first degree polynomial. By writing the general solution as ψ(z) = ϕ(z)y(z), we obtain a hypergeometric-type equation, λ d 2 y(z) τ (z) dy(z) + y(z) = 0. + 2 dz σ (z) dz σ (z)
(4)
The function ϕ(z) is defined as a logarithmic derivative, ϕ (z) π (z) = , ϕ(z) σ (z)
(5)
where y(z) is the hypergeometric-type function whose polynomial solutions are given by Rodriguez relation, yn (z) =
an d n n [σ (z)ρ(z)], ρ(z) dz n
(6)
where an is a normalization constant and ρ(z) is the weight function satisfying the following equation: (ρσ ) = τρ.
(7)
013506-3
Solution of Dirac equation
J. Math. Phys. 52, 013506 (2011)
The function π (z) and the parameter λ required for this method are defined as σ − τ˜ σ − τ˜ 2 π (z) = ± ( ) − σ˜ + kσ , 2 2 λ = k + π (z).
(8)
(9)
In the NU method, π (z) is a polynomial with the parameter z and the determination of k is the essential point in the calculation of π (z). For finding the value of k, the expression under the square root must be square of a polynomial, so we have a new eigenvalue equation, λ = λn = −τ −
n(n − 1) d 2 σ (z) , 2 dz 2
(10)
where the derivation of the function τ (z) = τ˜ (z) + 2π (z) should be negative, and by comparing (9) and (10), we obtain the energy eigenvalues. III. DIRAC EQUATION
The Dirac equation of a nucleon with mass M moving in a scalar field potential S(r ) and a vector field potential V (r ) can be written as ( = c = 1) [α. ˆ p + β(M + S(r ))]ψ(r ) = (E − V (r ))ψ(r ), where
αˆ = p = −i ∇,
0
σˆ i
σˆ i
0
, β=
I
0
0
−I
(11)
,
(12)
and p is the momentum operator, αˆ and β are 4 × 4 Dirac matrices, I is 2 × 2 unit matrix, and σˆ i (i = 1, 2, 3) are Pauli matrices. For spherical nuclei, the nucleon angular momentum J and Kˆ commute with the Dirac Hamiltonian. The operator Kˆ is the spin–orbit matrix operator and written in terms of the orbital angular momentum operator L as Kˆ = −β(σˆ . L + 1). The Dirac spinors can be labeled by the quantum number set (n, κ), where κ is the eigenvalue of the spin–orbit operator, and written as l 1 Fn,κ (r ) Y jm (θ, ϕ) , (13) ψn,κ = r G n,κ (r ) Y l˜ (θ, ϕ) jm
where n is the radial quantum number, m is the projection of angular momentum on the third l (θ, ϕ) and axis, Fn,κ (r ) and G n,κ (r ) are, respectively, the upper and lower components, and Y jm ˜
l Y jm (θ, ϕ) are the spherical harmonics function. The total angular momentum, the orbital angular momentum, and pseudoorbital angular momentum can be written in terms of the spin–orbit quantum number κ = ±1, ±2, ... , such as j = |κ| − 1/2, l = |κ + 1/2| − 1/2, and l˜ = |κ − 1/2| − 1/2, respectively. Substituting Eq. (13) into Eq. (11), we obtain two radial coupled Dirac equations for the spinor components,
(
κ d + )Fn,κ (r ) = [M + E n,κ − V (r ) + S(r )]G n,κ (r ), dr r
(
κ d − )G n,κ (r ) = [M − E n,κ + V (r ) + S(r )]Fn,κ (r ). dr r
(14)
013506-4
Goudarzi, Sohbati, and Zarrin
J. Math. Phys. 52, 013506 (2011)
To eliminate G n,κ (r ) and Fn,κ (r ) in Eq. (14), we obtain Schrodinger-like equations for the upper and the lower components, κ d d + d2 κ(κ + 1) dr dr r ˜ [ 2− ]Fn,κ (r ) = 0, − (M + E − )(M − E + ) + (15) n,κ n,κ dr r2 M + E n,κ −
[
κ(κ − 1) d2 − − (M + E n,κ − )(M − E n,κ 2 dr r2
d κ + dr r ˜ − ]G n,κ (r ) = 0, + ) M + E n,κ − ˜ d dr
(16)
˜ are assumed to be radial function, i.e., (r ) = V (r ) − S(r ) and (r ˜ ) = V (r ) + S(r ). where and We insert the anharmonic oscillator potential [Eq. (2)], into the above equations and in order to solve the above equations we apply the spin and pseudospin symmetries. IV. SPIN SYMMETRY SOLUTION
∼ V (r ) ( d(r ) = 0, i.e., (r ) = V (r ) − S(r ) In the case of exact spin symmetry S(r ) = dr = A =constant ), Eq. (15) can be approximately written as 2
d κ(κ + 1) ˜ − − M + E − A M − E + Fn,κ (r ) = 0, (17) n,κ n,κ dr 2 r2 where κ = l for κ < 0 and κ = −(l + l) for κ > 0, and the spin symmetry energy eigenvalues ˜ ) = 2V (r ) and depend on n and κ, i.e., E n,κ = E(n, κ(κ + 1)). In the last equation the choice of (r ˜ ) into Eq. (17), we obtain the Schrodinger-like equation in the spherical coordinates substituting (r for the upper-spinor component Fn,κ (r ) , d 2 Fn,κ n,κ r 2 + β 2r 4 + γ − Fn,κ = 0, dr 2 r2
(18)
where n,κ , β 2 and γ are defined as 2 n,κ = M 2 − E n,κ − A(M − E n,κ ),
β 2 = M(M + E n,κ − A),
γ = κ(κ + 1) +
α(M + E n,κ − A) . M
(19)
By introducing a new variable x = r 2 , Eq. (18) becomes n,κ x + β 2 x 2 + γ 1 d Fn,κ d 2 Fn,κ − + Fn,κ = 0. (20) 2 dx 2x d x 4x 2 In order to solve Eq. (20) by means of the NU method, we should compare it with Eq. (3). So the following expressions are obtained: τ˜ = 1 , σ = 2x , σ˜ = −(n,κ x + β 2 x 2 + γ ). Substituting these expressions into Eq. (8), we obtain 1 π (x) = ± 1/4 + n,κ x + β 2 x 2 + γ + 2kx. 2
(21)
(22)
According to the NU method, the constant parameter k can be determined from the condition that the expression under the square root must be the square of a polynomial of first degree, so we
013506-5
Solution of Dirac equation
J. Math. Phys. 52, 013506 (2011)
have
β 2 (4γ + 1) . k1,2 = 2 In this regard, we can find the four possible function for π (x) as
1 4γ + 1 −n,κ + β 2 (4γ + 1) π (x) = ± β x + for k1 = , 2 4β 2 2 −n,κ ±
4γ + 1 −n,κ − β 2 (4γ + 1) 1 . π (x) = ± β x − for k2 = 2 4β 2 2
(23)
(24)
(25)
Also according to the NU method, one of the four values of the polynomial π (x) is just proper to obtain the bound states because τ (x) has a negative derivative. Therefore k and π (x) are defined as
1 4γ + 1 −n,κ − β 2 (4γ + 1) π (x) = − β x − , (26) , k= 2 4β 2 2 and one can easily obtain τ = 2 − 2βx +
4γ + 1 , τ =
dτ = −2β < 0. dx
(27)
According to Eqs. (5) and (7) we have ϕ=x
√ 1+ 4γ +1 4
ρ=x
√ 4γ +1 2
exp (
−βx ), 2
exp (−βx),
(28)
and further inserting Eqs. (28) into Eq. (6), yn (x) can be found as follows: √ √ √ n 4γ +1 4γ +1 d n+ 4γ2 +1 2 (βx) , yn (x) = Bn exp (βx)x − 2 x (29) exp (−βx) ≈ L n n dx where L αn (x) is the generalized Laguerre polynomials. By using Fn,κ = ϕ (x) yn (x), the radial upper-spinor wave function can be found to be √ √ 4γ +1 1+ 4γ +1 −βx x 4 L n 2 (βx) , (30) Fn,κ (r ) = Cn exp 2 in which x = r 2 , on the other hand
Fn,κ (r ) = Cn exp
√ √ 4γ +1 1+ 4γ +1 −βr 2 r 2 L n 2 βr 2 . 2
From Eq. (26) and λ = k + π (x), we have
β 2 (4γ + 1) , 2
(32)
λ = λn = 2nβ , n = 0, 1, 2, ...
(33)
λ = −β −
n,κ −
(31)
and by using Eqs. (10) and (27) we have
Now, taking λ = λn , we can solve the above equations to obtain the energy equation for the anharmonic oscillator potential with spin symmetry in Dirac theory, (34) n,κ + 2β(2n + 1) + β 4γ + 1 = 0.
013506-6
Goudarzi, Sohbati, and Zarrin
J. Math. Phys. 52, 013506 (2011)
If we consider the case when S(r ) = V (r ), or A = 0, and taking γ = L(L + 1), then Eq. (34) can be reduced to the following form:
2 M 2 − E n,κ
M(M + E n,κ )
+ 4n + 2L + 3 = 0.
(35)
Further, the radial upper wave function turns to be − m(m + E n,κ ) 2 L+1 L+1/2 r r Fn,κ (r ) = Cn exp Ln m(m + E n,κ )r 2 . 2
(36)
V. PSEUDOSPIN SYMMETRY SOLUTION
In the presence of the pseudospin symmetry S(r ) ≈ −V (r ) (i.e., = constant), Eq. (17) can be exactly written as
˜ d dr
˜ )=B = 0, or (r
κ(κ − 1) d2 − − (M + E n,κ − )(M − E n,κ + B)]G n,κ (r ) = 0, (37) 2 dr r2 where the energy eigenvalues E n,κ depend on n and κ, i.e., E n,κ = E(n, κ(κ − 1)). In the last equation, the choice of = 2V (r ) and substituting (r ) into Eq. (37) we obtain an equation for the lower component G n,κ (r ), [
d 2 G n,κ (r ) ˜n,κ r 2 + β˜ 2r 4 + γ˜ − G n,κ (r ) = 0, dr 2 r2
(38)
where we have defined 2 ˜n,κ = M 2 − E n,κ + B(M + E n,κ ),
β˜ 2 = M(E n,κ − M − B),
γ˜ = κ(κ − 1) +
α(M − E n,κ + B) . M
(39)
To avoid repetition in the solution of Eq. (38), a first inspection for the relationship between ˜ γ˜ ) and the previous set (n,κ , β, γ ) tells us that the negative the present set of parameters (˜n,κ , β, energy solution for pseudospin symmetry, where S(r ) = −V (r ) can be obtained directly from those of the positive energy solution for spin symmetry using the parameter map: Fn,κ (r ) ↔ G n,κ (r ) , V (r ) → −V (r ) , E n,κ → −E n,κ , A → −A , β → β˜ = M(E n,κ − M − B). (40) Following the previous results with the above transformation, we finally arrive at the energy equation for the anharmonic oscillator potential with pseudospin symmetry, ˜ + 1) + β˜ 4γ˜ + 1 = 0, (41) ˜n,κ + 2β(2n and the lower-spinor wave function,
G n,κ (r ) = Cn exp
˜ 2 1+√4γ˜ +1 √4γ˜ +1 2 −βr ˜ . r 2 L n 2 βr 2
(42)
˜ L˜ + 1), then Eq. (42) can If we consider the case when S(r ) = −V (r ), or B = 0, and taking γ˜ = L( be reduced to the following form:
2 M 2 − E n,κ
M(E n,κ − M)
+ 4n + 2 L˜ + 3 = 0.
(43)
013506-7
Solution of Dirac equation
J. Math. Phys. 52, 013506 (2011)
Further, the radial lower-spinor wave function turns to be − M(E n,κ − M)r 2 ˜ ˜ M(E n,κ − M)r 2 . G n,κ (r ) = Cn exp r L+1 L nL+1/2 2
(44)
VI. CONCLUSIONS
To summarize, we have studied the bound state solution of Dirac equation with the anharmonic oscillator potential (Davidson potential), under the conditions of the spin symmetry and pseudospin symmetry. We have used the Nikiforov–Uvarov method to obtain the results. The obtained solutions of the wave function are being expressed in terms of the generalized Laguerre polynomial. So we have obtained the energy eigenvalue equations and the corresponding eigenfunctions of the anharmonic oscillator in the spin symmetry and pseudospin symmetry case, respectively. 1 J.
N. Ginocchio, Phys. Rev. Lett. 78, 436 (1997). Leviatan, Phys. Rev. Lett. 92, 202501 (2004). 3 J. N. Ginocchio, Phys. Rep. 414, 165 (2005). 4 J. N. Ginocchio and A. Leviatan, Phys. Rev. Lett. 87, 7 (2001). 5 S. M. Ikhdair and R. Sever, e-print arXiv:1001.3943v1 [math-ph] . 6 A. Bohr, I. Hamamoto, and B. R. Mottelson, Phys. Scr. 26, 267 (1982). 7 J. Dudek W. Nazarewicz, Z. Szymanski, and G. A. Leander, Phys. Rev. Lett. 59, 1405 (1987). 8 A. Arima, M. Harrey, and K. Shimizu, Phys. Lett. B 30, 517 (1969). 9 K. T. Hecht and A. Adler, Nucl. Phys. Rev. C 57, 1167 (1998). 10 D. Troltenier, C. Bahri, and J. P. Draayer, Nucl. Phys. A 586, 53 (1995). 11 J-You Guo, X-Zheng Fang, and Fu-Xin Xu, Nucl. Phys. A 757, 411 (2005). 12 J. Meng, K. Sugawara-Tanabe, S. Yamaji, P. Ring, and A. Arima, Phys. Rev. C 58, R628 (1998). 13 J. N. Ginocchio, Phys. Rev. C 69, 034318 (2004). 14 A. Soylu, O. Bayrak, and I. Boztosum, J. Math. Phys. A 40, 1677 (2007). 15 G. Chen, Mod. Phys. Lett. A 19, 2009 (2004). 16 A. D. Alhaidari, J. Phys. A 34, 9827 (2001). 17 A. D. Alhaidari, J. Phys. A 35, 6207 (2002). 18 W. C. Qiany, R. S. Zhou, and Y. Gao, J. Phys. A 40, 1677 (2007). 19 A. D. Alhaidari, Phys. Rev. Lett. 87, 210405 (2001). 20 A. D. Alhaidari, Phys. Rev. Lett. 88, 189901 (2002). 21 J. Y. Guo and Z. Q. Sheng, Phys. Lett. A 338, 90 (2005). 22 Q. Xu and S. J. Zhu, Nucl. Phys. A 768, 161 (2006). 23 R. Lisboa, M. Malheiro, A. S. Castro, P. Alberto, and M. Fiolhais, Phys. Rev. C 69, 024319 (2004). 24 A. S. Castro, P. Alberto, R. Lisboa, and M. Malheiro, Phys. Rev. C 73, 054309 (2006). 25 J. Y. Guo, J. C. Han, and R. D. Wany, Phys. Lett. A 353, 378 (2006). 26 C. Y. Chen and Y. W. Liu, High Energy Phys. Nucl. Phys. 23, 865 (1999). 27 C. Quesne, J. Phys. A 21, 3093 (1988). 28 D. S. Sun and C. Y. Chen, Acta Photon. Sin. 30, 539 (2001). 29 T. L. Cheng and C. Y. Chen, Acta Quantum Opt. Sin. 7, 67 (2001). 30 M. R. Setare and S. Haidari, Int. J. Theor. Phys. 48, 3249 (2001). 31 C. Bahri and D. J. Rowe, Nucl. Phys. A 662, 125 (2000). 32 C.-S. Jia , P. Gao, X.-L. Peng, J. Phys. A 39, 7737 (2006). 33 A. F. Nikiforov and V. B. Uvarov, Special Function of Mathematical Physics (Brikhauser, Bassel, 1988). 2 A.
