Approximate solutions of the Dirac equation for the Rosen-Morse potential including the spin-orbit centrifugal term Sameer M. Ikhdair1, ∗ 1
Department of Physics, Near East University, Nicosia, North Cyprus, Turkey
arXiv:0912.0619v1 [quant-ph] 3 Dec 2009
(Dated: December 4, 2009)
Abstract We give the approximate analytic solutions of the Dirac equations for the Rosen-Morse potential including the spin-orbit centrifugal term. In the framework of the spin and pseudospin symmetry concept, we obtain the analytic bound state energy spectra and corresponding two-component upper- and lower-spinors of the two Dirac particles, in closed form, by means of the NikiforovUvarov method. The special cases of the s-wave κ = ±1 (l = e l = 0) Rosen-Morse potential,
the Eckart-type potential, the PT-symmetric Rosen-Morse potential and non-relativistic limits are briefly studied. Keywords: Dirac equation, spin and pseudospin symmetry, bound states, Rosen-Morse potential, Nikiforov-Uvarov method. PACS numbers: 03.65.Pm; 03.65.Ge; 02.30.Gp
∗
E-mail:
[email protected];
[email protected]
1
I.
INTRODUCTION
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 are nearly equal, S(r) ∼ V (r) in the nuclei (i.e., when the difference potential ∆(r) = V (r) − S(r) = Cs = constant). However, the pseudospin symmetry occurs if S(r) ∼ −V (r) are nearly equal (i.e., when the sum potential Σ(r) = V (r) + S(r) = Cps = constant) [1-3]. The spin symmetry is relevant for mesons [4]. The pseudospin symmetry concept has been applied to many systems in nuclear physics and related areas [2-7] and used to explain features of deformed nuclei [8], the super-deformation [9] and to establish an effective nuclear shell-model scheme [5,6,10]. The pseudospin symmetry introduced in nuclear theory refers to a quasi-degeneracy of the single-nucleon doublets and can be characterized with the non-relativistic 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 [5,6]. The total angular momentum is given as j = e l+e s, where e l = l + 1 is a pseudo-
angular momentum and se = 1/2 is a pseudospin angular momentum. In real nuclei, the
pseudospin symmetry is only an approximation and the quality of approximation depends on
the pseudo-centrifugal potential and pseudospin orbital potential [11]. Alhaidari et al. [12] investigated in detail the physical interpretation on the three-dimensional Dirac equation in the context of spin symmetry limitation ∆(r) = 0 and pseudospin symmetry limitation Σ(r) = 0. Some authors have applied the spin and pseudospin symmetry on several physical potentials, such as the harmonic oscillator [12-15], the Woods-Saxon potential [16], the Morse potential [17,18], the Hulth´en potential [19], the Eckart potential [20-22], the molecular diatomic three-parameter potential [23], the P¨oschl-Teller potential [24], the Rosen-Morse potential [25] and the generalized Morse potential [26]. The exact solutions of the Dirac equation for the exponential-type potentials are possible only for the s-wave (l = 0 case). However, for l-states an approximation scheme has to be used to deal with the centrifugal and pseudo-centrifugal terms. Many authors have used different methods to study the partially exactly solvable and exactly solvable Schr¨odinger, Klein-Gordon (KG) and Dirac equations in 1D, 3D and/or any D-dimensional cases for different potentials [27-39]. In the context of spatially-dependent mass, we have also used 2
and applied a recently proposed approximation scheme [40] for the centrifugal term to find a quasi-exact analytic bound-state solution of the radial KG equation with spatially-dependent effective mass for scalar and vector Hulth´en potentials in any arbitrary dimension D and orbital angular momentum quantum number l within the framework of the NU method [40-42]. Another physical potential is the Rosen-Morse potential [43] expressed in the form V (r) = −V1 sec h2 αr + V2 tanh αr,
(1)
where V1 and V2 denote the depth of the potential and α is the range of the potential. This potential is useful for describing interatomic interaction of the linear molecules and helpful for disscussing polyatomic vibration energies such as the vibration states of NH3 molecule [43]. It is shown that the Rosen-Morse potential and its PT-symmetric version are the special cases of the five-parameter exponential-type potential model [44,45]. The exact energy spectrum of the trigonometric Rosen-Morse potential has been investigated by using supersymmetric and improved quantization rule methods [46,47]. Recently, many works have been done to solve the Dirac equation to obtain the energy equation and the two-component spinor wave functions. Jia et al. [48] employed an improved approximation scheme to deal with the pseudo-centrifugal term to solve the Dirac equation with the generalized P¨oschl-Teller potential for arbitrary spin-orbit quantum number κ. Zhang et al. [49] solved the Dirac equation with equal Scarf-type scalar and vector potentials by the method of the supersymmetric quantum mechanics (SUSYQM), shape invariance approach and the alternative method. Zou et al. [50] solved the Dirac equation with equal Eckart scalar and vector potentials in terms of SUSYQM method, shape invariance approach and function analysis method. Wei and Dong [51] obtained approximately the analytical bound state solutions of the Dirac equation with the Manning-Rosen for arbitrary spin-orbit coupling quantum number κ. Thylwe [52] presented the approach inspired by amplitude-phase method for analyzing the radial Dirac equation to calculate phase shifts by including the spin- and pseudo-spin symmetries of relativistic spectra. Alhaidari [53] solved Dirac equation by separation of variables in spherical coordinates for a large class of noncentral electromagnetic potentials. Berkdemir and Sever [54] investigated systematically the pseudospin symmetry solution of the Dirac equation for spin 1/2 particles moving within the Kratzer potential connected with an angle-dependent potential. Alberto et al. [55] 3
concluded that the values of energy spectra may not depend on the spinor structure of the particle, i.e., whether one has a spin-1/2 or a spin-0 particle. Also, they showed that a spin-1/2 or a spin-0 particle with the same mass and subject to the same scalar S(r) and vector V (r) potentials of equal magnitude, i.e., S(r) = ±V (r), will have the same energy spectrum (isospectrality), including both bound and scattering states. In the present paper, our aim is to study the analytic solutions of the Dirac equation for the Rosen-Morse potential with arbitrary spin-orbit quantum number κ by using a new approximation to deal with the centrifugal term. However, we use the approximation given in Ref. [56] which is quite different from the ones used in our previous works [39,40,42], i h 1 1 e−αr 2 where d = 0 or d = 12 ≈ α d + . The approximation given in [56] is convenient 2 r2 (1−e−αr )
for the Rosen-Morse type potential because one may propose a more reasonable physical
wave functions for this system. Under the conditions of the spin symmetry S(r) ∼ V (r) and pseudospin symmetry S(r) ∼ −V (r), we investigate the bound state energy eigenvalues and corresponding upper and lower spinor wave functions in the framework of the NU method. We also show that the spin and pseudospin symmetry Dirac solutions can be reduced to the S(r) = V (r) and S(r) = −V (r) in the cases of exact spin symmetry limitation ∆(r) = 0 and pseudospin symmetry limitation Σ(r) = 0, respectively. Furthermore, the solutions obtained for the Dirac equation can be easily reduced to the Schr¨odinger solutions when the appropriate map of parameters is used. The paper is structured as follows: In Sect. 2, we outline the NU method. Section 3 is devoted to the analytic bound state solutions of the (3 + 1)-dimensional Dirac equation for the Rosen-Morse quantum system obtained by means of the NU method. The spin symmetry and pseudospin symmetry solutions are investigated. In Sect. 4, we study the cases κ = ±1 (l = e l = 0, i.e., s-wave), the Eckart-type potential, the PT-symmetric Rosen-
Morse potential. Finally, the relevant conclusions are given in Sect. 5.
II.
NU METHOD
The NU method [41] is briefly outlined here. It was proposed to solve the second-order differential equation of hypergeometric-type: ψ ′′n (r) +
σ e(r) τe(r) ′ ψ n (r) + 2 ψ n (r) = 0, σ(r) σ (r) 4
(2)
where σ(r) and σ e(r) are polynomials, at most, of second-degree, and e τ (r) is a first-degree polynomial. In order to find a particular solution for Eq. (2), let us decompose the wavefunction ψ n (r) as follows:
ψ n (r) = φ(r)yn (r),
(3)
[σ(r)ρ(r)]′ = τ (r)ρ(r),
(4)
σ(r)yn′′ (r) + τ (r)yn′ (r) + λyn (r) = 0,
(5)
τ (r) = e τ (r) + 2π(r), τ ′ (r) < 0,
(6)
and use
to reduce Eq. (2) to the form
with
where the prime denotes the differentiation with respect to r. One is looking for a family of solutions corresponding to 1 λ = λn = −nτ ′ (r) − n (n − 1) σ ′′ (r), 2
n = 0, 1, 2, · · · .
(7)
The yn (r) can be expressed in terms of the Rodrigues relation: yn (r) =
Bn dn n [σ (r)ρ(r)] , ρ(r) dr n
(8)
where Bn is the normalization constant and the weight function ρ(r) is the solution of the differential equation (4). The other part of the wavefunction (3) must satisfy the following logarithmic equation π(r) φ′ (r) = . φ(r) σ(r)
(9)
k = λ − π ′ (r).
(10)
By defining
one obtains the polynomial 1 τ (r)] ± π(r) = [σ ′ (r) − e 2
r
1 ′ [σ (r) − e τ (r)]2 − σ e(r) + kσ(r), 4
(11)
where π(r) is a parameter at most of order 1. The expression under the square root sign in the above equation can be arranged as a polynomial of second order where its discriminant
5
is zero. Hence, an equation for k is being obtained. After solving such an equation, the k values are determined through the NU method. In this regard, we derive a parametric generalization version of the NU method valid for any solvable potential by the method. We begin by writting the hypergeometric equation in general parametric form as [r (c3 − c4 r)]2 ψ ′′n (r) + [r (c3 − c4 r) (c1 − c2 r)] ψ ′n (r) + −ξ 1 r 2 + ξ 2 r − ξ 3 ψ n (r) = 0, (12)
with
e τ (r) = c1 − c2 r,
(13)
σ(r) = r (c3 − c4 r) ,
(14)
σ e(r) = −ξ 1 r 2 + ξ 2 r − ξ 3 ,
(15)
where the coefficients ci (i = 1, 2, 3, 4) and the analytic expressions ξ j (j = 1, 2, 3). Furthermore, in comparing Eq. (12) with the counterpart Eq. (2), one obtains the appropriate analytic polynomials, energy equation and wave functions together with the associated coefficients expressed in general parameteric form as displayed in Appendix A.
III.
ANALYTIC SOLUTION OF THE DIRAC-ROSEN-MORSE PROBLEM
In spherical coordinates, the Dirac equation for fermionic massive spin- 21 particles interacting with arbitrary scalar potential S(r) and the time-component V (r) of a four-vector potential can be expressed as [26,57-60]
cα · p + β Mc2 + S(r) + V (r) − E ψ nκ (r) = 0, ψ nκ (r) = ψ(r, θ, φ),
(16)
where E is the relativistic energy of the system, M is the mass of a particle, p = −i~∇ is the momentum operator, and α and β are 4 × 4 Dirac matrices, i.e., 1 0 0 −i 0 1 I 0 0 σi , (17) , σ3 = , σ2= , σ1= , β = α = 0 −1 i 0 1 0 0 −I σi 0 where I denotes the 2 × 2 identity matrix and σ i are the three-vector Pauli spin matrices.
For a spherical symmetrical nuclei, the total angular momentum operator of the nuclei J and spin-orbit matrix operator K = −β (σ · L + I) commute with the Dirac Hamiltonian, where L is the orbital angular momentum operator. The spinor wavefunctions can be classified 6
according to the radial quantum number n and the spin-orbit quantum number κ and can be written using the Pauli-Dirac representation in the following forms: l F (r)Yjm(θ, φ) fnκ (r) , = 1 nκ ψ nκ (r) = e l r iGnκ (r)Y (θ, φ) gnκ (r)
(18)
jm
where Fnκ (r) and Gnκ (r) are the radial wave functions of the upper- and lower-spinor compoe
l l nents, respectively and Yjm (θ, φ) and Yjm (θ, φ) are the spherical harmonic functions coupled
to the total angular momentum j and it’s projection m on the z axis. The orbital and pseudoorbital angular momentum quantum numbers for spin symmetry l and pseudospin symmetry e l refer to the upper- and lower-spinor components, respectively, for which l(l +1) = κ (κ + 1)
and e l(e l + 1) = κ (κ − 1). The quantum number κ is related to the quantum numbers for spin symmetry l and pseudospin symmetry e l as − (l + 1) = − j + 1 , (s , p , etc.), j = l + 1 , aligned spin (κ < 0) , 1/2 3/2 2 2 κ= 1 1 (p1/2 , d3/2 , etc.), j = l − 2 , unaligned spin (κ > 0) , +l = + j + 2 , and the quasi-degenerate doublet structure can be expressed in terms of a pseudospin angular momentum se = 1/2 and pseudo-orbital angular momentum e l which is defined as −e l = − j + 21 , (s1/2 , p3/2 , etc.), j = e l − 1/2, aligned spin (κ < 0) , κ= + e l + 1/2, unaligned spin (κ > 0) , l + 1 = + j + 1 , (d3/2 , f5/2 , etc.), j = e 2
where κ = ±1, ±2, · · · . For example, (1s1/2 , 0d3/2 ) and (2p3/2 , 1f5/2 ) can be considered as pseudospin doublets. Thus, the substitution of Eq. (18) into Eq. (16) leads to the following two radial coupled
Dirac equations for the spinor components d κ Fnκ (r) = Mc2 + Enκ − ∆(r) Gnκ (r), (19a) + dr r d κ Gnκ (r) = Mc2 − Enκ + Σ(r) Fnκ (r), (19b) − dr r where ∆(r) = V (r) − S(r) and Σ(r) = V (r) + S(r) are the difference and sum potentials, respectively. Under the spin symmetry ( i.e., ∆(r) = Cs = constant), one can eliminate Gnκ (r) in Eq. (19a), with the aid of Eq. (19b), to obtain a second-order differential equation for the upper-spinor component as follows [16,26]: κ (κ + 1) 1 d2 2 + 2 2 Mc + Enκ − Cs Σ(r) Fnκ (r) − 2+ dr r2 ~ c 7
1 2 2 4 2 E − M c + C Mc − E Fnκ (r), (20) s nκ nκ ~ 2 c2 where κ (κ + 1) = l (l + 1) , κ = l for κ < 0 and κ = − (l + 1) for κ > 0. The spin symmetry =
energy eigenvalues depend on n and κ, i.e., Enκ = E(n, κ (κ + 1)). For l 6= 0, the states with j = l ± 1/2 are degenerate. Further, the lower-spinor component can be obtained from Eq. (19a) as
where Enκ
d κ 1 Fnκ (r), (21) + Gnκ (r) = Mc2 + Enκ − Cs dr r 6 −Mc2 , only real positive energy states exist when Cs = 0 (exact spin symme=
try). On the other hand, under the pseudospin symmetry ( i.e., Σ(r) = Cps = constant), one can eliminate Fnκ (r) in Eq. (19b), with the aid of Eq. (19a), to obtain a second-order differential equation for the lower-spinor component as follows [16,26]: d2 κ (κ − 1) 1 2 − 2+ − 2 2 Mc − Enκ + Cps ∆(r) Gnκ (r) dr r2 ~c 1 2 = 2 2 Enκ − M 2 c4 − Cps Mc2 + Enκ Gnκ (r), ~c and the upper-spinor component Fnκ (r) is obtained from Eq. (19b) as d κ 1 Gnκ (r), − Fnκ (r) = Mc2 − Enκ + Cps dr r
(22)
(23)
where Enκ 6= Mc2 , only real negative energy states exist when Cps = 0 (exact pseudospin
symmetry). From the above equations, the energy eigenvalues depend on the quantum numbers n and κ, and also the pseudo-orbital angular quantum number e l according to
κ(κ − 1) = e l(e l + 1), which implies that j = e l ± 1/2 are degenerate for e l 6= 0. The quantum
condition is obtained from the finiteness of the solution at infinity and at the origin point, i.e., Fnκ (0) = Gnκ (0) = 0 and Fnκ (∞) = Gnκ (∞) = 0. At this stage, we take the vector and scalar potentials in the form of Rosen-Morse potential model (see Eq. (1)). Equations (20) and (22) can be solved exactly for κ = 0, −1 and κ = 0, 1, respectively, because of the spin-orbit centrifugal and pseudo-centrifugal terms. Therefore, to find approximate solution for the radial Dirac equation with the Rosen-Morse potential, we have to use an approximation for the spin-orbit centrifugal term. For values of κ that are not large and vibrations of the small amplitude about the minimum, Lu [56] has introduced an approximation to the centrifugal term near the minimum point r = re as " 2 # − exp(−2αr) 1 − exp(−2αr) 1 , (24) ≈ 2 D0 + D1 + D2 r2 re 1 + exp(−2αr) 1 + exp(−2αr) 8
where
2 1 + exp(−2αre ) 8αre D0 = 1 − − (3 + 2αre ) , 2αre 1 + exp(−2αre ) 1 + exp(−2αre ) 1 + exp(−2αre ) − (3 + 2αre ) , D1 = −2 (exp(2αre ) + 1) 3 2αre 2αre 2 1 + exp(−2αre ) 4αre 2 D2 = (exp(2αre ) + 1) 3 + 2αre − , (25) 2αre 1 + exp(−2αre )
and higher order terms are neglected.
A.
Spin symmetry solution of the Rosen-Morse Problem
We take the sum potential in Eq. (20) as the Rosen-Morse potential model, i.e., Σ(r) = −4V1
(1 − exp(−2αr)) exp(−2αr) . 2 + V2 (1 + exp(−2αr)) (1 + exp(−2αr))
(26)
The choice of Σ(r) = 2V (r) → V (r) as mentioned in Ref. [12] enables one to reduce the resulting relativistic solutions into their non-relativistic limit under appropriate transformations. . Using the approximation given by Eq. (24) and introducing a new parameter change z(r) = − exp(−2αr), this allows us to decompose the spin-symmetric Dirac equation (20) into the Schr¨odinger-like equation in the spherical coordinates for the upper-spinor component Fnκ (r), 2 d (1 − z) d (−β 1 z 2 + β 2 z − ε2nκ ) Fnκ (z) = 0, Fnκ (0) = Fnκ (1) = 0, + + dz 2 z (1 − z) dz z 2 (1 − z)2 with 1 = 2α
s
ω 1 D0 + 2 2 (Mc2 + Enκ − Cs ) (Mc2 − Enκ + V2 ) > 0, 2 re ~ c 1 ω 1 2 2 (D0 + D1 + D2 ) + 2 2 Mc + Enκ − Cs Mc − Enκ − V2 , β1 = 2 4α re2 ~ c 1 2 ω 2 2 β2 = 2 (2D0 + D1 ) + 2 2 Mc + Enκ − Cs Mc − Enκ − 2V1 , 4α re2 ~c εnκ
(27)
(28a) (28b) (28c)
where ω = κ (κ + 1) .
In order to solve Eq. (27) by means of the NU method, we should compare it with Eq. (2) to obtain the following particular values for the parameters: e τ (z) = 1 − z, σ(z) = z (1 − z) , σ e(z) = −β 1 z 2 + β 2 z − ε2nκ . 9
(29)
Comparing Eqs. (13)-(15) with Eq. (29), we can easily obtain the coefficients ci (i = 1, 2, 3, 4) and the analytic expressions ξ j (j = 1, 2, 3). However, the values of the coefficients ci (i = 5, 6, · · · , 16) are found from the relations A1-A5 of Appendix A. Therefore, the specific values of the coefficients ci (i = 1, 2, · · · , 16) together with ξ j (j = 1, 2, 3) are displayed in Table 1. From the relations A6 and A7 of Appendix A together with the coefficients in Table 1, the selected forms of π(z) and k take the following particular values π(z) = εnκ − (1 + εnκ + δ) z,
(30)
k = β 2 − 2ε2nκ + (2δ + 1) εnκ ,
respectively, where 1 δ= 2
−1 +
(31)
s
! 4V1 ωD2 1 + 2 2 + 2 2 2 (Mc2 + Enκ − Cs ) , α re α ~ c
(32)
for bound state solutions. According to the NU method, the relations A8 and A9 of Appendix A give τ (z) = 1 + 2εnκ − (3 + 2εnκ + 2δ) z, τ ′ (r) = − (3 + 2εnκ + 2δ) < 0,
(33)
with prime denotes the derivative with respect to z. In addition, the relation A10 of Appendix A gives the energy equation for the Rosen-Morse potential in the Dirac theory as Mc2 + Enκ − Cs + α 2 ~ 2 c2
"
ωD0 Mc2 − Enκ + V2 = − 2 ~2 c2 re ω(D1 +D2 ) 4α2 re2
− 2α2V~22 c2 (Mc2 + Enκ − Cs ) + (n + δ + 1)
#2
− (n + δ + 1)
.
(34)
Further, for the exact spin symmetry case, V (r) = S(r) or Cs = 0, we obtain Mc2 + Enκ
with
+ α 2 ~ 2 c2
− 2α2V~22 c2
ωD0 Mc2 − Enκ + V2 = − 2 ~2 c2 re 2
(Mc + Enκ − Cs ) + n +e δ+1
ω(D1 +D2 ) 4α2 re2
e δ = δ(Cs → 0). 10
2 − n+e δ+1 ,
(35)
(36)
Let us now find the corresponding wave functions for this model. Referring to Table 1 and the relations A11 and A12 of Appendix A, we find the functions: ρ(z) = z 2εnκ (1 − z)2δ+1 ,
(37)
φ(r) = z εnκ (1 − z)δ+1 .
(38)
Hence, the relation A13 of Appendix A gives yn (z) = An z
−2εnκ
(1 − z)
−(2δ+1)
i dn h n+2εnκ n+2δ+1 ∼ Pn(2εnκ ,2δ+1) (1 − 2z), z ∈ [0, 1], z (1 − z) n dz (39) (µ,ν)
where the Jacobi polynomial Pn
(x) is defined only for µ > −1, ν > −1, and for the
argument x ∈ [−1, +1] . By using Fnκ (z) = φ(z)yn (z), we get the radial upper-spinor wave functions from the relation A14 as Fnκ (z) = Nnκ z εnκ (1 − z)δ+1 Pn(2εnκ ,2δ+1) (1 − 2z) = Nnκ (exp(−2αr))εnκ (1 − exp(−2αr))δ+1 × 2 F1 (−n, n + 2 (εnκ + δ + 1) ; 2εnκ + 1; − exp(−2αr)) .
(40)
The above upper-spinor component satisfies the restriction condition for the bound states, i.e., δ > 0 and εnκ > 0. The normalization constants Nnκ are calculated in Appendix B. Before presenting the corresponding lower-component Gnκ (r), let us recall a recurrence relation of hypergeometric function, which is used to solve Eq. (21) and present the corresponding lower component Gnκ (r), i ab d h F (a; b; c; z) = 2 F1 (a + 1; b + 1; c + 1; z) , dz 2 1 c
(41)
with which the corresponding lower component Gnκ (r) can be obtained as follows 2α (δ + 1) exp(−2αr) κ Nnκ (− exp(−2αr))εnκ (1 + exp(−2αr))δ+1 −2αεnκ − + Gnκ (r) = (Mc2 + Enκ − Cs ) (1 + exp(−2αr)) r "
× 2 F1 (−n, n + 2 (εnκ + δ + 1) ; 2εnκ + 1; − exp(−2αr))
2αn [n + 2 (εnκ + δ + 1)] (− exp(−2αr))εnκ +1 (1 + exp(−2αr))δ+1 +Nnκ (2εnκ + 1) (Mc2 + Enκ − Cs ) 3 ; 2 (εnκ + 1) ; − exp(−2αr) , × 2 F1 −n + 1; n + 2 εnκ + δ + 2 11
# (42)
where Enκ 6= −Mc2 for exact spin symmetry. Here, it should be noted that the hypergeometric series
2 F1
(−n, n + 2 (εnκ + δ + 1) ; 2εnκ + 1; − exp(−2αr)) does not terminate for
n = 0 and thus does not diverge for all values of real parameters δ and εnκ . For Cs > Mc2 + Enκ and Enκ < Mc2 + V2 or Cs < Mc2 + Enκ and Enκ > Mc2 + V2 , we note that parameters given in Eq. (28a) turn to be imaginary, i.e., ε2nκ < 0 in the s-state (κ = −1). As a result, the condition of existing bound states are εnκ > 0 and δ > 0, that
is to say, in the case of Cs < Mc2 + Enκ and Enκ < Mc2 + V2 , bound-states do exist for some quantum number κ such as the s-state (κ = −1). Of course, if these conditions are satisfied for existing bound-states, the energy equation and wave functions are the same as these given in Eq. (34) and Eqs. (40) and (42).
B.
Pseudospin symmetry solution of the Rosen-Morse Problem
Now taking the difference potential in Eq. (22) as the Rosen-Morse potential model, i.e., ∆(r) = −4V1
(1 − exp(−2αr)) exp(−2αr) , 2 + V2 (1 + exp(−2αr)) (1 + exp(−2αr))
(43)
leads us to obtain a Schr¨odinger-like equation for the lower-spinor component Gnκ (r), 2 e z2 + β e z −e − β ε 2 1 2 nκ d + (1 − z) d + Gnκ (z) = 0, Gnκ (0) = Gnκ (1) = 0, (44) 2 dz 2 z (1 − z) dz z 2 (1 − z) where e εnκ
1 = 2α
e = 1 β 1 4α2
e = 1 β 2 4α2
s ω e re2 ω e re2
ω e 1 2 − M 2 c4 − (Mc2 + E ) C + (Mc2 − E D0 − 2 2 [Enκ nκ ps nκ + Cps ) V2 ] > 0, 2 re ~c (45a) 1 2 2 4 2 2 (D0 + D1 + D2 ) − 2 2 Enκ − M c − Mc + Enκ Cps − Mc − Enκ + Cps V2 , ~c (45b) 2 2 (2D0 + D1 ) − 2 2 Enκ − M 2 c4 − Mc2 + Enκ Cps − 2 Mc2 − Enκ + Cps V1 , ~ c (45c)
and ω e = κ (κ − 1) . To avoid repetition in the solution of Eq. (44), a first inspection e ,β e ) and the previous for the relationship between the present set of parameters (e εnκ , β 1 2
set (εnκ , β 1 , β 2 ) tells us that the negative energy solution for pseudospin symmetry, where
S(r) = −V (r), can be obtained directly from those of the positive energy solution above for 12
spin symmetry using the parameter map [57-59]: Fnκ (r) ↔ Gnκ (r), V (r) → −V (r) (or V1 → −V1 and V2 → −V2 ), Enκ → −Enκ and Cs → −Cps . (46) Following the previous results with the above transformations, we finally arrive at the energy equation Mc2 − Enκ + Cps + α 2 ~ 2 c2
"
V2 2α2 ~2 c2
ω e D0 Mc2 + Enκ − V2 = − 2 ~2 c2 re
(Mc2 − Enκ + Cps ) +
ω e (D1 +D2 ) 4α2 re2
(n + δ 1 + 1)
where 1 δ1 = 2
−1 +
#2
− (n + δ 1 + 1)
,
s
! 4V1 ω e D2 1 + 2 2 − 2 2 2 (Mc2 − Enκ + Cps ) . α re α ~c
(47)
(48)
By using Gnκ (z) = φ(z)yn (z), we get the radial lower-spinor wave functions as enκ (exp(−2αr))eεnκ (1 − exp(−2αr))δ1 +1 P (2eεnκ ,2δ1 +1) (1 − 2 exp(−2αr)). Gnκ (r) = N n
(49)
The above upper-spinor component satisfies the restriction condition for the bound states, enl are calculated in Appendix B. i.e., δ 1 > 0 and e εnκ > 0. The normalization constants N IV.
DISCUSSIONS
In this section, we are going to study four special cases of the energy eigenvalues given by Eqs. (34) and (47) for the spin and pseudospin symmetry, respectively. First, let us study s-wave case l = 0 (κ = −1) and e l = 0 (κ = 1) case " #2 V2 2 (Mc + E − C ) n,−1 s 2 2 2 2 2 2 2 2 Mc + En,−1 − Cs Mc − En,−1 + V2 = α ~ c 2α ~ c + n + δ2 + 1 , n + δ2 + 1 (50) where 1 δ2 = 2
−1 +
! 4V1 1 + 2 2 2 (Mc2 + En,−1 − Cs ) . α ~ c
r
(51)
If one sets Cs = 0 into Eq. (50) and Cps = 0 into Eq. (47), we obtain for spin and pseudospin symmetric Dirac theory, Mc2 + En,−1
Mc2 − En,−1 + V2 = α2 ~2 c2
"
V2 2α2 ~2 c2
(Mc2 + En,−1 ) + n + δ −1 + 1 n + δ −1 + 1
#2
,
(52) 13
δ −1
1 = 2
−1 +
r
! 4V1 1 + 2 2 2 (Mc2 + En,−1 ) , α ~c
(53)
and Mc2 − En,+1
"
Mc2 + En,+1 − V2 = +α2 ~2 c2 − δ +1 =
1 2
−1 +
r
1−
4V1 2 α ~ 2 c2
V2 2α2 ~2 c2
(Mc2 − En,+1 ) + n + δ +1 + 1 n + δ +1 + 1
#2
,
(54)
!
(Mc2 − En,+1 ) .
(55)
respectively. The above solutions for the s-wave are found to be identical for spin and pseudospin cases S(r) = V (r) and S(r) = −V (r), respectively. Second, when we set V1 → −V1 and V2 → −V2 the potential reduces to the Eckart-type potential and energy eigenvalues are given by
2
Mc + En,−1
2
2 2 2
Mc − En,−1 − V2 = α ~ c δ−1 =
1 2
−1 +
r
1−
"
−
V2 2α2 ~2 c2
(Mc2 + En,−1 ) + n + δ −1 + 1 n + δ −1 + 1
4V1 2 α ~ 2 c2
#2
,
(56)
!
(Mc2 + En,−1 ) ,
(57)
for spin symmetry and Mc2 − En,+1
Mc2 + En,+1 + V2 = +α2 ~2 c2
δ+1 =
1 2
−1 +
r
1+
"
V2 2α2 ~2 c2
(Mc2 − En,+1 ) + n + δ +1 + 1 n + δ +1 + 1
4V1 2 α ~ 2 c2
#2
,
(58)
!
(Mc2 − En,+1 ) ,
(59)
for pseudospin symmetry. Third, let us now discuss the non-relativistic limit of the energy eigenvalues and wave functions of our solution. If we take Cs = 0 and put S(r) = V (r) = Σ(r), the non-relativistic limit of energy equation (34) and wave functions (40) under the following appropriate transformations (Mc2 + Enκ ) /~2 c2 → 2µ/~2 and Mc2 − Enκ → −Enl [26,57,42] become 2
Enl = V2 +
ωD0 2 ~ 2 ~ − α 2µre2 2µ
"
µ α2 ~2
(2V1 + V2 ) −
with 1 e δ0 = 2
−1 +
s
ωD1 4α2 re2
+ (n + 1)2 + (2n + 1) e δ0
n +e δ0 + 1
8µV1 ωD2 1+ 2 2 + 2 2 α ~ α re
14
!
,
#2
,
(60)
(61)
and the associated wave functions are e (2εnκ ,2eδ0 +1) (1 − 2 exp(−2αr)), Fnl (r) = Nnl (exp(−2αr))εnκ (1 − exp(−2αr))1+δ0 Pn
where 1 εnl = 2α
s
ω 2µ D0 + 2 (V2 − Enl ) > 0, ω = l(l + 1), 2 re ~
(62)
(63a)
which are identical with Ref. [25] in the solution of the Schr¨odinger equation. Finally, the Jacobi polynomials can be expressed in terms of the hypergeometric function as Pn(µ,ν) (1 − 2 exp(−2αr)) =
(µ + 1)n 2 F1 (−n, 1 + µ + ν + n; µ + 1; exp(−2αr)) , n!
(64)
where z ∈ [0, 1] which lie within or on the boundary of the interval [−1, 1] . Fourth, if we choose V2 → iV2 , the potential becomes the P T -symmetric Rosen-Morse potential, where P denotes parity operator and T denotes time reversal.
For a potential
V (r), making the transformation of r − r (or r → ξ − r) and i → −i, if we have the relation V (−r) = V ∗ (r)), the potential V (r) is said to be P T -symmetric [43]. In this case we obtain for spin-symmetric Dirac equation Mc2 + Enκ
ωD0 Mc2 − Enκ + iV2 = − 2 ~2 c2 re
2 2 − iV 2 2 2 (Mc + Enκ − Cs ) + 2 2 2 2α ~ c +α ~ c e n+δ+1
ω(D1 +D2 ) 4α2 re2
In the non-relativistic limit, it turns to become ~2 2 ω~2 D0 − α Enl = iV2 + 2µre2 2µ
"
µ α2 ~2
(2V1 + iV2 ) −
ωD1 4α2 re2
2 − n+e δ+1 .
+ (n + 1)2 + (2n + 1) e δ0
n+e δ0 + 1
(65)
#2
,
(66)
where real V1 > 0, which is identical to the results of Ref. [25]. If one sets l = 0 in the above equation, the result is identical with that of Refs. [44,45].
V.
CONCLUSIONS
We have obtained analytically the energy spectra and corresponding wave functions of the Dirac equation for the Rosen-Morse potential under the conditions of the spin symmetry and pseudospin symmetry in the context of the Nikiforov-Uvarov method. For any spin-orbit 15
coupling centrifugal term κ, we have found the explicit expressions for energy eigenvalues and associated wave functions in closed form. The most stringent interesting result is that the present spin and pseudospin symmetry cases can be easily reduced to the KG solution once S(r) = V (r) and S(r) = −V (r) (i.e., Cs = Cps = 0) [55]. The resulting solutions of the wave functions are being expressed in terms of the generalized Jacobi polynomials. Obviously, the relativistic solution can be reduced to it’s non-relativistic limit by the choice of appropriate mapping transformations. Also, in case when spin-orbit quantum number κ = 0, the problem reduces to the s-wave solution. The s-wave Rosen-Morse, the Eckarttype potential, the PT-symmetric Rosen-Morse potential.and the non-relativistic cases are briefly studied.
Acknowledgments
The partial support provided by the Scientific and Technological Research Council of ¨ ITAK) ˙ Turkey (TUB is highly appreciated. The author thanks the anonymous kind referees and editors for the very constructive comments and suggestions.
16
APPENDIX A: PARAMETRIC GENERALIZATION OF THE NU METHOD
Our systematical derivation holds for any potential form. (i) The relevant coefficients ci (i = 5, 6, · · · , 16) are given as follows: c5 =
1 1 (c3 − c1 ) , c6 = (c2 − 2c4 ) , c7 = c26 + ξ 1 , 2 2
c8 = 2c5 c6 − ξ 2 , c9 = c25 + ξ 3 , c10 = c4 (c3 c8 + c4 c9 ) + c23 c7 , 2 √ 2√ c9 , c12 = c10 , c3 c3 c4 √ 1 √ 1 = (c5 + c9 ) , c14 = ( c10 − c4 c5 − c3 c6 ) , c3 c3 c4 2√ 1 √ c15 = c10 , c16 = ( c10 − c4 c5 − c3 c6 ) . c3 c3 c11 =
c13
(A1) (A2) (A3) (A4) (A5)
(ii) The analytic results for the key polynomials: π(r) = c5 +
√
c9 −
√ √ 1 (c4 c9 + c10 − c3 c6 ) r, c3
√ 1 (c3 c8 + 2c4 c9 + 2 c9 c10 ) , 2 c3 √ √ √ 2 τ (r) = c3 + 2 c9 − (c3 c4 + c4 c9 + c10 ) r, c3 √ √ 2 τ ′ (r) = − (c3 c4 + c4 c9 + c10 ) < 0. c3 k=−
(A6) (A7) (A8) (A9)
(iii) The energy equation: c2 n−(2n + 1) c6 +
√ √ √ 1 1 (2n + 1) ( c10 + c4 c9 )+n (n − 1) c4 + 2 (c3 c8 + 2c4 c9 + 2 c9 c10 ) = 0. c3 c3 (A10)
(iv) The wave functions: ρ(r) = r c11 (c3 − c4 r)c12 ,
(A11)
φ(r) = r c13 (c3 − c4 r)c14 , c13 > 0, c14 > 0,
(A12)
ynκ (r) = Pn(c11 ,c12 ) (c3 − 2c4 r), c11 > −1, c12 > −1, r ∈ [(c3 − 1)/2c4 , (1 + c3 )/2c4 ] , (A13) ψ nκ (r) = φ(r)ynκ (r) = Nn r c13 (c3 − c4 r)c14 Pn(c11 ,c12 ) (c3 − 2c4 r), (a,b)
where Pn
(c3 − 2c4 r) are the Jacobi polynomials and Nn is a normalizing factor.
17
(A14)
When c4 = 0, the Jacobi polynomial turn to be the generalized Laguerre polynomial and the constants relevant to this polynomial change are lim Pn(c11 ,c12 ) (c3 − 2c4 r) = Lcn11 (c15 r),
(A15)
lim (c3 − c4 r)c14 = exp(−c16 r),
(A16)
ψ nκ (r) = Nn exp(−c16 r)Lcn11 (c15 r),
(A17)
c4 →0
c4 →0
where Lcn11 (c15 r) are the generalized Laguerre polynomials and Nn is a normalizing constant.
APPENDIX B: NORMALIZATION OF THE RADIAL WAVE FUNCTION
In order to find the normalization factor Nnκ , we start by writting the normalization condition:
2 Z 1 2 Nnκ z 2εnκ −1 (1 − z)2δ+2 Pn(2εnκ ,2δ+1) (1 − 2z) dz = 1. (B1) 2α 0 Unfortunately, there is no formula available to calculate this key integration. Neveretheless,
we can find the explicit normalization constant Nnl . For this purpose, it is not difficult to obtain the results of the above integral by using the following formulas [61-64] Z1
(1 − s)µ−1 sν−1 2 F1 (α, β; γ; as) ds =
Γ(µ)Γ(ν) F (ν, α, β; µ + ν; γ; a) , Γ(µ + ν) 3 2
(B2)
0
and
2 F1
(a, b; c; z) =
Γ(c) Γ(a)Γ(b)
∞ X Γ(a+p)Γ(b+p) Γ(c+p)
zp . p!
Hence, the normalization constants for the
p=0
upper-spinor component are #−1/2 " ∞ Γ(2δ + 3)Γ(2εnκ + 1) X (−1)m (n + 2(1 + εnκ + δ))m Γ(n + m) fnκ , (B3) Nnκ = 2αΓ(n) m! (m + 2εnκ )!Γ m + 2 εnκ + δ + 23 m=0 with
fnκ
3 ; 1 + 2εnκ ; 1 , (B4) = 3 F2 2εnκ + m, −n, n + 2(1 + εnκ + δ); m + 2 εnκ + δ + 2
where (x)m = Γ(x + m)/Γ(x). Also, the normalization constants for the lower-spinor component are #−1/2 " ∞ m X (−1) (n + 2(1 + e ε + δ )) Γ(n + m) Γ(2δ + 3)Γ(2e ε + 1) nκ 1 1 nκ m enκ = gnκ , N 3 2αΓ(n) m! (m + 2e ε )!Γ m + 2 e ε nκ nκ + δ 1 + 2 m=0 (B5) 18
with gnκ
3 = 3 F2 2e εnκ + m, −n, n + 2(1 + e εnκ + δ 1 ); m + 2 e εnκ + δ 1 + ; 1 + 2e εnκ ; 1 . (B6) 2
19
[1] J.N. Ginocchio, Phys. Rev. C 69, 034318 (2004). [2] J.N. Ginocchio, Phys. Rev. Lett. 78, 436 (1997). [3] J.N. Ginocchio, Phys. Rep. 414, 165 (2005). [4] P.R. Page, T. Goldman and J.N. Ginocchio, Phys. Rev. Lett. 86, 204 (2001). [5] A. Arima, M. Harvey and K. Shimizu, Phys. Lett. B 30, 517 (1969). [6] K.T. Hecht and A. Adler, Nucl. Phys. A 137, 129 (1969). [7] J.N. Ginocchio and D.G. Madland, Phys. Rev. C 57, 1167 (1998). [8] A. Bohr, I. Hamarnoto and B.R. Motelson, Phys. Scr. 26, 267 (1982). [9] J. Dudek, W. Nazarewicz, Z. Szymanski and G.A. Leander, Phys. Rev. Lett. 59, 1405 (1987). [10] D. Troltenier, C. bahri and J. P. Draayer, Nucl. Phys. A 586, 53 (1995). [11] J. Meng, K. Sugawara-Tanabe, S. Yamaji, P. Ring and A. Arima, Phys. Rev. C 58, R628 (1998). [12] A.D. Alhaidari, H. Bahlouli and A. Al-Hasan, Phys. Lett. A 349, 87 (2006). [13] J.N. Ginocchio, Phys. Rev. Lett. 95, 252501 (2005). [14] J.Y. Gou, X.Z. Fang and F.X. Xu, Nucl. Phys. A 757, 411 (2005). [15] R. Lisboa, M. Malheiro, A.S. De Castro, P. Alberto and M. Fiolhais, Phys. Rev. C 69, 024319 (2004); A.S. De Castro, P. Alberto, R. Lisboa and M. Malheiro, Phys. Rev. C 73, 054309 (2006). [16] J.Y. Gou and Z.Q. Sheng, Phys. Lett. A 338, 90 (2005); S.M. Ikhdair and R. Sever, [DOI:10.2478/s11534-009-0118-5] to appear in Cent. Eur. J. Phys. (2010). [17] W.C. Qiang, R.S. Zhou, Y. Gao and J. Phys. A: Math. Theor. 40, 1677 (2007). [18] O. Bayrak and I. Boztosun, J. Phys. A: Math. Theor. 40, 11119 (2007). [19] A. Soylu, O. Bayrak and I. Boztosun, J. Math. Phys. 48, 082302 (2007). [20] A. Soylu, O. Bayrak and I. Boztosun, J. Phys. A: Math. Theor. 41, 065308 (2008). [21] C.S. Jia, P. Gao and X.L. Peng, J. Phys. A: Math. Gen. 39, 7737 (2006). [22] L.H. Zhang, X.P. Li and C.S. Jia, Phys. Lett. A 372, 2201 (2008). [23] C.S. Jia, J.Y. Liu, L. He and L. Sun, Phys. Scr. 75, 388 (2007).
20
[24] C.S. Jia, P. Gao, Y.F. Diao, L.Z. Yi and X.J. Xie, Eur. Phys. J. A 34, 41 (2007). [25] F. Ta¸skin, Int. J. Theor. Phys. 48, 1142 (2009). [26] S.M. Ikhdair, Solutions of the Dirac equation for the generalized Morse potential by NikiforovUvarov method, submitted to Eur. Phys. J. D (2009). [27] R. Ko¸c and M. Koca, J. Phys. A: Math. Gen. 36, 8105 (2003). [28] C.-S. Jia and A. de Souza Dutra, Annals Phys. 323, 566 (2008); C.S. Jia, P.Q. Wang, J.Y. Liu and S. He, Int. J. Theor. Phys. 47, 2513 (2008). [29] J. Yu, S.-H. Dong and G.-H. Sun, Phys. Lett. A322, 290 (2004); C. Gang, Phys. Lett. A 329, 22 (2004); C. Berkdemir, Nucl. Phys. A 770, 32 (2006). [30] Y. Xu, S. He and C.-S. Jia, J. Phys. A: Math. Theor. 41, 255302 (2008); H. Akcay, J. Phys. A: Math. Theor. 42, 198001 (2009). [31] A. de Souza Dutra and C.-S. Jia, Phys. Lett. A 352, 484 (2006). [32] G. Chen and Z.D. Chen, Phys. Lett. A 331, 312 (2004). [33] R. Sever and C. Tezcan, Int. J. Mod. Phys. E 17, 1327 (2008). [34] S.M. Ikhdair and R. Sever, Int. J. Mod. Phys. C 20 (3), 361 (2009). [35] A. de Souza Dutra and C.A.S. Almeida, Phys. Lett. A 275, 25 (2000). [36] A.D. Alhaideri, Phys. Lett. A 322, 72 (2004). [37] I.O. Vakarchuk, J. Phys. A 38, 4727 (2005). [38] A. Arda, R. Sever and C. Tezcan, Phys. Scr. 79, 015006 (2009). [39] S.M. Ikhdair and R. Sever, Phys. Scr. 79, 035002 (2009). [40] S.M. Ikhdair, Eur. Phys. J. A 39, 307 (2009); S.M. Ikhdair, Eur. Phys. J. A 40 (2), 143 (2009). [41] A.F. Nikiforov and V.B. Uvarov, Special Functions of Mathematical Physics (Birkhauser, Bassel, 1988). [42] S.M. Ikhdair, Int. J. Mod. Phys. C 20 (1), 25 (2009); S.M. Ikhdair and R. Sever, J. Math. Chem. 42, 461 (2007); S.M. Ikhdair and R. Sever, Ann. Phys. (Leibzig) 16, 218 (2007); S.M. Ikhdair and R. Sever, Int. J. Theor. Phys. 46, 1643, 2384 (2007); S.M. Ikhdair and R. Sever, Int. J. Mod. Phys. C 19, 221 (2008); S.M. Ikhdair and R. Sever, Int. J. Mod. Phys. C 19, 1425 (2008); S.M. Ikhdair and R. Sever, Int. J. Mod. Phys. E 17, 1107 (2008); S.M. Ikhdair and R. Sever, Ann. Phys. (Berlin) 18, 189, 747 (2009); S.M. Ikhdair and R. Sever, J. Math. Chem. 41 (4), 329 (2007); 343 (2007); S.M. Ikhdair, Chin. J. Phys. 46, 291 (2008); S.M. Ikhdair and R. Sever, Cent. Eur. J. Phys. 6, 141 (2008); S.M. Ikhdair and R. Sever, J. Math. Chem. 45
21
(4), 1137 (2009). [43] N. Rosen and P.M. Morse, Phys. Rev. 42, 210 (1932). [44] C.-S. Jia, S.-C. Li, Y. Li and L.-T. Sun, Phys. Lett. A 300, 115 (2002). [45] C.-S. Jia, Y. Li, Y. Sun, J.-Y. Liu and L.-T. Sun, Phys. Lett. A 311, 115 (2003). [46] C.B. Compean and M. Kirchbach, J. Phys. A: Math. Gen. 39, 547 (2006). [47] Z.Q. Ma, A. Gonzalez-Cisneros, B.-W. Xu and S.H. Dong, Phys. Lett. A 371, 180 (2007). [48] C.-S. Jia, T. Chen and L.-G. Cui, Phys. Lett. A 373, 1621 (2009). [49] X.-C. Zhang, Q.-W. Liu, C.-S. Jia and L.-Z. Wang, Phys. Lett. A 340, 59 (2005. [50] X. Zou, L.-Z. Yi and C.-S. Jia, Phys. Lett. A 346, 54 (2005). [51] G.F. Wei and S.H. Dong, Phys. Lett. A 373, 49 (2008). [52] K.-E. Thylwe, Phys. Scr. 77, 065005 (2008). [53] A.D. Alhaidari, Annals Phys. 320, 453 (2005). [54] C. Berkdemir and R. Sever, J. Phys. A: Math. Theor. 41, 045302 (2008). [55] P. Alberto, A.S. de Castro and M. Malheiro, Phys. Rev. C 75, 047303 (2007). [56] J. Lu, Phys. Scr. 72, 349 (2005). [57] S.M. Ikhdair and R. Sever, to appear in J. Math. Chem. (2010). [58] C. Berkdemir and Y.-F. Cheng, Phys. Scr. 79, 035003 (2009). [59] A. De Souza Dutra and M. Hott, Phys. Lett. A 356, 215 (2006). [60] W. Greiner, Relativistic Quantum Mechanics (Springer, Verlag, 1981). [61] I.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series, and Products, 5th ed. (Academic, New York, 1994). [62] G. Sezgo, Orthogonal Polynomials (American Mathematical Society, New York, 1939). [63] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964). [64] S.M. Ikhdair, Int. J. Mod. Phys. C 20 (10), 1563 (2009); S.M. Ikhdair, Chem. Phys. 361 (1-2), 9 (2009).
22
TABLE I: The specific values for the parametric constants necessary for calculating the energy eigenvalues and eigenfunctions of the spin symmetry Dirac wave equation. Constant Analytic value Constant Analytic value c1
1
c2
1
c3
1
c4
1
c5
0
c6
− 12
c7
1 4
c8
−β 2
c9
ε2nκ
c10
c11
2εnκ
c12 = c15 2δ + 1
c13
εnκ
c14 = c16 δ + 1
ξ1
β1
ξ2
ξ3
ε2nκ
+ β1
23
δ+
β2
1 2 2
