Chinese Physics B
Solutions of the Duffin—Kemmer—Petiau equation
in the presence of Hulthén potential in (1+2)
dimensions for unity spin particles using the
asymptotic iteration method
To cite this article: Z. Molaee et al 2013 Chinese Phys. B 22 060306
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-Solutions of the Duffin Kemmer Petiau equation in the presence of
Hulth´en potential in (1+2) dimensions for unity spin particles using
the asymptotic iteration method
Z. Molaeea)†, M. K. Baharb)c), F. Yasukb), and H. Hassanabadid)
a)Department of Basic Sciences, Garmsar Branch, Islamic Azad University, Garmsar, Iran b)Department of Physics, Erciyes University, 38039, Kayseri, Turkey c)Department of Physics, Karamanoglu Mehmetbey University, 70100, Karaman,Turkey d)Department of Basic Sciences, Shahrood Branch, Islamic Azad University, Shahrood, Iran
(Received 28 June 2012; revised manuscript received 4 October 2012)
The relativistic Duffin–Kemmer–Petiau equation in the presence of Hulth´en potential in (1+2) dimensions for spin-one particles is studied. Hence, the asymptotic iteration method is used for obtaining energy eigenvalues and eigenfunctions. Keywords: Duffin–Kemmer–Petiau equation, Hulth´en potential, asymptotic iteration method
PACS: 03.65.Pm, 03.65.Ca, 98.80.cq. DOI:10.1088/1674-1056/22/6/060306
1. Introduction
The first order Duffin–Kemmer–Petiau (DKP) equation describes spin-zero and spin-one particles.[1–3]The DKP for-malism has been studied to analyze relativistic interactions of spin-one and spin-zero hadrons with nuclei. In analogy with Dirac phenomenology for proton–nucleus scattering, the DKP formalism is in better agreement with the experimental data for a phenomenological treatment of the elastic meson–nucleus scattering than Proca and KG theory.[4,5]There is a renewed
interest in considering the DKP equation in the presence of interactions. Recently, many articles have been devoted to investigating the DKP theory under different types of poten-tials, hence we may cite Refs. [6]–[17], also this equation ex-plains the quark confinement problem of quantum chromody-namics theory[18]and we may cite the papers on the meson– nuclear interaction.[19]Moreover, the relativistic model of α– nucleus elastic scattering has been treated by the formalism of the DKP theory[20] and the covariant Hamiltonian[21] in the causal approach.[22,23]In addition, there has been an increas-ing interest in the DKP oscillator.[24–29]Since the wave func-tion includes all the necessary informafunc-tion about considering systems, the energy eigenvalues and corresponding eigenfunc-tions between interacting systems in relativistic quantum me-chanics and in non-relativistic quantum meme-chanics have been studied more efficiently in recent years. The main purpose of the present work is to study the DKP equation in the pres-ence of the Hulth´en potential in (1+2)-dimensional space-time for spin-one particles by asymptotic iteration method (AIM) which is based on solving the second-order differential equa-tions, where it has been used for solving the Schr¨odinger, Dirac, DKP, and Klein–Gordon wave equations in the
pres-ence of different types of potentials. The Hulth´en potential is a short range potential and its form is[30]
V(r) = −ze2δ e
−δ r
1 − e−δ r,
where δ is a screening parameter that is used for determining the range of the Hulth´en potential and z is an atomic number. Therefore, the Hulth´en potential is a particular case of Eckart potential which explains the interaction between two atoms or molecular structure and nuclear interaction.[31–35]Moreover, the Hulth´en potential and its different forms are used in non-relativistic and non-relativistic spaces.[36–42]
The organization of the rest of this paper is as follows. In Section 2 , we explain the DKP equation and in Section 3 we discuss the DKP equation in the presence of interaction in (1+2) dimensions. Then, in Section 4, we introduce the basic equations of the AIM, and in Section 5 we derive the solutions of the DKP equation in the presence of Hulth´en potential by the AIM. Finally, concluding remarks and discussion are given in Section 6.
2. Duffin–Kemmer–Petiau equation
The one-dimensional DKP equation for non-interacting bosons of spin-zero and spin-one is (¯h = c = 1)[1–3]
(iβµ
∂µ− m)Ψ = 0, (1)
where βµare the DKP matrices that satisfy in this algebra
βµβνβλ+ βλβνβµ = gµ νβλ+ gλ νβµ. (2) Moreover
gµ ν= diag(1, −1, −1, −1), (gµ ν)2= 1, (3)
where we can use the following relations,[43]
β0βkβ0= 0, k = 1, 2, 3, (4)
†Corresponding author. E-mail:zhrmolaee@gmail.com
© 2013 Chinese Physical Society and IOP Publishing Ltd http://iopscience.iop.org/cpb http://cpb.iphy.ac.cn 060306-1
β03= β0, (5) ∂µb µ βν∂µb µ= ∂ µb µb ν, bµ= (b0, 𝑏), (6)
where the bµis a generic four-vector.
(𝛽 · 𝑏)β0(𝛽 · 𝑏) = 0. (7)
Multiplying Eq. (1) by βµ∂µ and using Eq. (6) we have
(iβµ∂ µ βµβµ∂ µ− mβ µ∂ µ βµ)Ψ = 0, (8) so, (i∂µβµ∂µ− mβµ∂µβµ)Ψ = 0, (9)
hence, equation (1) becomes
(m∂µ− mβµ∂µβµ)Ψ = 0, (10)
and we can obtain
∂µΨ = βµ∂µβµΨ . (11)
Multiplying Eq. (1) by β0, getting the zero component of
Eq. (11), multiplying the resulting equation by the imaginary unity, one determines, upon adding the results,[43]
i∂0+ ∂k(β0βk− βkβ0) − mβ0 Ψ = 0. (12) On the other hand, i∂0Ψ = HΨ , where H is the DKP Hamil-tonian,
H= −i𝛼 · ∇ + β0m= 𝛼 · 𝑃 + β0m,
αk≡ β0βk− βkβ0, k = 1, 2, 3, (13)
within the ten-dimensional representation of spin-one sector
β0= 0 ¯0 ¯0 ¯0 ¯0T 0 𝐼 0 ¯0T 𝐼 0 0 ¯0T 0 0 0 , βi= 0 ¯0 ei ¯0 ¯0T 0 0 −iS i −eT i 0 0 0 ¯0T −iS i 0 0 , (14)
with S matrices being 3 × 3 ones, (Si)jk= −iεi jk where εi jk
takes the values 1, −1, 0 for an even permutation, an odd per-mutation, and repeated indices, respectively; ei matrices are
1 × 3 ones with (ei)1 j= δi j, i.e.,
e1= (1, 0, 0) e2= (0, 1, 0) and e3= (0, 0, 1),
where 𝐼 and 0 respectively represent unit and null 3 × 3 ma-trices and ¯0s are 1 × 3 ones.
On the other hand, the DKP equation in the presence of the interaction is
(iβµ
∂µ− m −U)Ψ = 0, (15)
where the general form of the interaction is
U= S(r) + PSµ(r) + βµVµ(r) + βµPVpµ(r), (16)
furthermore, for an elastic scattering, U is[44]
U= S(r) + PSµ(r) + β0V(r) + β0PVP(r), (17)
where each term has a specific Lorentz character. Two Lorentz vectors may be written as βµand Pβµ by assuming rotational
invariance and parity conservation hence the projection
opera-tor is P=1
3(β
µ
βµ− 2) = diag(1, 1, 1, 1, 0, 0, 0, 0, 0, 0).
The DKP matrix has three irreducible representations: one-dimensional representation that is trivial, five-one-dimensional representation for spin-zero particles, and ten-dimensional representation for spin-one particles.[1–3]
3. The DKP equation in the presence of
interac-tion in (1+2)-dimensional space
The DKP equation in the presence of an interaction in (1+2) dimensions is
(Eβ0− β0PV(ρ) + iβ1
∂1+ iβ2∂2− m)Ψ = 0, (18)
of which a solution in the following form is obtained as Ψ (x, y, t) = exp(−iEn,λt)ψn,λ(x, y) (19)
by removing the time dependence. We pick up two quantum numbers and display our ten-component stationary spinor as
ψn,λT (x, y) = (ψn,λ(1), ψn,λ(2), ψn,λ(3), ψn,λ(4), ψn,λ(5), ψn,λ(6), ψn,λ(7), ψn,λ(8), ψn,λ(9), ψn,λ(10))T. (20) Substitution of the latter spinor into Eq. (19) yields the coupled equations i∂1ψ (5) n,λ(x, y) + i∂2ψ (6) n,λ(x, y) − mψ (1) n,λ(x, y) = 0, (21) En,λψ (5) n,λ(x, y) + i∂2ψ (10) n,λ (x, y) − mψ (2) n,λ(x, y) = 0, (22) En,λψ (6) n,λ(x, y) − i∂1ψ (10) n,λ (x, y) − mψ (3) n,λ(x, y) = 0, (23) En,λψn,λ(7)(x, y) + i∂1ψn,λ(9)(x, y) − i∂2ψn,λ(8)(x, y) − mψn,λ(4)(x, y) = 0, (24) En,λψ (2) n,λ(x, y) −V (ρ)ψ (2) n,λ(x, y) − i∂1ψ (1) n,λ(x, y) − mψn,λ(5)(x, y) = 0, (25) En,λψ (3) n,λ(x, y) −V (ρ)ψ (3) n,λ(x, y) − i∂2ψ (1) n,λ(x, y) − mψn,λ(6)(x, y) = 0, (26) En,λψn,λ(4)(x, y) −V (ρ)ψn,λ(4)(x, y) − mψn,λ(7)(x, y) = 0, (27) i∂2ψ (4) n,λ(x, y) − mψ (8) n,λ(x, y) = 0, (28) −i∂1ψ (4) n,λ(x, y) − mψ (9) n,λ(x, y) = 0, (29) i∂1ψn,λ(3)(x, y) − i∂2ψn,λ(2)(x, y) − mψn,λ(10)(x, y) = 0. (30) If we suppose ψn,λ(2)(x, y) = 0, it is simply seen that the other components of ψn,λ(3)(x, y) can be obtained from coupled equa-tions as
i∂1ψn,λ(5)(x, y) + i∂2ψn,λ(6)(x, y) − mψn,λ(1)(x, y) = 0, (31a) En,λψ (5) n,λ(x, y) + i∂2ψ (10) n,λ (x, y) = 0, (31b) En,λψ (6) n,λ(x, y) − i∂1ψ (10) n,λ (x, y) − mψ (3) n,λ(x, y) = 0, (31c) −i∂1ψn,λ(1)(x, y) − mψn,λ(5)(x, y) = 0, (31d) En,λψn,λ(3)(x, y) −V (r)ψn,λ(3)(x, y) − i∂2ψn,λ(1)(x, y)
− mψn,λ(6)(x, y) = 0, (31e) i∂1ψn,λ(3)(x, y) − mψn,λ(10)(x, y) = 0. (31f) Similarly we can obtain the ψn,λ(4)(x, y) from four coupled equa-tions as follows: En,λψn,λ(7)(x, y) + i∂1ψn,λ(9)(x, y) − i∂2ψn,λ(8)(x, y) − mψn,λ(4)(x, y) = 0, (32a) En,λψ (4) n,λ(x, y) −V (ρ)ψ (4) n,λ(x, y) − mψ (7) n,λ(x, y) = 0, (32b) i∂2ψn,λ(4)(x, y) − mψn,λ(8)(x, y) = 0, (32c) −i∂1ψn,λ(4)(x, y) − mψn,λ(9)(x, y) = 0. (32d) Elimination of other components in favor of the fourth one yields (En,λ(En,λ−V (ρ)) − m2)ψ (4) n,λ(x, y) + ∂ 2 1ψ (4) n,λ(x, y) + ∂22ψn,λ(4)(x, y) = 0. (33) Now, we rewrite Eq. (33) as
(∂12+ ∂22)ψn,λ(4)(x, y) = −(En,λ(En,λ−V (ρ)) − m2)
× ψn,λ(4)(x, y). (34) The first and second terms on the right-hand side in Eq. (34) lead to (∂12+ ∂22)ψn,λ(4)(x, y) = ∇2ψn,λ(4)(x, y), (35) where ∇2= ∂ 2 ∂ ρ2 +1 ρ ∂ ∂ ρ + 1 ρ2 ∂2 ∂ ϕ2 . (36)
By choosing the wave function as
ψn,λ(4)= ψn,λ(4)(ρ, ϕ) = R(4)n (ρ)Qλ(ϕ), (37)
we can easily obtain
Qλ(ϕ) = eiλ ϕ, (38) ∇2ψn,λ(4)= d2 dρ2+ 1 ρ d dρ− λ2 ρ2 R(4)n (ρ) eiλ ϕ. (39)
Equations (36)–(39) lead to the following first order differen-tial equation d2 dρ2+ 1 ρ d dρ− λ2 ρ2 R(4)n (ρ) + (En,λ2 − En,λV(ρ) − m2)R(4)n (ρ) = 0. (40)
To solve Eq. (40), we will introduce the asymptotic iteration method.
4. Basic equations of the AIM
We briefly outline the asymptotic iteration method here; the details can be found in Refs. [45]–[47]. The AIM was pro-posed to solve second-order differential equations of the form y00= λ0(x)y0+ s0(x)y, (41)
where λ0(x) 6= 0; s0(x) and λ0(x) are in C∞(a, b). The
vari-ables, s0(x) and λ0(x), are sufficiently differentiable. The
dif-ferential equation (41) has a general solution y(x) = exp− Z x α dx0 ×hC2+C1 Z x exp Z x0 [λ0(x00)+2α(x00)]dx00 dx0i,(42) if, for sufficiently large n,
sn λn = sn−1 λn−1 = α, (43) where λn(x) = λn−10 (x) + sn−1(x) + λ0(x)λn−1(x), sn(x) = s0n−1(x) + s0(x)λn−1(x), n = 1, 2, 3, . . . . (44)
The termination condition of the method together with Eq. (44) can also be written as
δ (x) = λn−1(x)sn(x) − λn(x)sn−1(x) = 0 (45)
for a given potential, the idea is to convert the relativistic wave equation into the form of Eq. (41). Then, s0and λ0 are
de-termined and snand λnparameters are calculated. The energy
eigenvalues are obtained by the termination condition given by Eq. (45). However, the exact eigenfunctions can be derived from the following wave function generator
yn(x) = C2exp − Z x αkdx0 . (46)
5. Solutions of the DKP equation in the presence
of Hulth´en potential
We investigate the Hulth´en potential in the following equation d2 dρ2+ 1 ρ d dρ− λ2 ρ2 R(4)n (ρ) + (En,λ2 − En,λV(ρ) − m2)R (4) n (ρ) = 0, (47)
where the Hulth´en potential V(ρ) = −V0
eα ρ− 1, (48)
with V and α being real parameters.
Hence for removing the first derivation in Eq. (47) we need to choose the the following wave function
R(4)n (ρ) = u(4)n (ρ) √ ρ . (49) So, we have d2 dρ2+ 1 4ρ2− λ2 ρ2 u(4)n (ρ) +En,λ2 + En,λV0 eα ρ− 1− m 2u(4) n (ρ) = 0. (50)
We use the following approximation α2
( eα ρ− 1)2 ≈
1
ρ2, (51)
which is valid for small α. Substituting this approximation into Eq. (50), we have
d2 dρ2+ 1 4 α2 ( eα ρ− 1)2− λ2α2 ( eα ρ− 1)2 u(4)n (ρ) 060306-3
+En,λ2 + En,λV0 eα ρ− 1− m
2u(4)
n (ρ) = 0. (52)
So, by using the change of variable z = e−αρ we can rewrite Eq. (52) as d2 dz2+ 1 z d dz+ ξ0 (1 − z)2− ξ1 z2+ ξ2 z(1 − z) u(4)n (z) = 0, (53) where ξ0= 1 − 4λ2 4 , ξ1= m2− E2 n,λ α2 and ξ2= En,λV0 α2 . In order to solve Eq. (53) with the aid of AIM, we should convert Eq. (53) into the form of Eq. (41). The wave func-tion should satisfy the boundary condifunc-tions, i.e., u(4)n (0) = 0 at
z= 0 for ρ → ∞ and u(4)n (1) = 0 at z = 1 for ρ → 0. Therefore,
the reasonable physical wave function is proposed as follows: u(4)n (z) = z
√
ξ1(1 − z)(1+ √
1−4ξ0)/2g(z). (54)
If Eq. (54) is inserted into Eq. (53), the second-order homoge-neous linear differential equation is obtained in the following form g00n(z) = −−1+z(2 + p 1−4ξ0+2 p ξ1)−2 p ξ1 z(−1+z) g 0 n(z) −1+ p 1−4ξ0+2(1+ p 1−4ξ0) p ξ1−2ξ2 2z(−1+z) gn(z),(55) where λ0(z) = 1 − z(2 +p1 − 4ξ0+ 2 p ξ1) + 2 p ξ1 z(−1 + z) , (56a) s0(s) = − 1 +p1 − 4ξ0+ 2(1 + p 1 − 4ξ0) p ξ1− 2ξ2 2z(−1 + z) . (56b)
By means of Eq. (44), we may calculate λn(s) and sn(s) as
follows: λ1(z) = 4 + 12pξ1+ 8ξ1 2z2(−1 + z)2 + 11 + 2(1 − 4ξ0) + p 1 − 4ξ0(9 + 6 p ξ1) + 18 p ξ1+ 8ξ1+ 2ξ2 2(−1 + z)2 −11 + p 1 − 4ξ0(3 + 6 p ξ1) + 30 p ξ1+ 16ξ1+ 2ξ2 2z(−1 + z)2 , (57a) s1(z) = −2(1 +pξ1) + z(4 + p 1 − 4ξ0+ 2 p ξ1) 2z2(−1 + z)2 · (1 +p1 − 4ξ0+ 2 p ξ1+ 2 p 1 − 4ξ0 p ξ1− 2ξ2 2z2(−1 + z)2 . (57b)
Combining these results obtained by the AIM with quantiza-tion condiquantiza-tion given by Eq. (45) yields:
s0λ1− s1λ0= 0 ⇒ ξ2,0,λ = 1 2+ p 1 − 4ξ0 1 2+ p ξ1 +pξ1, (58a) s1λ2− s2λ1= 0 ⇒ ξ2,1,λ = 5 2+ p 1 − 4ξ0 3 2+ p ξ1 + 3pξ1, (58b) s2λ3− s3λ2= 0 ⇒ ξ2,2,λ = 13 2 + p 1 − 4ξ0 5 2+ p ξ1 + 5pξ1. (58c)
When the above expressions are generalized, ξ0, ξ1, ξ2
state-ments containing the energy eigenvalues turn into ξ2n = 1 2+ n + n 2+p 1 − 4ξ0 1 2+ n + p ξ1 +pξ1+ 2n p ξ1. (59)
If the above ξ0, ξ1, ξ2 expressions are inserted, the energy
eigenvalues statement of the DKP equation for the Hulth´en potential is as follows: −En,λV0+ α2 1 2+ n + n 2+ λ 1 + 2n + 2 s m2− E2 n,λ α2 + 1 + 2n s m2− E2 n,λ α2 = 0. (60) 0 0.5 0.1 0.2 0.3 0.4 0 0.05 n=0 n=1 n=2 n=3 0.10 0.15 0.20 α Ε n ,λ
Fig. 1. (color online) Variations of energy eigenvalue with the α screening parameter. The parameters are in atomic units (¯h = c = m= λ = V0= 1).
In Fig.1we can deduce that with the decrease of param-eter α the values of energy become close together, and when α = 0 the energy eigenvalues have a same value, so a degen-eracy can be reached. A simple glance at the energies reveals that the energy difference between the levels decreases with principal quantum number increasing. For example,
E8,3− E8,2= 0.032923 fm−1, E7,3− E7,2= 0.034133 fm−1,
E6,3− E6,2= 0.035264 fm−1, E5,3− E5,2= 0.036297 fm−1,
E4,3− E4,2= 0.037213 fm−1, E3,3− E3,2= 0.037990 fm−1,
E2,3− E2,2= 0.038613 fm−1, E1,3− E1,2= 0.039065 fm−1,
E0,3− E0,2= 0.039335 fm−1.
the unit 1 fm = 10−15 m. Moreover, numerical solutions of the DKP equation without any approximation are obtained by
using the finite difference method[48]to check the exactness of our energy eigenvalue equation (60). These results are given in Table1.
Table 1. Energy eigenvalues for m = 1, V0= 0.5, α = 0.01.
|n, λ i Our calculation Numerical results |0, 0i 0.020096 0.027401 |0, 1i 0.060191 0.060898 |0, 2i 0.099999 0.098268 |0, 3i 0.139334 0.13946 |1, 0i 0.060390 0.070536 |1, 1i 0.100593 0.10801
Now, the corresponding eigenfunctions can be obtained for the DKP equation with Hulth´en potential by using the wave
function generator given by Eq. (46). The first lowest states are given as follows: g0(z) = 1, (61a) g1(z) = −C2(1 + 2 p ξ1) 1 −2 + 2 p ξ1+ p 1 − 4ξ0 1 + 2pξ1 z , (61b) g2(z) = C2(1+2 p ξ1)(2+2 p ξ1) ×h1−2(3+2 p ξ1+ p 1−4ξ0) (1+2pξ1) z +(3+2 p ξ1+ p 1−4ξ0)(4+2 p ξ1+ p 1−4ξ0) (1+2pξ1)(2+2 p ξ1) z2i, (61c) and f3(z) = −C2(1 + 2 p ξ1)(2 + 2 p ξ1)(3 + 2 p ξ1) h 1 −3(4 + 2 p ξ1+ p 1 − 4ξ0) (1 + 2pξ1) z +3(4 + 2 p ξ1+ p 1 − 4ξ0)(5 + 2 p ξ1+ p 1 − 4ξ0) (1 + 2pξ1)(2 + 2 p ξ1) z2+3(4 + 2 p ξ1+ p 1 − 4ξ0)(5 + 2 p ξ1+ p 1 − 4ξ0) (1 + 2pξ1)(2 + 2 p ξ1) z2 −(4 + 2 p ξ1+ p 1 − 4ξ0)(5 + 2 p ξ1+ p 1 − 4ξ0)(6 + 2 p ξ1+ p 1 − 4ξ0) (1 + 2pξ1)(2 + 2 p ξ1)(3 + 2 p ξ1) z3i. (62)
Thus, the wave function gn(z) can be written as
gn(z) = (−1)nC2 Γ(1 + 2pξ1+ n) Γ(1 + 2pξ1) 2F1(−n, 1 + 2 p ξ1+ p 1 − 4ξ0+ n; 1 + 2 p ξ1; z). (63)
Hence, we can write the total radial wavefunctions as follows: u(4)n (z) = Nz √ ξ1(1 − z)(1+ √ 1−4ξ0)/2 2F1(−n, 1 + 2 p ξ1+ p 1 − 4ξ0+ n; 1 + 2 p ξ1; z), (64) ψ(4) n,λ(ρ, φ ) = Ne−αρ √ ξ1(1 − e−αρ)(1+ √ 1−4ξ0)/2 2F1(−n, 1 + 2 p ξ1+ p 1 − 4ξ0+ n; 1 + 2 p ξ1; e−αρ) √ ρ e iλ φ. (65)
Since ψn,λ(4)(ρ, φ ) is found, the ten-component stationary spinor is also obtained ψ(3) n,λ(ρ, φ ) = Ne−αρ √ ξ1(1 − e−αρ)(1+ √ 1−4ξ0)/2 2F1(−n, 1 + 2 p ξ1+ p 1 − 4ξ0+ n; 1 + 2 p ξ1; e−αρ) √ ρ e iλ φ, (66) ψn,λ(1)(ρ, φ ) = i En,λ sin φ d dρ+ sin φ ρ d dφ ×hNe −αρ√ξ1(1 − e−αρ)(1+ √ 1−4ξ0)/2 2F1(−n, 1 + 2 p ξ1+ p 1 − 4ξ0+ n; 1 + 2 p ξ1; e−αρ) √ ρ e iλ φi, (67) ψ(5) n,λ(ρ, φ ) = 1 im cos φ d dρ− sin φ ρ d dφ n i En,λ sin φ d dρ+ sin φ ρ d dφ ×Ne −αρ√ξ1(1 − e−αρ)(1+ √ 1−4ξ0)/2 2F1(−n, 1 + 2 p ξ1+ p 1 − 4ξ0+ n; 1 + 2 p ξ1; e−αρ) √ ρ e iλ φo, (68) ψn,λ(6)(ρ, φ ) = 1 m n (En,m−V (ρ)) ×hNe −αρ√ξ1(1 − e−αρ)(1+ √ 1−4ξ0)/2 2F1(−n, 1 + 2 p ξ1+ p 1 − 4ξ0+ n; 1 + 2 p ξ1; e−αρ) √ ρ e iλ φi + 1 En,λ sin φ d dρ+ sin φ ρ d dφ cos φ d dρ− sin φ ρ d dφ ×hNe −αρ√ξ1(1 − e−αρ)(1+ √ 1−4ξ0)/2 2F1(−n, 1 + 2 p ξ1+ p 1 − 4ξ0+ n; 1 + 2 p ξ1; e−αρ) √ ρ e iλ φio, (69) 060306-5
ψ(7) n,λ(ρ, φ ) = N(En,λ−V (ρ)) e−αρ√ξ1(1− e−αρ)(1+ √ 1−4ξ0)/2 2F1(−n, 1+2 p ξ1+ p 1−4ξ0+n; 1+2 p ξ1; e−αρ) m√ρ e iλ φ, (70) ψ(8) n,λ(ρ, φ ) = i m sin φ d dρ+ sin φ ρ d dφ ×hNe −αρ√ξ1(1 − e−αρ)(1+ √ 1−4ξ0)/2 2F1(−n, 1 + 2 p ξ1+ p 1 − 4ξ0+ n; 1 + 2 p ξ1; e−αρ) √ ρ e iλ φi, (71) ψ(9) n,λ(ρ, φ ) = 1 im cos φ d dρ− sin φ ρ d dφ ×hNe −αρ√ξ1(1 − e−αρ)(1+ √ 1−4ξ0)/2 2F1(−n, 1 + 2 p ξ1+ p 1 − 4ξ0+ n; 1 + 2 p ξ1; e−αρ) √ ρ e iλ φi, (72) ψ(10) n,λ (ρ, φ ) = i m cos φ d dρ− sin φ ρ d dφ ×hNe −αρ√ξ1(1 − e−αρ)(1+ √ 1−4ξ0)/2 2F1(−n, 1 + 2 p ξ1+ p 1 − 4ξ0+ n; 1 + 2 p ξ1; e−αρ) √ ρ e iλ φi. (73)
Now we check the energy relation for J/ψ. Choosing α = 0.2 fm−1, V0= 4.75466 fm−1, and mc= 7 fm−1, we
deter-mine Mtheor= 1.485 fm−1 which is in acceptable agreement
with its experimental value Mexp= 15.485 fm−1.
6. Concluding remarks
We obtained energy eigenvalues and corresponding eigenfunctions by the asymptotic iteration method. On the other hand, we investigated spin-one particles by using the Duffin–Kemmer–Petiau equation with the Hulth´en potential in (1+2) dimensions. This study presents a different approach, the AIM, to the calculation of the bound state solutions of the DKP equation with the Hulth´en potential in (1+2) dimensions for unity spin particles. For an arbitrary quantum number n state, we obtained the energy eigenvalues and corresponding eigenfunctions in the 1/ρ2approximation by AIM. The ad-vantage of the AIM is that it gives the eigenvalues directly by converting the second-order differential equation into the form of g00= λ0(x)g0+ s0(x)g. The non-normalized wavefunctions
are easily constructed by iterating the values of s0and λ0. The
method presented in this study is general and worth extending to the solution of other interactions. Also we elucidated the energy spectra for various quantum numbers and the behav-iors of the energy spectrum versus some parameters were also investigated. The results of this study are definitely useful for investigating a wide range of physical problems, from meson spectroscopy to cosmology.
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