arXiv:0704.0573v1  [quant-ph]  4 Apr 2007
Relativistic treatment in D-dimensions to a spin-zero particle with
noncentral equal scalar and vector ring-shaped Kratzer potential
Sameer M. Ikhdair∗ and Ramazan Sever†
∗Department of Physics, Near East University, Nicosia, North Cyprus, Mersin 10, Turkey
†Department of Physics, Middle East Technical University, 06531 Ankara, Turkey.
(November 4, 2018)
Abstract
The Klein-Gordon equation in D-dimensions for a recently proposed Kratzer
potential plus ring-shaped potential is solved analytically by means of the
conventional Nikiforov-Uvarov method. The exact energy bound-states and
the corresponding wave functions of the Klein-Gordon are obtained in the
presence of the noncentral equal scalar and vector potentials. The results
obtained in this work are more general and can be reduced to the standard
forms in three-dimensions given by other works.
Keywords: Energy eigenvalues and eigenfunctions, Klein-Gordon equa-
tion, Kratzer potential, ring-shaped potential, non-central potentials, Niki-
forov and Uvarov method.
PACS numbers: 03.65.-w; 03.65.Fd; 03.65.Ge.
∗sikhdair@neu.edu.tr
†sever@metu.edu.tr
http://arxiv.org/abs/0704.0573v1
I. INTRODUCTION
In various physical applications including those in nuclear physics and high energy physics
[1,2], one of the interesting problems is to obtain exact solutions of the relativistic equations
like Klein-Gordon and Dirac equations for mixed vector and scalar potential. The Klein-
Gordon and Dirac wave equations are frequently used to describe the particle dynamics in
relativistic quantum mechanics. The Klein-Gordon equation has also been used to under-
stand the motion of a spin-0 particle in large class of potentials. In recent years, much
efforts have been paid to solve these relativistic wave equations for various potentials by
using different methods. These relativistic equations contain two objects: the four-vector
linear momentum operator and the scalar rest mass. They allow us to introduce two types
of potential coupling, which are the four-vector potential (V) and the space-time scalar
potential (S).
Recently, many authors have worked on solving these equations with physical potentials
including Morse potential [3], Hulthen potential [4], Woods-Saxon potential [5], Pösch-Teller
potential [6], reflectionless-type potential [7], pseudoharmonic oscillator [8], ring-shaped har-
monic oscillator [9], V0 tanh
2(r/r0) potential [10], five-parameter exponential potential [11],
Rosen-Morse potential [12], and generalized symmetrical double-well potential [13], etc. It
is remarkable that in most works in this area, the scalar and vector potentials are almost
taken to be equal (i.e., S = V ) [2,14]. However, in some few other cases, it is considered
the case where the scalar potential is greater than the vector potential (in order to guar-
antee the existence of Klein-Gordon bound states) (i.e., S > V ) [15-19]. Nonetheless, such
physical potentials are very few. The bound-state solutions for the last case is obtained for
the exponential potential for the s-wave Klein-Gordon equation when the scalar potential is
greater than the vector potential [15].
The study of exact solutions of the nonrelativistic equation for a class of non-central po-
tentials with a vector potential and a non-central scalar potential is of considerable interest
in quantum chemistry [20-22]. In recent years, numerous studies [23] have been made in
analyzing the bound states of an electron in a Coulomb field with simultaneous presence
of Aharanov-Bohm (AB) [24] field, and/or a magnetic Dirac monopole [25], and Aharanov-
Bohm plus oscillator (ABO) systems. In most of these works, the eigenvalues and eigen-
functions are obtained by means of seperation of variables in spherical or other orthogonal
curvilinear coordinate systems. The path integral for particles moving in non-central poten-
tials is evaluated to derive the energy spectrum of this system analytically [26]. In addition,
the idea of SUSY and shape invariance is also used to obtain exact solutions of such non-
central but seperable potentials [27,28]. Very recently, the conventional Nikiforov-Uvarov
(NU) method [29] has been used to give a clear recipe of how to obtain an explicit exact
bound-states solutions for the energy eigenvalues and their corresponding wave functions in
terms of orthogonal polynomials for a class of non-central potentials [30].
Another type of noncentral potentials is the ring-shaped Kratzer potential, which is a
combination of a Coulomb potential plus an inverse square potential plus a noncentral angu-
lar part [31,32]. The Kratzer potential has been used to describe the vibrational-rotational
motion of isolated diatomic molecules [33] and has a mixed-energy spectrum containing both
bound and scattering states with bound-states have been widely used in molecular spec-
troscopy [34]. The ring-shaped Kratzer potential consists of radial and angular-dependent
potentials and is useful in studying ring-shaped molecules [22]. In taking the relativistic
effects into account for spin-0 particle in the presence of a class of noncentral potentials, Ya-
suk et al [35] applied the NU method to solve the Klein-Gordon equation for the noncentral
Coulombic ring-shaped potential [21] for the case V = S. Further, Berkdemir [36] also used
the same method to solve the Klein-Gordon equation for the Kratzer-type potential.
Recently, Chen and Dong [37] proposed a new ring-shaped potential and obtained the
exact solution of the Schrödinger equation for the Coulomb potential plus this new ring-
shaped potential which has possible applications to ring-shaped organic molecules like cyclic
polyenes and benzene. This type of potential used by Chen and Dong [37] appears to be
very similar to the potential used by Yasuk et al [35]. Moreover, Cheng and Dai [38],
proposed a new potential consisting from the modified Kratzer’s potential [33] plus the
new proposed ring-shaped potential in [37]. They have presented the energy eigenvalues
for this proposed exactly-solvable non-central potential in three dimensional (i.e., D = 3)-
Schrödinger equation by means of the NU method. The two quantum systems solved by
Refs [37,38] are closely relevant to each other as they deal with a Coulombic field interaction
except for a slight change in the angular momentum barrier acts as a repulsive core which
is for any arbitrary angular momentum ℓ prevents collapse of the system in any dimensional
space due to the slight perturbation to the original angular momentum barrier. Very recently,
we have also applied the NU method to solve the Schrödinger equation in any arbitrary D-
dimension to this new modified Kratzer-type potential [39,40].
The aim of the present paper is to consider the relativistic effects for the spin-0 parti-
cle in our recent works [39,40]. So we want to present a systematic recipe to solving the
D-dimensional Klein-Gordon equation for the Kratzer plus the new ring-shaped potential
proposed in [38] using the simple NU method. This method is based on solving the Klein-
Gordon equation by reducing it to a generalized hypergeometric equation.
This work is organized as follows: in section II, we shall present the Klein-Gordon
equation in spherical coordinates for spin-0 particle in the presence of equal scalar and
vector noncentral Kratzer plus the new ring-shaped potential and we also separate it into
radial and angular parts. Section III is devoted to a brief description of the NU method.
In section IV, we present the exact solutions to the radial and angular parts of the Klein-
Gordon equation in D-dimensions. Finally, the relevant conclusions are given in section
II. THE KLEIN-GORDON EQUATION WITH EQUAL SCALAR AND VECTOR
POTENTIALS
In relativistic quantum mechanics, we usually use the Klein-Gordon equation for de-
scribing a scalar particle, i.e., the spin-0 particle dynamics. The discussion of the relativistic
behavior of spin-zero particles requires understanding the single particle spectrum and the
exact solutions to the Klein Gordon equation which are constructed by using the four-vector
potential Aλ (λ = 0, 1, 2, 3) and the scalar potential (S). In order to simplify the solution
of the Klein-Gordon equation, the four-vector potential can be written as Aλ = (A0, 0, 0, 0).
The first component of the four-vector potential is represented by a vector potential (V ), i.e.,
A0 = V. In this case, the motion of a relativistic spin-0 particle in a potential is described by
the Klein-Gordon equation with the potentials V and S [1]. For the case S ≥ V, there exist
bound-state (real) solutions for a relativistic spin-zero particle [15-19]. On the other hand,
for S = V, the Klein-Gordon equation reduces to a Schrödinger-like equation and thereby
the bound-state solutions are easily obtained by using the well-known methods developed
in nonrelativistic quantum mechanics [2].
The Klein-Gordon equation describing a scalar particle (spin-0 particle) with scalar
S(r, θ, ϕ) and vector V (r, θ, ϕ) potentials is given by [2,14]
2 − [ER − V (r, θ, ϕ)/2]
+ [µ+ S(r, θ, ϕ)/2]
ψ(r, θ, ϕ) = 0, (1)
where ER,P and µ are the relativistic energy, momentum operator and rest mass of the
particle, respectively. The potential terms are scaled in (1) by Alhaidari et al [14] so that
in the nonrelativistic limit the interaction potential becomes V.
In this work, we consider the equal scalar and vector potentials case, that is, S(r, θ, ϕ) =
V (r, θ, ϕ) with the recently proposed general non-central potential taken in the form of the
Kratzer plus ring-shaped potential [38-40]:
V (r, θ, ϕ) = V1(r) +
V2(θ)
V3(ϕ)
r2 sin2 θ
, (2)
V1(r) = −
, V2(θ) = Cctg
2θ, V3(ϕ) = 0, (3)
where A = 2a0r0, B = a0r
0 and C is positive real constant with a0 is the dissociation energy
and r0 is the equilibrium internuclear distance [33]. The potentials in Eq. (3) introduced
by Cheng-Dai [38] reduce to the Kratzer potential in the limiting case of C = 0 [33]. In fact
the energy spectrum for this potential can be obtained directly by considering it as special
case of the general non-central seperable potentials [30].
In the relativistic atomic units (h̄ = c = 1), the D-dimensional Klein-Gordon equation
in (1) becomes [41]
sin θ
sin θ
sin2 θ
− (ER + µ)
V1(r) +
V2(θ)
V3(ϕ)
r2 sin2 θ
E2R − µ
ψ(r, θ, ϕ) = 0. (4)
with ψ(r, θ, ϕ) being the spherical total wave function separated as follows
ψnjm(r, θ, ϕ) = R(r)Y
j (θ, ϕ), R(r) = r
−(D−1)/2g(r), Y mj (θ, ϕ) = H(θ)Φ(ϕ). (5)
Inserting Eqs (3) and (5) into Eq. (4) and using the method of separation of variables, the
following differential equations are obtained:
dR(r)
j(j +D − 2)
+ α22
α21 −
R(r) = 0, (6)
sin θ
sin θ
m2 + Cα22 cos
sin2 θ
+ j(j +D − 2)
H(θ) = 0, (7)
d2Φ(ϕ)
+m2Φ(ϕ) = 0, (8)
where α21 = µ−ER, α
2 = µ+ER, m and j are constants and with m
2 and λj = j(j+D−2)
are the separation constants.
For a nonrelativistic treatment with the same potential, the Schrödinger equation in
spherical coordinates is
sin θ
sin θ
sin2 θ
ENR − V1(r)−
V2(θ)
V3(ϕ)
r2 sin2 θ
ψ(r, θ, ϕ) = 0. (9)
where µ and ENR are the reduced mass and the nonrelativistic energy, respectively. Besides,
the spherical total wave function appearing in Eq. (9) has the same representation as in Eq.
(5) but with the transformation j → ℓ. Inserting Eq. (5) into Eq. (9) leads to the following
differential equations [39,40]:
dR(r)
ENR +
R(r) = 0, (10)
sin θ
sin θ
m2 + 2µC cos2 θ
sin2 θ
+ ℓ(ℓ+D − 2)
H(θ) = 0, (11)
d2Φ(ϕ)
+m2Φ(ϕ) = 0, (12)
where m2 and λℓ = ℓ(ℓ + D − 2) are the separation constants. Equations (6)-(8) have the
same functional form as Eqs (10)-(12). Therefore, the solution of the Klein-Gordon equation
can be reduced to the solution of the Schrödinger equation with the appropriate choice of
parameters: j → ℓ, α21 → −ENR and α
2 → 2µ.
The solution of Eq. (8) is well-known periodic and must satisfy the period boundary
condition Φ(ϕ + 2π) = Φ(ϕ) which is the azimuthal angle solution:
Φm(ϕ) =
exp(±imϕ), m = 0, 1, 2, ..... (13)
Additionally, Eqs (6) and (7) are radial and polar angle equations and they will be solved
by using the Nikiforov-Uvarov (NU) method [29] which is given briefly in the following
section.
III. NIKIFOROV-UVAROV METHOD
The NU method is based on reducing the second-order differential equation to a gener-
alized equation of hypergeometric type [29]. In this sense, the Schrödinger equation, after
employing an appropriate coordinate transformation s = s(r), transforms to the following
form:
ψ′′n(s) +
τ̃(s)
ψ′n(s) +
σ̃(s)
σ2(s)
ψn(s) = 0, (14)
where σ(s) and σ̃(s) are polynomials, at most of second-degree, and τ̃ (s) is a first-degree
polynomial. Using a wave function, ψn(s), of the simple ansatz:
ψn(s) = φn(s)yn(s), (15)
reduces (14) into an equation of a hypergeometric type
σ(s)y′′n(s) + τ(s)y
n(s) + λyn(s) = 0, (16)
where
σ(s) = π(s)
φ′(s)
, (17)
τ(s) = τ̃(s) + 2π(s), τ ′(s) < 0, (18)
and λ is a parameter defined as
λ = λn = −nτ ′(s)−
n (n− 1)
σ′′(s), n = 0, 1, 2, .... (19)
The polynomial τ(s) with the parameter s and prime factors show the differentials at first
degree be negative. It is worthwhile to note that λ or λn are obtained from a particular
solution of the form y(s) = yn(s) which is a polynomial of degree n. Further, the other
part yn(s) of the wave function (14) is the hypergeometric-type function whose polynomial
solutions are given by Rodrigues relation
yn(s) =
[σn(s)ρ(s)] , (20)
where Bn is the normalization constant and the weight function ρ(s) must satisfy the con-
dition [29]
w(s) =
w(s), w(s) = σ(s)ρ(s). (21)
The function π and the parameter λ are defined as
π(s) =
σ′(s)− τ̃ (s)
σ′(s)− τ̃ (s)
− σ̃(s) + kσ(s), (22)
λ = k + π′(s). (23)
In principle, since π(s) has to be a polynomial of degree at most one, the expression under
the square root sign in (22) can be arranged to be the square of a polynomial of first degree
[29]. This is possible only if its discriminant is zero. In this case, an equation for k is
obtained. After solving this equation, the obtained values of k are substituted in (22). In
addition, by comparing equations (19) and (23), we obtain the energy eigenvalues.
IV. EXACT SOLUTIONS OF THE RADIAL AND ANGLE-DEPENDENT
EQUATIONS
A. Separating variables of the Klein-Gordon equation
We seek to solving the radial and angular parts of the Klein-Gordon equation given
by Eqs (6) and (7). Equation (6) involving the radial part can be written simply in the
following form [39-41]:
d2g(r)
(M − 1)(M − 3)
− α22
+ α21α
g(r) = 0, (24)
where
M = D + 2j. (25)
On the other hand, Eq. (7) involving the angular part of Klein-Gordon equation retakes the
simple form
d2H(θ)
+ ctg(θ)
dH(θ)
m2 + Cα22 cos
sin2 θ
− j(j +D − 2)
H(θ) = 0. (26)
Thus, Eqs (24) and (26) have to be solved latter through the NU method in the following
subsections.
B. Eigenvalues and eigenfunctions of the angle-dependent equation
In order to apply NU method [29,30,33,35,36,38-40,42-44], we use a suitable transforma-
tion variable s = cos θ. The polar angle part of the Klein Gordon equation in (26) can be
written in the following universal associated-Legendre differential equation form [38-40]
d2H(s)
1− s2
dH(s)
(1− s2)2
j(j +D − 2)(1− s2)−m2 − Cα22s
H(θ) = 0. (27)
Equation (27) has already been solved for the three-dimensional Schrödinger equation
through the NU method in [38]. However, the aim in this subsection is to solve with different
parameters resulting from the D-space-dimensions of Klein-Gordon equation. Further, Eq.
(27) is compared with (14) and the following identifications are obtained
τ̃(s) = −2s, σ(s) = 1− s2, σ̃(s) = −m′2 + (1− s2)ν ′, (28)
where
ν ′ = j′(j′ +D − 2) = j(j +D − 2) + Cα22, m
′2 = m2 + Cα22. (29)
Inserting the above expressions into equation (22), one obtains the following function:
π(s) = ±
(ν ′ − k)s2 + k − ν ′ +m′2, (30)
Following the method, the polynomial π(s) is found in the following possible values
π(s) =


m′s for k1 = ν
′ −m′2,
−m′s for k1 = ν ′ −m′2,
m′ for k2 = ν
−m′ for k2 = ν ′.
Imposing the condition τ ′(s) < 0, for equation (18), one selects
k1 = ν
′ −m′2 and π(s) = −m′s, (32)
which yields
τ(s) = −2(1 +m′)s. (33)
Using equations (19) and (23), the following expressions for λ are obtained, respectively,
λ = λn = 2ñ(1 +m
′) + ñ(ñ− 1), (34)
λ = ν ′ −m′(1 +m′). (35)
We compare equations (34) and (35), the new angular momentum j values are obtained as
j = −
(D − 2)
(D − 2)2 + (2ñ+ 2m′ + 1)2 − 4Cα22 − 1, (36)
j′ = −
(D − 2)
(D − 2)2 + (2ñ+ 2m′ + 1)2 − 1. (37)
Using Eqs (15)-(17) and (20)-(21), the polynomial solution of yn is expressed in terms of
Jacobi polynomials [39,40] which are one of the orthogonal polynomials:
Hñ(θ) = Nñ sin
m′(θ)P
(m′,m′)
(cos θ), (38)
where Nñ =
(ñ+m′)!
(2ñ+2m′+1)(ñ+2m′)!ñ!
is the normalization constant determined by
[Hñ(s)]
ds = 1 and using the orthogonality relation of Jacobi polynomials [35,45,46].
Besides
ñ = −
(1 + 2m′)
(2j + 1)2 + 4j(D − 3) + 4Cα22, (39)
where m′ is defined by equation (29).
C. Eigensolutions of the radial equation
The solution of the radial part of Klein-Gordon equation, Eq. (24), for the Kratzer’s
potential has already been solved by means of NU-method in [39]. Very recently, using
the same method, the problem for the non-central potential in (2) has been solved in three
dimensions (3D) by Cheng and Dai [36]. However, the aim of this subsection is to solve
the problem with a different radial separation function g(r) in any arbitrary dimensions. In
what follows, we present the exact bound-states (real) solution of Eq. (24). Letting
ε2 = α21α
2, 4γ
2 = (M − 1)(M − 3) + 4Bα22, β
2 = Aα22, (40)
and substituting these expressions in equation (24), one gets
d2g(r)
−ε2r2 + β2r − γ2
g(r) = 0. (41)
To apply the conventional NU-method, equation (41) is compared with (14), resulting in
the following expressions:
τ̃ (r) = 0, σ(r) = r, σ̃(r) = −ε2r2 + β2r − γ2. (42)
Substituting the above expressions into equation (22) gives
π(r) =
4ε2r2 + 4(k − β2)r + 4γ2 + 1. (43)
Therefore, we can determine the constant k by using the condition that the discriminant of
the square root is zero, that is
k = β2 ± ε
4γ2 + 1, 4γ2 + 1 = (D + 2j − 2)2 + 4Bα22. (44)
In view of that, we arrive at the following four possible functions of π(r) :
π(r) =


εr + 1
4γ2 + 1
for k1 = β
2 + ε
4γ2 + 1,
εr + 1
4γ2 + 1
for k1 = β
2 + ε
4γ2 + 1,
εr − 1
4γ2 + 1
for k2 = β
2 − ε
4γ2 + 1,
εr − 1
4γ2 + 1
for k2 = β
2 − ε
4γ2 + 1.
The correct value of π(r) is chosen such that the function τ(r) given by Eq. (18) will have
negative derivative [29]. So we can select the physical values to be
k = β2 − ε
4γ2 + 1 and π(r) =
4γ2 + 1
, (46)
which yield
τ(r) = −2εr + (1 +
4γ2 + 1), τ ′(r) = −2ε < 0. (47)
Using Eqs (19) and (23), the following expressions for λ are obtained, respectively,
λ = λn = 2nε, n = 0, 1, 2, ..., (48)
λ = δ2 − ε(1 +
4γ2 + 1). (49)
So we can obtain the Klein Gordon energy eigenvalues from the following relation:
1 + 2n+
(D + 2j − 2)2 + 4(µ+ ER)B
µ−ER = A
µ+ ER, (50)
and hence for the Kratzer plus the new ring-shaped potential, it becomes
1 + 2n+
(D + 2j − 2)2 + 4a0r20(µ+ ER)
µ−ER = 2a0r0
µ+ ER, (51)
with j defined in (36). Although Eq. (51) is exactly solvable for ER but it looks to be little
complicated. Further, it is interesting to investigate the solution for the Coulomb potential.
Therefore, applying the following transformations: A = Ze2, B = 0, and j = ℓ, the central
part of the potential in (3) turns into the Coulomb potential with Klein Gordon solution for
the energy spectra given by
ER = µ
2q2e2
q2e2 + (2n+ 2ℓ+D − 1)2
, n, ℓ = 0, 1, 2, ..., (52)
where q = Ze is the charge of the nucleus. Further, Eq. (52) can be expanded as a series in
the nucleus charge as
ER = µ−
2µq2e2
(2n + 2ℓ+D − 1)2
2µq4e4
(2n+ 2ℓ+D − 1)4
−O(qe)6, (53)
The physical meaning of each term in the last equation was given in detail by Ref. [36].
Besides, the difference from the conventional nonrelativistic form is because of the choice of
the vector V (r, θ, ϕ) and scalar S(r, θ, ϕ) parts of the potential in Eq. (1).
Overmore, if the value of j obtained by Eq.(36) is inserted into the eigenvalues of the
radial part of the Klein Gordon equation with the noncentral potential given by Eq. (51),
we finally find the energy eigenvalues for a bound electron in the presence of a noncentral
potential by Eq. (2) as
1 + 2n+
(2j′ +D − 2)2 + 4(a0r20 − C)(µ+ ER)
µ−ER = 2a0r0
µ+ ER, (54)
where m′ =
m2 + C(µ+ ER) and ñ is given by Eq. (39). On the other hand, the solution
of the Schrödinger equation, Eq. (9), for this potential has already been obtained by using
the same method in Ref. [39] and it is in the Coulombic-like form:
ENR = −
8µa20r
2n+ 1 +
(2ℓ′ +D − 2)2 + 8µ(a0r20 − C)
]2 , n = 0, 1, 2, ... (55)
2ℓ′ +D − 2 =
(D − 2)2 + (2ñ + 2m′ + 1)2 − 1, (56)
wherem′ =
m2 + 2µC. Also, applying the following appropriate transformation: µ+ER →
2µ, µ−ER → − ENR, j → ℓ to Eq. (54) provides exactly the nonrelativistic limit given by
Eq. (55).
In what follows, let us now turn attention to find the radial wavefunctions for this
potential. Substituting the values of σ(r), π(r) and τ(r) in Eqs (42), (45) and (47) into Eqs.
(17) and (21), we find
φ(r) = r(ζ+1)/2e−εr, (57)
ρ(r) = rζe−2εr, (58)
where ζ =
4γ2 + 1. Then from equation (20), we obtain
ynj(r) = Bnjr
−ζe2εr
rn+ζe−2εr
, (59)
and the wave function g(r) can be written in the form of the generalized Laguerre polyno-
mials as
g(ρ) = Cnj
)(1+ζ)/2
e−ρ/2Lζn(ρ), (60)
where for Kratzer’s potential we have
(D + 2j − 2)2 + 4a0r20(µ+ ER), ρ = 2εr. (61)
Finally, the radial wave functions of the Klein-Gordon equation are obtained
R(ρ) = Cnj
)(ζ+2−D)/2
e−ρ/2Lζn(ρ), (62)
where Cnj is the normalization constant to be determined below. Using the normalization
condition,
R2(r)rD−1dr = 1, and the orthogonality relation of the generalized Laguerre
polynomials,
zη+1e−z [Lηn(z)]
(2n+η+1)(n+η)!
, we have
Cnj =
µ2 − E2R
)1+ ζ
(2n+ ζ + 1) (n+ ζ)!
. (63)
Finally, we may express the normalized total wave functions as
ψ(r, θ, ϕ) =
µ2 − E2R
)1+ ζ
(ñ+m′)!
√√√√(2ñ+ 2m
′ + 1)(ñ+ 2m′)!ñ!n!
2π (2n+ ζ + 1) (n + ζ)!
(ζ+2−D)
2 exp(−
µ2 − E2Rr)L
µ2 − E2Rr) sin
m′(θ)P (m
′,m′)
n (cos θ) exp(±imϕ). (64)
where ζ is defined in Eq. (61) and m′ is given after the Eq. (54).
V. CONCLUSIONS
The relativistic spin-0 particle D-dimensional Klein-Gordon equation has been solved
easily for its exact bound-states with equal scalar and vector ring-shaped Kratzer potential
through the conventional NU method. The analytical expressions for the total energy levels
and eigenfunctions of this system can be reduced to their conventional three-dimensional
space form upon setting D = 3. Further, the noncentral potentials treated in [30] can
be introduced as perturbation to the Kratzer’s potential by adjusting the strength of the
coupling constant C in terms of a0, which is the coupling constant of the Kratzer’s potential.
Additionally, the radial and polar angle wave functions of Klein-Gordon equation are found
in terms of Laguerre and Jacobi polynomials, respectively. The method presented in this
paper is general and worth extending to the solution of other interaction problems. This
method is very simple and useful in solving other complicated systems analytically without
given a restiction conditions on the solution of some quantum systems as the case in the
other models. We have seen that for the nonrelativistic model, the exact energy spectra can
be obtained either by solving the Schrödinger equation in (9) (cf. Ref. [39] or Eq. (55)) or by
applying appropriate transformation to the relativistic solution given by Eq. (54). Finally,
we point out that these exact results obtained for this new proposed form of the potential
(2) may have some interesting applications in the study of different quantum mechanical
systems, atomic and molecular physics.
ACKNOWLEDGMENTS
This research was partially supported by the Scientific and Technological Research Coun-
cil of Turkey. S.M. Ikhdair wishes to dedicate this work to his family for their love and
assistance.
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  The Klein-Gordon equation in D-dimensions for a recently proposed Kratzer
potential plus ring-shaped potential is solved analytically by means of the
conventional Nikiforov-Uvarov method. The exact energy bound-states and the
corresponding wave functions of the Klein-Gordon are obtained in the presence
of the noncentral equal scalar and vector potentials. The results obtained in
this work are more general and can be reduced to the standard forms in
three-dimensions given by other works.

arXiv:0704.0573v1  [quant-ph]  4 Apr 2007
Relativistic treatment in D-dimensions to a spin-zero particle with
noncentral equal scalar and vector ring-shaped Kratzer potential
Sameer M. Ikhdair∗ and Ramazan Sever†
∗Department of Physics, Near East University, Nicosia, North Cyprus, Mersin 10, Turkey
†Department of Physics, Middle East Technical University, 06531 Ankara, Turkey.
(November 4, 2018)
Abstract
The Klein-Gordon equation in D-dimensions for a recently proposed Kratzer
potential plus ring-shaped potential is solved analytically by means of the
conventional Nikiforov-Uvarov method. The exact energy bound-states and
the corresponding wave functions of the Klein-Gordon are obtained in the
presence of the noncentral equal scalar and vector potentials. The results
obtained in this work are more general and can be reduced to the standard
forms in three-dimensions given by other works.
Keywords: Energy eigenvalues and eigenfunctions, Klein-Gordon equa-
tion, Kratzer potential, ring-shaped potential, non-central potentials, Niki-
forov and Uvarov method.
PACS numbers: 03.65.-w; 03.65.Fd; 03.65.Ge.
∗sikhdair@neu.edu.tr
†sever@metu.edu.tr
http://arxiv.org/abs/0704.0573v1
I. INTRODUCTION
In various physical applications including those in nuclear physics and high energy physics
[1,2], one of the interesting problems is to obtain exact solutions of the relativistic equations
like Klein-Gordon and Dirac equations for mixed vector and scalar potential. The Klein-
Gordon and Dirac wave equations are frequently used to describe the particle dynamics in
relativistic quantum mechanics. The Klein-Gordon equation has also been used to under-
stand the motion of a spin-0 particle in large class of potentials. In recent years, much
efforts have been paid to solve these relativistic wave equations for various potentials by
using different methods. These relativistic equations contain two objects: the four-vector
linear momentum operator and the scalar rest mass. They allow us to introduce two types
of potential coupling, which are the four-vector potential (V) and the space-time scalar
potential (S).
Recently, many authors have worked on solving these equations with physical potentials
including Morse potential [3], Hulthen potential [4], Woods-Saxon potential [5], Pösch-Teller
potential [6], reflectionless-type potential [7], pseudoharmonic oscillator [8], ring-shaped har-
monic oscillator [9], V0 tanh
2(r/r0) potential [10], five-parameter exponential potential [11],
Rosen-Morse potential [12], and generalized symmetrical double-well potential [13], etc. It
is remarkable that in most works in this area, the scalar and vector potentials are almost
taken to be equal (i.e., S = V ) [2,14]. However, in some few other cases, it is considered
the case where the scalar potential is greater than the vector potential (in order to guar-
antee the existence of Klein-Gordon bound states) (i.e., S > V ) [15-19]. Nonetheless, such
physical potentials are very few. The bound-state solutions for the last case is obtained for
the exponential potential for the s-wave Klein-Gordon equation when the scalar potential is
greater than the vector potential [15].
The study of exact solutions of the nonrelativistic equation for a class of non-central po-
tentials with a vector potential and a non-central scalar potential is of considerable interest
in quantum chemistry [20-22]. In recent years, numerous studies [23] have been made in
analyzing the bound states of an electron in a Coulomb field with simultaneous presence
of Aharanov-Bohm (AB) [24] field, and/or a magnetic Dirac monopole [25], and Aharanov-
Bohm plus oscillator (ABO) systems. In most of these works, the eigenvalues and eigen-
functions are obtained by means of seperation of variables in spherical or other orthogonal
curvilinear coordinate systems. The path integral for particles moving in non-central poten-
tials is evaluated to derive the energy spectrum of this system analytically [26]. In addition,
the idea of SUSY and shape invariance is also used to obtain exact solutions of such non-
central but seperable potentials [27,28]. Very recently, the conventional Nikiforov-Uvarov
(NU) method [29] has been used to give a clear recipe of how to obtain an explicit exact
bound-states solutions for the energy eigenvalues and their corresponding wave functions in
terms of orthogonal polynomials for a class of non-central potentials [30].
Another type of noncentral potentials is the ring-shaped Kratzer potential, which is a
combination of a Coulomb potential plus an inverse square potential plus a noncentral angu-
lar part [31,32]. The Kratzer potential has been used to describe the vibrational-rotational
motion of isolated diatomic molecules [33] and has a mixed-energy spectrum containing both
bound and scattering states with bound-states have been widely used in molecular spec-
troscopy [34]. The ring-shaped Kratzer potential consists of radial and angular-dependent
potentials and is useful in studying ring-shaped molecules [22]. In taking the relativistic
effects into account for spin-0 particle in the presence of a class of noncentral potentials, Ya-
suk et al [35] applied the NU method to solve the Klein-Gordon equation for the noncentral
Coulombic ring-shaped potential [21] for the case V = S. Further, Berkdemir [36] also used
the same method to solve the Klein-Gordon equation for the Kratzer-type potential.
Recently, Chen and Dong [37] proposed a new ring-shaped potential and obtained the
exact solution of the Schrödinger equation for the Coulomb potential plus this new ring-
shaped potential which has possible applications to ring-shaped organic molecules like cyclic
polyenes and benzene. This type of potential used by Chen and Dong [37] appears to be
very similar to the potential used by Yasuk et al [35]. Moreover, Cheng and Dai [38],
proposed a new potential consisting from the modified Kratzer’s potential [33] plus the
new proposed ring-shaped potential in [37]. They have presented the energy eigenvalues
for this proposed exactly-solvable non-central potential in three dimensional (i.e., D = 3)-
Schrödinger equation by means of the NU method. The two quantum systems solved by
Refs [37,38] are closely relevant to each other as they deal with a Coulombic field interaction
except for a slight change in the angular momentum barrier acts as a repulsive core which
is for any arbitrary angular momentum ℓ prevents collapse of the system in any dimensional
space due to the slight perturbation to the original angular momentum barrier. Very recently,
we have also applied the NU method to solve the Schrödinger equation in any arbitrary D-
dimension to this new modified Kratzer-type potential [39,40].
The aim of the present paper is to consider the relativistic effects for the spin-0 parti-
cle in our recent works [39,40]. So we want to present a systematic recipe to solving the
D-dimensional Klein-Gordon equation for the Kratzer plus the new ring-shaped potential
proposed in [38] using the simple NU method. This method is based on solving the Klein-
Gordon equation by reducing it to a generalized hypergeometric equation.
This work is organized as follows: in section II, we shall present the Klein-Gordon
equation in spherical coordinates for spin-0 particle in the presence of equal scalar and
vector noncentral Kratzer plus the new ring-shaped potential and we also separate it into
radial and angular parts. Section III is devoted to a brief description of the NU method.
In section IV, we present the exact solutions to the radial and angular parts of the Klein-
Gordon equation in D-dimensions. Finally, the relevant conclusions are given in section
II. THE KLEIN-GORDON EQUATION WITH EQUAL SCALAR AND VECTOR
POTENTIALS
In relativistic quantum mechanics, we usually use the Klein-Gordon equation for de-
scribing a scalar particle, i.e., the spin-0 particle dynamics. The discussion of the relativistic
behavior of spin-zero particles requires understanding the single particle spectrum and the
exact solutions to the Klein Gordon equation which are constructed by using the four-vector
potential Aλ (λ = 0, 1, 2, 3) and the scalar potential (S). In order to simplify the solution
of the Klein-Gordon equation, the four-vector potential can be written as Aλ = (A0, 0, 0, 0).
The first component of the four-vector potential is represented by a vector potential (V ), i.e.,
A0 = V. In this case, the motion of a relativistic spin-0 particle in a potential is described by
the Klein-Gordon equation with the potentials V and S [1]. For the case S ≥ V, there exist
bound-state (real) solutions for a relativistic spin-zero particle [15-19]. On the other hand,
for S = V, the Klein-Gordon equation reduces to a Schrödinger-like equation and thereby
the bound-state solutions are easily obtained by using the well-known methods developed
in nonrelativistic quantum mechanics [2].
The Klein-Gordon equation describing a scalar particle (spin-0 particle) with scalar
S(r, θ, ϕ) and vector V (r, θ, ϕ) potentials is given by [2,14]
2 − [ER − V (r, θ, ϕ)/2]
+ [µ+ S(r, θ, ϕ)/2]
ψ(r, θ, ϕ) = 0, (1)
where ER,P and µ are the relativistic energy, momentum operator and rest mass of the
particle, respectively. The potential terms are scaled in (1) by Alhaidari et al [14] so that
in the nonrelativistic limit the interaction potential becomes V.
In this work, we consider the equal scalar and vector potentials case, that is, S(r, θ, ϕ) =
V (r, θ, ϕ) with the recently proposed general non-central potential taken in the form of the
Kratzer plus ring-shaped potential [38-40]:
V (r, θ, ϕ) = V1(r) +
V2(θ)
V3(ϕ)
r2 sin2 θ
, (2)
V1(r) = −
, V2(θ) = Cctg
2θ, V3(ϕ) = 0, (3)
where A = 2a0r0, B = a0r
0 and C is positive real constant with a0 is the dissociation energy
and r0 is the equilibrium internuclear distance [33]. The potentials in Eq. (3) introduced
by Cheng-Dai [38] reduce to the Kratzer potential in the limiting case of C = 0 [33]. In fact
the energy spectrum for this potential can be obtained directly by considering it as special
case of the general non-central seperable potentials [30].
In the relativistic atomic units (h̄ = c = 1), the D-dimensional Klein-Gordon equation
in (1) becomes [41]
sin θ
sin θ
sin2 θ
− (ER + µ)
V1(r) +
V2(θ)
V3(ϕ)
r2 sin2 θ
E2R − µ
ψ(r, θ, ϕ) = 0. (4)
with ψ(r, θ, ϕ) being the spherical total wave function separated as follows
ψnjm(r, θ, ϕ) = R(r)Y
j (θ, ϕ), R(r) = r
−(D−1)/2g(r), Y mj (θ, ϕ) = H(θ)Φ(ϕ). (5)
Inserting Eqs (3) and (5) into Eq. (4) and using the method of separation of variables, the
following differential equations are obtained:
dR(r)
j(j +D − 2)
+ α22
α21 −
R(r) = 0, (6)
sin θ
sin θ
m2 + Cα22 cos
sin2 θ
+ j(j +D − 2)
H(θ) = 0, (7)
d2Φ(ϕ)
+m2Φ(ϕ) = 0, (8)
where α21 = µ−ER, α
2 = µ+ER, m and j are constants and with m
2 and λj = j(j+D−2)
are the separation constants.
For a nonrelativistic treatment with the same potential, the Schrödinger equation in
spherical coordinates is
sin θ
sin θ
sin2 θ
ENR − V1(r)−
V2(θ)
V3(ϕ)
r2 sin2 θ
ψ(r, θ, ϕ) = 0. (9)
where µ and ENR are the reduced mass and the nonrelativistic energy, respectively. Besides,
the spherical total wave function appearing in Eq. (9) has the same representation as in Eq.
(5) but with the transformation j → ℓ. Inserting Eq. (5) into Eq. (9) leads to the following
differential equations [39,40]:
dR(r)
ENR +
R(r) = 0, (10)
sin θ
sin θ
m2 + 2µC cos2 θ
sin2 θ
+ ℓ(ℓ+D − 2)
H(θ) = 0, (11)
d2Φ(ϕ)
+m2Φ(ϕ) = 0, (12)
where m2 and λℓ = ℓ(ℓ + D − 2) are the separation constants. Equations (6)-(8) have the
same functional form as Eqs (10)-(12). Therefore, the solution of the Klein-Gordon equation
can be reduced to the solution of the Schrödinger equation with the appropriate choice of
parameters: j → ℓ, α21 → −ENR and α
2 → 2µ.
The solution of Eq. (8) is well-known periodic and must satisfy the period boundary
condition Φ(ϕ + 2π) = Φ(ϕ) which is the azimuthal angle solution:
Φm(ϕ) =
exp(±imϕ), m = 0, 1, 2, ..... (13)
Additionally, Eqs (6) and (7) are radial and polar angle equations and they will be solved
by using the Nikiforov-Uvarov (NU) method [29] which is given briefly in the following
section.
III. NIKIFOROV-UVAROV METHOD
The NU method is based on reducing the second-order differential equation to a gener-
alized equation of hypergeometric type [29]. In this sense, the Schrödinger equation, after
employing an appropriate coordinate transformation s = s(r), transforms to the following
form:
ψ′′n(s) +
τ̃(s)
ψ′n(s) +
σ̃(s)
σ2(s)
ψn(s) = 0, (14)
where σ(s) and σ̃(s) are polynomials, at most of second-degree, and τ̃ (s) is a first-degree
polynomial. Using a wave function, ψn(s), of the simple ansatz:
ψn(s) = φn(s)yn(s), (15)
reduces (14) into an equation of a hypergeometric type
σ(s)y′′n(s) + τ(s)y
n(s) + λyn(s) = 0, (16)
where
σ(s) = π(s)
φ′(s)
, (17)
τ(s) = τ̃(s) + 2π(s), τ ′(s) < 0, (18)
and λ is a parameter defined as
λ = λn = −nτ ′(s)−
n (n− 1)
σ′′(s), n = 0, 1, 2, .... (19)
The polynomial τ(s) with the parameter s and prime factors show the differentials at first
degree be negative. It is worthwhile to note that λ or λn are obtained from a particular
solution of the form y(s) = yn(s) which is a polynomial of degree n. Further, the other
part yn(s) of the wave function (14) is the hypergeometric-type function whose polynomial
solutions are given by Rodrigues relation
yn(s) =
[σn(s)ρ(s)] , (20)
where Bn is the normalization constant and the weight function ρ(s) must satisfy the con-
dition [29]
w(s) =
w(s), w(s) = σ(s)ρ(s). (21)
The function π and the parameter λ are defined as
π(s) =
σ′(s)− τ̃ (s)
σ′(s)− τ̃ (s)
− σ̃(s) + kσ(s), (22)
λ = k + π′(s). (23)
In principle, since π(s) has to be a polynomial of degree at most one, the expression under
the square root sign in (22) can be arranged to be the square of a polynomial of first degree
[29]. This is possible only if its discriminant is zero. In this case, an equation for k is
obtained. After solving this equation, the obtained values of k are substituted in (22). In
addition, by comparing equations (19) and (23), we obtain the energy eigenvalues.
IV. EXACT SOLUTIONS OF THE RADIAL AND ANGLE-DEPENDENT
EQUATIONS
A. Separating variables of the Klein-Gordon equation
We seek to solving the radial and angular parts of the Klein-Gordon equation given
by Eqs (6) and (7). Equation (6) involving the radial part can be written simply in the
following form [39-41]:
d2g(r)
(M − 1)(M − 3)
− α22
+ α21α
g(r) = 0, (24)
where
M = D + 2j. (25)
On the other hand, Eq. (7) involving the angular part of Klein-Gordon equation retakes the
simple form
d2H(θ)
+ ctg(θ)
dH(θ)
m2 + Cα22 cos
sin2 θ
− j(j +D − 2)
H(θ) = 0. (26)
Thus, Eqs (24) and (26) have to be solved latter through the NU method in the following
subsections.
B. Eigenvalues and eigenfunctions of the angle-dependent equation
In order to apply NU method [29,30,33,35,36,38-40,42-44], we use a suitable transforma-
tion variable s = cos θ. The polar angle part of the Klein Gordon equation in (26) can be
written in the following universal associated-Legendre differential equation form [38-40]
d2H(s)
1− s2
dH(s)
(1− s2)2
j(j +D − 2)(1− s2)−m2 − Cα22s
H(θ) = 0. (27)
Equation (27) has already been solved for the three-dimensional Schrödinger equation
through the NU method in [38]. However, the aim in this subsection is to solve with different
parameters resulting from the D-space-dimensions of Klein-Gordon equation. Further, Eq.
(27) is compared with (14) and the following identifications are obtained
τ̃(s) = −2s, σ(s) = 1− s2, σ̃(s) = −m′2 + (1− s2)ν ′, (28)
where
ν ′ = j′(j′ +D − 2) = j(j +D − 2) + Cα22, m
′2 = m2 + Cα22. (29)
Inserting the above expressions into equation (22), one obtains the following function:
π(s) = ±
(ν ′ − k)s2 + k − ν ′ +m′2, (30)
Following the method, the polynomial π(s) is found in the following possible values
π(s) =


m′s for k1 = ν
′ −m′2,
−m′s for k1 = ν ′ −m′2,
m′ for k2 = ν
−m′ for k2 = ν ′.
Imposing the condition τ ′(s) < 0, for equation (18), one selects
k1 = ν
′ −m′2 and π(s) = −m′s, (32)
which yields
τ(s) = −2(1 +m′)s. (33)
Using equations (19) and (23), the following expressions for λ are obtained, respectively,
λ = λn = 2ñ(1 +m
′) + ñ(ñ− 1), (34)
λ = ν ′ −m′(1 +m′). (35)
We compare equations (34) and (35), the new angular momentum j values are obtained as
j = −
(D − 2)
(D − 2)2 + (2ñ+ 2m′ + 1)2 − 4Cα22 − 1, (36)
j′ = −
(D − 2)
(D − 2)2 + (2ñ+ 2m′ + 1)2 − 1. (37)
Using Eqs (15)-(17) and (20)-(21), the polynomial solution of yn is expressed in terms of
Jacobi polynomials [39,40] which are one of the orthogonal polynomials:
Hñ(θ) = Nñ sin
m′(θ)P
(m′,m′)
(cos θ), (38)
where Nñ =
(ñ+m′)!
(2ñ+2m′+1)(ñ+2m′)!ñ!
is the normalization constant determined by
[Hñ(s)]
ds = 1 and using the orthogonality relation of Jacobi polynomials [35,45,46].
Besides
ñ = −
(1 + 2m′)
(2j + 1)2 + 4j(D − 3) + 4Cα22, (39)
where m′ is defined by equation (29).
C. Eigensolutions of the radial equation
The solution of the radial part of Klein-Gordon equation, Eq. (24), for the Kratzer’s
potential has already been solved by means of NU-method in [39]. Very recently, using
the same method, the problem for the non-central potential in (2) has been solved in three
dimensions (3D) by Cheng and Dai [36]. However, the aim of this subsection is to solve
the problem with a different radial separation function g(r) in any arbitrary dimensions. In
what follows, we present the exact bound-states (real) solution of Eq. (24). Letting
ε2 = α21α
2, 4γ
2 = (M − 1)(M − 3) + 4Bα22, β
2 = Aα22, (40)
and substituting these expressions in equation (24), one gets
d2g(r)
−ε2r2 + β2r − γ2
g(r) = 0. (41)
To apply the conventional NU-method, equation (41) is compared with (14), resulting in
the following expressions:
τ̃ (r) = 0, σ(r) = r, σ̃(r) = −ε2r2 + β2r − γ2. (42)
Substituting the above expressions into equation (22) gives
π(r) =
4ε2r2 + 4(k − β2)r + 4γ2 + 1. (43)
Therefore, we can determine the constant k by using the condition that the discriminant of
the square root is zero, that is
k = β2 ± ε
4γ2 + 1, 4γ2 + 1 = (D + 2j − 2)2 + 4Bα22. (44)
In view of that, we arrive at the following four possible functions of π(r) :
π(r) =


εr + 1
4γ2 + 1
for k1 = β
2 + ε
4γ2 + 1,
εr + 1
4γ2 + 1
for k1 = β
2 + ε
4γ2 + 1,
εr − 1
4γ2 + 1
for k2 = β
2 − ε
4γ2 + 1,
εr − 1
4γ2 + 1
for k2 = β
2 − ε
4γ2 + 1.
The correct value of π(r) is chosen such that the function τ(r) given by Eq. (18) will have
negative derivative [29]. So we can select the physical values to be
k = β2 − ε
4γ2 + 1 and π(r) =
4γ2 + 1
, (46)
which yield
τ(r) = −2εr + (1 +
4γ2 + 1), τ ′(r) = −2ε < 0. (47)
Using Eqs (19) and (23), the following expressions for λ are obtained, respectively,
λ = λn = 2nε, n = 0, 1, 2, ..., (48)
λ = δ2 − ε(1 +
4γ2 + 1). (49)
So we can obtain the Klein Gordon energy eigenvalues from the following relation:
1 + 2n+
(D + 2j − 2)2 + 4(µ+ ER)B
µ−ER = A
µ+ ER, (50)
and hence for the Kratzer plus the new ring-shaped potential, it becomes
1 + 2n+
(D + 2j − 2)2 + 4a0r20(µ+ ER)
µ−ER = 2a0r0
µ+ ER, (51)
with j defined in (36). Although Eq. (51) is exactly solvable for ER but it looks to be little
complicated. Further, it is interesting to investigate the solution for the Coulomb potential.
Therefore, applying the following transformations: A = Ze2, B = 0, and j = ℓ, the central
part of the potential in (3) turns into the Coulomb potential with Klein Gordon solution for
the energy spectra given by
ER = µ
2q2e2
q2e2 + (2n+ 2ℓ+D − 1)2
, n, ℓ = 0, 1, 2, ..., (52)
where q = Ze is the charge of the nucleus. Further, Eq. (52) can be expanded as a series in
the nucleus charge as
ER = µ−
2µq2e2
(2n + 2ℓ+D − 1)2
2µq4e4
(2n+ 2ℓ+D − 1)4
−O(qe)6, (53)
The physical meaning of each term in the last equation was given in detail by Ref. [36].
Besides, the difference from the conventional nonrelativistic form is because of the choice of
the vector V (r, θ, ϕ) and scalar S(r, θ, ϕ) parts of the potential in Eq. (1).
Overmore, if the value of j obtained by Eq.(36) is inserted into the eigenvalues of the
radial part of the Klein Gordon equation with the noncentral potential given by Eq. (51),
we finally find the energy eigenvalues for a bound electron in the presence of a noncentral
potential by Eq. (2) as
1 + 2n+
(2j′ +D − 2)2 + 4(a0r20 − C)(µ+ ER)
µ−ER = 2a0r0
µ+ ER, (54)
where m′ =
m2 + C(µ+ ER) and ñ is given by Eq. (39). On the other hand, the solution
of the Schrödinger equation, Eq. (9), for this potential has already been obtained by using
the same method in Ref. [39] and it is in the Coulombic-like form:
ENR = −
8µa20r
2n+ 1 +
(2ℓ′ +D − 2)2 + 8µ(a0r20 − C)
]2 , n = 0, 1, 2, ... (55)
2ℓ′ +D − 2 =
(D − 2)2 + (2ñ + 2m′ + 1)2 − 1, (56)
wherem′ =
m2 + 2µC. Also, applying the following appropriate transformation: µ+ER →
2µ, µ−ER → − ENR, j → ℓ to Eq. (54) provides exactly the nonrelativistic limit given by
Eq. (55).
In what follows, let us now turn attention to find the radial wavefunctions for this
potential. Substituting the values of σ(r), π(r) and τ(r) in Eqs (42), (45) and (47) into Eqs.
(17) and (21), we find
φ(r) = r(ζ+1)/2e−εr, (57)
ρ(r) = rζe−2εr, (58)
where ζ =
4γ2 + 1. Then from equation (20), we obtain
ynj(r) = Bnjr
−ζe2εr
rn+ζe−2εr
, (59)
and the wave function g(r) can be written in the form of the generalized Laguerre polyno-
mials as
g(ρ) = Cnj
)(1+ζ)/2
e−ρ/2Lζn(ρ), (60)
where for Kratzer’s potential we have
(D + 2j − 2)2 + 4a0r20(µ+ ER), ρ = 2εr. (61)
Finally, the radial wave functions of the Klein-Gordon equation are obtained
R(ρ) = Cnj
)(ζ+2−D)/2
e−ρ/2Lζn(ρ), (62)
where Cnj is the normalization constant to be determined below. Using the normalization
condition,
R2(r)rD−1dr = 1, and the orthogonality relation of the generalized Laguerre
polynomials,
zη+1e−z [Lηn(z)]
(2n+η+1)(n+η)!
, we have
Cnj =
µ2 − E2R
)1+ ζ
(2n+ ζ + 1) (n+ ζ)!
. (63)
Finally, we may express the normalized total wave functions as
ψ(r, θ, ϕ) =
µ2 − E2R
)1+ ζ
(ñ+m′)!
√√√√(2ñ+ 2m
′ + 1)(ñ+ 2m′)!ñ!n!
2π (2n+ ζ + 1) (n + ζ)!
(ζ+2−D)
2 exp(−
µ2 − E2Rr)L
µ2 − E2Rr) sin
m′(θ)P (m
′,m′)
n (cos θ) exp(±imϕ). (64)
where ζ is defined in Eq. (61) and m′ is given after the Eq. (54).
V. CONCLUSIONS
The relativistic spin-0 particle D-dimensional Klein-Gordon equation has been solved
easily for its exact bound-states with equal scalar and vector ring-shaped Kratzer potential
through the conventional NU method. The analytical expressions for the total energy levels
and eigenfunctions of this system can be reduced to their conventional three-dimensional
space form upon setting D = 3. Further, the noncentral potentials treated in [30] can
be introduced as perturbation to the Kratzer’s potential by adjusting the strength of the
coupling constant C in terms of a0, which is the coupling constant of the Kratzer’s potential.
Additionally, the radial and polar angle wave functions of Klein-Gordon equation are found
in terms of Laguerre and Jacobi polynomials, respectively. The method presented in this
paper is general and worth extending to the solution of other interaction problems. This
method is very simple and useful in solving other complicated systems analytically without
given a restiction conditions on the solution of some quantum systems as the case in the
other models. We have seen that for the nonrelativistic model, the exact energy spectra can
be obtained either by solving the Schrödinger equation in (9) (cf. Ref. [39] or Eq. (55)) or by
applying appropriate transformation to the relativistic solution given by Eq. (54). Finally,
we point out that these exact results obtained for this new proposed form of the potential
(2) may have some interesting applications in the study of different quantum mechanical
systems, atomic and molecular physics.
ACKNOWLEDGMENTS
This research was partially supported by the Scientific and Technological Research Coun-
cil of Turkey. S.M. Ikhdair wishes to dedicate this work to his family for their love and
assistance.
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