# Bound-State Solutions of the Klein-Gordon Equation withq-Deformed Equal Scalar and Vector Eckart Potential Using a Newly Improved Approximation Scheme

Authors: Ita O. Akpan; Akaninyene D. Antia; Akpan N. Ikot
Journal: ISRN High Energy Physics
 (2012)

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## Abstract

We present the analytical solutions of the Klein-Gordon equation forq-deformed equal vector and scalar Eckart potential for arbitrary l-state. We obtain the energy spectrum and the corresponding unnormalized wave function expressed in terms of the Jacobi polynomial. We also discussed the special cases of the potential.


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## Body

## 1. Introduction

The study of exactly solvable potentials has attracted much attention since the early development of quantum mechanics. For example, the exact solutions of the Klein-Gordon equation for an hydrogen atom and for a harmonic oscillator in 3D represent two typical examples [1–3]. When a particle is in a strong potential field, the relativistic effect must be considered, leading to the relativistic quantum mechanical description of such particle [4–7]. In the relativistic limit, the particle motions are commonly described using either the Klein-Gordon or the Dirac equations [4, 6] depending on the spin character of the particle. The spin-zero particles like the mesons are described by the Klein-Gordon equation. On the other hand, the spin-half particles such as electrons are described satisfactorily by the Dirac equation. One of the interesting problems in nuclear and high energy physics is to obtain exact solution of the Klein-Gordon and the Dirac equation. In recent years, many studies have been carried out to explore the relativistic energy eigenvalues and the corresponding wave functions of the Klein-Gordon and the Dirac equation [8–11].These relativistic equations contain two objects: the vector potentialV(r) and the scalar potential S(r). The Klein-Gordon equation with the vector and scalar potentials can be written as follows:
(1.1)[-(i∂∂t-V(r))2-∇2+(S(r)+M)2]ψ(r,θ,φ)=0,
where M is the rest mass, i(∂/∂t)=E is the energy eigenvalues, and V(r) and S(r) are the vector and scalar potentials, respectively.Recently, some authors have assumed that the scalar potential is equal to the vector potential and obtained the bound state of the Klein-Gordon and the Dirac equations with some potentials of interest such as Woods-Saxon’s potential [12], Hartman’s potential [13], Coulomb-like potentials [14], ring-shape pseudoharmonic potential [15], Kratzer’s potential [16, 17], and Pöschl-Teller and Rosen Morse potential [18]. Different methods such as the asymptotic iteration method (AIM) [19], supersymmetric quantum mechanics (SUSSY) [20],and Nikiforov-Uvarov (NU) method [12, 21] have been used to solve the differential equation arising from these considerations.However, the analytical solutions of the Klein-Gordon equation are possible only in thes-wave case with the angular momentum l=0 for some well-known potential models [22, 23]. Conversely, when l≠0, one can only solve approximately the Klein-Gordon equation and the Dirac equation for some potentials using a suitable approximation scheme [24].The purpose of this work is to solve approximately the arbitraryl-state Klein-Gordon equation with q-deformed equal scalar and vector Eckart potential. This paper is organized as follows. In Section 2, we present the review of the NU method and its parametric form. Section 3 is devoted to the factorization method for the Klein-Gordon equation. Solution to the radial equation is presented in Section 4. Discussion of the result is given in Section 5. Finally, a brief conclusion is presented in Section 6.


## 2. Review of the Nikiforov-Uvarov (NU) Method and Its  Parametric Form

The NU method [25] is based on the solution of a generalized second-order linear differential equation into the equation of hypergeometric type. The Schrödinger equation
(2.1)ψ′′(x)+(E-V(x))ψ(x)=0
can be solved by this method. This can be done by transforming this equation into equation of hypergeometric type with appropriate transformation, s=s(x):
(2.2)ψ′′(s)+τ~(s)σ(s)ψ′(s)+σ~(s)σ2(s)ψ(s)=0.
In order to find the exact solution to (2.2), we set the wave function as
(2.3)ψ(s)=ϕ(s)χ(s),
and substituting (2.3) into (2.2)  reduces (2.2) into hypergeometric-type equation:
(2.4)σ(s)χ′′(s)+τ(s)χ′(s)+λχ(s)=0,
where the wave function ϕ(s) is defined as the logarithmic derivative [25]:
(2.5)ϕ′(s)ϕ(s)=π(s)σ(s),
where π(s) is at most first-order polynomials.Likewise, the hypergeometric type functionχ(s) in (2.4) for a fixed n is given by the Rodriques relation as
(2.6)χn(s)=Bnρ(s)dndsn[σn(s)ρ(s)],
where Bn is the normalization constant and the weight function ρ(s) must satisfy the condition
(2.7)dds(σ(s)ρ(s))=τ(s)ρ(s)
with
(2.8)τ(s)=τ~(s)+2π(s).In order to accomplish the condition imposed on the weight functionρ(s), it is necessary that the classical orthogonal polynomials τ(s) be equal to zero to some point of an interval (a,b) and its derivative at this interval at σ(s)>0 will be negative; that is,
(2.9)dτ(s)ds<0.
Therefore, the function π(s) and the parameters λ required for the NU method are defined as follows:
(2.10)π(s)=σ′-τ~2±(σ′-τ~2)2-σ~+kσ,(2.11)λ=k+π′(s).
The k-values in (2.10) are possible to evaluate if the expression under the square root must be square of polynomials. This is possible, if and only if its discriminant is zero. With this condition, the new eigenvalues’ equation becomes
(2.12)λ=λn=-ndτds-n(n-1)2d2σds2,n=0,1,2,….On comparing (2.11) and (2.12), we obtain the energy eigenvalues.The parametric generalization of the NU method is given by the generalized hypergeometric-type equation as [26]
(2.13)ψ′′(s)+(α1-α2s)s(1-α3s)ψ′(s)+1s2(1-α3s)2[-ξ1s2+ξ2s-ξ3]ψ(s)=0.
Comparing (2.13) with (2.2), the following polynomials are obtained:
(2.14)τ~(s)=(α1-α2s),σ(s)=s(1-α3s),σ~(s)=-ξ1s2+ξ2s-ξ3.
Now substituting (2.14) into (2.10), we find
(2.15)π(s)=α4+α5s±[(α6-α3k±)s2+(α7+k±)s+α8]1/2,
where
(2.16)α4=12(1-α1),α5=12(α2-2α3),α6=α52+ξ1,α7=2α4α5-ξ2,α8=α42+ξ3.The resulting value ofk in (2.15) is obtained from the condition that the function under the square root must be square of a polynomials, and it yields
(2.17)k±=-(α7+2α3α8)±2α8α9,
where
(2.18)α9=α3α7+α22α8+α6.
The new π(s) for each k becomes
(2.19)π(s)=α4+α5s-[(α9+α3α8)s-α8],
for the k- value
(2.20)k-=-(α7+2α3α8)-2α8α9.
Using (2.8), we obtain
(2.21)τ(s)=α1+2α4-(α2-2α5)s-2[(α9+α3α8)s-α8].
The physical condition for the bound-state solution is τ′<0, and thus
(2.22)τ′(s)=-2α3-2(α9+α3α8)<0.
With the aid of (2.11) and (2.12), we derive the energy equation as
(2.23)(α2-α3)n+α3n2-(2n+1)α5+(2n+1)(α9+α3α8)+α7+2α3α8+2α8α9=0.
The weight function ρ(s) is obtained from (2.7) as
(2.24)ρ(s)=sα10-1(1-α3s)(α11/α3)-α10-1,
and together with (2.6), we have
(2.25)χn(s)=Pn(α10-1,(α11/α3)-α10-1)(1-2α3s),
where
(2.26)α10=α1+2α4+2α8,α11=α2-2α5+2(α9+α3α8),
and Pn(α,β)(s) are the Jacobi polynomials. The second part of the wave function is obtained from (2.5) as
(2.27)ϕ(s)=sα12(1-α3s)-α12-(α13/α3),
where
(2.28)α12=α4+α8,α13=α5-(α9+α3α8).
Thus, the total wave function becomes
(2.29)ψ(s)=Nnlsα12(1-α3s)-α12-(α13/α3)Pn(α10-1,(α11/α3)-α10-1)(1-2α3s),
whose Nnl is the normalization constant.


## 3. Factorization Method for the Klein-Gordon Equation

The three-dimensional Klein-Gordon equation with mixed vector and scalar potentials can be written as(3.1)[∇2+(V(r)-E)2-(S(r)+M)2]ψ(r,θ,φ)=0,
where M is the rest mass, E is the relativistic energy, and S(r) and V(r) are the scalar and vector potentials, respectively. ∇2 is the Laplace operator, c is the speed of light, and ħ is the reduced Planck’s constant which have been set to unity. In spherical coordinates, the Klein-Gordon equation for a particle in the present of Eckart potential V(r) becomes
(3.2)[1r2∂∂r(r2∂∂r)+1r2sinθ∂∂θ(sinθ∂∂θ)+1r2sin2θ∂2∂φ2-2(EV(r)+MS(r))+V2(r)-S2(r)+E2-M21r2∂∂r(r2∂∂r)+1r2sinθ∂∂θ(sinθ∂∂θ)+1r2sin2θ∂2∂φ2]ψ(r,θ,φ)=0.
If one assigns the corresponding spherical total wave function as
(3.3)ψ(r,θ,φ)=R(r)rYlm(θ,φ),
where
(3.4)Ylm(θ,φ)=Θ(θ)Φ(φ),
then the wave equation in (3.2) is separated into variables and the following equations are obtained:
(3.5)d2R(r)dr2+[E2-M2-2(EV(r)+MS(r))+V2(r)-S2(r)-λr2]R(r)=0,(3.6)d2Θ(θ)dθ2+cotθdΘ(θ)dθ(λ-m2sin2θ)Θ(θ)=0,d2Φ(φ)dφ2+m2Φ(φ)=0,
where  m2 and λ=l(l+1) are the separation constants.Equation (3.6) are spherical harmonic functions whose solutions are well known [27].


## 4. Solution of the Radial Equation

Theq-deformed Eckart potential is defined from [23, 28, 29] as
(4.1)V(r)=-V1e-r/b(1-qe-r/b)+V2e-r/b(1-qe-r/b)2,
where V1,  V2 are the potential depth, q is the deformation parameter, b=1/2α  is the parameter, and α is the range of the potential. The radial part of the Klein-Gordon equation in (3.5) for special case V(r)=S(r) is written as
(4.2)d2R(r)dr2+[E2-M2-2(E+M)V(r)-λr2]R(r)=0.
Substituting (4.1) into (4.2), we obtain
(4.3)d2R(r)dr2+[E2-M2+2(E+M)V1e-r/b(1-qe-r/b)-2(E+M)V2e-r/b(1-qe-r/b)2-l(l+1)r2]R(r)=0.
Obviously, this equation cannot be solved analytically for l≠0 due to the centrifugal term. Therefore, (4.3) can be evaluated by using a newly improved approximation scheme [30]:
(4.4)1r2≈1b2(C0+C1e-r/b(1-qe-r/b)+C2e-2r/b(1-qe-r/b)2),
where C0,  C1,  and  C2 are three adjustable parameters.Substituting (4.4) into (4.3), we obtain
(4.5)d2R(r)dr2+[E2-M2+2(E+M)V1e-r/b(1-qe-r/b)-2(E+M)V2e-r/b(1-qe-r/b)2-l(l+1)b2(C0+C1e-r/b(1-qe-r/b)+C2e-2r/b(1-qe-r/b)2)]R(r)=0.
Using a new variable s=e-r/b and substituting in (4.5), we have the following hypergeometric equation:
(4.6)d2R(r)ds2+(1-qs)s(1-qs)dR(s)ds+1s2(1-qs)2[(ε2q2+A)s2+(-2ε2q+B)s+(ε2+C)]R(s)=0,
where
(4.7)ε2=b2(E2-M2),A=-2(E+M)b2V1q-l(l+1)C0q2+l(l+1)C1q-l(l+1)C2,B=2(E+M)b2V1-2b2(E+M)V2+2l(l+1)C0q-l(l+1)C1,C=-l(l+1)C0.
Comparing (4.6) with (2.13), we obtain the parameter set
(4.8)ξ1=-ε2q2-A,ξ2=-2ε2q+B,ξ3=-ε2-C,α1=1,α2=α3=q,α4=0,α5=-q2,α6=q24-q2ε2-A,α7=2ε2q-B,α8=-(ε2+C),α9=q24-A-qB-q2C,α10=1+2i(ε2+C),α11=2q+2(q24-A-qB-q2C+iq(ε2+C)),α12=i(ε2+C),α13=-q2-(q24-A-qB-q2C+iq(ε2+C)).
Substituting (4.8) into (2.15), we obtain the polynomials π(s) as
(4.9)π(s)=-qs2±[(q24-ε2q2-A-qk±)s2+(2ε2q-B+k±)s-(ε2+C)]1/2.
Substituting (4.8) into (2.17), we obtain k± as
(4.10)k±=B+2qC±2(ε2+C)(A+qB+q2(C-14)).
Using (2.19), (2.20), and (4.8), we can obtain π(s) and k- suitable for the NU method as
(4.11)π(s)=-qs2-[(q24-A-qB-q2C+iq(ε2+C))s-i(ε2+C)],k-=B+2qC-2(ε2+C)(A+qB+q2(C-14)).
Substituting (4.8) into (2.22), we obtain
(4.12)τ′(s)=-2q-2(q24-A-qB-q2C+iq(ε2+C))<0,
which is the essential condition for bound-state (real) solution.Substituting (4.8) into (2.23), we obtain the energy eigenvalues of the q-deformed Eckart potential as
(4.13)E2-M2=-14b2[γ+((n+12)+β)2((n+12)+β)]2+l(l+1)C0b2,
where γ=(-2(E+M)b2V1/q)+(l(l+1)C1/q)-(l(l+1)C2/q2) and β=(1/4)+(l(l+1)C2/q2)+(2b2(E+M)V2/q).The weight functionρ(s) in (2.24) is obtained as
(4.14)ρ(s)=sμ(1-qs)ϑ,
where μ=2i(ε2+C) and  ϑ=2(1/4)-(A/q2)-(B/q)-C which gives the first part of the wave function in (2.3) using (2.25) as
(4.15)χn(s)=Pn(μ,ϑ)(1-2qs).
The other wave function ϕ(s) is obtained from (2.27) as
(4.16)ϕ(s)=sμ/2(1-qs)(1+ϑ)/2.
The radial wave function R(r) expressed in terms of the Jacobi polynomials is obtained from (2.29):
(4.17)R(r)=Nnl(e-r/b)μ/2(1-qe-r/b)(1+ϑ)/2Pn(μ,ϑ)(1-2qe-r/b),
where  Nnl is the normalization constant.Hence, the total wave functionψ(r,θ,φ) for the q-deformed Eckart potential is obtained using (3.3) as
(4.18)ψ(r,θ,φ)=Nnl1r(e-r/b)μ/2(1-qe-r/b)(1+ϑ)/2Pn(μ,ϑ)(1-2qe-r/b)Ylm(θ,φ).


## 5. Discussion

By setting some potential parameters into (4.1), we obtain some well-known potentials.


### 5.1. Hulthen’s Potential

If we setV2=0,  V1=V0, and q=1 in (4.1), we obtain the Hulthen potential [31]
(5.1)V(r)=-V0e-r/b1-e-r/b.
Substituting these parameters into (4.13) and (4.18) we obtain the energy eigenvalues and the corresponding wave function as
(5.2)E2-M2=-14b2[γ~+((n+(1/2))+(1/4)+l(l+1)c2)2((n+(1/2))+(1/4)+l(l+1)c2)]2+l(l+1)C0b2,
where γ~=-2(E+M)b2V0+l(l+1)c1-l(l+1)c2 and
(5.3)ψ(r)=Nnl1r(e-r/b)μ/2(1-e-r/b)(1+ϑ~)/2Pn(μ,ϑ~)(1-2e-r/b)Ylm(θ,φ),
where
(5.4)ϑ~=214-A~-B~-C,A~=-2(E+M)b2V0-l(l+1)c0+l(l+1)c1-l(l+1)c2,B~=2(E+M)b2V0+2l(l+1)c0-l(l+1)c1.


### 5.2.  Modified Pöschl-Teller Potential

SettingV1=0,  q=-1, and V2=-4V0 in (4.1), we obtain the modified Poschl-Teller potential of the form [32–35]
(5.5)V(r)=-V0cosh2(r/2b).
Substituting these parameters into (4.13) and (4.18), we obtain the energy spectrum and the corresponding eigen function as
(5.6)E2-M2=-14b2[-l(l+1)(c1+c2)+((n+(1/2))+(1/4)+l(l+1)c2+8b2(E+M)V0)2((n+(1/2))+(1/4)+l(l+1)c2+8b2(E+M)V0)]2+l(l+1)C0b2,ψ(r)=Nnl1r(e-r/b)μ/2(1+e-r/b)(1+ϑ1)/2Pn(μ,ϑ1)(1+2e-r/b)Ylm(θ,φ),
where
(5.7)ϑ1=214-A1+B1-C,A1=-l(l+1)(c0+c1+c2),B1=8b2(E+M)V0-l(l+1)(2c0+c1),
respectively.


### 5.3. Morse’s Potential

If we setq=0, V1=0 and V2=V0 into (4.1), we obtain the Morse potential of the form [36]
(5.8)V(r)=V0e-r/b.
Substituting these parameters into (4.13) and (4.18), we obtain energy eigenvalues and wave function as
(5.9)E2-M2=-14b2(n+1)2+l(l+1)C0b2,ψ(r)=Nnl1r(e-r/b)μ/2Pn(μ,ϵ)Ylm(θ,φ),
where
(5.10)ϵ=214-C,
respectively.


## 5.1. Hulthen’s Potential

If we setV2=0,  V1=V0, and q=1 in (4.1), we obtain the Hulthen potential [31]
(5.1)V(r)=-V0e-r/b1-e-r/b.
Substituting these parameters into (4.13) and (4.18) we obtain the energy eigenvalues and the corresponding wave function as
(5.2)E2-M2=-14b2[γ~+((n+(1/2))+(1/4)+l(l+1)c2)2((n+(1/2))+(1/4)+l(l+1)c2)]2+l(l+1)C0b2,
where γ~=-2(E+M)b2V0+l(l+1)c1-l(l+1)c2 and
(5.3)ψ(r)=Nnl1r(e-r/b)μ/2(1-e-r/b)(1+ϑ~)/2Pn(μ,ϑ~)(1-2e-r/b)Ylm(θ,φ),
where
(5.4)ϑ~=214-A~-B~-C,A~=-2(E+M)b2V0-l(l+1)c0+l(l+1)c1-l(l+1)c2,B~=2(E+M)b2V0+2l(l+1)c0-l(l+1)c1.


## 5.2.  Modified Pöschl-Teller Potential

SettingV1=0,  q=-1, and V2=-4V0 in (4.1), we obtain the modified Poschl-Teller potential of the form [32–35]
(5.5)V(r)=-V0cosh2(r/2b).
Substituting these parameters into (4.13) and (4.18), we obtain the energy spectrum and the corresponding eigen function as
(5.6)E2-M2=-14b2[-l(l+1)(c1+c2)+((n+(1/2))+(1/4)+l(l+1)c2+8b2(E+M)V0)2((n+(1/2))+(1/4)+l(l+1)c2+8b2(E+M)V0)]2+l(l+1)C0b2,ψ(r)=Nnl1r(e-r/b)μ/2(1+e-r/b)(1+ϑ1)/2Pn(μ,ϑ1)(1+2e-r/b)Ylm(θ,φ),
where
(5.7)ϑ1=214-A1+B1-C,A1=-l(l+1)(c0+c1+c2),B1=8b2(E+M)V0-l(l+1)(2c0+c1),
respectively.


## 5.3. Morse’s Potential

If we setq=0, V1=0 and V2=V0 into (4.1), we obtain the Morse potential of the form [36]
(5.8)V(r)=V0e-r/b.
Substituting these parameters into (4.13) and (4.18), we obtain energy eigenvalues and wave function as
(5.9)E2-M2=-14b2(n+1)2+l(l+1)C0b2,ψ(r)=Nnl1r(e-r/b)μ/2Pn(μ,ϵ)Ylm(θ,φ),
where
(5.10)ϵ=214-C,
respectively.


## 6. Conclusion

In this paper, we have studied the Klein-Gordon equation subject to equalq-deformed scalar and vector Eckart potentials. The energy and wave functions for bound states have been obtained by parametric form of the Nikiforov-Uvarov method. We also discussed some special cases of the potential.



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