# Exact Solutions of the Mass-Dependent Klein-Gordon Equation with the Vector Quark-Antiquark Interaction and Harmonic Oscillator Potential

Authors: M. K. Bahar; F. Yasuk
Journal: Advances in High Energy Physics
 (2013)

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## Abstract

Using  the asymptotic iteration and wave function ansatz method, we present exact solutions of the Klein-Gordon equation for the quark-antiquark interaction and harmonic oscillator potential in the case of the position-dependent mass.


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## Body

## 1. Introduction

The movement of a particle with harmonic oscillations around an equilibrium position creates one of the most fundamental problems of physics. The basic structure of different systems such as vibration of the diatomic molecules, oscillations of atoms in crystal lattices, or nucleons in core   is a harmonic oscillator problem [1]. Further, the quantum theory of electromagnetic fields is closely related to the  examples of the harmonic oscillator [2, 3]. When a particle is in a strong potential field, the relativistic effect must be considered. However, in relativistic and nonrelativistic quantum mechanics, many authors have adequately the harmonic oscillator problem [4, 5]. But, it has never been investigated for relativistic particles with position-dependent mass (pdm). Quantum mechanical review of relativistic spin-0 particles with pdm in the harmonic oscillator potential is very important in terms of understanding physical behavior of systems as the above-mentioned systems. The studies of quantum and relativistic quantum systems with pdm have received increasing attention in the literature. Systems with pdm have been found to be very useful in studying the physical and electronic properties of semiconductors, quantum wells and quantum dots, quantum liquids, 3He clusters, graded alloys, and semiconductor heterostructures [6]. To study the quasi-exactly solvable and exactly solvable nonrelativistic Schrödinger, relativistic Klein-Gordon and Dirac equations in the presence of pdm having a suitable mass distribution functions in one, three, and/or any arbitrary D-dimensional cases for different potentials have been used in different methods by many authors [7–21].In this study, in addition to the examination of the harmonic oscillator potential, relativistic spin-0 particles with pdm have also been investigated in the quark-antiquark interaction potential. This type of potential and some of central potentials have recently been studied using different techniques [22–25]. The quark-antiquark interaction potential consists of harmonic, linear, and Coulomb potential terms. As we know, the quark-antiquark interaction potentials are a spherically symmetrical potential. The spherically symmetrical potential model also presents a good description of heavy quarkonium mass spectra such as charmonium and bottomonium. The interaction potentials for such systems are of a confining type called the Cornell potential. The Cornell potential consists of two terms, namely, the Coulomb and linear terms. The Coulomb term is responsible for the interaction at small distances and the linear term leads to the confinement. This type of interaction potential is also supported by lattice quantum chromodynamics calculations [26]. The quark-antiquark interaction has also been studied using the Coulomb term plus the power potential [27].The organization of this paper is as follows. In the second section, the AIM is given shortly. In the third section, the general formalism of the Klein-Gordon equation for spin-0 particles with pdm has been considered. The relativistic energy eigenvalues and corresponding eigenfunctions have been presented for the harmonic oscillator and quark-antiquark interaction potentials preserved in the fourth and fifth section, respectively. Finally, conclusions are given in the last section.


## 2. Basic Equations of the AIM

We briefly outline the AIM here; the details can be found in [28–30]. The AIM was proposed to solve second-order differential equations of the form:
(1)y′′=λ0(x)y′+s0(x)y,
where λ0(x)≠0 and s0(x) are in C∞(a,b), and these variables are sufficiently differentiable. The differential equation (1) has a general solution as follows:
(2)y(x)=exp(-∫xαdx′)×[C2+C1∫xexp(∫x′[λ0(x′′)+2α(x′′)]dx′′)dx′]
if n>0, for sufficiently large n,
(3)snλn=sn-1λn-1=αk,
where
(4)λn(x)=λn-1′(x)+sn-1(x)+λ0(x)λn-1(x),sn(x)=sn-1′(x)+s0(x)λn-1(x),n=1,2,3,…
The termination condition of the method together with  (4) can be also written as follows:
(5)δ(x)=λn+1(x)sn(x)-λn(x)sn+1(x)=0.
For a given potential, the idea is to convert the relativistic wave equation to the form of  (1). Then, s0 and λ0 are determined and sn and λn parameters are calculated. The energy eigenvalues are obtained by the termination condition given by (5). However, the exact eigenfunctions can be derived from the following wave function generator:
(6)yn(x)=C2exp(-∫xαkdx′),
where n=0,1,2,… and k is the iteration step number, and it is greater than n.


## 3. Formalism of the Klein-Gordon Equation with Pdm

In the relativistic quantum mechanics, for spin-0 particles with pdm, the Klein-Gordon equation is defined as follows:(7)∇2ψ(r)+1ℏ2c2[(Enl-V(r))2-(m(r)c2+S(r))2]ψ(r)=0,∇2=∑i=13∂2∂xi2,
where V(r) and S(r) are Lorentz vector and scalar potential, respectively, m(r) is mass function, and Enl is the energy of particle. Let us decompose the radial wave function ψ(r) as follows:
(8)ψ(r)=u(r)rYml(r^),
where u(r) is the radial wave function and Yml(r^) is the angular dependent spherical harmonics, and this reduces (8) into the following Schrödinger-like equation with position-dependent mass:
(9)d2u(r)dr2+1ℏ2c2×([Enl-V(r)]2-[m(r)c2+S(r)]2-l(l+1)r2)×u(r)=0.


## 4. In Case of Harmonic Oscillator Potential

### 4.1. The Eigenvalues

In case of harmonic oscillator to investigate spin-0 particles with pdm, we should solve (9). In this solution, we use atomic units ℏ=c=1. However, in (9), we prefer to use mass function similar to type of harmonic oscillator potential as follows:
(10)m(r)=m0+12kr2,
where m0 and k are positive constants. The selection as in (10) of position dependent mass function is more suitable both physically and mathematical. Already, in physical applications, the position-dependent mass creates a new effective potential by shifting potential profile of the system.In this study, in the absence of scalar potential, vector harmonic oscillator potential is defined as(11)V(r)=12m(r)ω2r2,
where ω=k/m(r) is the angular frequency and k is elastic coefficient.In the presence of vector potential and by takingℏ=c=1, if (10) and (11) are inserted into (9), it is obtained that
(12)[d2dr2+ξ0-ξ1r2-ξ2r2]u(r)=0,
where ξ0=Enl2-m02, ξ1=k(Enl-m0),  ξ2=l(l+1).In (12), while r approaches zero and infinite, solving of (12) are r(1/2)(1+1+4ξ2), e-r2ξ1/2, respectively. Therefore, the reasonable physical wave function is proposed as follows:
(13)unl(r)=r(1/2)(1+1+4ξ2)e-r2ξ1/2fnl(r).
Substituting (13) into (12), we have second-order homogeneous linear differential equation:
(14)f′′(r)=(-1+1+4ξ2r+2r)f′(r)-(ξ0-ξ1(2+1+4ξ2))f(r).
Defining a new variable z=ξ1r2, so doing, we have solvable differential equation by AIM as follows:
(15)f′′(z)=(1-2+1+4ξ22z)f′(z)+ξ1(2+1+4ξ2)-ξ04ξ1zf(z).
By comparing (15) with (1), λ0(z) and s0(z) values are  obtained, and using (4) we calculate λn(z) and sn(z). In this way,
(16)λ0(z)=(1-2+1+4ξ22z),s0(z)=ξ1(2+1+4ξ2)-ξ04ξ1z,λ1(z)=(4ξ1(2-2z+1+4ξ2)2+4ξ1(-2zξ0+2ξ1(2+1+4ξ2))+zξ1(2+1+4ξ2)(2-4ξ12z+1+4ξ2)2)×(16z2ξ1)-1,s1(z)=((2-ξ1(2+1+4ξ2))×(4ξ1+2ξ1(2-2z+1+4ξ2)))×(16z2ξ1)-1s1(z)=⋮
Combining these results obtained by the AIM with quantization condition given by (5) yields(17a)s0λ1-s1λ0=0⇒ξ00=12ξ1(2+1+4ξ2),forn=0(17b)s1λ2-s2λ1=0⇒ξ01=12ξ1(6+1+4ξ2),forn=1(17c)s2λ3-s3λ2=0⇒ξ02=12ξ1(10+1+4ξ2),forn=2s1(z)=⋮If the set of equations(17a), (17b), and (17c) are generalized, the indirect energy eigenvalues statement turns out to be
(18)ξ0n=12ξ1(4n+2+1+4ξ2).
When (18) and ξ0=Enl2-m02 are compared, it is found for energy eigenvalues that
(19)Enl2=m02+k(Enl-m0)2(4n+2+1+4l(l+1)).


### 4.2. The Eigenfunctions

The exact eigenfunctions can be derived from the following generator:(20)fn(z)=C2exp(-∫zαkdz′).
Using (3) and (20), the eigenfunctions are obtained as follows:
(21)f0(z)=1,f1(z)=(z-(2+1+4ξ2)2),f2(z)=(2+1+4ξ22)(1+2+1+4ξ22)-2(1+2+1+4ξ22)z+z2,s1(z)=⋮
Finally, the following general formula for the exact solutions fn(z) is acquired as
(22)fn(z)=(-1)n(2+1+4ξ22)n×F11(-n,2+1+4ξ22;z).
Hence, we write the total radial wavefunction as follows:
(23)unl(r)=Nr(1/2)(1+1+4ξ2)e-r2ξ1/2(2+1+4ξ22)n×F11(-n,2+1+4ξ22;ξ1r2),
where N is normalization constant.


## 4.1. The Eigenvalues

In case of harmonic oscillator to investigate spin-0 particles with pdm, we should solve (9). In this solution, we use atomic units ℏ=c=1. However, in (9), we prefer to use mass function similar to type of harmonic oscillator potential as follows:
(10)m(r)=m0+12kr2,
where m0 and k are positive constants. The selection as in (10) of position dependent mass function is more suitable both physically and mathematical. Already, in physical applications, the position-dependent mass creates a new effective potential by shifting potential profile of the system.In this study, in the absence of scalar potential, vector harmonic oscillator potential is defined as(11)V(r)=12m(r)ω2r2,
where ω=k/m(r) is the angular frequency and k is elastic coefficient.In the presence of vector potential and by takingℏ=c=1, if (10) and (11) are inserted into (9), it is obtained that
(12)[d2dr2+ξ0-ξ1r2-ξ2r2]u(r)=0,
where ξ0=Enl2-m02, ξ1=k(Enl-m0),  ξ2=l(l+1).In (12), while r approaches zero and infinite, solving of (12) are r(1/2)(1+1+4ξ2), e-r2ξ1/2, respectively. Therefore, the reasonable physical wave function is proposed as follows:
(13)unl(r)=r(1/2)(1+1+4ξ2)e-r2ξ1/2fnl(r).
Substituting (13) into (12), we have second-order homogeneous linear differential equation:
(14)f′′(r)=(-1+1+4ξ2r+2r)f′(r)-(ξ0-ξ1(2+1+4ξ2))f(r).
Defining a new variable z=ξ1r2, so doing, we have solvable differential equation by AIM as follows:
(15)f′′(z)=(1-2+1+4ξ22z)f′(z)+ξ1(2+1+4ξ2)-ξ04ξ1zf(z).
By comparing (15) with (1), λ0(z) and s0(z) values are  obtained, and using (4) we calculate λn(z) and sn(z). In this way,
(16)λ0(z)=(1-2+1+4ξ22z),s0(z)=ξ1(2+1+4ξ2)-ξ04ξ1z,λ1(z)=(4ξ1(2-2z+1+4ξ2)2+4ξ1(-2zξ0+2ξ1(2+1+4ξ2))+zξ1(2+1+4ξ2)(2-4ξ12z+1+4ξ2)2)×(16z2ξ1)-1,s1(z)=((2-ξ1(2+1+4ξ2))×(4ξ1+2ξ1(2-2z+1+4ξ2)))×(16z2ξ1)-1s1(z)=⋮
Combining these results obtained by the AIM with quantization condition given by (5) yields(17a)s0λ1-s1λ0=0⇒ξ00=12ξ1(2+1+4ξ2),forn=0(17b)s1λ2-s2λ1=0⇒ξ01=12ξ1(6+1+4ξ2),forn=1(17c)s2λ3-s3λ2=0⇒ξ02=12ξ1(10+1+4ξ2),forn=2s1(z)=⋮If the set of equations(17a), (17b), and (17c) are generalized, the indirect energy eigenvalues statement turns out to be
(18)ξ0n=12ξ1(4n+2+1+4ξ2).
When (18) and ξ0=Enl2-m02 are compared, it is found for energy eigenvalues that
(19)Enl2=m02+k(Enl-m0)2(4n+2+1+4l(l+1)).


## 4.2. The Eigenfunctions

The exact eigenfunctions can be derived from the following generator:(20)fn(z)=C2exp(-∫zαkdz′).
Using (3) and (20), the eigenfunctions are obtained as follows:
(21)f0(z)=1,f1(z)=(z-(2+1+4ξ2)2),f2(z)=(2+1+4ξ22)(1+2+1+4ξ22)-2(1+2+1+4ξ22)z+z2,s1(z)=⋮
Finally, the following general formula for the exact solutions fn(z) is acquired as
(22)fn(z)=(-1)n(2+1+4ξ22)n×F11(-n,2+1+4ξ22;z).
Hence, we write the total radial wavefunction as follows:
(23)unl(r)=Nr(1/2)(1+1+4ξ2)e-r2ξ1/2(2+1+4ξ22)n×F11(-n,2+1+4ξ22;ξ1r2),
where N is normalization constant.


## 5. In Case of Quark-Antiquark Interaction Potential

### 5.1. The Eigenvalues and Corresponding Eigenfunctions

In this section, we present the solution of the Klein-Gordon equation for the quark-antiquark interaction potential. This potential is defined as(24)V(r)=ar2+br-cr,a>0,
where a, b, and c are constants. In atomic units ℏ=c=1, we solve (9) in the absence of scalar potential. Consider the position-dependent mass function:
(25)m(r)=m0+ar2+br-cr.
On substituting (24) and (25) into (9), we find that
(26)[d2dr2+ε-ε0r2-ε1r+ε2r-l(l+1)r2]ul(r)=0,
where ε=Enl2-m02, ε0=2a(Enl+m0), ε1=2b(Enl+m0), and ε2=2c(Enl+m0).To solve (26), applying an ansatz to the radial wavefunction ul(r),
(27)ul(r)=eαr+(γ/2)r2∑n=0anrn+δ.
If (27) is inserted into (26), it is obtained that
(28)∑n=0an[γ+2γ(n+δ)+ε]︸Anrn+δ+∑n=0an[2αγ-ε1]︸2αγ=ε1rn+δ+1+∑n=0an[α2-ε0+γ2]︸ε0=α2+γ2rn+δ+2+∑n=0an[2α(n+δ)+ε2]︸Bnrn+δ-1+∑n=0an[(n+δ)(n+δ-1)-l(l+1)]︸Cnrn+δ-2=0,(29)ε1=2αγ,ε0=α2+γ2α and γ can be obtained with the help of (29). Then, α=(-ε0+ε1-ε0-ε1)/2 and γ=(-ε0+ε1+ε0-ε1)/2. Editing (28),
(30)∑n=0anAnrn+δ+∑n=0anBnrn+δ-1+︸n→n+1∑n=0anCnrn+δ-2︸n→n+2=0,(31)(a0B0+a1C1)︸0rδ-1+a0C0︸0rδ-2+∑n=0(anAn+an+1Bn+1+an+2Cn+2)︸0rn+δ=0.
In (31), if the first nonvanishing coefficient is a0≠0, C0 should be equal to zero:
(32)C0=δ(δ-1)-l(l+1)=0.
We choose δ=l+1 as a physically acceptable solution from (32). Moreover, if the pth nonvanishing coefficient is ap≠0, but ap+1=ap+2=⋯=0, then, from (31), it has to be Ap=0. At that rate,
(33)γ+2γ(p+δ)+ε=0.
Using together (29) and (33), we obtain the energy eigenvalues. Namely,
(34)b(Enl+m0)α+2b(Enl+m0)α(p+l+1)+Enl2-m02=0.An,Bn, and Cn must satisfy the determinant relation for a nontrivial solution:
(35)det|B0C1………0A0B1C2⋯⋯.⋮⋮⋮⋱⋮⋮0000Ap-1Bp|=0.
In order to appreciate this method, we present the exact solutions for the cases p=0,1 as follows.Ifp=0, det|B0|=0 and B0=0. So,
(36)B0=0⇒2(E0+m0)c=-2(l+1)α.
We will obtain energy eigenvalues by using (34). But, we cannot ignore (36) Because it is a restriction on the parameters of the potential and the l quantum number.The corresponding eigenfunction forp=0 is given as
(37)ul0(r)=a0exp[-ε0+ε1+ε0-ε12r-ε0+ε1-ε0-ε14r2]rδ,
where a0 is the normalization constant.Ifp=1, det|B0C1A0B1|=0. In this case, it is obtained that
(38)(ε12α+ε1α(l+1)+ε)2(l+1)-(ε2+2α(l+1))(ε2+2α(l+2))︸forE1=0,
which is a restriction on the parameters of the potential and the angular momentum quantum number.The corresponding eigenfunction forp=1 is given as
(39)ul1(r)=(a0+a1r)exp[-ε0+ε1+ε0-ε12r-ε0+ε1-ε0-ε14r2]rδ,
where a0 is the normalization constant.Following in this way, we can generate a class of exact solutions by settingp=1,2,…. Generally, if  ap≠0, but ap+1=ap+2=⋯=0. So, the energy eigenvalues Ep is obtained by using (34). The corresponding eigenfunction is
(40)ulp(r)=(a0+a1r+⋯+aprp)×exp[-ε0+ε1+ε0-ε12r-ε0+ε1-ε0-ε14r2]rδ,
where a0,a1,…,ap are normalization constants.


## 5.1. The Eigenvalues and Corresponding Eigenfunctions

In this section, we present the solution of the Klein-Gordon equation for the quark-antiquark interaction potential. This potential is defined as(24)V(r)=ar2+br-cr,a>0,
where a, b, and c are constants. In atomic units ℏ=c=1, we solve (9) in the absence of scalar potential. Consider the position-dependent mass function:
(25)m(r)=m0+ar2+br-cr.
On substituting (24) and (25) into (9), we find that
(26)[d2dr2+ε-ε0r2-ε1r+ε2r-l(l+1)r2]ul(r)=0,
where ε=Enl2-m02, ε0=2a(Enl+m0), ε1=2b(Enl+m0), and ε2=2c(Enl+m0).To solve (26), applying an ansatz to the radial wavefunction ul(r),
(27)ul(r)=eαr+(γ/2)r2∑n=0anrn+δ.
If (27) is inserted into (26), it is obtained that
(28)∑n=0an[γ+2γ(n+δ)+ε]︸Anrn+δ+∑n=0an[2αγ-ε1]︸2αγ=ε1rn+δ+1+∑n=0an[α2-ε0+γ2]︸ε0=α2+γ2rn+δ+2+∑n=0an[2α(n+δ)+ε2]︸Bnrn+δ-1+∑n=0an[(n+δ)(n+δ-1)-l(l+1)]︸Cnrn+δ-2=0,(29)ε1=2αγ,ε0=α2+γ2α and γ can be obtained with the help of (29). Then, α=(-ε0+ε1-ε0-ε1)/2 and γ=(-ε0+ε1+ε0-ε1)/2. Editing (28),
(30)∑n=0anAnrn+δ+∑n=0anBnrn+δ-1+︸n→n+1∑n=0anCnrn+δ-2︸n→n+2=0,(31)(a0B0+a1C1)︸0rδ-1+a0C0︸0rδ-2+∑n=0(anAn+an+1Bn+1+an+2Cn+2)︸0rn+δ=0.
In (31), if the first nonvanishing coefficient is a0≠0, C0 should be equal to zero:
(32)C0=δ(δ-1)-l(l+1)=0.
We choose δ=l+1 as a physically acceptable solution from (32). Moreover, if the pth nonvanishing coefficient is ap≠0, but ap+1=ap+2=⋯=0, then, from (31), it has to be Ap=0. At that rate,
(33)γ+2γ(p+δ)+ε=0.
Using together (29) and (33), we obtain the energy eigenvalues. Namely,
(34)b(Enl+m0)α+2b(Enl+m0)α(p+l+1)+Enl2-m02=0.An,Bn, and Cn must satisfy the determinant relation for a nontrivial solution:
(35)det|B0C1………0A0B1C2⋯⋯.⋮⋮⋮⋱⋮⋮0000Ap-1Bp|=0.
In order to appreciate this method, we present the exact solutions for the cases p=0,1 as follows.Ifp=0, det|B0|=0 and B0=0. So,
(36)B0=0⇒2(E0+m0)c=-2(l+1)α.
We will obtain energy eigenvalues by using (34). But, we cannot ignore (36) Because it is a restriction on the parameters of the potential and the l quantum number.The corresponding eigenfunction forp=0 is given as
(37)ul0(r)=a0exp[-ε0+ε1+ε0-ε12r-ε0+ε1-ε0-ε14r2]rδ,
where a0 is the normalization constant.Ifp=1, det|B0C1A0B1|=0. In this case, it is obtained that
(38)(ε12α+ε1α(l+1)+ε)2(l+1)-(ε2+2α(l+1))(ε2+2α(l+2))︸forE1=0,
which is a restriction on the parameters of the potential and the angular momentum quantum number.The corresponding eigenfunction forp=1 is given as
(39)ul1(r)=(a0+a1r)exp[-ε0+ε1+ε0-ε12r-ε0+ε1-ε0-ε14r2]rδ,
where a0 is the normalization constant.Following in this way, we can generate a class of exact solutions by settingp=1,2,…. Generally, if  ap≠0, but ap+1=ap+2=⋯=0. So, the energy eigenvalues Ep is obtained by using (34). The corresponding eigenfunction is
(40)ulp(r)=(a0+a1r+⋯+aprp)×exp[-ε0+ε1+ε0-ε12r-ε0+ε1-ε0-ε14r2]rδ,
where a0,a1,…,ap are normalization constants.


## 6. Conclusions

This paper has presented a different approach, the AIM, to calculate the bound state solutions of the relativistic Klein-Gordon with the harmonic oscillator potential in the case of the pdm. For arbitrary quantum numberl state, we have exactly obtained the energy eigenvalues and corresponding eigenfunctions for the case of mass function by AIM. The advantage of the AIM is that it gives the eigenvalues directly by transforming the second-order differential equation into a form of y′′=λ0(r)y′+s0(r)y. The exact wavefunctions are easily constructed by iterating the values of s0 and λ0. The method presented in this study is general and worth extending to the solution of other interactions. For the quark-antiquark interaction potential, to solve Klein-Gordon equation with pdm, we have used wave function ansatz method. While using this method, the most important factor to be considered is a corresponding restriction on the parameters of the quark-antiquark potential and the l quantum number.



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