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

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.

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: 𝑦^′′=𝜆₀(𝑥)𝑦^′+𝑠₀(𝑥)𝑦,(1) 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: 𝑦(𝑥)=exp−^𝑥𝛼𝑑𝑥^′×𝐶₂+𝐶₁^𝑥exp^𝑥′𝜆₀𝑥^′′+2𝛼𝑥^′′𝑑𝑥^′′𝑑𝑥^′(2) if 𝑛>0, for sufficiently large 𝑛, 𝑠_𝑛𝜆_𝑛=𝑠_𝑛−1𝜆_𝑛−1=𝛼_𝑘,(3) where 𝜆_𝑛(𝑥)=𝜆^′_𝑛−1(𝑥)+𝑠_𝑛−1(𝑥)+𝜆₀(𝑥)𝜆_𝑛−1(𝑥),𝑠_𝑛(𝑥)=𝑠^′_𝑛−1(𝑥)+𝑠₀(𝑥)𝜆_𝑛−1(𝑥),𝑛=1,2,3,…(4) The termination condition of the method together with (4) can be also written as follows: 𝛿(𝑥)=𝜆_𝑛+1(𝑥)𝑠_𝑛(𝑥)−𝜆_𝑛(𝑥)𝑠_𝑛+1(𝑥)=0.(5) 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 𝑠_𝑛 and 𝜆_𝑛 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: 𝑦_𝑛(𝑥)=𝐶₂exp−^𝑥𝛼_𝑘𝑑𝑥^′,(6) 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: ∇²𝜓(𝑟)+1ℏ²𝑐²𝐸_𝑛𝑙−𝑉(𝑟)²−𝑚(𝑟)𝑐²+𝑆(𝑟)²𝜓(𝑟)∇=0,₂=³_𝑖=1𝜕²𝜕𝑥²_𝑖,(7) where 𝑉(𝑟) and 𝑆(𝑟) are Lorentz vector and scalar potential, respectively, 𝑚(𝑟) is mass function, and 𝐸_𝑛𝑙 is the energy of particle. Let us decompose the radial wave function 𝜓(𝑟) as follows: 𝜓(𝑟)=𝑢(𝑟)𝑟𝑌^𝑙_𝑚(𝑟)̂,(8) where 𝑢(𝑟) is the radial wave function and 𝑌^𝑙_𝑚(𝑟)̂ is the angular dependent spherical harmonics, and this reduces (8) into the following Schrödinger-like equation with position-dependent mass: 𝑑²𝑢(𝑟)𝑑𝑟²+1ℏ²𝑐²×𝐸_𝑛𝑙−𝑉(𝑟)²−𝑚(𝑟)𝑐²+𝑆(𝑟)²−𝑙(𝑙+1)𝑟²×𝑢(𝑟)=0.(9)

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: 𝑚(𝑟)=𝑚₀+12𝑘𝑟²,(10) 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 𝑉(𝑟)=12𝑚(𝑟)𝜔²𝑟²,(11) where 𝜔=𝑘/𝑚(𝑟) is the angular frequency and 𝑘 is elastic coefficient.

In the presence of vector potential and by taking ℏ=𝑐=1, if (10) and (11) are inserted into (9), it is obtained that 𝑑²𝑑𝑟²+𝜉₀−𝜉₁𝑟²−𝜉₂𝑟²𝑢(𝑟)=0,(12) where 𝜉₀=𝐸²_𝑛𝑙−𝑚²₀, 𝜉₁=𝑘(𝐸_𝑛𝑙−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: 𝑢_𝑛𝑙(𝑟)=𝑟^{(1/2)(1+1+4𝜉2)}𝑒^{−𝑟2𝜉1/2}𝑓_𝑛𝑙(𝑟).(13) Substituting (13) into (12), we have second-order homogeneous linear differential equation: 𝑓^′′(𝑟)=−1+1+4𝜉₂𝑟+2𝑟𝑓^′(𝑟)−𝜉₀−𝜉₁2+1+4𝜉₂𝑓(𝑟).(14) Defining a new variable 𝑧=𝜉₁𝑟², so doing, we have solvable differential equation by AIM as follows: 𝑓^′′(𝑧)=1−2+1+4𝜉₂𝑓2𝑧^′(𝑧)+𝜉₁2+1+4𝜉₂−𝜉₀4𝜉₁𝑧𝑓(𝑧).(15) By comparing (15) with (1), 𝜆₀(𝑧) and 𝑠₀(𝑧) values are obtained, and using (4) we calculate 𝜆_𝑛(𝑧) and 𝑠_𝑛(𝑧). In this way, 𝜆₀(𝑧)=1−2+1+4𝜉₂,𝑠2𝑧₀(𝑧)=𝜉₁2+1+4𝜉₂−𝜉₀4𝜉₁𝑧,𝜆₁(𝑧)=4𝜉₁2−2𝑧+1+4𝜉₂²+4𝜉₁−2𝑧𝜉₀+2𝜉₁2+1+4𝜉₂+𝑧𝜉₁2+1+4𝜉₂×16𝑧²𝜉₁⁻¹,𝑠₁(𝑧)=2−𝜉₁2+1+4𝜉₂×4𝜉₁+2𝜉₁2−2𝑧+1+4𝜉₂×16𝑧²𝜉₁⁻¹⋮(16) Combining these results obtained by the AIM with quantization condition given by (5) yields𝑠₀𝜆₁−𝑠₁𝜆₀=0⇒𝜉₀₀=12𝜉₁2+1+4𝜉₂()𝑠,for𝑛=017a₁𝜆₂−𝑠₂𝜆₁=0⇒𝜉₀₁=12𝜉₁6+1+4𝜉₂()𝑠,for𝑛=117b₂𝜆₃−𝑠₃𝜆₂=0⇒𝜉₀₂=12𝜉₁10+1+4𝜉₂⋮,for𝑛=2(17c)

If the set of equations (17a), (17b), and (17c) are generalized, the indirect energy eigenvalues statement turns out to be 𝜉_0𝑛=12𝜉₁4𝑛+2+1+4𝜉₂.(18) When (18) and 𝜉₀=𝐸²_𝑛𝑙−𝑚²₀ are compared, it is found for energy eigenvalues that 𝐸²_𝑛𝑙=𝑚²₀+𝑘𝐸_𝑛𝑙−𝑚₀24𝑛+2+1+4𝑙(𝑙+1).(19)

4.2. The Eigenfunctions

The exact eigenfunctions can be derived from the following generator: 𝑓_𝑛(𝑧)=𝐶₂exp−^𝑧𝛼_𝑘𝑑𝑧^′.(20) Using (3) and (20), the eigenfunctions are obtained as follows: 𝑓₀(𝑧)𝑓=1,₁(𝑧)=𝑧−2+1+4𝜉₂2,𝑓₂(𝑧)=2+1+4𝜉₂21+2+1+4𝜉₂2−21+2+1+4𝜉₂2𝑧+𝑧²,⋮(21) Finally, the following general formula for the exact solutions 𝑓_𝑛(𝑧) is acquired as 𝑓_𝑛(𝑧)=(−1)^𝑛2+1+4𝜉₂2_𝑛×₁𝐹₁−𝑛,2+1+4𝜉₂2;𝑧.(22) Hence, we write the total radial wavefunction as follows: 𝑢_𝑛𝑙(𝑟)=𝑁𝑟^{(1/2)(1+1+4𝜉2)}𝑒^{−𝑟2𝜉1/2}2+1+4𝜉₂2_𝑛×₁𝐹₁−𝑛,2+1+4𝜉₂2;𝜉₁𝑟²,(23) 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 𝑉(𝑟)=𝑎𝑟²𝑐+𝑏𝑟−𝑟,𝑎>0,(24) 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: 𝑚(𝑟)=𝑚₀+𝑎𝑟²𝑐+𝑏𝑟−𝑟.(25) On substituting (24) and (25) into (9), we find that 𝑑²𝑑𝑟²+𝜀−𝜀₀𝑟²−𝜀₁𝜀𝑟+²𝑟−𝑙(𝑙+1)𝑟²𝑢_𝑙(𝑟)=0,(26) where 𝜀=𝐸²_𝑛𝑙−𝑚²₀, 𝜀₀=2𝑎(𝐸_𝑛𝑙+𝑚₀), 𝜀₁=2𝑏(𝐸_𝑛𝑙+𝑚₀), and 𝜀₂=2𝑐(𝐸_𝑛𝑙+𝑚₀).

To solve (26), applying an ansatz to the radial wavefunction 𝑢_𝑙(𝑟), 𝑢_𝑙(𝑟)=𝑒^{𝛼𝑟+(𝛾/2)𝑟2}_𝑛=0𝑎_𝑛𝑟^𝑛+𝛿.(27) If (27) is inserted into (26), it is obtained that _𝑛=0𝑎_𝑛𝛾+2𝛾(𝑛+𝛿)+𝜀_𝐴𝑛𝑟^𝑛+𝛿+_𝑛=0𝑎_𝑛2𝛼𝛾−𝜀_{12𝛼𝛾=𝜀1}𝑟^𝑛+𝛿+1+_𝑛=0𝑎_𝑛𝛼²−𝜀₀+𝛾²_{𝜀0=𝛼2+𝛾2}𝑟^𝑛+𝛿+2+_𝑛=0𝑎_𝑛2𝛼(𝑛+𝛿)+𝜀_2𝐵𝑛𝑟^{𝑛+𝛿−1}+_𝑛=0𝑎_𝑛[(𝑛+𝛿)(𝑛+𝛿−1)−𝑙(𝑙+1)]_𝐶𝑛𝑟^{𝑛+𝛿−2}𝜀=0,(28)₁=2𝛼𝛾,𝜀₀=𝛼²+𝛾²(29)𝛼 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), _𝑛=0𝑎_𝑛𝐴_𝑛𝑟^𝑛+𝛿+_𝑛=0𝑎_𝑛𝐵_𝑛𝑟^{𝑛+𝛿−1}+_{𝑛→𝑛+1𝑛=0}𝑎_𝑛𝐶_𝑛𝑟^{𝑛+𝛿−2}_{𝑛→𝑛+2}𝑎=0,(30)₀𝐵₀+𝑎₁𝐶₁₀𝑟^𝛿−1+𝑎₀𝐶₀₀𝑟^𝛿−2+_𝑛=0𝑎_𝑛𝐴_𝑛+𝑎_𝑛+1𝐵_𝑛+1+𝑎_𝑛+2𝐶_𝑛+20𝑟^𝑛+𝛿=0.(31) In (31), if the first nonvanishing coefficient is 𝑎₀≠0, 𝐶₀ should be equal to zero: 𝐶₀=𝛿(𝛿−1)−𝑙(𝑙+1)=0.(32) We choose 𝛿=𝑙+1 as a physically acceptable solution from (32). Moreover, if the 𝑝th nonvanishing coefficient is 𝑎_𝑝≠0, but 𝑎_𝑝+1=𝑎_𝑝+2=⋯=0, then, from (31), it has to be 𝐴_𝑝=0. At that rate, 𝛾+2𝛾𝑝+𝛿+𝜀=0.(33) Using together (29) and (33), we obtain the energy eigenvalues. Namely, 𝑏𝐸_𝑛𝑙+𝑚₀𝛼+2𝑏𝐸_𝑛𝑙+𝑚₀𝛼𝑝+𝑙+1+𝐸²_𝑛𝑙−𝑚²₀=0.(34)𝐴_𝑛,𝐵_𝑛, and 𝐶_𝑛 must satisfy the determinant relation for a nontrivial solution: 𝐵det⁰𝐶₁𝐴………0₀𝐵₁𝐶₂⋯⋯.⋮⋮⋮⋱⋮⋮0000𝐴_𝑝−1𝐵_𝑝=0.(35) In order to appreciate this method, we present the exact solutions for the cases 𝑝=0,1 as follows.

If 𝑝=0, det|𝐵₀|=0 and 𝐵₀=0. So, 𝐵₀=0⇒2𝐸₀+𝑚₀𝑐=−2(𝑙+1)𝛼.(36) 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 for 𝑝=0 is given as 𝑢⁰_𝑙(𝑟)=𝑎₀−√exp𝜀₀+𝜀₁+√𝜀₀−𝜀₁2𝑟−√𝜀₀+𝜀₁−√𝜀₀−𝜀₁4𝑟²𝑟^𝛿,(37) where 𝑎₀ is the normalization constant.

If 𝑝=1, det^𝐵0𝐶1_𝐴0𝐵1=0. In this case, it is obtained that 𝜀₁+𝜀2𝛼𝛼¹(𝑙+1)+𝜀2(𝑙+1)−𝜀₂+2𝛼(𝑙+1)𝜀₂+2𝛼(𝑙+2_for𝐸1)=0,(38) which is a restriction on the parameters of the potential and the angular momentum quantum number.

The corresponding eigenfunction for 𝑝=1 is given as 𝑢¹_𝑙(𝑟)=𝑎₀+𝑎₁𝑟−√exp𝜀₀+𝜀₁+√𝜀₀−𝜀₁2𝑟−√𝜀₀+𝜀₁−√𝜀₀−𝜀₁4𝑟²𝑟^𝛿,(39) where 𝑎₀ is the normalization constant.

Following in this way, we can generate a class of exact solutions by setting 𝑝=1,2,…. Generally, if  𝑎_𝑝≠0, but 𝑎_𝑝+1=𝑎_𝑝+2=⋯=0. So, the energy eigenvalues 𝐸_𝑝 is obtained by using (34). The corresponding eigenfunction is 𝑢^𝑝_𝑙(𝑟)=𝑎₀+𝑎₁𝑟+⋯+𝑎_𝑝𝑟^𝑝−√×exp𝜀₀+𝜀₁+√𝜀₀−𝜀₁2𝑟−√𝜀₀+𝜀₁−√𝜀₀−𝜀₁4𝑟²𝑟^𝛿,(40) where 𝑎₀,𝑎₁,…,𝑎_𝑝 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 number 𝑙 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.

Acknowledgment

The authors wish to thank Professor Shi-Hai Dong.