# Analytical Bound State Solutions of the Klein-Fock-Gordon Equation for the Sum of Hulthén and Yukawa Potential within SUSY Quantum Mechanics

Authors: A. I. Ahmadov; S. M. Aslanova; M. Sh. Orujova; S. V. Badalov
Journal: Advances in High Energy Physics
 (2021)

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## Abstract

The relativistic wave equations determine the dynamics of quantum fields in the context of quantum field theory. One of the conventional tools for dealing with the relativistic bound state problem is the Klein-Fock-Gordon equation. In this work, using a developed scheme, we present how to surmount the centrifugal part and solve the modified Klein-Fock-Gordon equation for the linear combination of Hulthén and Yukawa potentials. In particular, we show that the relativistic energy eigenvalues and corresponding radial wave functions are obtained from supersymmetric quantum mechanics by applying the shape invariance concept. Here, both scalar potential conditions, which are whether equal and nonequal to vector potential, are considered in the calculation. The energy levels and corresponding normalized eigenfunctions are represented as a recursion relation regarding the Jacobi polynomials for arbitraryl states. Beyond that, a closed form of the normalization constant of the wave functions is found. Furthermore, we state that the energy eigenvalues are quite sensitive with potential parameters for the quantum states. The nonrelativistic and relativistic results obtained within SUSY QM overlap entirely with the results obtained by ordinary quantum mechanics, and it displays that the mathematical implementation of SUSY quantum mechanics is quite perfect.


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## Body

## 1. Introduction

The exactly solvable problems for quantum systems have long been a subject of intense study in many branches of quantum physics. The main aim of an analytical solution of wave equations for this attention is that the wave function contains all the requisite information for the full description of a quantum system [1–6]. In physics, especially the relativistic quantum mechanical applications to particle and nuclear physics, the relativistic wave equations predict particles’ reaction at high energies [5–7]. The analytical solution of the Klein-Fock-Gordon (KFG) equation with physical potentials plays a central role in relativistic quantum mechanics since this wave equation perfectly defines the spinless pseudoscalar pions and Higgs boson.In principle, numerous methods were developed, and they are still successfully implemented in solving the nonrelativistic and relativistic wave equations with some familiar potentials. The Nikiforov-Uvarov method [8], factorization method [9], Laplace transform approach [10], and the path integral method [11] and shifted 1/N expansion approach [12, 13] are used for solving radial and azimuthal parts of the wave equations exactly or quasiexactly in l≠0 for various potentials. Additionally, there are numerous interesting research works about the KFG equation with physical potentials by using different methods in the literature [14–26]. Among them, as an example, in Ref. [22], the s-wave KFG equation with the vector Hulthén type potential was treated by the standard method. As reported by Talukdar et al., the scattering state solutions of the s-wave KFG equation with the vector and scalar Hulthén potentials were obtained for the irregular and regular boundary conditions [25]. Besides, the supersymmetry method (SUSY) was also proposed for solving the wave equations analytically [27–30]. Nonetheless, Okon et al. reported analytical solutions of the Schrödinger equation for the Hulthén-Yukawa plus inversely quadratic potential. [31] In Ref. [32–39], the scalar potential, which is nonequal and equal to the vector potential, was supposed to get the bound states of the KFG equation for some typical potential from the ordinary quantum mechanics. Furthermore, KFG equation with the ring-shaped potential was investigated by Dong et al. [36]. If the condition where the interaction potential is insufficient to form antiparticle-particle pairs is considered, the KFG and Dirac equations can be utilized for the investigation of zero- and 1/2-spin particles, respectively.When a particle is in a strong field, the relativistic wave equations should be considered in the quantum system. In any case, it can be corrected quickly for nonrelativistic quantum mechanics. The Hulthén potential is one of the essential short-range potentials in physics, extensively used to describe the continuum and bound states of the interaction systems. It has been applied to several research areas such as nuclear and particle, atomic, chemical, and condensed matter physics, so analyzing relativistic effects for a particle under this potential could become significant, especially for strong coupling. The Hulthén potential is defined as(1)VHr=−Ze2a·e−r/a1−e−r/a,where Z and a are the atomic number and the screening parameter, respectively. They determine the range for the Hulthén potential [40]. The Yukawa potential was proposed in 1935 as an operative potential to describe the strong interactions between nucleons [41]. It takes the following form:
(2)VYr=−Ae−krr,where A describes the strength of the interaction and 1/k, its range. Unfortunately, for arbitrary l states (l≠0), the KFG equation cannot get an exact solution with these potentials due to the centrifugal term of potentials. The numerous research works reveal the SUSY QM method’s power and simplicity in solving wave equations of the central and noncentral potentials for arbitrary l states [42–49].In principle, the radial function nature at the origin was investigated particularly for singular potentials by Khelashvili et al. [50, 51]. While the Laplace operator is portrayed in spherical coordinates, the radial wave equation’s exact derivation demonstrates the perspective of a delta function term. Thus, the delta function term of the Laplace operator yields an essential contribution to the energy level. Although the various research attempts have provided satisfactory bound state energies using Hulthén and Yukawa potentials separately [1, 52–57], we first considered these potentials under the linear combination form. [58] It is also worth mentioning that this potential can be used in nuclear physics to investigate the interaction between the deformed pair of the nucleus and spin-orbit coupling for the particle motion in the potential fields. Another fascinating perspective of this potential can be used as a mathematical model in the description of vibrations on the hadronic system’s side, and it can constitute a convenient model for other physical situations. The investigation of the relativistic bound states in the arbitrary l-wave KFG equation with the linear combination of Hulthén and Yukawa potentials is quite interesting, and it can provide the deeper and accurate appreciations of the physical properties of the wave functions and energies in the continuum and bound states of the interacting systems. Inspired by all developments and works, in this paper, we present the solution of the relativistic radial KFG equation for the linear combination of Hulthén and Yukawa potentials, defined as
(3)Vr=−V0e−2δr1−e−2δr−Ae−δrr,where V0=2δZe2 and δ is the screening parameter.To study the system, we use an improved scheme to overcome the centrifugal term and the SUSY quantum mechanics [59, 60]. Despite our previous research effort on this potential [58], the investigation of this potential still needs to be clarified in detail. Accordingly, the main goal is to solve the KFG equation for the linear combination of Hulthén and Yukawa potentials by considering two cases, i.e., the scalar potential which is equal and unequal to vector potential by using SUSY QM. Thereby, the energy eigenvalues and corresponding radial wave functions are found for any l orbital angular momentum case. Then, we compare the obtained results with the results obtained by the NU method in ordinary quantum mechanics to present the legitimacy and feasibility of this SUSY QM method. The remainder of the paper is structured as follows. In Section 2, we introduce the analytical solution of the radial KFG equation for the linear combination of Hulthén and Yukawa potentials from SUSY quantum mechanics. Next, the analysis of the results is presented in Section 3. Finally, Section 4 contains the conclusions.


## 2. Bound State Solution of the Radial Klein-Fock-Gordon Equation

### 2.1. Implementation SUSY Quantum Mechanics

Two different types of potential can be introduced into the KFG equation, which contains two objects: (i) the four-vector linear momentum operator and (ii) the scalar rest mass. Hence, the first one is a vector potentialV, which is introduced via minimal coupling, and the second one is a scalar potential S, which is introduced via scalar coupling [5]. At this moment, they allow one to introduce two types of potential coupling: the vector potential V and the space-time scalar potential S. The natural units (ℏ=c=1) are set throughout this study. In the spherical coordinates systems, the KFG equation with vector potential Vr,θ and scalar potential Sr,θ has the form
(4)−∇2+M+Sr2ψr,θ,ϕ=E−Vr2ψr,θ,ϕ,where E is the relativistic energy and M denotes the rest mass of the system’s scalar particle. For the separation of the angular and radial parts of the wave function, in the stationary KFG equation with the linear combination of Hulthén and Yukawa potentials, the wave function should utilize the following wave function:
(5)ψr,θ,ϕ=χrrΘθeimϕ,m=0,±1,±2,±3⋯and substituting this into Equation (4), the radial KFG equation is defined in the following form:
(6)χ′′r+E2−M2−2M·Sr+E·Vr+V2r−S2r−ll+1r2χr=0.As it is known that the KFG equation with this potential can be solved exactly using a suitable approximation scheme to surmount the centrifugal term. To solve Equation (6) for l≠0, we ought to approximate the centrifugal term of the Yukawa potential in this system. As a result of this, while δr<<1, the improved approximation scheme [61–65] must be used as
(7)1r≈2δe−δr1−e−2δr,1r2≈4δ2e−2δr1−e−2δr2.Next, the vector and scalar potential forms for the general Hulthén and Yukawa potentials can be considered in the following forms:(8)VHr=−V0e−2δr1−e−2δr,SHr=−S0e−2δr1−e−2δr,VYr=−V0′e−2δr1−e−2δr,SYr=−S0′e−2δr1−e−2δr.Then, Equation (6) becomes
(9)χ′′r+E2−M2+2MS0+S0′+EV0+V0′1−e−2δre−2δr+V0+V0′2−S0+S0′21−e−2δr2e−4δr−4ll+1δ2e−2δr1−e−2δr2χr=0.Thereby, the effective potential of the Hulthén and Yukawa potential linear combination has the following form:(10)Veffr=−4δ2α2+β2e−2δr1−e−2δr−4δ2γ2−ρ2e−4δr1−e−2δr2+4ll+1δ2e−2δr1−e−2δr2,where
(11)ε=M2−E22δ>0,α=2EV0+2MS02δ>0,β=2EV0′+2MS0′2δ>0,γ=V0+V0′2δ>0,ρ=S0+S′02δ>0.For investigation in detail, the nonrelativistic limit of the formula must be studied for the energy level. WhenVr=Sr, Equation (4) reduces to a Schrödinger equation for the potential 2Vr. Based on supersymmetric quantum mechanics, the eigenfunction of ground state χ0r in Equation (6) should be in the following form:
(12)χ0r=Nexp−∫Wrdr,where N and Wr are normalised constant and superpotential, respectively. The connection between the supersymmetric partner potentials V−r and V+r of the superpotential Wr is as follows [27, 28]:
(13)V−r=W2r−W′r,V+r=W2r+W′r.The particular solution of the Riccati equation (Equation (13)) must be in the following form:
(14)Wr=−F+Ge−2δr1−e−2δr,where G and F are unknown constants. Having inserted Equation (14) into Equation (13) and taking into account that V−r=Veffr−E2−M2, we obtain
(15)F2+2FGe−2δr1−e−2δr+G2e−4δr1−e−2δr2−2δGe−2δr1−e−2δr−2δGe−4δr1−e−2δr2=−4δ2α2+β2e−2δr1−e−2δr−4δ2γ2−ρ2e−4δr1−e−2δr2+4ll+1δ2e−2δr1−e−2δr2−E2−M2.After small simplification, it can be rewritten as(16)F2+2FG−2δGe−2δr1−e−2δr+G2−2δGe−4δr1−e−2δr2=4δ2ε2−4δ2α2+β2e−2δr1−e−2δr−4δ2γ2−ρ2e−4δr1−e−2δr2+4ll+1δ2e−2δr1−e−2δr+e−4δr1−e−2δr2.From comparison of compatible quantities in the left and right sides of the equation (Equation (16)), we find the following relations for G and F constants:
(17)F2=4δ2ε2,(18)2FG−2δG=4δ2ll+1−4δ2α2+β2,(19)G2−2δG=−4δ2γ2−ρ2+4δ2ll+1.Considering extremity conditions for wave functions, we obtainG>0 and F<0. Solving Equation (19) yields
(20)G=δ±2δl+122−γ2+ρ2,and considering G>0 from Equations (18) and (19), we find that
(21)F=G2−2δ2α2+β2−γ2+ρ2G.From Equation (17) and Equation (21), we find that
(22)ε2=14δ2δ+2δl+1/22−γ2+ρ22−2δα2+β2−γ2+ρ21+2l+1/22−γ2+ρ22.After inserting (22) into (11) for the definitions of the energy eigenvalue of ground state for general case Vr≠Sr, we obtain the following energy level equation:
(23)M2−E02=δ+2δl+1/22−γ2+ρ22−2δα2+β2−γ2+ρ21+2l+1/22−γ2+ρ22.Whenr→∞, the chosen superpotential Wr→-F. Inserting Equation (14) into Equation (13), the supersymmetric partner potentials V−r and V+r can be found in the following forms:
(24)V−r=F2+2FG−2δGe−2δr1−e−2δr+G2−2δGe−4δr1−e−2δr2,V+r=F2+2FG+2δGe−2δr1−e−2δr+G2+2δGe−4δr1−e−2δr2.By using the superpotentialWr from Equation (14), we can find χ0r radial eigenfunction of ground state in the following form:
(25)χ0r=NeFr1−e−2δrG/2δwhere r→0; χ0r→0, G>0, and r→∞; χ0r→0,F<0. Two partner potentials V−r and V+r which differ from each other with additive constants and have the same functional form are called the invariant potentials [59, 60]. Hence, for the partner potentials V−r and V+r given with Equations (13) and (14), the invariant forms are defined as
(26)RG1=V+G,r−V−G1,r=F2−F12=G2−2δ2α2+β2−γ2+ρ2G2−G+2δ2−2δ2α2+β2−γ2+ρ2G+2δ2,(27)RGi=V+G+i−12δ,r−V−G+i2δ,r=G+i−1·2δ2−2α2+β2−γ2+ρ2δ2G+i−1·2δ2−G+i·2δ2−2α2+β2−γ2+ρ2δ2G+i·2δ2.where the reminder RGi is independent of r. If we keep going this procedure and make the following substitution Gnr=Gnr−1+2δ=G+2nrδ, the whole discrete level of Hamiltonian H−G can be written as
(28)Enr2=E02+∑i=1nRGi,and we obtain the following form:
(29)Enrl2=M2−G+2δnr2−2δ2α2+β2−γ2+ρ2G+2δnr2.In the following, we obtain the energy level equation in ordinary quantum mechanics(30)M2−Enr,l2=α2+β2−l+1/22−nr+1/22−2nr+1/2l+1/22−γ2+ρ2nr+1/2+l+1/22−γ2+ρ2·δ2.As seen from Equation (30), it is in a perfect agreement with the result obtained in Equation (27) of Ref. [49]. If we consider Equation (11) into Equation (30) and do some simple algebraic derivation, we can obtain the energy eigenvalue equation in the simplest form
(31)M2−Enrl2=δnr+12+l+122−γ2+ρ2−γEnrl+ρM+δρ2−γ2nr+1/2+l+1/22−γ2+ρ22with α2+β2=γEnrl+ρM/δ.Based on the SUSY QM method and knowing the ground state eigenvaluesE0 and eigenfunctions χ0, all energy eigenvalues Enrl and eigenfunctions χnrl can be easily obtained. Briefly, using the following equation:
(32)χnrr,a0=A+r,a0χnr−1r,a1,χnrl can be easily obtained in terms of the ground state wave functions. The superpotential Wr depends on two parameters a0=F,G and the first partner potential has like that parameter a1=F1,G1. Hence, Equation (32) will be in the following form:
(33)χnrr,a0=−ddr−F−Ge−2δr1−e−2δrχnr−1r,a1,We define a new variables=e−2δr∈0,1 and factoring out the ground state wavefunction
(34)χnrs,a0=χ0s,a0Rnrs,a0.Substituting into Equation (33) and using the ground state wavefunction (Equation (25)), we get
(35)Rnrs;ε,K=s1−sddsRnr−1s;ε,K+1+2ε−2ε+2K+1Rnr−1s;ε,K+1.Based on comparison, it with the recursion relation in Ref [66]:
(36)Pnrα,β1−2s=s1−sddsPnr−1α+1,β+11−2s+α+1−α+β+2sPnr−1α+1,β+11−2s,it is seen that Rnrs,a0 is proportional to the Jacobi polynomial Pnr2ε,2K−11−2s. Thus, the normalized eigenfunction for this potential is taken in the following form:
(37)χnrls=Cnrlsε1−sKPnr2ε,2K−11−2s,(38)χnrls=Cnrlsε1−sKΓnr+2ε+1nr!Γ2ε+1·2F1−nr,2ε+2K+nr,1+2ε;s,where K=1/2+l+1/22−γ2−ρ2. The normalization constant Cnrl can be found by using the normalization condition
(39)∫0∞Rr2r2dr=∫0∞χr2dr=12δ∫011sχs2ds=1,by utilizing the following integral formula in Ref. [66]:
(40)∫011−z2δ+1z2λ−12F1−nr,2δ+λ+1+nr,2λ+1;z2dz=nr+δ+1nr!Γnr+2δ+2Γ2λΓ2λ+1nr+δ+λ+1Γnr+2λ+1Γ2δ+λ+1+nr,where λ>0 and δ>−3/2. After making simple calculations, we arrive at the following expression for the normalization constant:
(41)Cnrl=2δnr!nr+K+εΓ2ε+1Γnr+2ε+2Knr+KΓ2εΓnr+2KΓnr+2ε+1.


## 2.1. Implementation SUSY Quantum Mechanics

Two different types of potential can be introduced into the KFG equation, which contains two objects: (i) the four-vector linear momentum operator and (ii) the scalar rest mass. Hence, the first one is a vector potentialV, which is introduced via minimal coupling, and the second one is a scalar potential S, which is introduced via scalar coupling [5]. At this moment, they allow one to introduce two types of potential coupling: the vector potential V and the space-time scalar potential S. The natural units (ℏ=c=1) are set throughout this study. In the spherical coordinates systems, the KFG equation with vector potential Vr,θ and scalar potential Sr,θ has the form
(4)−∇2+M+Sr2ψr,θ,ϕ=E−Vr2ψr,θ,ϕ,where E is the relativistic energy and M denotes the rest mass of the system’s scalar particle. For the separation of the angular and radial parts of the wave function, in the stationary KFG equation with the linear combination of Hulthén and Yukawa potentials, the wave function should utilize the following wave function:
(5)ψr,θ,ϕ=χrrΘθeimϕ,m=0,±1,±2,±3⋯and substituting this into Equation (4), the radial KFG equation is defined in the following form:
(6)χ′′r+E2−M2−2M·Sr+E·Vr+V2r−S2r−ll+1r2χr=0.As it is known that the KFG equation with this potential can be solved exactly using a suitable approximation scheme to surmount the centrifugal term. To solve Equation (6) for l≠0, we ought to approximate the centrifugal term of the Yukawa potential in this system. As a result of this, while δr<<1, the improved approximation scheme [61–65] must be used as
(7)1r≈2δe−δr1−e−2δr,1r2≈4δ2e−2δr1−e−2δr2.Next, the vector and scalar potential forms for the general Hulthén and Yukawa potentials can be considered in the following forms:(8)VHr=−V0e−2δr1−e−2δr,SHr=−S0e−2δr1−e−2δr,VYr=−V0′e−2δr1−e−2δr,SYr=−S0′e−2δr1−e−2δr.Then, Equation (6) becomes
(9)χ′′r+E2−M2+2MS0+S0′+EV0+V0′1−e−2δre−2δr+V0+V0′2−S0+S0′21−e−2δr2e−4δr−4ll+1δ2e−2δr1−e−2δr2χr=0.Thereby, the effective potential of the Hulthén and Yukawa potential linear combination has the following form:(10)Veffr=−4δ2α2+β2e−2δr1−e−2δr−4δ2γ2−ρ2e−4δr1−e−2δr2+4ll+1δ2e−2δr1−e−2δr2,where
(11)ε=M2−E22δ>0,α=2EV0+2MS02δ>0,β=2EV0′+2MS0′2δ>0,γ=V0+V0′2δ>0,ρ=S0+S′02δ>0.For investigation in detail, the nonrelativistic limit of the formula must be studied for the energy level. WhenVr=Sr, Equation (4) reduces to a Schrödinger equation for the potential 2Vr. Based on supersymmetric quantum mechanics, the eigenfunction of ground state χ0r in Equation (6) should be in the following form:
(12)χ0r=Nexp−∫Wrdr,where N and Wr are normalised constant and superpotential, respectively. The connection between the supersymmetric partner potentials V−r and V+r of the superpotential Wr is as follows [27, 28]:
(13)V−r=W2r−W′r,V+r=W2r+W′r.The particular solution of the Riccati equation (Equation (13)) must be in the following form:
(14)Wr=−F+Ge−2δr1−e−2δr,where G and F are unknown constants. Having inserted Equation (14) into Equation (13) and taking into account that V−r=Veffr−E2−M2, we obtain
(15)F2+2FGe−2δr1−e−2δr+G2e−4δr1−e−2δr2−2δGe−2δr1−e−2δr−2δGe−4δr1−e−2δr2=−4δ2α2+β2e−2δr1−e−2δr−4δ2γ2−ρ2e−4δr1−e−2δr2+4ll+1δ2e−2δr1−e−2δr2−E2−M2.After small simplification, it can be rewritten as(16)F2+2FG−2δGe−2δr1−e−2δr+G2−2δGe−4δr1−e−2δr2=4δ2ε2−4δ2α2+β2e−2δr1−e−2δr−4δ2γ2−ρ2e−4δr1−e−2δr2+4ll+1δ2e−2δr1−e−2δr+e−4δr1−e−2δr2.From comparison of compatible quantities in the left and right sides of the equation (Equation (16)), we find the following relations for G and F constants:
(17)F2=4δ2ε2,(18)2FG−2δG=4δ2ll+1−4δ2α2+β2,(19)G2−2δG=−4δ2γ2−ρ2+4δ2ll+1.Considering extremity conditions for wave functions, we obtainG>0 and F<0. Solving Equation (19) yields
(20)G=δ±2δl+122−γ2+ρ2,and considering G>0 from Equations (18) and (19), we find that
(21)F=G2−2δ2α2+β2−γ2+ρ2G.From Equation (17) and Equation (21), we find that
(22)ε2=14δ2δ+2δl+1/22−γ2+ρ22−2δα2+β2−γ2+ρ21+2l+1/22−γ2+ρ22.After inserting (22) into (11) for the definitions of the energy eigenvalue of ground state for general case Vr≠Sr, we obtain the following energy level equation:
(23)M2−E02=δ+2δl+1/22−γ2+ρ22−2δα2+β2−γ2+ρ21+2l+1/22−γ2+ρ22.Whenr→∞, the chosen superpotential Wr→-F. Inserting Equation (14) into Equation (13), the supersymmetric partner potentials V−r and V+r can be found in the following forms:
(24)V−r=F2+2FG−2δGe−2δr1−e−2δr+G2−2δGe−4δr1−e−2δr2,V+r=F2+2FG+2δGe−2δr1−e−2δr+G2+2δGe−4δr1−e−2δr2.By using the superpotentialWr from Equation (14), we can find χ0r radial eigenfunction of ground state in the following form:
(25)χ0r=NeFr1−e−2δrG/2δwhere r→0; χ0r→0, G>0, and r→∞; χ0r→0,F<0. Two partner potentials V−r and V+r which differ from each other with additive constants and have the same functional form are called the invariant potentials [59, 60]. Hence, for the partner potentials V−r and V+r given with Equations (13) and (14), the invariant forms are defined as
(26)RG1=V+G,r−V−G1,r=F2−F12=G2−2δ2α2+β2−γ2+ρ2G2−G+2δ2−2δ2α2+β2−γ2+ρ2G+2δ2,(27)RGi=V+G+i−12δ,r−V−G+i2δ,r=G+i−1·2δ2−2α2+β2−γ2+ρ2δ2G+i−1·2δ2−G+i·2δ2−2α2+β2−γ2+ρ2δ2G+i·2δ2.where the reminder RGi is independent of r. If we keep going this procedure and make the following substitution Gnr=Gnr−1+2δ=G+2nrδ, the whole discrete level of Hamiltonian H−G can be written as
(28)Enr2=E02+∑i=1nRGi,and we obtain the following form:
(29)Enrl2=M2−G+2δnr2−2δ2α2+β2−γ2+ρ2G+2δnr2.In the following, we obtain the energy level equation in ordinary quantum mechanics(30)M2−Enr,l2=α2+β2−l+1/22−nr+1/22−2nr+1/2l+1/22−γ2+ρ2nr+1/2+l+1/22−γ2+ρ2·δ2.As seen from Equation (30), it is in a perfect agreement with the result obtained in Equation (27) of Ref. [49]. If we consider Equation (11) into Equation (30) and do some simple algebraic derivation, we can obtain the energy eigenvalue equation in the simplest form
(31)M2−Enrl2=δnr+12+l+122−γ2+ρ2−γEnrl+ρM+δρ2−γ2nr+1/2+l+1/22−γ2+ρ22with α2+β2=γEnrl+ρM/δ.Based on the SUSY QM method and knowing the ground state eigenvaluesE0 and eigenfunctions χ0, all energy eigenvalues Enrl and eigenfunctions χnrl can be easily obtained. Briefly, using the following equation:
(32)χnrr,a0=A+r,a0χnr−1r,a1,χnrl can be easily obtained in terms of the ground state wave functions. The superpotential Wr depends on two parameters a0=F,G and the first partner potential has like that parameter a1=F1,G1. Hence, Equation (32) will be in the following form:
(33)χnrr,a0=−ddr−F−Ge−2δr1−e−2δrχnr−1r,a1,We define a new variables=e−2δr∈0,1 and factoring out the ground state wavefunction
(34)χnrs,a0=χ0s,a0Rnrs,a0.Substituting into Equation (33) and using the ground state wavefunction (Equation (25)), we get
(35)Rnrs;ε,K=s1−sddsRnr−1s;ε,K+1+2ε−2ε+2K+1Rnr−1s;ε,K+1.Based on comparison, it with the recursion relation in Ref [66]:
(36)Pnrα,β1−2s=s1−sddsPnr−1α+1,β+11−2s+α+1−α+β+2sPnr−1α+1,β+11−2s,it is seen that Rnrs,a0 is proportional to the Jacobi polynomial Pnr2ε,2K−11−2s. Thus, the normalized eigenfunction for this potential is taken in the following form:
(37)χnrls=Cnrlsε1−sKPnr2ε,2K−11−2s,(38)χnrls=Cnrlsε1−sKΓnr+2ε+1nr!Γ2ε+1·2F1−nr,2ε+2K+nr,1+2ε;s,where K=1/2+l+1/22−γ2−ρ2. The normalization constant Cnrl can be found by using the normalization condition
(39)∫0∞Rr2r2dr=∫0∞χr2dr=12δ∫011sχs2ds=1,by utilizing the following integral formula in Ref. [66]:
(40)∫011−z2δ+1z2λ−12F1−nr,2δ+λ+1+nr,2λ+1;z2dz=nr+δ+1nr!Γnr+2δ+2Γ2λΓ2λ+1nr+δ+λ+1Γnr+2λ+1Γ2δ+λ+1+nr,where λ>0 and δ>−3/2. After making simple calculations, we arrive at the following expression for the normalization constant:
(41)Cnrl=2δnr!nr+K+εΓ2ε+1Γnr+2ε+2Knr+KΓ2εΓnr+2KΓnr+2ε+1.


## 3. Results and Discussion

In this section, we present the numerical evaluation for the bound state solutions of thel-wave KFG equation with the vector and scalar form of the linear combination of Hulthén and Yukawa potentials. To study the property of the energy levels regarding potential parameters in some quantum states, (see Figure 1) we take M=10, V0=0.01, V′0=0.05, S0=0.025, and S′0=0.035 and M=10, V0=0.02, V′0=0.06, S0=0.035, and S′0=0.045. The little difference (~0.01) in model potential parameters V0, V′0, S0, and S′0 is quite sufficient, in order to see the energy level of quantum states displaye completely different behavior. In Figures 1(a) and 1(b), the energy levels E of quantum states first are decreasing until some of small δ values (~0.075-0.1), then the energy levels E increase in the δ>0.1. In Figures 1(c) and 1(d), the energy levels E of quantum states have very little variation for an interval of δ∈0,0.1, and it causes the degenerate of all quantum states, then the energy levels E of quantum states continue to gradually increase with increments of δ. These behaviors are better recognized in higher quantum states.Figure 1
(Color online). The variation of energy level as a function of screening parameterδ for quantum states in (a, b) the parameters M=10, V0=0.01, V′0=0.05, S0=0.025, and S′0=0.035 and (c, d) the parameters M=10, V0=0.02, V′0=0.06, S0=0.035, and S′0=0.045.
(a)(b)(c)(d)Behind that of these results, we can investigate some special cases.(i)
In caseS0=V0, and S0′=V0′, namely, γ=ρ, we obtain(42)M2−Enrl2=δnr+l+1−γEnrl+Mnr+l+12,where γ=V0+V0′/2δ(ii)
IfV0′=0, S0′=0, we obtain the energy level equation for Hulthén potential case(43)M2−Enrl2=δnr+12+l+122−γ′2+ρ′2−γ′Enrl+ρ′M+δρ′2−γ′2nr+1/2+l+1/22−γ′2+ρ′22,where γ′=V0/2δ and ρ′=S0/2δ. This result is in good agreement with the expression obtained in Equation (A.1) of Ref [58]
(iii)
In caseV0=0, S0=0, but V0′≠S0′, we obtain the energy level equation for Yukawa potential, which is defined as the following form:(44)M2−Enrl2=δnr+12+l+122−γ′′2+ρ′′2−γ′′Enrl+ρ′′M+δρ′′2−γ′′2nr+1/2+l+1/22−γ′′2+ρ′′22,where γ′′=V′0/2δ, and ρ′′=S′0/2δ. This result is in good agreement with the expression obtained in Equation (A.3) of Ref [58]. Furthermore, this result is also the same with the expression for the constant mass case obtained in Ref [67]. One can easily see this by setting q=1 and α→δ in Equation (39) of Ref [67]
(iv)
Also, ifV0=−S0, and V0′=−S0′, namely, γ=−ρ(45)M2−Enrl2=δnr+l+1−γEnrl−Mnr+l+12,where γ=V0+V0′/2δ(v)
Ifδ→0 and S0=V0=2δZe2 or γ′=ρ′=V0/2δ=Ze2 in Equation (31), the potential reduces to Coulomb potential, Vcr=−Ze2/r, and the corresponding energy spectrum is obtained as(46)Enrl=nr+l+12−Z2e4nr+l+12+Z2e4M,and this result is the same with Equation (A.2) of Ref [67]
(vi)
If we takel=0 (the s-wave case), the centrifugal term in Equation (9) disappears because 4ll+1δ2e−2δr/1−e−2δr2=0 and the equation turns to the s-wave KFG equation. By setting l=0 in Equation (31), its energy spectrum equation is the following form:(47)M2−Enrl2=δnr+12+14−γ2+ρ2−γEnrl+ρM+δρ2−γ2nr+1/2+1/4−γ2+ρ22(vii)
If we takeS0=V0 and S′0=V′0, and based on the following transformations: Enrl–M→EnrlNR, Enrl+M→2M, V0→V0/2, and V′0→V′0/2, we obtain the energy level equation of Equation (31) for the nonelativistic case. Briefly, because of the following relation:(48)M2−Enrl2=M−EnrlM+Enrl=−EnrlNR·2M,γ=ρ=V0+V0′4δ=V0+2δA4δ,γEnrl+M=2Mγ=MV0+2δA2δ,the energy level equation of Equation (31) for the nonrelativistic case, can be written the following form:
(49)EnrlNR=−12Mδnr+l+1+MV0+2δA2δnr+l+12,which is good agreement with the result in Equation (28) (If B and C are considered zero as a special case) of Ref [31]. Generally, it is obviously seen from Equation. (31) that the bound states show more stability in the case of the linear combination of Hulthén and Yukawa potentials than Hulthén and Yukawa potential cases. Furthermore, the energy eigenvalues of the quantum states are considerably sensitive with depending potential parameters


## 4. Conclusion

To conclude, we admit that the SUSY QM method was presented to solve the KFG equation for the linear combination of Hulthén and Yukawa potentials. Hence, the energy eigenvalues and corresponding eigenfunctions of a mentioned quantum system were analytically obtained for arbitraryl angular momentum and nr radial quantum numbers. Next, a closed form of the normalization constant of the wave functions was also found. Beyond that, it was also shown that the energy eigenvalues are considerably sensitive respecting quantum states. Finally, the results obtained within SUSY QM are in excellent agreement with the results obtained by ordinary quantum mechanics, and it confirms that the mathematical application of SUSY quantum mechanics is ideal for similar systems.It is worth mentioning that the main results of this paper are the explicit and closed form expressions for the energy eigenvalues and the normalized wave functions. The method presented in this study is systematic, and in many cases, one of the most definite works in this field. In particular, the linear combination of Hulthén and Yukawa potentials can be one of the essential exponential potentials, and it probably provides a promising avenue in many branches of physics, especially in hadronic and nuclear physics.



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