Analytical bound-state solutions of the Klein-Fock-Gordon equation for the sum of Hulthén and Yukawa potential within SUSY quantum mechanics
A. I. Ahmadov, S. M. Aslanova, M. Sh. Orujova, S. V. Badalov
aa r X i v : . [ h e p - t h ] J a n Analytical bound-state solutions of the Klein-Fock-Gordon equation for the sum ofHulth´en and Yukawa potential within SUSY quantum mechanics
A. I. Ahmadov,
1, 2, ∗ S. M. Aslanova, M. Sh. Orujova, and S. V. Badalov † Department of Theoretical Physics, Baku State University, 1148 Baku, Azerbaijan Institute for Physical Problems, Baku State University, 1148, Baku, Azerbaijan Azerbaijan State University of Economics, Istiglaliyyat st. 22, 1001 Baku, Azerbaijan Lehrstuhl f¨ur Theoretische Materialphysik, Universit¨at Paderborn, 33095 Paderborn, Germany (Dated: February 1, 2021)
Abstract
The relativistic wave equations determine the dynamics of quantum fields in the context of quantumfield theory. One of the conventional tools for dealing with the relativistic bound-state problem is theKlein-Fock-Gordon equation. In this work, using a developed scheme, we present how to surmountthe centrifugal part and solve the modified Klein-Fock-Gordon equation for the linear combinationof Hulth´en and Yukawa potentials. In particular, we show that the relativistic energy eigenvaluesand corresponding radial wave functions are obtained from supersymmetric quantum mechanics byapplying the shape invariance concept. Here, both scalar potential conditions, which are whetherequal and non-equal to vector potential, are considered in the calculation. The energy levels andcorresponding normalized eigenfunctions are represented as a recursion relation regarding the Jacobipolynomials for arbitrary l states. Beyond that, a closed-form of the normalization constant of thewave functions is found. Furthermore, we state that the energy eigenvalues are quite sensitive withpotential parameters for the quantum states. The non-relativistic and relativistic results obtainedwithin SUSY QM overlap entirely with the results obtained by ordinary quantum mechanics, andit displays that the mathematical implementation of SUSY quantum mechanics is quite perfect. PACS numbers: 03.65.GeKeywords: Hulth´en and Yukawa potential, Supersymmetric Quantum Mechanics
I. INTRODUCTION
The exactly solvable problems for quantum systems have long been a subject of intense study in many branchesof quantum physics. The main aim of an analytical solution of wave equations for this attention is that the wavefunction contains all the requisite information for the full description of a quantum system.[1–6] In physics, especiallythe relativistic quantum mechanical applications to particle and nuclear physics, the relativistic wave equationspredict particles’ reaction at high energies.[5–7] The analytical solution of the Klein-Fock-Gordon (KFG) equationwith physical potentials plays a central role in relativistic quantum mechanics since this wave equation perfectlydefines the spinless pseudo scalar pions and Higgs boson.In principle, numerous methods were developed, and they are still successfully implemented in solving the non-relativistic and relativistic wave equations with some familiar potentials. The Nikiforov-Uvarov method,[8] factor-ization method,[9] Laplace transform approach,[10] and the path integral method,[11] and shifted 1/N expansionapproach[12, 13] for solving radial and azimuthal parts of the wave equations exactly or quasi-exactly in l = 0 forvarious potentials. Additionally, there are numerous interesting research works about the KFG equation with physicalpotentials by using different methods in the literature.[14–26] Among them, as an example, in Ref.[22], the s-waveKFG equation with the vector Hulth´en 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 Hulth´en potentials wereobtained for the irregular and regular boundary conditions.[25] Besides, the supersymmetry method (SUSY) was alsoproposed for solving the wave equations analytically.[27–30] Nonetheless, Okon et al. reported analytical solutions ofSchr¨odinger equation for the Hulth´en-Yukawa plus inversely quadratic potential.[31] In Ref.[32–39], the scalar poten-tial, which is non-equal and equal to the vector potential, was supposed to get the bound states of the KFG equation ∗ Electronic address: [email protected] † Electronic address: [email protected] for some typical potential from the ordinary quantum mechanics. Furthermore, KFG equation with the Ring-Shapedpotential was investigated by Dong et al .[36] If the condition where the interaction potential is insufficient to formantiparticle-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 non-relativistic quantum mechanics. The Hulth´en potential is one ofthe essential short-range potentials in physics, extensively using to describe the continuum and bound states of theinteraction systems. It has been applied to several research areas such as nuclear and particle, atomic, chemical, andcondensed matter physics, so analyzing relativistic effects for a particle under this potential could become significant,especially for strong coupling. The Hulth´en potential is defined as V H ( r ) = − Ze a · e − r/a − e − r/a (1.1)where Z and a are the atomic number and the screening parameter, respectively. They determine the range for theHulth´en potential.[40] The Yukawa potential was proposed in 1935 as an operative potential to describe the stronginteractions between nucleons.[41] It takes the following form V Y ( r ) = − Ae − kr r , (1.2)where A describing the strength of the interaction and 1 /k its range. Unfortunately, for an arbitrary l -states ( l = 0),the KFG equation cannot get an exact solution with these potentials due to the centrifugal term of potentials. Thenumerous research works reveal the SUSY QM method’s power and simplicity in solving wave equations of the centraland non-central potentials for arbitrary l states.[42–49]In principle, the radial function nature at the origin was investigated particularly for singular potentials by Khe-lashvili et al. [50, 51]. While the Laplace operator is portrayed in spherical coordinates, the radial wave equation’sexact derivation demonstrates the perspective of a delta function term. Thus, the delta function term of the Laplaceoperator yields an essential contribution to the energy level. Although the various research attempts have providedsatisfactory bound state energies using Hulth´en and Yukawa potentials separately,[1, 52–57] we first considered thesepotentials under the linear combination form.[58] It is also worth mentioning that this potential can be use in nuclearphysics to investigate the interaction between the deformed pair of the nucleus and spin-orbit coupling for the particlemotion in the potential fields. Another fascinating perspective of this potential can be used as a mathematical modelin the description of vibrations on the hadronic system’s side, and it can constitute a convenient model for otherphysical situations. The investigation of the relativistic bound states in the arbitrary l -wave KFG equation with thelinear combination of Hulth´en and Yukawa potentials is quite interesting, and it can provide the deeper and accurateappreciations of the physical properties of the wave functions and energies in the continuum and bound states of theinteracting systems. Inspired by all developments and works, in this paper, we present the solution of the relativisticradial KFG equation for the linear combination of Hulth´en and Yukawa potentials, defined as V ( r ) = − V e − δr − e − δr − Ae − δr r , (1.3)where V = 2 δZe , and δ is the screening parameter.To study the system, we use an improved scheme to overcome the centrifugal term and the SUSY quantummechanics[59, 60]. Despite our previous research effort on this potential,[58] the investigation of this potential stillneeds to be clarified in detail. Accordingly, the main goal is to solve the KFG equation for the linear combinationof Hulth´en and Yukawa potentials by considering two cases, i.e., the scalar potential which is equal and unequal tovector potential by using SUSY QM. Thereby, the energy eigenvalues and corresponding radial wave functions arefound for any l orbital angular momentum case. Then, we compare the obtained results with the results obtained bythe 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 II, we introduce the analytical solution of the radialKFG equation for the linear combination of Hulth´en and Yukawa potentials from SUSY quantum mechanics. Next,the analysis of the results is presented in Section III. Finally, Section IV contains the conclusions. II. BOUND STATE SOLUTION OF THE RADIAL KLEIN-FOCK-GORDON EQUATIONA. Implementation SUSY Quantum Mechanics
Two different types of potential can be introduced into 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 potential V , whichintroduce via minimal coupling, and the second one is a scalar potential S , which introduce via scalar coupling[5]. Atthis moment, they allow one to introduce two types of potential coupling: the vector potential V and the space-timescalar potential S . The natural units ( ~ = c = 1) are set throughout this study. In the spherical coordinates systems,the KFG equation with vector potential V ( r, θ ) and scalar potential S ( r, θ ) has the form[ −∇ + ( M + S ( r )) ] ψ ( r, θ, φ ) = [ E − V ( r )] ψ ( r, θ, φ ) , (2.1)where E is the relativistic energy and M denotes the rest mass of the system’s scalar particle. For the separationof the angular and radial parts of the wave function, in the stationary KFG equation with the linear combination ofHulth´en and Yukawa potentials, the wave function should be utilize the following wave function ψ ( r, θ, φ ) = χ ( r ) r Θ( θ ) e imφ , m = 0 , ± , ± , ± ... (2.2)and substituting this into Eq.(2.1), the radial KFG equation is defined in the following form χ ′′ ( r ) + [( E − M ) − M · S ( r ) + E · V ( r )) + ( V ( r ) − S ( r )) − l ( l + 1) r ] χ ( r ) = 0 . (2.3)As it is known that the KFG equation with this potential can be solved exactly using a suitable approximation schemeto surmount the centrifugal term. To solve Eq.(2.3) for l = 0 , we ought to approximate the centrifugal term of theYukawa potential in this system. As a result of this, while δr <<
1, the improved approximation scheme,[61–65] mustbe used as 1 r ≈ δe − δr − e − δr , r ≈ δ e − δr (1 − e − δr ) . (2.4)Next, the vector and scalar potential forms for the general Hulth´en and Yukawa potentials can be considered in thefollowing forms V H ( r ) = − V e − δr − e − δr , S H ( r ) = − S e − δr − e − δr V Y ( r ) = − V ′ e − δr − e − δr , S Y ( r ) = − S ′ e − δr − e − δr . (2.5)Then, Eq.(2.3) becomes as χ ′′ ( r )+ (cid:20) ( E − M )+2 (cid:18) M ( S + S ′ ) + E ( V + V ′ )1 − e − δr (cid:19) e − δr + (cid:18) ( V + V ′ ) − ( S + S ′ ) (1 − e − δr ) (cid:19) e − δr − l ( l + 1) δ e − δr (1 − e − δr ) (cid:21) χ ( r ) = 0 . (2.6)Thereby, the effective potential of the Hulth´en and Yukawa potentials linear combination has the following form V eff ( r ) = − δ ( α + β ) e − δr − e − δr − δ ( γ − ρ ) e − δr (1 − e − δr ) + 4 l ( l + 1) δ e − δr (1 − e − δr ) , (2.7)where ε = √ M − E δ > , α = √ EV + 2 M S δ > ,β = p EV ′ + 2 M S ′ δ > , γ = V + V ′ δ > ,ρ = S + S ′ δ > . (2.8)For investigation in detail, the non-relativistic limit of the formula must be studied for the energy level. When V ( r ) = S ( r ), the Eq.(2.1) reduces to a Schr¨odinger equation for the potential 2 V ( r ). Based on supersymmetricquantum mechanics, the eigenfunction of ground state χ ( r ) in Eq.(2.3) should be in the following form χ ( r ) = N exp ( − Z W ( r ) dr ) , (2.9)where N and W ( r ) are normalised constant and superpotential, respectively. The connection between the supersym-metric partner potentials V − ( r ) and V + ( r ) of the superpotential W ( r ) is as follows[27, 28] V − ( r ) = W ( r ) − W ′ ( r ) ,V + ( r ) = W ( r ) + W ′ ( r ) . (2.10)The particular solution of the Riccati equation Eq.(2.10) must be in the following form W ( r ) = − ( F + Ge − δr − e − δr ) , (2.11)where G and F are unknown constants. Having inserted Eq.(2.11) into Eq.(2.10) and taking into account that V − ( r ) = V eff ( r ) − ( E − M ), we obtain F + 2 F Ge − δr − e − δr + G e − δr (1 − e − δr ) − δGe − δr − e − δr − δGe − δr (1 − e − δr ) == − δ ( α + β ) e − δr − e − δr − δ ( γ − ρ ) e − δr (1 − e − δr ) + 4 l ( l + 1) δ e − δr (1 − e − δr ) − ( E − M ) (2.12)After small simplification, it can be rewritten as F + (2 F G − δG ) e − δr − e − δr + ( G − δG ) e − δr (1 − e − δr ) == 4 δ ε − δ ( α + β ) e − δr − e − δr − δ ( γ − ρ ) e − δr (1 − e − δr ) + 4 l ( l + 1) δ [ e − δr − e − δr + e − δr (1 − e − δr ) ] (2.13)From comparison of compatible quantities in the left and right sides of the equation Eq.(2.13), we find the followingrelations for G and F constants F = 4 δ ε , (2.14)2 F G − δG = 4 δ l ( l + 1) − δ ( α + β ) , (2.15) G − δG = − δ ( γ − ρ ) + 4 δ l ( l + 1) . (2.16)Considering extremity conditions for wave functions, we obtain G >
F <
0. Solving Eq.(2.16) yields G = δ ± δ r ( l + 12 ) − γ + ρ , (2.17)and considering G > F = G − δ ( α + β − γ + ρ ) G . (2.18)From Eq.(2.14) and Eq.(2.18), we find that ε = 14 δ [ δ + 2 δ q ( l + ) − γ + ρ − δ ( α + β − γ + ρ )1 + 2 q ( l + ) − γ + ρ ] . (2.19)After inserting the (2.19) into the (2.8) for the definitions the energy eigenvalue of ground state for general case V ( r ) = S ( r ), we obtain the following energy level equation M − E = [ δ + 2 δ q ( l + ) − γ + ρ − δ ( α + β − γ + ρ )1 + 2 q ( l + ) − γ + ρ ] . (2.20)When r → ∞ , the chosen superpotential W ( r ) → - F . Inserting the Eq.(2.11) into Eq.(2.10), the supersymmetricpartner potentials V − ( r ) and V + ( r ) can be found in the following forms V − ( r ) = F + (2 F G − δG ) e − δr − e − δr + ( G − δG ) e − δr (1 − e − δr ) ,V + ( r ) = F + (2 F G + 2 δG ) e − δr − e − δr + ( G + 2 δG ) e − δr (1 − e − δr ) . (2.21)By using the superpotential W ( r ) from Eq.(2.11), we can find χ ( r ) radial eigenfunction of ground state in thefollowing form χ ( r ) = N e
F r (1 − e − δr ) G δ (2.22)where r → χ ( r ) → G >
0, and r → ∞ ; χ ( r ) → , F <
0. Two partner potentials V − ( r ) and V + ( r ) which differfrom 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 Eq.(2.10) and Eq.(2.11), the invariant forms are definedas R ( G ) = V + ( G, r ) − V − ( G , r ) = F − F = [ G − δ ( α + β − γ + ρ ) G ] − [ G + 2 δ − δ ( α + β − γ + ρ ) G + 2 δ ] , (2.23) R ( G i ) = V + [ G + ( i − δ, r ] − V − [ G + i δ, r ] == ( G + ( i − · δ − α + β − γ + ρ ) δ G + ( i − · δ ) − ( G + i · δ − α + β − γ + ρ ) δ G + i · δ ) . (2.24)where the reminder R ( G i ) is independent of r . If we keep going this procedure and make the following substitution G n r = G n r − + 2 δ = G + 2 n r δ , the whole discrete level of Hamiltonian H − ( G ) can be written as E n r = E + n X i =1 R ( G i ) , (2.25)and we obtain the following form E n r l = M − ( G + 2 δn r − δ ( α + β − γ + ρ ) G + 2 δn r ) . (2.26)In the following, we obtain the energy level equation in ordinary quantum mechanics M − E n r ,l = [ α + β − ( l + ) − ( n r + ) − n r + ) q ( l + ) − γ + ρ n r + + q ( l + ) − γ + ρ · δ ] . (2.27)As seen from Eq.(2.27), it is in a perfect agreement with the result obtained in Eq.(27) of Ref.[49]. If we considerEq.(2.8) into Eq.(2.27) and do some simple algebraic derivation, we can obtain the energy eigenvalues equation in thesimplest form M − E n r l = [ δ ( n r + 12 + r ( l + 12 ) − γ + ρ ) − γE n r l + ρM + δ ( ρ − γ ) n r + + q ( l + ) − γ + ρ ] (2.28) -10-9.95-9.9-9.85-9.8-9.75-9.7-9.65-9.6 δ E n e r gy (a) -10.02-10.01-10-9.99-9.98-9.97-9.96-9.95-9.94-9.93-9.92
1s 2s 2p 3s 3p 3d 4s E n e r gy δ = 0.04δ = 0.3 (b) -10-9.95-9.9-9.85-9.8-9.75-9.7-9.65-9.60 0.05 0.1 0.15 0.2 0.25 δ E n e r gy (c) E n e r gy
1s 2s 2p 3s 3p 3d 4s-10.02-10.01-10-9.99-9.98-9.97-9.96-9.95-9.94-9.93-9.92 δ = 0.04δ = 0.3 (d)
Figure 1: (Color online) The variation of energy level as a function of screening parameter δ for quantum states in (a,b) theparameters M =10, V =0.01, V ′ =0.05, S =0.025, S ′ =0.035 and (c,d) the parameters M =10, V =0.02, V ′ =0.06, S =0.035, S ′ =0.045. with α + β = γE nrl + ρMδ .Based on the SUSY QM method and knowing the ground state eigenvalues E and eigenfunctions χ , all energyeigenvalues E n r l and eigenfunctions χ n r l can be easily obtained. Briefly, using the following equation χ n r ( r, a ) = A + ( r, a ) χ n r − ( r, a ) , (2.29) χ n r l can be easily obtained in terms of the ground state wave functions. The superpotential W ( r ) depends on twoparameters a = ( F, G ) and the first partner potential has like that parameter a = ( F , G ). Hence, Eq.(2.29) willbe in the following form χ n r ( r, a ) = ( − ddr − F − Ge − δr − e − δr ) χ n r − ( r, a ) , (2.30)We define a new variable s = e − δr ∈ [0 ,
1] and factoring out the ground state wavefunction χ n r ( s, a ) = χ ( s, a ) R n r ( s, a ) . (2.31)Substituting into Eq.(2.30) and using the ground state wavefunction Eq.(2.22), we get R n r ( s ; ǫ, K ) = s (1 − s ) dds R n r − ( s ; ǫ, K + 1) + [2 ǫ − (2 ǫ + 2 K + 1)] R n r − ( s ; ǫ, K + 1) . (2.32)Based on comparison it with the recursion relation in Ref:[66] P ( α,β ) n r (1 − s ) = s (1 − s ) dds P ( α +1 ,β +1) n r − (1 − s ) + [ α + 1 − ( α + β + 2) s )] P ( α +1 ,β +1) n r − (1 − s ) , (2.33)it is seen that R n r ( s, a ) is proportional to the Jacobi polynomial P (2 ǫ, K − n r (1 − s ). Thus, the normalized eigen-function for this potential is taken in the following form χ n r l ( s ) = C n r l s ε (1 − s ) K P (2 ǫ, K − n r (1 − s ) , (2.34)or χ n r l ( s ) = C n r l s ε (1 − s ) K Γ( n r + 2 ε + 1) n r !Γ(2 ε + 1) · F ( − n r , ε + 2 K + n r , ε ; s ) , (2.35)where K = 1 / p ( l + 1 / − γ − ρ . The normalization constant C n r l can be found by using the normalizationcondition ∞ Z | R ( r ) | r dr = ∞ Z | χ ( r ) | dr = 12 δ Z s | χ ( s ) | ds = 1 , (2.36)by utilizing the following integral formula in Ref.[66]: Z (1 − z ) δ +1) z λ − F ( − n r , δ + λ + 1) + n r , λ + 1; z ) dz == ( n r + δ + 1) n r !Γ( n r + 2 δ + 2)Γ(2 λ )Γ(2 λ + 1)( n r + δ + λ + 1)Γ( n r + 2 λ + 1)Γ(2( δ + λ + 1) + n r ) , (2.37)where λ > δ > − . After making simple calculations, we arrive at the following expression for the normalizationconstant C n r l = s δn r !( n r + K + ε )Γ(2 ε + 1)Γ( n r + 2 ε + 2 K )( n r + K )Γ(2 ε )Γ( n r + 2 K )Γ( n r + 2 ε + 1) . (2.38) III. RESULTS AND DISCUSSION
In this section, we present the numerical evaluation for the bound state solutions of the l-wave KFG equation withthe vector and scalar form of the linear combination of Hulth´en and Yukawa potentials. To study the property ofthe energy levels regarding potential parameters in some quantum states, (see Figure 1) we take M =10, V =0.01, V ′ =0.05, S =0.025, S ′ =0.035 and M =10, V =0.02, V ′ =0.06, S =0.035, S ′ =0.045. The little difference ( ∼ V , V ′ , S , S ′ is quite sufficient, in order to see the energy level of quantum statesdisplayed completely different behavior. In the Figure 1 (a, b), the energy levels E of quantum states first aredecreasing until some of small δ values ( ∼ E increase in the δ ¿0.1. In the Figure1 (c, d), the energy levels E of quantum states have very little variation for an interval of δ ∈ [0 , . E of quantum states continue to gradually increase withincrements of δ . These behaviors are better recognized in higher quantum states.Behind that of these results, we can investigate some special cases.i) In case S = V , and S ′ = V ′ , namely γ = ρ , we obtain as M − E n r l = [ δ ( n r + l + 1) − γ ( E n r l + M ) n r + l + 1 ] , (3.1)where γ = V + V ′ δ .ii) If V ′ = 0, S ′ = 0, we obtain the energy level equation for Hulth´en potential case. M − E n r l = [ δ ( n r + 12 + r ( l + 12 ) − γ ′ + ρ ′ ) − γ ′ E n r l + ρ ′ M + δ ( ρ ′ − γ ′ ) n r + + q ( l + ) − γ ′ + ρ ′ ] , (3.2)where γ ′ = V δ and ρ ′ = S δ . This result is in good agreement with the expression obtained in Eq.(50) of Ref:[58].iii) In case V = 0, S = 0, but V ′ = S ′ , we obtain the energy level equation for Yukawa potential, which definedas following form M − E n r l = [ δ ( n r + 12 + r ( l + 12 ) − γ ′′ + ρ ′′ ) − γ ′′ E n r l + ρ ′′ M + δ ( ρ ′′ − γ ′′ ) n r + + q ( l + ) − γ ′′ + ρ ′′ ] , (3.3)where γ ′′ = V ′ / δ , and ρ ′′ = S ′ / δ . This result is in good agreement with the expression obtained in Eq.(52) ofRef:[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 Eq.(39) of Ref:[67].iv) Also, if V = − S , and V ′ = − S ′ , namely γ = − ρ . M − E n r l = [ δ ( n r + l + 1) − γ ( E n r l − M ) n r + l + 1 ] , (3.4)where γ = V + V ′ δ .v) If δ → S = V = 2 δZe or γ ′ = ρ ′ = V δ = Ze in Eq.(2.28), the potential reduces to Coulomb potential, V c ( r ) = − Ze /r , and the corresponding energy spectrum is obtained as E n r l = ( n r + l + 1) − Z e ( n r + l + 1) + Z e M (3.5)and this result is the same with Eq.(51) of Ref:[67].vi) If we take l = 0 (the s-wave case), the centrifugal term in Eq.(2.6) disappears because l ( l +1) δ e − δr (1 − e − δr ) = 0 andthe equation turns to the s-wave KFG equation. By setting l = 0 in Eq.(2.28), its energy spectrum equation is thefollowing form M − E n r l = [ δ ( n r + 12 + r − γ + ρ ) − γE n r l + ρM + δ ( ρ − γ ) n r + + r − γ + ρ ] (3.6)vii) If we take S = V and S ′ = V ′ , and based on the following transformations E n r l – M → E NR n r l , E n r l + M → M , V → V , and V ′ → V ′ , we obtain the energy level equation of Eq.(2.28) for the non-relativistic case. Briefly, becauseof the following relation M − E n r l = ( M − E n r l )( M + E n r l ) = − E NR n r l · M ,γ = ρ = V + V ′ δ = V + 2 δA δ ,γ ( E n r l + M ) = 2 M γ = M ( V + 2 δA )2 δ , (3.7)the energy level equation of Eq.(2.28) for the non-relativistic case, can be written the following form E NR n r l = − M (cid:20) δ ( n r + l + 1) + M ( V + 2 δA )2 δ ( n r + l + 1) (cid:21) , (3.8)which is good agreement with the result in Eq.(28) (If B and C are considered zero as a special case) of Ref:[31].Generally, it is obviously seen from Eq.(2.28) the bound states show more stability in the case of the linear combinationof Hulth´en and Yukawa potentials than Hulth´en and Yukawa potentials cases. Furthermore, the energy eigenvaluesof the quantum states are considerably sensitive with depending potential parameters. IV. CONCLUSION
To conclude, we admit that the SUSY QM method was presented to solve the KFG equation for the linear com-bination of Hulth´en and Yukawa potentials. Hence the energy eigenvalues and corresponding eigenfunctions of amentioned quantum system were analytically obtained for arbitrary l angular momentum and n r radial quantumnumbers. Next, a closed-form of the normalization constant of the wave functions was also found. Beyond that, itwas also shown that the energy eigenvalues are considerably sensitive respecting quantum states. Finally, the resultsobtained within SUSY QM are in excellent agreement with the results obtained by ordinary quantum mechanics, andit 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 theenergy eigenvalues and the normalized wave functions. The method presented in this study is systematic, and inmany cases, one of the most definite works in this field. In particular, the linear combination of Hulth´en and Yukawapotentials can be one of the essential exponential potentials, and it probably provides a promising avenue in manybranches of physics, especially in hadronic and nuclear physics. Appendix A: SUSYQM Method
For N = 2 in SUSYQM, it is possible to define two nilpotent operators, Q and Q † . They satisfy the followinganti-commutation relations: { Q , Q } = 0 , { Q † , Q † } = 0 , { Q, Q † } = H. (A1)Here H is the supersymmetric Hamiltonian operator and conventionally Q = (cid:18) A − (cid:19) and Q † = (cid:18) A + (cid:19) . The Q and Q † are also known as the supercharges operators. Here A − is bosonic operator and A + is its adjoint. In termsof these operators, the Hamiltonian H can be defined as [27, 28]: H = (cid:18) A + A − A − A + (cid:19) = (cid:18) H − H + (cid:19) , (A2)where the H ± are named as the Hamiltonian of supersymmetric-partner. Note also that Q and Q † operators commutewith H . If we have zero ground state energy for H (i.e. E = 0), we can always represent the Hamiltonian as aproduct of a linear differential operators pairs in a factorable form. Therefore, the ground state ψ ( x ) obeys theSchr¨odinger equation as follows: Hψ o ( x ) = − ~ m d ψ dx + V ( x ) ψ ( x ) = 0 , (A3)hence V ( x ) = ~ m ψ ′′ ( x ) ψ ( x ) . (A4)This result makes us possible to globally reconstruct the above potential from the information of its ground statewave function that contain zero nodes. Hence, factorizing of H is quite easy by using the following ansatz [27, 28]: H − = − ~ m d dx + V ( x ) = A + A − (A5)where A − = ~ √ m ddx + W ( x ) , A + = − ~ √ m ddx + W ( x ) . (A6)After that, the Riccati equation for W ( x ) can be written as V − ( x ) = W ( x ) − ~ √ m W ′ ( x ) . (A7)0Solving for W ( x ) from this equation, we can express it in terms of ψ ( x ) by W ( x ) = − ~ √ m ψ ′ ( x ) ψ ( x ) . (A8)We obtain this solution by noticing that when A − ψ ( x ) = 0 is satisfied, we have Hψ = A + A − ψ = 0 . We thenintroduce the operator H + = A − A + which is written by reversing the order of the H − components. After a bitsimplification, we find that H + is nothing but the Hamiltonian for new potential V + ( x ). H + = − ~ m d dx + V + ( x ) , V + ( x ) = W ( x ) + ~ √ m W ′ ( x ) . (A9)We call V ± ( x ) as supersymmetric partner potentials. For example, when the ground state energy of H is E witheigenfunction ψ , from Eq.(A.5) we can always write H = − ~ m d dx + V ( x ) = A + A − + E , (A10)where A − = ~ √ m ddx + W ( x ) , A +1 = − ~ √ m ddx + W ( x ) ,V ( x ) = W ( x ) − ~ √ m W ′ ( x ) + E , W ( x ) = − ~ √ m d ln ψ dx . (A11)The SUSY partner Hamiltonian is defined by [27, 28] H = A − A +1 + E = − ~ m d dx + V ( x ) , (A12)where V ( x ) = W ( x ) + ~ √ m W ′ ( x ) + E = V ( x ) + 2 ~ √ m W ′ ( x ) = V ( x ) − ~ m d dx (ln ψ (1)0 ) . (A13)Using Eq.(A.12), for H and H , the energy eigenvalues and eigenfunctions are obtained as E n = E n +1 , ψ n = [ E n +1 − E ] − A − ψ n +1 , ψ n +1 = [ E n − E ] − A + ψ n . (A14)Here E mn represents the energy eigenvalue, where n and m denote the energy level and the m ’th Hamiltonian H m ,respectively. Hence, it is clear that if H has p ≥ E n , as well aseigenfunctions ψ n defined in 0 < n < p , then we can always generate a hierarchy of ( p −
1) Hamiltonians, i.e., H , H , ..., H p such that the ( H m ) has the same spectrum of eigenvalue as H , apart from the fact that the first( m −
1) eigenvalues of H are absent in H [27, 28]: H m = A + m A − m + E − = − ~ m d dx + V m ( x ) , (A15)where A − m = ~ √ m ddx + W m ( x ) , W m ( x ) = − ~ √ m d ln ψ ( m )0 dx , ( m = 2 3 4 , · · · p ) . (A16)We also have E ( m ) n = E ( m − n +1 = · · · = E n + m − ,ψ ( m ) n = [ E n + m − − E m − ] − · · · [ E n + m − − E ] − A − m − · · · A − ψ n + m − ,V m ( x ) = V ( x ) − ~ m d dx ln( ψ (1)0 · · · ψ ( m − ) . (A17)such that, by knowing all the eigenfunctions and eigenvalues of H we also obtain the corresponding eigenfunctions ψ n and energy eigenvalues E n of the ( p −
1) Hamiltonians ( H , H , ..., H p ).1 Conflict of Interest Statement
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