# Bound State Solutions of Three-Dimensional Klein-Gordon Equation for Two Model Potentials by NU Method

Authors: N. Tazimi; A. Ghasempour
Journal: Advances in High Energy Physics
 (2020)

---

## Abstract

In this study, we investigate the relativistic Klein-Gordon equation analytically for the Deng-Fan potential and Hulthen plus Eckart potential under the equal vector and scalar potential conditions. Accordingly, we obtain the energy eigenvalues of the molecular systems in different states as well as the normalized wave function in terms of the generalized Laguerre polynomials function through the NU method, which is an effective method for the exact solution of second-order linear differential equations.


---

## Body

## 1. Introduction

The exact solution is of paramount importance in quantum mechanics as it carries essential information on the quantum systems under investigation. It is possible only for quantum systems such asH2 and harmonic oscillator. As for the majority of quantum systems, the approximation method needs to be used. In most quantum systems, for the analytical solution, methods such as the Nikiforov-Uvarov method [1], quantization rules [2], ansatz method [3], supersymmetry (SUSY) method [4], and series expansion [5] have been used for any arbitrary l state.Recently, the bound state of the Schrödinger equation has been solved by the Deng-Fan potential [6], modified Morse potential [7], and Eckart potential [8] by approximation to the centrifugal term, and the wave function and energy level for bound states in any arbitrary l state have been identified. The bound state solutions of the Klein-Gordon equation with the Deng-Fan molecular potential are solved by Dong [9]. Wei et al. investigated the relativistic scattering states of the Hulthen potential by taking the same approximation [10]. Wei and Dong examined the approximate solution of the bound state of the Dirac equation with the second Pöschl-Teller potential under spin symmetry conditions and with scalar and vector modified potentials under pseudospin symmetry conditions [11, 12]. They also solved the Dirac equation with the scalar and vector Manning-Rosen potentials under pseudospin symmetry conditions by using the function analysis method and algebraic formalism [13].In our previous works, we solved the Schrödinger equation for different potentials for few-quark systems [14–17]. However, in the present work, we make use of the NU method to solve the Klein-Gordon equation for a diatomic molecule analytically. The NU method has recently been exploited in a variety of physical fields, including the Schrödinger equation with a spherically harmonic oscillatory ring-shaped potential [18] or the second Pöschl-Teller-like potential by the Nikiforov-Uvarov method [19].In this paper, first we describe the Nikiforov-Uvarov method. In Review of Nikiforov-Uvarov (NU) Method, we consider the Deng-Fan potential and calculate the energy eigenvalue for different diatomic molecules. Next, we solve the Klein-Gordon equation analytically for the Eckart plus Hulthen potential through the NU method and obtain the energy eigenvalue. And finally, we present the results, discussion, and conclusion.


## 2. Review of Nikiforov-Uvarov (NU) Method

The Schrödinger equation can be converted into a second-order differential equation as follows:(1)σ′′s+d2ψsds2+σsτ~sdψsds+σ~sψs=0,where σs and σ~s denote polynomials at most of the second degree and τ~s is a first-degree polynomial. We use the following form to find the solution:
(2)Ψs=ψsφs.By introducing Equation (3) into Equation (2), we arrive at
(3)σsd2φsds2+τsdφsds+λφs0,where φs in terms of the Rodriguez formula appears as
(4)φns=Bnρsdndsnσnsρs.The weight functionρsholds in the following formula:
(5)dψsds+σsρsdρsds=τs.The other solution factor is defined as(6)πs=12σ′s−τ~s±σ′s−τ~s2−σ~s+kσs1/2.In this method, the polynomialπs and parameter k are defined as
(7)k=λ−π′s,whereψs is defined as
(8)1ψsdψsds=πsψs.By substitutingk into Equation (7):
(9)τs=τ~s+2πs,and λ is defined as:
(10)λ=λn=−nτ′−nn−1σ′′2,n=0,1,2,⋯.The general form of the Schrödinger equation including any potential is(11)d2ψsds2+α1−α2ss1−α3sdψsds+−As2+Bs−Cs21−α3s2ψs=0.Comparing Equation (12) with Equation (2), we get the parameters
(12)τ~s=α1−α2,σs=s1−α3s,σ~s=−As2+Bs−C.Based on the equations, the constant parameters are defined as(13)α4=121−α1,α5=12α2−2α3,α6=α52+A,α7=2α4α5−B,α8=α42+C,α9=α3α7+α32α8+α6,α10=α1+2α4+2α8,α11=α2−2α5+2α9+α3α8,α12=α4+α8,α13=α5−α9+α3α8.The energy equation is obtained from(14)α2n−2n+1α5+2n+1α9+α3α8+nn−1α3+α7+2α3α8+2α8α9=0.Now we consider the eigenfunctions of the problem with any potential. We obtain the second part of the solution from Equation (4).
(15)ϕns=Pnα10−1,α11/α3−α10−11−2α3s.From the explicit form of the weight function obtained from Equation (5), we arrive at
(16)ρs=sα10−11−α3sα11/α3−α10−1.Pnα,β1−2α3s is the Jacobi polynomial. From Equation (6), we arrive at
(17)ψs=sα121−α3s−α12−α13/α3.Then the general solutionΨsψsϕs becomes
(18)ψs=sα121−α3s−α12−α13/α3×Pnα10−1,α11/α3−α10−11−2α3s.


## 3. Solving the Klein-Gordon Equation for the Ground State

The potential that is selected for molecular spectroscopy and molecular dynamics is of paramount importance. The Deng-Fan oscillator potential [20] is a simple potential model for diatomic molecules. It has the correct physical boundary conditions at r=0 and ∞. It is defined by
(19)Vr=D1−beαr−12,b=eαre−1,r∈0,∞,where D stands for the dissociation energy, re for the equilibrium bond length, and α for the potential range. The shifted Deng-Fan potential is of the following form (Figure 1)
(20)Vr=D1−beαr−12−D=D−2beαr−1+b2eαr−12.Figure 1
The Deng-Fan potentialeV in terms of rA0 for H2 diatomic molecule.The radial part of the Klein-Gordon equation for a particle with a massm and potential Vr is
(21)d2φrdr2+1ℏc2E2−m2−2E+mVr−ll+1r2φr=0.Because of the term1/r2 in Equation (21), it cannot be analytically solved except for l=0. Therefore, a suitable approximation to the centrifugal term is required, as used in [9, 21, 22]:
(22)1r2=α2d0+1eαr−1+1eαr−12,in which αr≪1 and d0=1/12. As illustrated in Figure 2, this approximation is very close to the term 1/r2. By introducing the potential and the approximation 1/r2 and ℏc=1, Equation (21) becomes
(23)d2φrdr2+E2−m2+4bDE+meαr−1−2b2DE+meαr−12−ll+1α2d0−ll+1α2eαr−1−ll+1α2eαr−12φr=0.Figure 2
Comparision between1/r2 of the Deng-Fan potential and the approximation scheme as function of r for H2 diatomic molecule.By using the variable changes=e−αr which maps the half-line 0,∞ into the interval 0,1, Equation (23) becomes
(24)d2φsds2+1−ss1−sdφds+E2−m2α2s2+4bDE+mα2s1−s−2b2DE+mα21−s2−ll+1d0s2−ll+1s1−s−ll+11−s2φs=0.By comparing Equations (11) and (24), the following coefficients are obtained:
(25)γ2=−E2−m2α2,A=γ2+β2+ν2+ll+1d0,β2=4bDE+mα2,B=2γ2+β2+2d0−1ll+1ν2=2b2DE+mα2,C=γ2+ll+1d0.By combining Equations (23) and (11), the following quantities are obtained:
(26)α1=α2=α3=1,α4=0,α5=−12,α6=14+A,α7=−B,α8=C,α9=A−B+C+14,α10=1+2C,α11=2+2A−B+C+14+C,α12=C,α13=−12−A−B+C+14+C.Finally, we find the energy eigenvalue as(27)E2−m2=−α2−n2+β2−ll+1−2n+1δ′2n+δ′2+ll+1d0α2,in which
(28)δ′=121+4ν2+2l+12.In Table1, we present potential parameters adopted from [23–25]. Furthermore, by the following data, we obtain the energy eigenvalue for diatomic molecules mentioned in Table 1.
(29)ℏc=1973.27evA0,1amu=931.494028Mev/C2d0=112,Vr=Vr2,E−m=Enl,E+m=2μℏ2.Table 1
The potential model parameters for some diatomic molecules [18–20].
Molecule
μamu
αA0−1
reA0
Dcm−1
H2
0.50391
1.9426
0.7416
38,266
LiH
0.8801221
1.1280
1.5956
20,287
CO
6.8606719
2.2994
1.1283
90,540
HCl
0.9801045
1.8677
1.2746
37,255In Table2, we calculate energy levels for different n and l states and compare them with other findings in [23, 24, 26, 27]. The radial wave function is of the following form:
(30)Rnlr=Nnle−Cαr1−e−αrδlPn2C,2δl−11−2e−αr=Nnl2C+1nn!e−Cαr1−e−αrδl2F1−n,n+2C+2δl;1+2C;e−αr,where
(31)2F1−n,n+ν+μ+1;ν+1;1−x2=n!ν+1nPnν,μxν+1n=Γn+ν+1Γν+1.Table 2
The energy levels for a few diatomic molecules obtained from the sDF and Morse oscillator potentials.
n
l
−EnleVNU
−EnleVAP21,22
−EnleV19,20
H2
0
0
4.39444
4.39444
4.47601
5
4.17644
4.18054
4.25880
10
3.62165
3.63782
3.72194
5
0
1.75835
1.75835
2.22052
5
1.61731
1.62548
2.04355
10
1.26034
1.29257
1.60391
7
0
1.07756
1.07756
1.53744
5
0.96174
0.97232
1.37565
10
0.66976
0.71172
0.97581
LiH
0
0
2.41195
2.41195
2.42886
5
2.38348
2.38458
2.40133
10
2.30815
2.31229
2.32884
5
0
1.51628
1.51628
1.64771
5
1.49278
1.49429
1.62377
10
1.43062
1.43627
1.56074
7
0
1.22340
1.22340
1.37756
5
1.20173
1.20344
1.35505
10
1.14444
1.15083
1.29580
CO
0
0
11.08068
11.08068
11.0915
5
11.07247
11.07354
11.0844
10
11.05057
11.05449
11.0653
5
0
9.68809
9.68809
9.79518
5
9.68017
9.68130
9.78833
10
9.65905
9.66321
9.77009
7
0
9.15911
9.15911
9.29918
5
9.15131
9.15247
9.29246
10
9.13050
9.13476
9.27455
HCl
0
0
4.41705
4041705
4.43556
5
4.37403
4.37843
4.39682
10
4.25973
4.27591
4.29408
5
0
2.66574
2.66574
2.80506
5
2.62859
2.63411
2.77209
10
2.52989
2.55027
2.68471
7
0
2.09652
2.09652
2.25701
5
2.06161
2.06768
2.22634
10
1.96888
1.99127
2.14511Nnl is the normalization constant. Pnν,μ denotes the Jacobi polynomial and 2F1 stands for the hypergeometric function. The Nnl constant is defined as
(32)∫0∞Rnlr2dr=∫01Rnls2dsαs=1,Nnl2∫01s2C−11−s2δl2F1−n,n+2C+2δl;2C+1;s2ds=αn!Γ2C+1Γ2C+n+12.By using the following formula [28, 29]:
(33)∫01s2a−11−s2b+12F1−n,n+2a+b+1;2a+1;s2ds=n+b+1n!Γn+2b+1Γ2aΓ2a+1n+a+b+1Γn+2a+1Γn+2a+b+1a>−1/2,b>−3/2,the normalization constant is obtained as
(34)Nnl=2Cαn!n+C+δlΓn+2C+δln+δlΓn+2C+1Γn+2δl,and for the ground state n=0
(35)N0l=αC+δlδlB2C,2δl,B2C,2δl=Γ2C+1Γ2δl2CΓ2C+2δl.We plot wavefunction of the Deng-Fan potential (eV) as a function ofrA0 for the H2 diatomic molecule in n=l=0 in Figure 3.Figure 3
Wave functioneV in terms of rA0 of Deng-Fan potential for H2 diatomic molecule (n=l=0) in atomic units ÿ=μ=1.


## 4. Eckart plus Hulthen Potential

The Eckart plus Hulthen potential has been used for the analytical solution of the Schrödinger equation. This potential as a diatomic molecular potential model has been utilized in applied physics and chemical physics. The NU method has been exploited to solve the Schrödinger equation for the Eckart plus Hulthen potential [30]. However, in the present work, we make use of the NU method to solve the Klein-Gordon equation for the Eckart plus Hulthen potential. The Eckart plus Hulthen potential runs as shown in Figure 4:
(36)Vr=cosech2ar+cothar+V01−e−2αr−V11−e−2αr2,where V0 and V1 stand for the depths of potential well and α for the inverse of the potential range. The hyperbolic functions are defined as
(37)sinhαr=eαr−e−αr2,coshαr=eαr+e−αr2,tanhαr=eαr−e−αreαr+e−αr.Figure 4
The Eckart plus Hulthen potentialeV in terms of rA0 for V0=V1=1 and α=0.05,0.1.In this way, the potential is obtained as(38)Vr=4e−2αr1−e−2αr2+1+e−2αr1−e−2αr+V01−e−2αr−V11−e−2αr2.The radial Klein-Gordon equation by using the Eckart plus Hulthen potential inl=0 is of the following form:
(39)d2φrdr2+E2−m2−4E+me−2αr1−e−2αr2−E+m1+e−2αr1−e−2αr−E+mV01−e−2αr+E+mV11−e−2αr2φr=0.With the variable changes=e−2αr, Equation (40) is transformed to
(40)d2φsds2+1−ss1−sdφds+1s21−s2E2−m21−s24α2−E+msα2−E+m4α21+s1−s−E+mV01−s4α2+E+mV14α2φs=0.With a simple comparison, the following quantities are obtained:(41)γ2=−E2−m24α2,A=γ2+β2,β2=−E+m4α2,B=2γ2−ν2,ν2=E+m4−V04α2,C=γ2−β2−ξ2,ξ2=E+mV1−V04α2.Accordingly, theαi parameters are obtained through Equation (24). Moreover, by using Equation (14) and
(42)δ′=121+1+4ν2−4ξ2,the energy eigenvalue is identified as
(43)E2−m2=−4α2−n2−ν2+2β2+2ξ2−2n+1δ′2n+δ′2+β2+ξ2.The radial wave function is of the following form:(44)Rnlr=Nnle−2ηαr1−e−2αrδlPn2η,2δl−11−2e−2αr=Nnl2η+1nn!e−2ηαr1−e−2αrδl2F1−n,n+2η+2δl;1+2η;e−2αr.With the variable changeη=C and the parameterrs of Equation (42), we obtain
(45)η=12α−E+mE−m+2V1−V0−1.The normalization constant is obtained as(46)Nnl=2ηαn!n+η+δlΓn+2η+δln+δlΓn+2η+1Γn+2δlWe plot wave function of the Eckart plus Hulthen potential (eV) as a function ofrA0 for the H2 diatomic molecule in n=l=0 in Figure 5.Figure 5
Wave functioneV in terms of rA0 of the Eckart plus Hulthen potential for n=l=0, α=0.05, and V0=V1=1 in atomic units ÿ=μ=1.


## 5. Results and Discussion

In this study, we examined the solution of the Klein-Gordon equation for two different potentials. Accordingly, we calculated the energy eigenvalues and normalized wave function for the diatomic molecules for differentn and l states through the NU method. In order to solve the Klein-Gordon equation, we utilized the NU method, which can also be used to identify the wave function and energy eigenvalue for any particular potential. However, with certain potentials, the wave equation fails to furnish the boundary conditions of the method.The parameters related to the spectroscopic constants of these molecules, taken from [19–21] appear in Table 1. Table 2 shows the energy levels for diatomic molecules by using the Morse oscillator potential and the shifted Deng-Fan (sDF) potential. The findings comply well with refs. [23, 24] and also with the energies calculated from AP (amplitude phase) [26, 27].Based on the results, as the quantum numbern increases, the energy value decreases. Furthermore, an increase in l makes the particle less bound. It can further be inferred that in higher dimensions, the energy value decreases. To show the accuracy of our results, we have calculated the eigenvalues numerically for arbitrary n with l=0, α=0.05, and α=0.1. Tables 3 and 4 show the energy levels of the Eckart plus Hulthen potential in atomic units ℏ=μ=1 with l=0, α=0.05, and α=0.1, respectively. It is observed that as α increases, the energy eigenvalue decreases.Table 3
The energy levels of the Eckart plus Hulthen potential for several states withl=0,α=0.05, in atomic units ℏ=μ=1.
V0
V1
n
EnleVNU
1
1
0
-1.10125
1
-1.14085
2
-1.18559
3
-1.23510
4
-1.23906
5
-1.34726
2
2
0
-1.00125
1
-1.01056
2
-1.02800
3
-1.05246
4
-1.08371
5
-1.12063
3
3
0
-1.20005
1
-1.12267
2
-1.06792
3
-1.03138
4
-1.00976
5
-1.00061
-1
1
0
-1.78125
1
-1.84500
2
-1.91125
3
-1.98000
4
-2.05125
5
-2.12500
-1
2
0
-2.14453
1
-2.18625
2
-2.23187
3
-2.28117
4
-2.23395
5
-2.39006Table 4
The energy levels of the Eckart plus Hulthen potential for several states withl=0,α=0.1, in atomic units ℏ=μ=1.
V0
V1
n
EnleVNU
1
1
0
-1.12099
1
-1.21053
2
-1.31852
3
-1.44288
4
-1.58244
5
-1.73523
2
2
0
-1.00500
1
-1.03993
2
-1.10217
3
-1.18677
4
-1.29046
5
-1.41096
3
3
0
-1.15616
1
-1.04664
2
-1.00352
3
-1.00655
4
-1.04379
5
-1.10788
-1
1
0
-1.81378
1
-1.94636
2
-2.08894
3
-2.24151
4
-2.40409
5
-2.57667
-1
2
0
-2.16567
1
-2.25700
2
-2.36265
3
-2.48139
4
-2.61241
5
-2.75513


## 6. Conclusion

Considering the importance of the molecular Deng-Fan potential and the Eckart plus Hulthen potential in molecular physics, chemical physics, molecular spectroscopy, and other related areas, we investigated the bound state solution of the relativistic wave equation. We provided exact solutions of the Klein-Gordon equation for these potentials by means of the Nikiforov-Uvarov (NU) method. We formulated the eigenvalues equation and the corresponding wave function in terms of hypergeometric functions via the NU method within an approximation to the centrifugal potential term.As we know, there is no analytical solution for the radial equation forl≠0. Therefore, the Klein-Gordon equation is transformed into a differential Schrödinger-like equation through a suitable coordinate transformation. The obtained energies are very close to the energies reported in other studies [19–22]. We preferred to calculate the energy eigenvalue of H2, CO, LiH, and HCl as diatomic molecules. The main advantage of these molecules is that their spectroscopic values are already known [31]. This feature has made them suitable candidates for working with in other studies, too, e.g., [32]. They also serve different purposes in various aspects of both physics and chemistry [22, 33, 34].This method of approximation is simple and practical. It can be applied to different quantum models to enhance the accuracy of the energy eigenvalues for some potential models of exponential-type, such as the hyperbolical potential and the Manning-Rosen potential [35, 36]. Our findings in this section are significant not only in theoretical and chemical physics but also in experimental physics since we obtained general results which are useful for studying nuclear charge radius, spin, and nuclear scattering. In the future, we plan to improve the approximation to solve the Bethe-Salpeter equation with different potentials.



---

*Source: 2541837-2020-04-17.xml*