# The Application of Minimal Length in Klein-Gordon Equation with Hulthen Potential Using Asymptotic Iteration Method

Authors: Isnaini Lilis Elviyanti; Beta Nur Pratiwi; A. Suparmi; C. Cari
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

The application of minimal length formalism in Klein-Gordon equation with Hulthen potential was studied in the case of scalar potential that was equal to vector potential. The approximate solution was used to solve the Klein-Gordon equation within the minimal length formalism. The relativistic energy and wave functions of Klein-Gordon equation were obtained by using the Asymptotic Iteration Method. By using the Matlab software, the relativistic energies were calculated numerically. The unnormalized wave functions were expressed in hypergeometric terms. The results showed the relativistic energy increased by the increase of the minimal length parameter. The unnormalized wave function amplitude increased for the larger minimal length parameter.


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## Body

## 1. Introduction

The relativistic effect gives the correction in the nonrelativistic quantum mechanics by applying the strong potential field in the particles dynamic. The particles dynamic in relativistic effect can be described by using Klein-Gordon equation, particularly for a zero spin particle. The Klein-Gordon equation is formed by two potentials coupling which are the four-vector potentialVr and scalar potential Sr. From these two potentials, the Klein-Gordon equation has two framework conditions, which are as follows: if the scalar potential was equal to vector potential Sr=Vr and if the scalar potential was equal to minus vector potential Sr=-Vr. This condition appears in nuclear and high energy physics problem [1–3]. Some of researchers have investigated both these conditions with the certainty vector potential. The main case of that research was how to reduce the Klein-Gordon equation into the Schrodinger-like equation, so we can solve it by using the certainty suitable methods. The methods which are usually used are such as Supersymmetric Quantum Mechanics (SUSY) [4], Nikiforov-Uvarov [1, 5], and Asymptotic Iteration Method [6, 7]. Various potentials are used to explain the dynamic particle, such as the harmonic potential [8], Makarov potential [2], Hulthen potential [1, 5], Kratzer potential [6], and Trigonometric Poschl-Teller potential [7].The particles dynamic in quantum mechanics corresponds to the position and momentum of particles. And, as we have known, the study of commutation relations between position and momentum operators is explained using Heisenberg uncertainty principle [9], which is given by(1)X^,P^≥iħwhere X^ is position operator, P^ is momentum operator,i is imaginer number, and ħ=h/2π withh being Planck constant. The presence of a quantum gravity on quantum mechanics has the consequence of the existence of a minimal observable distance on the order of the Planck length. Therefore the Heisenberg uncertainty principle gets additional correction due to the presence of a quantum gravity, which is well known as Generalized Uncertainty Principle (GUP) [10, 11], given by(2)X^,P^≥iħ1+αMLP2where αML is minimal length parameter that has value 0≤αML≤1 andPis magnitude of the momentum [11]. When the energy is much smaller than the Planck mass, αML goes to zero and we recover Heisenberg uncertainty principle [12].In 2009, Jana and Roy have solved Klein-Gordon equation in the presence of minimal length for scalar potential was equal to vector potential using Algebraic Approach [13]. The Klein-Gordon equation in the presence of a minimal length is solved by using Feynman for scalar potential was equal to vector potential [14]. In addition, the hypergeometric method is used to solve Klein-Gordon equation using hyperbolic cotangent potential [15] and Asymptotic Iteration Method in trigonometric cotangent potential [16, 17].In this paper, we solved the minimal length formalism in radial part of Klein-Gordon equation for the conditionSr=Vr with Hulthen potential by using approximate solution. The approximate solution is used by Chabab et al. to solve the Bohr Mottelson Hamiltonian in the presence of a minimal length formalism by introducing the new wave function [11]. The minimal length formalism in the Klein-Gordon equation is reduced into second-order differential equation. The relativistic energy equation and wave functions of Klein-Gordon equation are obtained by using Asymptotic Iteration Method.The study is organized as follows. In Section2, the approximate form of Klein-Gordon equation within minimal length formalism is presented. The Hulthen potential is introduced in Section 3. We describe Asymptotic Iteration Method in Section 4. The result and discussion are presented in Section 5. At last, conclusion is given in Section 6.


## 2. Approximate Form of Klein-Gordon Equation within Minimal Length Formalism

Generalized Uncertainty Principle is called the minimal length that is deformed from the commutation relations between position and momentum operators in quantum mechanics. In (2), the commutation relations can be rewritten as follows [9, 12]:(3)X^i=x^i(4)P^i=1+αMLp^2p^iwhere P^i and p^i are momentum operators at high and low energy, respectively. The magnitude of p^i is expressed byp. The occurrence of minimal length is in string theory, black hole, quantum gravity, and noncommutative geometry, which yield new correction to Heisenberg uncertainty principle and imply a finite minimal uncertainty in position measurements, e.g., at the Planck scale [14].The general Klein-Gordon equation with scalar potentialSr and vector potential Vr is given as(5)E-Vr2ψr,θ,φ=P2c2+Moc2+Sr2ψr,θ,φwhereE is relativistic energy andMo is rest mass. By setting Sr=Vr in (5) and substituting (4) into (5), with p^=-iħ∇and c=ħ=1 (natural unit), we have(6)-Δ-2αMLΔ2ψr,θ,φ-E2-Mo2-E+MoVrψr,θ,φ=0and here we have set Vr→1/2Vr.Accordingly, it would be natural to scale the potential term in (6), so that the nonrelativistic energy is reproduced [18]. The new wave function that is used to get the approximate form of Klein-Gordon equation is given [11]:(7)ψr,θ,φ=1+2αMLΔϕr,θ,φEquation (7) is modification of (19) in [11]. The modification is proposed to eliminate quadratic of Laplacian factor such that we get (20) in [11]. By substituting (7) into (6), we obtain(8)-Δ-E2-Mo2-E+MoVr1+2αMLE2-Mo2-E+MoVrϕr,θ,φ=0Equation (8) is the minimal length formalism in Klein-Gordon equation within the approximate form. The component Δ3 is eliminated due to the value of αML2 which goes to zero, and value of αML is very small. Here, we have used the properties that Δ is scalar differential operator. If Δ operates to scalar fields ϕ at a point r,θ,φ, Δϕ will result in another scalar field. Here, Δϕ is also called scalar Laplacian. Inserting (7) into (6) leads us to do the multiplication operation among Laplacian. By using the property that the multiplication operation is commutative for scalar differential operator with constant coefficient, then the multiplication operator among Laplacian also has commutative properties. Therefore, here we can eliminate Δ2 when (7) is inserted into (6). To get simple solution of (8), binomial expansion is used for small αML; then (8) becomes(9)Δϕr,θ,φ+E2-Mo2-E+MoVr-2αMLE2-Mo2-E+MoVr2ϕr,θ,φ=0Equation (9) is obtained by setting αML2 which goes to zero, so αML2 is ignored. Applying spherical Laplacian operator, as(10)Δ=1r2∂∂rr2∂∂r+1r21sin⁡θ∂∂θsin⁡θ∂∂θ+1sin2θ∂2∂φ2into (9), we use variable separable method by setting the new wave function ϕr,θ,φ=RrΘθΦφ, and we have a polar part and radial part of Klein-Gordon equation in the presence of minimal length. The polar part is given:(11)-1Θθ1sin⁡θ∂∂θsin⁡θ∂Θθ∂θ+1Φφ1sin2θ∂2Φφ∂φ2=λand the radial part is as follows:(12)1r2∂∂rr2∂Rr∂r+E2-Mo2-E+MoVr-2αMLE2-Mo22-2E2-Mo2E+MoVr+E+Mo2V2rRr=λr2Rrwhere λ is a constant of variable separable method which corresponds to angular momentum (L). By applying Rr=UHr/r and λ=LL+1 into (12), it yields(13)d2UHrdr2-LL+1r2UHr+E2-Mo2-E+MoVr-2αMLE2-Mo22-2E2-Mo2E+MoVr+E+Mo2V2rUHr=0Equation (13) is the minimal length formalism in Klein-Gordon equation with Hulthen potential in the form of one-dimensional Schrodinger-like equation.


## 3. Asymptotic Iteration Method

Asymptotic Iteration Method is method to solve the second-order differential equation in form [19–21] (14)yn′′u=λ0uyn′u+s0uynuwhere λ0u≠0 and s0u are the coefficients of a differential equation andn is a quantum number. To obtain solution, we derive (14).(15)ynz+1u=λz-1uyn′u+sz-1uynuand here(16)λzu=λ′z-1u+sz-1u+λ0uλz-1uszu=s′z-1u+s0uλz-1u,z=1,2,3,…The eigenvalue is obtained from the quantization condition which is given by(17)Δzu=λzusz-1u-λz-1uszu=0To obtain the wave function, (14) is reduced into the formalism, as follows(18)yn′′u=2auN+11-buN+2-t+1uyn′u-wuN1-buN+2ynuEquation (18) is one-dimensional Schrodinger-like equation that has solution which is expressed in hypergeometric term, given as(19)ynu=-1nC′N+2nσnF21-n,ρ+n;σ;buN+2where(20)σn=Γσ+nΓσ;σ=2t+N+3N+2;ρ=2t+1b+2aN+2bC′ is normalization constant and2F1 is a hypergeometric function. The unnormalized wave functions of Klein-Gordon equation are obtained by using (19)-(20) [19–21].


## 4. Hulthen Potential

The Hulthen potential is one of short range potentials in physics. The Hulthen potential has been used in particle physics, atomic physics, nuclear physics, solid-state physics, and chemical physics. The Hulthen wave functions have been used in solid-state physics problems [22]. The Hulthen-like wave functions have been found to investigate atomic problems [22]. The general Hulthen potential is given by [23](21)Vr=-VHωHe-2ωHr1-e-2ωHrwhere ωH is a screening parameter and VH is potential depth. The value of screening potential is 0.025 for low screening and 0.15 for high screening [22]. To get simple solution, (21) was changed in hyperbolic trigonometric term [23], given as(22)Vr=VHωH21-coth⁡ωHrBy setting VH=7 and ωH=0.1, the visualization of Hulthen potential is expressed in Figure 1.Figure 1
The visualization of Hulthen potential.Figure1 shows the visualization of Hulthen potential inr function. The Hulthen potential for different value ofr is approximately from 0 until 0.05μ/eV (natural unit). The Hulthen potential has negative value in a very small value ofr, while the Hulthen potential inclines to be constant for higher value ofr.


## 5. Result and Discussion

Equation (13) can not be solved exactly unless we use the approximation to the function of 1/r2. The approximation of function 1/r2 is given as [23](23)1r2=ωH2sinh2ωHr=frfor small value of ωH or ωHr≪1. The visualization of that approximation is expressed in Figure 2.Figure 2
The visualization of approximation, withωH=0.1 and value ofr from 0 until 0.05 μ/eV (natural unit).Figure2 shows the red line as a function of 1/r2 and light blue line as a function of ωH2/sinh2ωHr. It is seen that the two lines overlap with each other; then the centrifugal term 1/r2 is approximated by ωH2/sinh2ωHr also as in [23].To obtain the exact solution of (13) the approximate term of 1/r2 in (23) is inserted into (13) and together with (22); then we get(24)d2UHrdr2-ωH2LL+1+αMLE+Mo2VHωH2/2sinh2ωHrUHr++E+MoVHωH2+αMLE+Mo2VHωH2-2αMLE2-Mo2E+MoVHωHcoth⁡ωHr+E2-Mo2-E+MoVHωH2+2αMLE2-Mo2E+MoVHωH-αMLE+Mo2VHωH2-2αMLE2-Mo22UHr=0Equation (24) is the Klein-Gordon equation with the minimal length for Hulthen potential which can be rewritten as(25)d2UHrdr2-vPHvPH-1sinh2ωHr-2qPHcoth⁡ωHr+κPH2UHr=0with(26)vPHvPH-1=ωH2LL+1+αMLE+Mo2VHωH22(27)2qPH=E+MoVHωH2+αMLE+Mo2VHωH2-2αMLE2-Mo2E+MoVHωH(28)-κPH2=E2-Mo2-E+MoVHωH2+2αMLE2-Mo2E+MoVHωH-αMLE+Mo2VHωH2-2αMLE2-Mo22Equation (25) is second-order differential equation that will be reduced to hypergeometric differential equation type; by letting coth⁡ωHr=1-2z, we get(29)z1-zd2UHrdz2+1-2zdUHrdz+vPH′vPH′-1--2qPH′+κPH′24z-2qPH′+κPH′241-zUHr=0and then we set(30)2qPH′=2qPHωH2;κPH′2=κPH2ωH2vPH′vPH′-1=vPHvPH-1ωH2;-2qPH′+κPH′2=4αPH2;2qPH′+κPH′2=4βPH2in (29), so we have(31)z1-zd2UHrdz2+1-2zdUHrdz+vPH′vPH′-1-4αPH24z-4βPH241-zUHr=0By setting UHr=zαPH1-zβPHfHz, then (31) is reduced to AIM-type differential equation that is similar to (14).(32)fH′′z+2αPH+1-2αPH+2βPH+2yz1-zfH′z+vPH′vPH′-1-αPH+βPHαPH+βPH+1z1-zfHz=0By comparing (14) and (32), we have(33)λ0PH=-2αPH+1z+2βPH+11-z(34)s0PH=αPH+βPHαPH+βPH+1-vPH′vPH′-1z+αPH+βPHαPH+βPH+1-vPH′vPH′-11-zTo obtain eigenvalue, we use (15)-(17) and (33)-(34), which yields(35)vPH′vPH′-1=αPH+βPH+nαPH+βPH+n+1By inserting (27)-(28) and (30) into (35), we obtain the relativistic energy equation Klein-Gordon equation with minimal length for Hulthen potential, as follows: (36)E2-Mo2=-ωH2vPH′vPH′-1+14-n-122+ζPH4ωH2vPH′vPH′-1+1/4-n-1/22+E+MoVH2+τPHwhere(37)v′PH=12+ωH2LL+1+αMLE+Mo2VHωH2/2ωH2+14(38)ζPH=E+MoVHωH2+αMLE+Mo2VHωH2-2αMLE2-Mo2E+MoVHωH(39)τPH=αMLE+Mo2VHωH2+2αMLE2-Mo22-2αMLE2-Mo2E+MoVHωHandn is quantum number. Equation (36) is relativistic energy equation of the minimal length formalism in Klein-Gordon equation with Hulthen potential. The relativistic energies were calculated numerically by using Matlab software. The results of relativistic energies are listed in Table 1.Table 1
Relativistic energy withVH=7, ωH=0.1,Mo=1, and L=0.
n E(eV) α M L = 0 α M L = 0.001 α M L = 0.01 α M L = 0.1 0 -1.9770 -1.9654 -1.8759 -1.5259 1 -1.8629 -1.8537 -1.7801 -1.4725 2 -1.7473 -1.7402 -1.6818 -1.4171 3 -1.6298 -1.6248 -1.5808 -1.3594 4 -1.5102 -1.5071 -1.4767 -1.2990 5 -1.3879 -1.3865 -1.3690 -1.2357Table1 shows that the presence of minimal length and Hulthen potential in Klein-Gordon equation gives influence to the relativistic energy value. The relativistic energy value without minimal length parameter αML=0 is lower than the relativistic energy value with minimal length parameter. Then, the relativistic energy value increases for the larger minimal length parameter and for the larger quantum number (n). The influence of Hulthen potential gives negative value in the relativistic energy value. If we set αML=0, without the presence of the minimal length parameter in relativistic energy equation (36), it was reduced to the relativistic energy equation which is in agreement with [5]. In [5] it was shown that the relativistic energy equation for the Klein-Gordon equation for Hulthen potential without the minimal length depends on the squared of quantum number (n).To get the unnormalized wave function, we used (19)-(20), so we have(40)NPH=-1;aPH=βPH+12;bPH=1;tPH=αPH-12;σPH=2αPH+1;ρPH=2αPH+2βH+1Inserting (40) into (19) yields(41)fHz=-1nC′1n2αPH+1nF21-n,2αPH+2βPH+1+n,2αPH+1,zEquation (41) is substituted into UHr=zαPH1-zβPHfHz, so we have the unnormalized wave function given as(42)UHr=zαPH1-zβPH-1nC′1n2αPH+1nF21-n,2αPH+2βPH+1+n,2αPH+1,zBy applying coth⁡ωHr=1-2z in (42), we obtain the unnormalized wave function of minimal length formalism in Klein-Gordon equation, given as(43)UH0z=C′1-coth⁡ωHr2αPH1+coth⁡ωHr2βPH(44)UH1z=-C′1-coth⁡ωHr2αPH1+coth⁡ωHr2βPH2αPH+11+-12αPH+2βPH+21-coth⁡ωHr/22αPH+11!(45)UH2r=-C′1-coth⁡ωHr2αPH1+coth⁡ωHr2βPH2αPH+12αPH+21+-12αPH+2βPH+21-coth⁡ωHr/22αPH+11!+-2-12αPH+2βPH+32αPH+2βPH+41-coth⁡ωHr/222αPH+12αPH+22!Equations (43), (44), and (45) are the unnormalized wave functionsn=0 for ground state,n=1 for energy level 1, andn=2 for energy level 2, respectively. The visualization of the unnormalized wave functions is shown in Figures 3 and 4.Figure 3
The unnormalized wave functions withαML=0 for (a)n=0, (b)n=1, and (c)n=2.
(a) (b) (c)Figure 4
The unnormalized wave functions withαML=0.1 for (a)n=0, (b)n=1, and (c)n=2.
(a) (b) (c)By inspecting Figures3 and 4, we can see that the influence of minimal length parameter increases the amplitude of the unnormalized wave function.


## 6. Conclusion

The investigation of the minimal length formalism in the Klein-Gordon equation is obtained by approximate solution. The minimal length in Klein-Gordon equation for Hulthen potential is solved using Asymptotic Iteration Method. The Asymptotic Iteration Method is used to obtain the relativistic energy and unnormalized wave functions. The results show that the relativistic energies value increases for the larger minimal length parameter and for the larger quantum number (n). Then, the influence of minimal length parameter exerts effect in increasing the amplitude value of the unnormalized wave functions.



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