# The Information-Theoretic Treatment of Spinless Particles with the Assorted Diatomic Molecular Potential

Authors: Muhammad Roshanzamir
Journal: Advances in High Energy Physics
 (2022)

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## Abstract

The relativistic solutions of the Klein-Gordon equation comprising an interaction of the generalized inversely quadratic Yukawa potential mixed linearly with the hyperbolic Schiöberg molecular potential is achieved employing the idea of parametric Nikiforov-Uvarov and the Greene-Aldrich approximation scheme. The energy spectra and the corresponding normalized wave functions are derived regarding the hypergeometric function in a closed form for arbitraryℓ-state. Numerical results of the energy eigenvalue are proposed. Moreover, special circumstances of this potential are reviewed, and their energy eigenvalues were assessed. Subsequently, the Tsallis entropy and Rényi entropy both in position and momentum spaces are defined under the desired potential. The impacts of these entropies on the angular momentum quantum number are explored in detail.


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## Body

## 1. Introduction

In recent years, different researches have centered on the investigations of the quantum mechanical issues with various physical potentials because they incorporate all of the basic data required to correctly characterize physical quantum models [1–3]. It is worthy of note that the investigation of the relativistic wave equation especially the Klein-Gordon equation has raised fundamentally more thought of numerous researchers as well as its considerable impact in quantum mechanics. These equations may explicitly predict the particles’ treatment, which lead to the study of relativistic effects in various branches of physics and chemistry [4, 5]. The Klein-Gordon equation is a notable relativistic wave equation that depicts the movement of spinless particles because of its square terms [6–8]. It is also realized that the specific feasible actual possibilities via the Klein-Gordon equations are uncommon, with the exception of some outstanding solvable quantum systems, for example, the hydrogen atom and harmonic oscillator, while the case of states with arbitrary angular momentum, which do not exhibit accurate solutions, is understood either mathematically or by approximation methods [9, 10]. In this manner, numerous quantum modes should be considered by approximation schemes, and the decision of these utilized approximation layouts relies upon their adaptability to the desired potential [11, 12]. Because of their various applications, solutions of the relativistic and nonrelativistic wave equations have been utilized in various quantum potential interactions utilizing different techniques [13, 14]. These methods incorporate 1/N shifted expansion procedure [15], Nikiforov-Uvarov approach [16, 17], asymptotic iteration method [18], factorization method [19, 20], formula technique [21], supersymmetric approach [22, 23], ansatz methodology [24], Laplace transform approach [25, 26], functional analysis approach [27, 28], proper quantization rule [29], and others [30, 31].The hypergeometric Nikiforov-Uvarov method has indicated its capacity to ascertain the specific energy levels of bound states for many solvable quantum frameworks [32]. The parametric adaptation of this approach is employed to acquire the bound-state solutions of the D-dimensional radial Klein-Gordon equation [33]. It is supported by its acceptable execution through comparisons with various techniques and a direct, precise, and user-friendly approach.Using exponential-type potentials, various quantum mechanical wave equations have been extensively studied [34, 35]. It has been contended that these types of potentials are much more efficient than their competitors, which exist in Coulomb, or inverse forms and have significant roles in solid-state physics, nuclear physics, and other fascinating domains [36, 37]. In the study of potentials with hypergeometric wave functions, a suitable form of a composed exponential-type potential is presumed, which grants obtaining various exponential potentials. Such potentials can be comprehensively utilized as interaction models of diatomic molecules. The desired combined potential encompasses the generalized inversely quadratic Yukawa potential with a molecular hyperbolic potential, which is given by
(1)VSGIQYP=−Τe−2σr2r2+2−2Qr+Q2eσreσr+2Ηϱe2σr+1−e2σr+12e2σr−12,where Τ, σ, and Q are the coupling strength, the screening parameter, and a constant (an inverse length), respectively, where σ<1 and Η along with ϱ are two adjustable positive parameters representing the properties of the potential where 0≤ϱ≤1.This combination might describe the effective potential of a many-body system since it incorporates the notable potential models as special cases. This model can be utilized in a variety of fields such as atomic, solid-state, plasma, and quantum field theory, where it has a minimum value0 and can attain infinity when r=0. Hence, this potential represents better treatment than other exponential potentials in some cases and is able to modify comparatively the efficiency of other potentials. In this manner, the study of quantum systems linked with these forms of potentials seems substantial in proposing several frameworks for quantum chemistry and molecular physics. The current investigation was inspired by the idea that this type of potential could be used as a mathematical example to impressively depict the characteristics of diatomic molecules, despite the fact that there has been little research into this linear combination of diatomic molecular potential energy functions, particularly within the framework of the Klein-Gordon equation. Subsequently, the accomplishment of quantum mechanics in the depiction of the physics of atomic and subatomic particles cannot be overstated [13, 36, 38–42].Spin and pseudo-spin symmetry have made a profit in recent studies due to their applications in nuclear physics and other related areas [43–46]. Recently, a large number of studies have demonstrated their advantages in acquiring the solutions of the Dirac equations for various generic potentials within the spin and pseudo-spin symmetric limit [47–52], but only a few investigations have been conducted within the framework of the Klein-Gordon structure. The objective of this research is to figure out the energy spectrum and wave function visualization for the Klein-Gordon equation at the spin symmetry limit using the Nikiforov-Uvarov technique and the Greene-Aldrich approximation scheme.In this context, what is required in this paper is categorized as follows. In Section2, the parametric Nikiforov-Uvarov method for a generally second-order differential equation is briefly discussed. Section 3 employs the basic concept of the parametric Nikiforov-Uvarov approach via the Greene-Aldrich approximation scheme to obtain ℓ-wave approximate analytical solutions of the D-dimensional radial Klein-Gordon equation and the normalized radial wave functions in terms of generalized hypergeometric functions for the desired potential within the spin symmetry limit. Entropic measures propose well-organized implementations to smooth the path for realizing connections in quantum frameworks [53–55]. Two basic theoretic quantities, such as the Rényi entropy [56, 57] and the Tsallis entropy [58, 59], are commonly used in information theory as a measure of entanglement. They have numerous purposes in optical lattice theory, statistical mechanics, and various realms of mathematics and physics and are calculated in Section 4. Section 5 is devoted to a brief discussion, while the conclusions are outlined in the last section.


## 2. Parametric Nikiforov-Uvarov Methodology

A numerous category of differential equations can be solved by the Nikiforov-Uvarov approach in mathematical physics, which relies on attenuating a second-order linear differential equation to a generalized equation of hypergeometric type [16]. Many accomplishments have been documented in the successful application of this technique to obtain viable results for various wave equations in a quantum framework [60, 61]. The parametric form of this approach was introduced in order to have an efficient application [17] and can easily solve any differential equations of the form
(2)d2dp2++ω1−ω2pp1−ω3pddp−1p1−ω3p2ℊ1p2−ℊ2p+ℊ3ψ=0.According to this approach, the eigenfunctions are expressed as(3)ψp=Nn,ℓpω121−ω3p−ω12−ω13/ω3Pnω10−1,ω11/ω3−ω10−11−2ω3p,where Pnα,βx and Nn,ℓ are the orthogonal Jacobi polynomial [62] and the normalization constant, respectively. Furthermore, the parametric energy eigenvalues are given as
(4)ω2n−2n+1ω5+2n+1ω9+ω3ω8+nn+1ω3+2ω3ω8+2ω8ω9+ω7=0,where
(5)ω4=121−ω1,ω5=12ω2−2ω3,ω6=ω52+ℊ1,ω7=2ω4ω5−ℊ2,ω8=ω42+ℊ3,ω9=ω3ω7+ω32ω8+ω6,ω10=ω1+2ω4+2ω8,ω11=ω2−2ω5+2ω9+ω3ω8,ω12=ω4+ω8,ω13=ω5−ω9+ω3ω8.


## 3.D-Dimensional Klein-Gordon Equation

The Klein-Gordon equation with a scalar potentialSr and a vector potential Vr in D-dimensions is given by [63, 64]
(6)ℏ2c2∇D2−μc2+Sr2+En,ℓ−Vr2Ψr,ΩD=0,where ∇D2 represents the Laplace operator in D-dimensions and is defined as
(7)∇D2=1rD−1∂∂rrD−1∂∂r−1r2ΚD2ΩD,whereD represents the spatial dimensionality where D≥2, ℏ is the reduced Planck constant, and μ, c, and En,ℓ are the rest mass of the particle, the speed of light, and the relativistic energy of the system, respectively. The attractive scalar potential is usually observed under static field conditions, whereas the repulsive vector potential is observed under dynamic conditions. The scalar potential Sr is regarded as an extra quantity added to the particle mass μ. Also, the total wave function in D-dimension can be given as Ψr,ΩD=r1−D/2Un,ℓrYℓ,mΩD, where Yℓ,mΩD is the generalized spherical harmonic function. Employing the eigenvalues of the generalized angular momentum operator, KD2ΩD, where KD2ΩDYℓ,mΩD=ℓℓ+D−2Yℓ,mΩD, the radial part of the D-dimensional Klein-Gordon Equation (6) is written as
(8)ℏ2c2d2dr2+En,ℓ2−μ2c4−2μc2Sr+En,ℓVr+V2r−S2r−℘℘+1ℏ2c2r2Un,ℓr=0,where ℘=ℓ+D−3/2, and ℓ represents the angular momentum quantum number. With the equal scalar and vector potentials being taken as the determined potential Equation (1), where Q=1, the following second-order Schrödinger-like equation may be obtained as
(9)d2dr2+En,ℓ2−μ2c4ℏ2c2+2μc2+En,ℓℏ2c2Τr2e−2σr−2Τre−σr+Τ−Ηϱe2σr+1−e2σr+12e2σr+12−℘℘+1r2Un,ℓr=0,which cannot be solved analytically even for the s-wave case. In order to solve Equation (9) for arbitrary ℓ-state and also to consider the Columbic characteristic 1/r of the Yukawa potential, the following Greene and Aldrich approximation scheme for σ≪1 might be used as [10, 65]
(10)1r2≈4σ2e−2σr1−e−2σr2.As comparison, the behavior of1/r2 and the recommended approximate term are shown in Figure 1. It is obvious that this approximation is most suitable for the low values of σ that are considered in this work. Consequently, this approximation will be used throughout the present study.Figure 1
Plot of1/r2 and its approximation in Equation (10) with σ=0.01, 0.05, 0.1, and 0.3.Substituting Equation (10) into Equation (9) and introducing coordinate transformation of the form h=e−2σr, the generalized hypergeometric-type equation reads as
(11)d2Un,ℓhdh2+1hdUn,ℓhdh+1h21−h2−ℊ1h2+ℊ2h−ℊ3Un,ℓh=0,where
(12)ℊ1=−4Tσ2−4Tσ+ϱ+12H+1/2c2μ−En,ℓ−Tc2μ+En,ℓ2c2ℏ2σ2,ℊ2=12c2ℏ2σ2c4μ2+−2℘ℏ2℘+1σ2−4Tμσ−2μϱ2H−H+Tc2−2En,ℓϱ2H+2Tσ−H+En,ℓ2+T,ℊ3=ϱ−12H+1/2c2μ−En,ℓ−Tc2μ+En,ℓ2c2ℏ2σ2.By comparing Equation (11) with Equation (2), one can obtain the following parametric coefficients as
(13)ω1=ω2=ω3=1,ω4=0,ω5=−12,ω6=14+ℊ1,ω7=−ℊ2,ω8=ℊ3,ω9=ℊ1−ℊ2+ℊ3+14,ω10=1+2ℊ3,ω11=21+ℊ3+ℊ1−ℊ2+ℊ3+14,ω12=ℊ3,ω13=−12−ℊ3−ℊ1−ℊ2+ℊ3+14.TheD-dimensional relativistic energy spectra for the Schiöberg-generalized inversely quadratic Yukawa potential model can be obtained explicitly by substituting Equation (13) into Equation (4) as
(14)12c2ℏ2σ2μc2+En,ℓ2ϱ−12Η+μc2−En,ℓ+2T+2σn+12cℏ×4℘+122ℏ2−8Tμc2−8En,ℓTσ2+8ϱ2μc2+En,ℓΗ+2σn+12ℏcμc2+En,ℓ2ϱ−12Η+μc2−En,ℓ−2T+2℘℘+1+nn+1+12ℏ2σ2+4μTσ+Ηϱϱ−1c2+4En,ℓTσ+Ηϱϱ−1=0.In addition, the corresponding wave function for the proposed potential is obtained by substituting Equation (13) into Equation (3) as
(15)Unh=Nnhζ1−hρΡn2ζ,2ρ−11−2h,where
(16)ζ=μc2−En,ℓ+2ϱ−12Η−2Tμc2+En,ℓ2cℏσ,ρ=cℏσ+4℘+1/22ℏ2−8Tμc2−8En,ℓTσ2+8ϱ2μc2+En,ℓΗ2cℏσ,where Ρnχ,ς denotes the Jacobi polynomials, which have been used extensively in mathematical analysis and practical applications. Using the orthogonality property of the Jacobi polynomials [62],
(17)∫−11Paχ,ςPbχ,ς1−ϖχ1+ϖςdx=2χ+ς+12n+χ+ς+1Γn+χ+1Γn+ς+1n!Γn+χ+ς+1δab,and the normalization coefficient Nn can be expressed as
(18)Nn=ζσρ+n+ζΓn+1Γρ+n+2ζ−1n+ρΓn+2ρ1/2.The solution of the Schrödinger equation can be obtained directly from the solution of the relativistic Klein-Gordon equation by making some certain transformations. Thus, employing the following mapping within the nonrelativistic limits [25], where σ<1 and 0≤ϱ≤1,
(19)μc2+En,ℓ⟶2μc2,En,ℓ−μc2⟶En,ℓN.R.,the corresponding D-dimensional nonrelativistic energy relation for the potential model Equation (1) is attained as
(20)EnJN.R.=−2ℏ2℘℘+1+nn+1+12σ2+4Tμ+n+12ℏ×4℘+122ℏ2−16μTσ2+16ϱ2Hμσ+4ϱμHϱ−122ℏσ2n+1+4℘+122ℏ2−16μTσ2+16ϱ2Hμ2μ−1+2ϱ−12H−T.


## 4. Information-Theoretic Measures

The concept of information theory plays a significant role in contemporary theoretical physics, while it has not been much discussed in the current research union. As has been demonstrated, the information-theoretic technique, which offers novel subjects, proposes wide insight into many other physical problems [66, 67]. The use of these problem-solving viewpoints, which best characterize the energy characterisation of these systems, exposes the information-based demonstration of molecular states to debate. According to statistical mechanics, the entropic uncertainty relation operates similarly to the Heisenberg uncertainty [68]. Information-theoretic measures have been used to explore numerous unique distributions that define the quantum mechanical states of various possible models [69]. Among them, the Rényi entropy [56, 68] and the Tsallis entropy [59, 68] via the given potential are the subjects of our study in this paper. It is known that suggested entropies in the position space represent the uncertainties in the localization of a particle in space, whereas the momentum space entropies measure the uncertainties in estimating the momentum of the particle. These entropies are tightly related to basic and experimentally measurable variables like kinetic energy and magnetic susceptibility, making them useful in studying the structure and dynamics of atomic and molecular systems [70]. One of the primary distinctions is that the Rényi entropy is additive, whereas the Tsallis entropy is nonadditive (or nonextensive). Supplementary comparison analyses may be found, for example, in [70] between these entropies.


### 4.1. Rényi Entropy

As a beneficial extension of the Shannon entropy, the Rényi entropy [56] was offered by Rényi in 1960 as a class of information-theoretic measures, which included many of its features and demonstrated numerous purposes. Its applications in physics include, but are not limited to, in, quantum entanglement [71, 72], quantum communication protocols [73], and quantum measurement [74]. Applying the transformation h=e−2σr, the Rényi entropy is given as [56]
(21)SηRpδ=1η−1Ln2πσ∫10δhη1hdh,where η denotes the entropic index. The Rényi entropy measures the sensitivity of the system to the deviation from the equilibrium. If the η parameter is larger than unity, the related entropy decreases, implying that the object is provided with more information on such a configuration than its equilibrium equivalent. The rate of change of the entropy with the Rényi parameter simply illustrated the sensitivity of the system to the degree of nonequilibricity. The Rényi entropy in the position space is mathematically specified as
(22)SηRpδ=11−ηLn2πσ∫01δzη11−zdz,where z=1−e−2σr and δz denotes the probability density as
(23)δz=Nn21−z2ζz2ρPn2ζ,2ρ−12z−12.By inserting the probability density, Equation (23), into Equation (22), the Rényi entropy in the position space is modified as
(24)SηRpδ=11−ηLn2πσ+ηLn22ζ+ρ+1ζσρ+n+ζ2ρ+nΓ2ζ+nΓ2ζ+ρ+n+1n+ρ2n+2ρ+2ζ−1Γ2ζ+ρ+n.In order to obtain the corresponding Rényi entropy in the momentum space, one can taket=1−2e−2σr. Therefore, Equation (21) is more precisely written as
(25)SηRmγ=11−ηLn2πσ∫−11γtη11−tdt.By considering the appropriate probability densityγt, Equation (23), and substituting it into Equation (25), the compact form of the Rényi entropy in the momentum space is explicitly represented by
(26)SηRmγ=11−ηLnπσ4σn+2ρρ+n+ζΓ2ζ+n+1Γ2ζ+ρ+n+1Γn+2ρ+2ζ+1ρ+nη.It is worth mentioning that whenη=1, the Shannon entropy and Kullback–Leibler divergence are recovered [75].


### 4.2. Tsallis Entropy

In order to generalize the Boltzmann-Gibbs theory [76] and open the way of describing quantum subjects such as quantum optics and quantum communication [77], the Tsallis entropy was reported by Tsallis [75] in 1988. Tsallis entropy is often used in nonextensive systems, including nonergodic structures and processes; see, e.g., reference [73] for more details. By implementing the conversion h=e−2σr, the Tsallis entropy becomes
(27)SηTpδ=1η−11−2πσ∫01δhη1hdh,and is indicated in position space by
(28)SηTpδ=1η−11−2πσ∫01δzη11−zdz,where z=1−e−2σr. By taking the probability density Equation (23) and inserting it into Equation (28), one can obtain the analogous entropy as
(29)SηTpδ=1η−11−2πσ∙22ζ+ρ+1σn+ρ+ζn+ρ2ζ+2ρ+2n+1∙2ζn+2ρΓ2ζ+nΓ2ζ+ρ+n+1Γ2ζ+ρ+nη.Introducingt=1−2e−2σr and substituting Equation (23) into Equation (27), the corresponding Tsallis entropy in the momentum space may further be represented as
(30)SηTmγ=1η−11−πσ×4σn+2ρn+ρ+ζΓ2ζ+n+1Γ2ζ+ρ+n+1n+ρΓ2ζ+2ρ+n+1η.It should be pointed out that, subject to the limit atη=1, one can resume the usual Boltzmann-Gibbs entropy [59].


## 4.1. Rényi Entropy

As a beneficial extension of the Shannon entropy, the Rényi entropy [56] was offered by Rényi in 1960 as a class of information-theoretic measures, which included many of its features and demonstrated numerous purposes. Its applications in physics include, but are not limited to, in, quantum entanglement [71, 72], quantum communication protocols [73], and quantum measurement [74]. Applying the transformation h=e−2σr, the Rényi entropy is given as [56]
(21)SηRpδ=1η−1Ln2πσ∫10δhη1hdh,where η denotes the entropic index. The Rényi entropy measures the sensitivity of the system to the deviation from the equilibrium. If the η parameter is larger than unity, the related entropy decreases, implying that the object is provided with more information on such a configuration than its equilibrium equivalent. The rate of change of the entropy with the Rényi parameter simply illustrated the sensitivity of the system to the degree of nonequilibricity. The Rényi entropy in the position space is mathematically specified as
(22)SηRpδ=11−ηLn2πσ∫01δzη11−zdz,where z=1−e−2σr and δz denotes the probability density as
(23)δz=Nn21−z2ζz2ρPn2ζ,2ρ−12z−12.By inserting the probability density, Equation (23), into Equation (22), the Rényi entropy in the position space is modified as
(24)SηRpδ=11−ηLn2πσ+ηLn22ζ+ρ+1ζσρ+n+ζ2ρ+nΓ2ζ+nΓ2ζ+ρ+n+1n+ρ2n+2ρ+2ζ−1Γ2ζ+ρ+n.In order to obtain the corresponding Rényi entropy in the momentum space, one can taket=1−2e−2σr. Therefore, Equation (21) is more precisely written as
(25)SηRmγ=11−ηLn2πσ∫−11γtη11−tdt.By considering the appropriate probability densityγt, Equation (23), and substituting it into Equation (25), the compact form of the Rényi entropy in the momentum space is explicitly represented by
(26)SηRmγ=11−ηLnπσ4σn+2ρρ+n+ζΓ2ζ+n+1Γ2ζ+ρ+n+1Γn+2ρ+2ζ+1ρ+nη.It is worth mentioning that whenη=1, the Shannon entropy and Kullback–Leibler divergence are recovered [75].


## 4.2. Tsallis Entropy

In order to generalize the Boltzmann-Gibbs theory [76] and open the way of describing quantum subjects such as quantum optics and quantum communication [77], the Tsallis entropy was reported by Tsallis [75] in 1988. Tsallis entropy is often used in nonextensive systems, including nonergodic structures and processes; see, e.g., reference [73] for more details. By implementing the conversion h=e−2σr, the Tsallis entropy becomes
(27)SηTpδ=1η−11−2πσ∫01δhη1hdh,and is indicated in position space by
(28)SηTpδ=1η−11−2πσ∫01δzη11−zdz,where z=1−e−2σr. By taking the probability density Equation (23) and inserting it into Equation (28), one can obtain the analogous entropy as
(29)SηTpδ=1η−11−2πσ∙22ζ+ρ+1σn+ρ+ζn+ρ2ζ+2ρ+2n+1∙2ζn+2ρΓ2ζ+nΓ2ζ+ρ+n+1Γ2ζ+ρ+nη.Introducingt=1−2e−2σr and substituting Equation (23) into Equation (27), the corresponding Tsallis entropy in the momentum space may further be represented as
(30)SηTmγ=1η−11−πσ×4σn+2ρn+ρ+ζΓ2ζ+n+1Γ2ζ+ρ+n+1n+ρΓ2ζ+2ρ+n+1η.It should be pointed out that, subject to the limit atη=1, one can resume the usual Boltzmann-Gibbs entropy [59].


## 5. Discussion

Figure2 depicts the variation of the generalized inversely quadratic Yukawa potential blended directly with the hyperbolic Schiöberg molecular potential in terms of internuclear separation with respect to the different screening parameters. It should be noted that around r=1, the figure decreases swiftly while for r>1, it gradually increases.Figure 2
Sketch of the composed-potential as a function ofrr>0 for Η=20, T=1, and ϱ=0.1 with σ=0.1, 0.15, and 0.2.In Figure1, the treatment of 1/r2 and the suggested approximation are figured with different screening parameters, σ. The pattern tends to gather asymptotically for various values of the screening parameters, which serves as a good approximation of the system.For the behavior of the energies in terms of the quantum numbern for different screening parameters, the dimension of the system and the potential molecular parameter are plotted, respectively, in Figures 3–5. In all cases, the energy curves increase as the quantum number increases. It can be observed that when each of the potential parameters increases, a molecule impacted by this type of potential exhibits almost identical behavior.Figure 3
Behavior of energy spectral with respect ton with μ=c=ℏ=1, Η=20, ϱ=0.1, Τ=15, and D=5 for σ=0.07, 0.09, and 0.1.Figure 4
Variation of the energyEn,ℓ in terms of n with μ=c=ℏ=1, Η=20, ϱ=0.1, Τ=10, and σ=0.1 for D=3, 5, and 7.Figure 5
The function of energy spectral with regard ton with μ=c=ℏ=1, Η=20, Τ=15, D=3, and σ=0.1 for ϱ=0.1, 0.15, and 0.20.Figure6 depicts the fluctuation of the energy eigenvalue with regard to the potential parameter T, where the energy of the system decreases monotonically with the increase in the potential parameter.Figure 6
The contribution of the coupling strength to the energy spectra withμ=c=ℏ=1, Η=10, σ=0.1, and ϱ=0.1 for D=3, 5, and 7.Figure7 indicates the approximate energy Enℓ, as far as the screening parameter, σ, within the proposed potential. It can be noticed that as the screening parameter, σ, expands, the approximate energy increases monotonically with increasing n.Figure 7
Contribution of theσ-parameters to the energy spectra with μ=c=ℏ=1, Η=40, D=3, T=0.5, and ϱ=0.5 for n=1, 2, 3, 4, and 5.It can be observed from Figure8 that the energy increases smoothly with increasing σ as the dimension of the system is increased D=3, 5, and 7.Figure 8
Effects of the screening parameterσ on the energy spectra with μ=c=ℏ=1, Η=40, T=0.5, and ϱ=0.5 for D=3, 5, and 7.Figures9 and 10 illustrate the Rényi entropy in position space and momentum space in terms of the angular momentum quantum number for D=3, respectively. In both cases, the Rényi entropy grows as the angular momentum quantum number increases.Figure 9
Behavior of the Rényi entropy for the position spaceSpRδ in terms of the angular momentum quantum number ℓ with μ=c=ℏ=1, Η=20, T=10, ϱ=0.1, and σ=0.01 for D=3.Figure 10
Variation of the Rényi entropy for the momentum spaceSmRγ with respect to the angular momentum quantum number ℓ with μ=c=ℏ=1, Η=20, T=10, ϱ=0.1, and σ=0.01 for D=3.It can be concluded from Figures11 and 12 that the Tsallis entropy decreases with increasing angular momentum quantum number in both position space and momentum space.Figure 11
Effects of the Tsallis entropy for the position spaceSpTδ with respect to the angular momentum quantum number ℓ with μ=c=ℏ=1, Η=20, T=10, and ϱ=σ=0.1 for D=3.Figure 12
Contribution of the angular momentum quantum numberℓ to the Tsallis entropy for the momentum space SmTγ with μ=c=ℏ=1, Η=20, T=10, and ϱ=σ=0.1 for D=3.One can acquire from Table1 that the energy eigenvalue in the nonrelativistic regime increases as the dimension of the system increments for practically all the states, while the energy turns out to be more negative in value and bounded as the screening parameter decreases in each state. It is fascinating to consider that the presented potential, Equation (1), shows some different highlights.Table 1
The energy spectraEn,ℓ as a function of σ with μ=c=ℏ=1, Η=20, T=10, and ϱ=0.1 for D=3, 5, and 7.
StatesσEn,ℓD=3D=5D=72p0.01-14.70968032-14.69313982-14.668359900.04-3.50480000-3.34087012-3.099883220.084.989815175.374293205.898479750.107.595000008.076470518.679940973p0.01-13.98344446-13.96744208-13.943468120.04-1.83644800-1.69193926-1.479392920.086.633086716.930346527.337173220.109.020000009.362324719.794565213d0.01-13.96744208-13.94346812-13.911558100.04-1.69193926-1.47939292-1.203598900.086.930346527.337173227.817324940.109.362324719.7945652110.256811414p0.01-13.28072216-13.26523528-13.242033420.04-0.36290651-0.23497208-0.046715690.087.910326058.143249768.462946600.1010.0396281010.2873367410.601403864𝒹0.01-13.26523528-13.24203342-13.211150640.04-0.23497208-0.046715690.197716950.088.143249768.462946608.841606340.1010.2873367410.6014038610.938514464𝒻0.01-13.24203342-13.21115064-13.172632080.04-0.046715690.197716950.493153650.088.462946608.841606349.251682990.1010.6014038610.9385144611.26515817The peculiar cases of Equation (1), obtained by selecting the appropriate controlling parameters, can readily suggest distinguished interaction potential patterns. Based on this, one can realize that at Η=0, and Equation (1) simplifies to the short-range generalized inversely quadratic Yukawa potential. This is regarded as a compelling model with applications in different branches of condensed matter and nuclear physics due to its ability to characterize the strong interaction between nucleons [78]. While the radial variable r approaches to zero, it behaves similarly to a Yukawa potential, but the first term in R.H.S. in Equation (1) moderates its treatment at the origin. In this manner, the approximate analytical solutions of ℓ-state D-dimensional Klein-Gordon equation in the relativistic framework are obtained as
(31)2σn+12ℏc4ℏ2c2℘+122−8Tc2μ+En,ℓ+2n+12ℏc−c2μ+En,ℓ−c2μ+En,ℓ+2T+σ22σℏ2c2−℘+1/22ℏ22+Tμc2+En,ℓT−c2μ+En,ℓ+2Tc2μ+En,ℓ+2℘+1℘+n+1n+12ℏ2+4Tμc2+4En,ℓT=0.In particular, the results, when settingH=0, have good accuracy with those obtained in [53] in the nonrelativistic limit for D=3.However, whenΗ=0 and in the low screening region where σ⟶0, the Mie-type potential [79] is obtained, whose energy spectra, in accordance with Equation (31), are given by
(32)−12cℏc2μ−En,ℓ−Tℏc4ℏ2c2℘+1/22−4Tc2μ+En,ℓℏ2c2+2n+1×c2μ−En,ℓ−T+2Tc2μ+En,ℓ=0.In addition, the outcomes are in good agreement with those provided in [80], taking D=3 in the nonrelativistic limit for the ℓ-wave case, and they agree with those given in [81] for the Mie-type potential.SettingΤ=0 yields the modified hyperbolic-type potential, suggested by Schiöberg as an important diatomic molecular potential [82]. This is strongly associated with the Morse [83], Kratzer [84], Coulomb [85, 86], harmonic oscillator [87, 88], and other potential functions as specific cases and is widely used in various applications such as quantum statistical theory [89], conformal field theory [90], nuclear structure [91], and chemical physics [92]. Thus, given this condition, the energy eigen-spectrum may be calculated as
(33)12c2ℏ2σ2c2μ+En,ℓ2ϱ−12H+c2μ−En,ℓ+2σn+12cℏ8ϱ2c2μ+En,ℓH+4σ2c2ℏ2℘+122+2ℏc2μ+En,ℓσn+12c2ϱ−12H+c2μ−En,ℓ+2ℏ2℘2+n2+℘+n+12σ2+4Hμϱϱ−1c2+4HEn,ℓϱϱ−1=0.It is shown that the result given in this manner is consistent with the one obtained by Ortakaya in [93] for the Schrödinger equation within the hyperspherical coordinates.


## 6. Conclusions

The approximateℓ-state solutions for the D-dimensional Klein-Gordon equation with a Schiöberg-generalized inversely quadratic Yukawa potential are investigated in this paper. The treatment of the respective energy eigenvalues is discussed as a function of various potential parameters. Particular samples of the potential and their analogous energy relations were concluded, which are compatible with those discussed in the literature. Two fundamental theoretic quantities, the Rényi entropy and the Tsallis entropy, are obtained which are useful in describing the synthetic reactivity of particles.



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