# Approximate Solutions, Thermal Properties, and Superstatistics Solutions to Schrödinger Equation

Authors: Ituen Okon; Clement Onate; Ekwevugbe Omugbe; Uduakobong Okorie; Akaninyene Antia; Michael Onyeaju; Chen Wen-Li; Judith Araujo
Journal: Advances in High Energy Physics
 (2022)

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## Abstract

In this work, we apply the parametric Nikiforov-Uvarov method to obtain eigensolutions and total normalized wave function of Schrödinger equation expressed in terms of Jacobi polynomial using Coulomb plus Screened Exponential Hyperbolic Potential (CPSEHP), where we obtained the probability density plots for the proposed potential for various orbital angular quantum number, as well as some special cases (Hellmann and Yukawa potential). The proposed potential is best suitable for smaller values of the screening parameterα. The resulting energy eigenvalue is presented in a close form and extended to study thermal properties and superstatistics expressed in terms of partition function Z and other thermodynamic properties such as vibrational mean energy U, vibrational specific heat capacity C, vibrational entropy S, and vibrational free energy F. Using the resulting energy equation and with the help of Matlab software, the numerical bound state solutions were obtained for various values of the screening parameter (α) as well as different expectation values via Hellmann-Feynman Theorem (HFT). The trend of the partition function and other thermodynamic properties obtained for both thermal properties and superstatistics were in excellent agreement with the existing literatures. Due to the analytical mathematical complexities, the superstatistics and thermal properties were evaluated using Mathematica 10.0 version software. The proposed potential model reduces to Hellmann potential, Yukawa potential, Screened Hyperbolic potential, and Coulomb potential as special cases.


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## Body

## 1. Introduction

The approximate analytical solutions of one-dimensional radial Schrödinger equation with a multiple potential function have been studied using a suitable approximation scheme to the centrifugal term within the frame work of the parametric Nikiforov-Uvarov method [1]. The solutions to the wave equations in quantum mechanics and applied physics play a crucial role in understanding the importance of physical systems [2]. The two most important parts in studying Schrödinger equations are the total wave function and energy eigenvalues [3]. The analytic solutions of wave equations for some physical potentials are possible for l=0. For l≠0, special approximation schemes like the Greene-Aldrich and Pekeris approximations are employed to deal with the centrifugal barrier in order to obtain approximate bound state solutions [4–6]. The Greene-Aldrich approximation scheme is mostly applicable for short range potentials [7]. Eigensolutions for both relativistic and nonrelativistic wave equations have been studied with different methods which include the following: Exact quantisation, WKB, Nikiforov-Uvarov method (NU), Laplace transform technique, asymptotic iteration method, proper quantisation, supersymmetric quantum mechanics approach, vibrational approach, formula method, factorisation method, and Shifted 1/N-expansion method [8–13]. Bound state solutions obtained from the Schrödinger equation has practical applications in investigating tunnelling rate of quantum mechanical systems [14] and mass spectra of quarkonia systems [15–19]. Among other goals achieved in this research article is to apply the Hellmann-Feynman Theorem (HFT) to eigenequation of the Schrödinger wave equations to obtain expectation values of <r−1>nl, <r−2>nl, <T>nl, and <p2>nl analytically. The Hellmann-Feynman Theorem gives an insight about chemical bonding and other forces existing among atoms of molecules [20–25]. To engage HFT in calculating the expectation values, one needs to promote the fixed parameter which appears in the Hamiltonian to be a continuous variable in order to ease the mathematical purpose of taking the derivative [26]. Similarly, the application of Hellmann-Feynman Theorem provides a less mathematical approach of obtaining expectation values of a quantum mechanical systems [27, 28]. Some of the potential models considered within the framework of relativistic and nonrelativistic wave equations are Hulthen-Yukawa Inversely quadratic potential [29], noncentral Inversely quadratic potential [30], Modified Hylleraas potential [31], Yukawa, Hulthen, Eckart, Deng-Fan, Pseudoharmonic, Kratzer, Woods-Saxon, double ring shape, Coulomb, Tietz–Wei, Tietz-Hua, Deng-Fan, Manning-Rosen, trigonometric Rosen-Morse, hyperbolic scalar, and vector potential and exponential type potentials among others [32–50]. Coulomb, hyperbolic, and screened exponential type potentials have been of interest to researchers in recent times because of their enormous applications in both chemical and physical sciences. In view of this, Parmar [51] studied ultrageneralized exponential hyperbolic potential where he obtained energy eigenvalues, unnormalized wave function and the partition function. This potential reduces to Yukawa potential, Screened cosine Kratzer potential, Manning-Rosen potential, Hulthen plus Inversely quadratic exponential Mie-type potential, and many others. Diaf et al. [52], in their studies, obtained eigensolutions to the Schrödinger equation with trigonometric Inversely quadratic plus Coulombic hyperbolic potential where they obtained energy eigenvalue and normalized wave function using the Nikiforov-Uvarov method. Onate [53] examined bound state solutions of the Schrödinger equation with second Pöschl-Teller-like potential where he obtained vibrational partition function, mean energy, vibrational specific heat capacity, and mean free energy. In that work, the Pöschl-Teller-like potential was expressed in a hyperbolic form. The practical application of energy eigenvalue of Schrödinger equation in investigating the partition function, thermodynamic properties, and superstatistics arouses the interest of many researchers. Recently, Okon et al. [54] obtained the thermodynamic properties and bound state solutions of the Schrödinger equation using Mobius square plus screened Kratzer potential for two diatomic systems (carbon(II) oxide and scandium fluoride) within the framework of the Nikiforov-Uvarov method. Their results were in agreement to semiclassical WKB among others. They presented energy eigenvalue in a close form in order to obtain partition function and other thermodynamic properties. Omugbe et al. [55] recently studied the unified treatment of the nonrelativistic bound state solutions, thermodynamic properties, and expectation values of exponential-type potentials where they obtained the thermodynamic properties within the framework of semiclassical WKB approach. The authors studied the special cases of the potential as Eckart, Manning-Rosen, and Hulthen potentials. Besides, Oyewumi et al. [56] studied the thermodynamic properties and the approximate solutions of the Schrödinger equation with shifted Deng-Fan potential model within the framework of asymptotic Iteration method where they apply Pekeris-type approximation to centrifugal term to obtain rotational-vibrational energy eigenvalues for selected diatomic systems. A lot of researches have been carried out by Ikot et al. These can be seen in Refs. [57–60]. Also, Boumali and Hassanabadi [61] studied thermal properties of a two-dimensional Dirac oscillator under an external magnetic field where they obtained relativistic spin-1\2 fermions subject to Dirac oscillator coupling and a constant magnetic field in both commutative and noncommutative spaces.In this work, we propose a novel potential called Coulomb plus Screened Exponential Hyperbolic Potential to study bound state solutions, expectation values, superstatistics, and thermal properties within the framework of the parametric Nikiforov-Uvarov method [62]. This article is divided into 9 sections. The introduction is given in Section 1. The parametric Nikiforov-Uvarov method is presented in Section 2. The solutions of the radial Schrödinger equation are presented in Section 3. The application of Hellmann-Feynman Theorem to obtain expectation values is presented in Section 4. The thermodynamic properties and superstatistics formulations are presented in Sections 5 and 6, respectively. Numerical results and discussion are presented Sections 7 and 8, respectively, and the article is concluded in Section 9.The propose Coulomb plus Screened Hyperbolic Exponential Potential (CPSHEP) is given as(1)Vr=−v1r+Br−v2coshαr2e−αr,where v1 and v2 are the potential depths, B is a real constant parameter, and α is the adjustable screening parameter. The Pekeris-like approximation to the centrifugal term is given as
(2)1r2=α21−e−αr2⇒1r=α1−e−αr.The graph of Pekeris approximation to centrifugal term is given in Figure1.Figure 1
The graph of Pekeris approximation for various values ofα.


## 2. Parametric Nikiforov-Uvarov (NU) Method

The NU method is based on reducing second order linear differential equation to a generalized equation of hypergeometric type and provides exact solutions in terms of special orthogonal functions like Jacobi and Laguerre as well as corresponding energy eigenvalues [63–70]. The reference equation for parametric NU method according to Tezcan and Sever [71] is given as
(3)Ψ″s+c1−c2ss1−c3sΨ′s+1s21−c3s2−Ω1s2+Ω2s−Ω3Ψs=0.The condition for energy equation is given as [70].
(4)c2n−2n+1c5+2n+1c9+c3c8+nn−1c3+c7+2c3c8+2c8c9=0.The total wave function is given as(5)ψnls=Nnlsc121−c3s−c12−c13/c3Pnc10−1,c11/c3−c10−11−2c3s.The parametric constants are obtained as follows:(6)c1=c2=c3=1,c4=121−c1,c5=12c2−2c3,c6=c52+Ω1,c7=2c4c5−Ω2,c8=c42+Ω3,c9=c3c7+c32c8+c6,c10=c1+2c4+2c8,c11=c2−2c5+2c9+c3c8,c12=c4+c8,c13=c5−c9+c3c8.


## 3. The Radial Solution of Schrödinger Wave Equation

The radial Schrödinger wave equation with the centrifugal term is given as(7)d2Rrdr2+2μℏ2E−Vr−ℏ2ll+12μr2Rr=0.Equation (7) can only be solved analytically to obtain exact solution if the angular orbital quantum number l=0. However, for l>0, equation (7) can only be solve by using the approximations in (2) to the centrifugal term. Substituting equation (1) into (7) gives
(8)d2Rrdr2+2μℏ2Enl+v1r−Be−αrr+v2e−αrcoshαr2−ℏ2ll+12μr2Rr=0.By substituting equation (2) into (8) gives the following equation:
(9)d2Rrdr2+2μℏ2Enl+v1α1−e−αr−Bαe−αr1−e−αr+v2α2e−αrcoshα1−e−αr2−ℏ2α2ll+12μ1−e−αr2Rr=0.By definings=e−αr and with some simple algebraic simplification, equation (9) can be presented in the form
(10)d2Rsds2+1−ss1−sdRds+1s21−s2−ε2−χ1s2+2ε2−δ2−χ1+χ2s−ε2−δ2+ll+1Rs=0,where
(11)ε2=−2μEnlℏ2α2,δ2=2μv1ℏ2α,χ1=2μBℏ2α,χ2=2μv2coshαℏ2.Comparing equation (10) to (3), the following polynomials were obtain:
(12)Ω1=ε2−χ1,Ω2=2ε2−δ2−χ1+χ2,Ω3=ε2−δ2+ll+1.Using equation (6), other parametric constants are obtained as follows:
(13)c1=c2=c3=1;c4=0,c5=−12,c6=14+ε2−χ1,c7=−2ε2+δ2+χ1−χ2,c8=ε2−δ2+ll+1,c9=14−χ2+ll+1,c10=1+2ε2−δ2+ll+1,c11=2+1−4χ2+4ll+1+2ε2−δ2+ll+1,c12=ε2−δ2+ll+1,c13=−12−121−4χ2+4ll+1+ε2−δ2+ll+1.Using equations (4), (12), and (13) with much algebraic simplification, the energy eigenvalue for the proposed potential is given as
(14)Enl=ℏ2α2ll+12μ−v1α+ℏ2α22μn2+n+1/2+n+1/21+4ll+1−8v2μcoshα/ℏ2+2μB/ℏ2α−2μv1/ℏ2α−2v2μcoshα/ℏ2+2ll+12n+1+1+4ll+1−2v2μcoshα/ℏ22.Using equation (5), the total unnormalized wave function is given as
(15)Ψnls=Nnlsε2−δ2+ll+11−s1/2+1/21+4ll+1−4χ2Pn2ε2−δ2+ll+1,1+4ll+1−4χ21−2s.To obtain the normalization constant of equation (15), we employ the normalization condition
(16)∫0∞Rnlr2dr=1⇒∫0∞Nnlsβ1−sηPn2β,2η−11−2s2ds=1,where
(17)β=ε2−δ2+ll+1,(18)η=12+121+4ll+1−4χ2.The wave function is assumed to be in bound atr∈0,∞ and s=e−αr∈1,0.Equation (15) reduces to
(19)−Nnl2α∫10s2β1−s2ηPn2β,2η−11−2s2dss=1.Letz=1−2s such that the boundary of integration of equation (19) changes from s∈1,0 to z∈−1,1. Then, equation (19) reduces to
(20)Nnl22α∫−111−z22β−11+z22ηPn2β,2η−1z2dz=1.Using the standard integral,(21)∫−111−w2x1+w2yPnx,y−1w2dw=2x+y+1Γx+n+1Γy+n+1n!Γx+y+n+1Γx+y+2n+1.Letz=w,x=2β−1,y=2η. Then, using equation (20), the normalization constant can be obtained as
(22)Nnl=2αn!Γ2β+2η+nΓ2β+2η+2n22β+2ηΓ2β+nΓ2η+n+1.Hence, the total normalized wave function is given as(23)Rn,ls=2αn!Γ2β+2η+nΓ2β+2η+2n22β+2ηΓ2β+nΓ2η+n+1sβ1−sηPn2β,2η−11−2s.


## 4. Expectation Values Using Hellmann-Feynman Theorem

In this section, some expectation values are obtain using Hellmann-Feynman Theorem (HFT). According to the Hellmann-Feynman Theorem, the HamiltonianH for a particular quantum mechanical system is expressed as a function of some parameters q. Let Eq and Ψq be the eigenvalues and eigenfunction of the Hamiltonian. Then,
(24)∂Enl∂q=Ψq∂Hq∂qΨq.For the purpose of clarity, the Hamiltonian for the propose potential using HFT is(25)H=−ℏ22μddr2+ℏ2ll+12μr2−v1r+Br−v2coshαr2e−αr.


### 4.1. Expectation Value for<r−2>nl

To obtain the expectation value for<r−2>nl, we set q=l, to have
(26)r−2n,l=α2−α22l+12122l+1l+122−2v2μcoshαℏ2−1/2+Q6Q7,where
(27)Q6=2l+1n+1/2+l+1/22−2v2μcoshα/ℏ2−2μB/ℏ2α−2μv1/ℏ2α+ll+11/22l+1l+1/22−2v2μcoshα/ℏ2−1/2n+1/2+l+1/22−2v2μcoshα/ℏ22,Q7=n+1/2+l+1/22−2v2μcoshαℏ2+2μB/ℏ2α−2μv1/ℏ2α+ll+1n+1/2+l+1/22−2v2μcoshα/ℏ2.


### 4.2. Expectation Value for<r−1>nl

Settingq=B,, then, the expectation value for <r−1>nl becomes
(28)<r−1>=−ℏ2α2Q7×eαr4μ2μ/ℏ2α2n+1/2+l+1/22−2v2μcoshα/ℏ2n+1/2+l+1/22−2v2μcoshα/ℏ22.


### 4.3. Expectation Value for<T>nl

Settingq=μ, then,
(29)<T>nl=ℏ2α2ll+12μ−ℏ2α2Q728μ+ℏ2α2Q74−v2coshαℏ2l+1/22−2v2μcoshα/ℏ2+2/ℏ2αB−v1l+1/22−2v2μcoshα/ℏ2+v2coshα/ℏ22μB/ℏ2α−2μv1/ℏ2α+ll+1n+1/2+l+1/22−2v2μcoshα/ℏ22.


### 4.4. Expectation Value for<p2>nl

The relationship betweenT and p2 is given as T=p2/2μ, therefore
(30)<p2>nl=ℏ2α2ll+1−ℏ2α2Q724+ℏ2α2μQ72−v2coshαℏ2l+1/22−2v2μcoshα/ℏ2+2/ℏ2αB−v1l+1/22−2v2μcoshα/ℏ2+v2coshα/ℏ22μB/ℏ2α−2μv1/ℏ2α+ll+1n+1/2+l+1/22−2v2μcoshα/ℏ22.


## 4.1. Expectation Value for<r−2>nl

To obtain the expectation value for<r−2>nl, we set q=l, to have
(26)r−2n,l=α2−α22l+12122l+1l+122−2v2μcoshαℏ2−1/2+Q6Q7,where
(27)Q6=2l+1n+1/2+l+1/22−2v2μcoshα/ℏ2−2μB/ℏ2α−2μv1/ℏ2α+ll+11/22l+1l+1/22−2v2μcoshα/ℏ2−1/2n+1/2+l+1/22−2v2μcoshα/ℏ22,Q7=n+1/2+l+1/22−2v2μcoshαℏ2+2μB/ℏ2α−2μv1/ℏ2α+ll+1n+1/2+l+1/22−2v2μcoshα/ℏ2.


## 4.2. Expectation Value for<r−1>nl

Settingq=B,, then, the expectation value for <r−1>nl becomes
(28)<r−1>=−ℏ2α2Q7×eαr4μ2μ/ℏ2α2n+1/2+l+1/22−2v2μcoshα/ℏ2n+1/2+l+1/22−2v2μcoshα/ℏ22.


## 4.3. Expectation Value for<T>nl

Settingq=μ, then,
(29)<T>nl=ℏ2α2ll+12μ−ℏ2α2Q728μ+ℏ2α2Q74−v2coshαℏ2l+1/22−2v2μcoshα/ℏ2+2/ℏ2αB−v1l+1/22−2v2μcoshα/ℏ2+v2coshα/ℏ22μB/ℏ2α−2μv1/ℏ2α+ll+1n+1/2+l+1/22−2v2μcoshα/ℏ22.


## 4.4. Expectation Value for<p2>nl

The relationship betweenT and p2 is given as T=p2/2μ, therefore
(30)<p2>nl=ℏ2α2ll+1−ℏ2α2Q724+ℏ2α2μQ72−v2coshαℏ2l+1/22−2v2μcoshα/ℏ2+2/ℏ2αB−v1l+1/22−2v2μcoshα/ℏ2+v2coshα/ℏ22μB/ℏ2α−2μv1/ℏ2α+ll+1n+1/2+l+1/22−2v2μcoshα/ℏ22.


## 5. Thermodynamic Properties

In this section, we present the thermodynamic properties for the potential model. The thermodynamic properties of quantum systems can be obtained from the exact partition function given by(31)Zβ=∑n=0λe−βEn,where λ is an upper bound of the vibrational quantum number obtained from the numerical solution of dEn/dn=0, given as λ=−δ+Q3, β=1/kT, where k and T are Boltzmann constant and absolute temperature, respectively. In the classical limit, the summation in equation (31) can be replaced with an integral:
(32)Zβ=∫0λe−βEndn.In order to obtain the partition function, the energy equation (14) can be presented in a close and compact form as
(33)Enl=ℏ2α2ll+12μ−v1α−ℏ2α28μn+12+l+122−2v2μcoshαℏ2+2μB/ℏ2α−2μv1/ℏ2α+ll+1n+1/2+l+1/22−2v2μcoshα/ℏ22.Equation (33) can further be simplified to
(34)Enl=Q1−Q2n+δ+Q3n+δ2,where
(35)Q1=ℏ2α2ll+12μ−v1α,Q2=ℏ2α28μ,Q3=2μBℏ2α−2μv1ℏ2α+ll+1,δ=12+l+122−2v2μcoshαℏ2,and equation (34) can be represented in the form
(36)Enl=Q1−Q2ρ+Q3ρ2⇒−Q2ρ2+Q2Q32ρ2−2Q2Q3−Q1,where
(37)ρ=n+δ.(i)
Partition function is obtained as follows:Substituting equation (36) into equation (32) taking note of changes in the integration boundaries using equation (37) gives the partition function as
(38)Zβ=eβ2Q2Q3−Q1∫δλ+δeβQ2ρ2+Q2Q32/ρ2dρ.Using Mathematica 10.0 version, the partition function of equation (38) is given as
(39)Zβ=e−βQ1+2βQ2Q3−2ℵ−βQ2π2erfℵ0−ℵ/δ4−βQ2.(ii)
Vibrational mean energy is given as(40)Uβ=−∂lnZβ∂β=2Q2e2ℵ−βQ2−eβQ2δ4+Q32/δ2δ+eβQ2δ+λ4+Q32/δ+λ2λ+2Q2e2ℵ−βQ2πerfℵℵ2−ℵ0−ℵ/δπerfℵ0−ℵ/δ−e4ℵ−βQ2erfℵ0+ℵ/δ+erfℵ1−ℵ/λ+δ+e4ℵ−βQ2erfℵ1+ℵ/λ+δ.(iii)
Vibrational entropy is given as(41)Sβ=klnZβ−kβ∂lnZβ∂β=kln14−βQ2e−βQ1+2βQ2Q3−2ℵ−βQ2πerfℵ0−ℵδ+e4ℵ−βQ2erfℵ1+ℵλ+δ−ℵ0−ℵδ+erfℵ1−ℵλ+δ−kβ2Q2e2−βQ2−βQ2Q32−eβQ2δ4+Q32/δ2δ+eβQ2δ+λ4+Q32/δ+λ2λ+2Q2e2ℵ−βQ2πerfℵℵ2−ℵ0−ℵ/δπ2erfℵ0−ℵ/δ.(iv)
Vibrational free energy is given as(42)Fβ=−1βlnZβ=−1βlne−βQ1+2βQ2Q3−2ℵ−βQ2π4−βQ2erfℵ1−ℵλ+δ+ℵ0−ℵδ+e4ℵ−βQ2erfℵ1+ℵλ+δ−ℵ0−ℵδ.(v)
Vibrational specific heat capacity is given as(43)Cβ=kβ2∂2lnZβ∂β2=k−βQ2e2ℵ−βQ2×−erfΛ1−e4ℵ−βQ2erfΛ2+erfΛ3+e4ℵ−βQ2erfΛ4ℵ4+ℵ42/4π−βQ2−4ℵerfΛ1−Λ3+e4ℵ−βQ2erfΛ2−Λ4−βQ2ℵ5−ℵ8ℵ6/δλ+δ−βQ22+Λ5−δ2−βQ2+2ℵ−2βQ2ℵΛ6+δλ+δℵ7πerfΛ1+e4ℵ−βQ2erfΛ1−erfΛ3−e4ℵ−βQ2erfΛ42,where
(44)Λ1=ℵ0−ℵδ,Λ2=ℵ0+ℵδ,Λ3=ℵ1−ℵλ+δ,Λ4=ℵ1+ℵλ+δ,Λ5=eβQ2δ4+Q32/δ2δ,Λ6=eβQ2δ+λ4+Q32/δ+λ2δ,,ℵ=−βQ2Q32,ℵ0=−βQ2δ,ℵ1=−βQ2λ+δ,ℵ2=ℵ0δ+2δλ−βQ2+λ2−βQ2+ℵλ+δ,ℵ3=2ℵe2ℵ−βQ2π,ℵ4=−Λ5+2Λ6+ℵ3erfΛ8−Λ1,ℵ5=Λ5−2Λ6+ℵ3erfΛ2−Λ8,ℵ6=Q22Q32−e2ℵ−βQ2πδλ+δerfΛ8−Λ2ℵ−βQ2,ℵ7=erfΛ6δ2−βQ2+2δλ−βQ2+λ2−βQ2−2ℵ,ℵ8=−erfΛ3+e4ℵ−βQ2erfΛ2−Λ3−4πβ2e4ℵ−βQ2δλ+δerfΛ8.


## 6. Superstatistics Formulation

Superstatistics is the superposition of two different statistics which is applicable to driven nonequilibrium systems to statistical intensive parameter (β) fluctuation [72]. This intensive parameter which undergoes spatiotemporal fluctuations includes chemical potential and energy fluctuation which is basically describe in terms of effective Boltzmann factor [73]. According to Edet et.al [74], the effective Boltzmann factor is given as
(45)BE=∫0∞e−β′Efβ′,βdβ′,where fβ′,β=δβ−β′ is the Dirac delta function. However, the generalized Boltzmann factor expressed in terms of deformation parameter q is given as
(46)BE=e−βE1+q2β2E2.The partition function for superstatistics formalism is then given as(47)Zs=∫0∞BEdn.Substituting equation (34) into equation (46) gives the generalized Boltzmann factor equation as
(48)BE=1+q2β2−Q2ρ2+Q2Q32ρ2−2Q2Q3−Q12e−β−Q2ρ2+Q2Q32/ρ2−2Q2Q3−Q1.Using equation (47), the superstatistics partition function equation is given as
(49)Zs=eβ2Q2Q3−Q1∫0∞1+q2β2−Q2ρ2+Q2Q32ρ2−2Q2Q3−Q12eβQ2ρ2+Q2Q32/ρ2dρ.Using Mathematica 10.0 version, the partition obtain from equation (47) is
(50)Zs=116−βQ2Q32βQ2e−βQ1+2βQ2Q3−2−βQ2−βQ2Q32π−32qβ3Q23Q33+8+3q+4qβQ1+4qβ2Q12−βQ2−βQ2Q32+8q1+2βQ1−βQ23/2Q3−βQ2Q32+8q2β2Q22Q32×1+2βQ1+4−βQ2−βQ2Q32.We use the same procedure of thermodynamics (Section5) to obtain superstatistics vibrational mean energy (Us), vibrational specific heat capacity (Cs), vibrational entropy (Ss), and vibrational free energy (Fs) from the partition equation (41). However, this solution is not included in the article because of the lengthy and bulky analytical equations.


## 7. Numerical Results

Using Matlab 10.0 version, the numerical bound state solutions for the proposed potential were calculated using (14) for different quantum state. Also, using equations (18), (20), (21), and (22), the expectation values for <r−2>nl, <r−1>nl, <T>nl, and <p2>nl, respectively, were calculated as shown in Tables 1–5.Table 1
Numerical bound state solutions for CPSHEP:B=0.2,v1=0.1V,v2=0.2V,ℏ=μ=1.
nlEnl,α=0.01 (eV)Enl,α=0.02 (eV)Enl,α=0.03 (eV)Enl,α=0.04 (eV)00-0.03061781-0.033096257-0.035560349-0.03801011510-0.01893207-0.021509527-0.024138612-0.0268193161-0.00842605-0.011467973-0.014626685-0.01790216820-0.00927784-0.012006400-0.014887315-0.0179205741-0.00586094-0.008981149-0.012380582-0.0160592052-0.00490303-0.008288097-0.012015183-0.01608426930-0.00619246-0.009146159-0.012402307-0.0159608921-0.00477947-0.008148623-0.001201250-0.0163710672-0.00434223-0.008001341-0.012261886-0.0171238363-0.00414695-0.008039833-0.012612300-0.01786435240-0.04924686-0.008178413-0.011934614-0.0161932761-0.00427845-0.008007979-0.012498900-0.0177511592-0.00408585-0.008147119-0.013123860-0.0190160343-0.00401576-0.008352259-0.013722838-0.0201274664-0.00400000-0.008568537-0.014266689-0.02109443550-0.00434190-0.007970639-0.012351858-0.0174855401-0.00405927-0.008241005-0.013501346-0.0198402192-0.00400216-0.008568666-0.014417121-0.0215474803-0.00400991-0.008901268-0.015236763-0.0230163654-0.00404605-0.009216668-0.015965876-0.0242936435-0.00409483-0.009507448-0.016612192-0.025409045Table 2
Expectation values for<r−2>nl: B=0.2,v1=0.1V,v2=0.2V,ℏ=μ=1.
nl<r−2>nlα=0.01Ȧ−2<r−2>nlα=0.03Ȧ−2<r−2>nlα=0.03Ȧ−2<r−2>nlα=0.04Ȧ−200-0.180845600-0.19312890-0.20559450-0.2182366010-0.007804320-0.01146716-0.01567526-0.0204271710.0315342200.031932520.031123500.0291076320-0.003148090-0.00654506-0.01084855-0.0160558010.0105617200.008552890.00383228-0.0035994220.0570174700.043106010.01225219-0.0355420330-0.002633730-0.00636279-0.01136044-0.0176226310.004432820-0.00033896-0.00987264-0.0241672120.021856240-0.01200783-0.07495907-0.1669948030.051656280-0.03959544-0.20947110-0.4579665040-0.002613291-0.00679560-0.01260871-0.0200473210.001361432-0.00701901-0.02276403-0.0458721620.003256901-0.05268207-0.01526563-0.0296662230.002156285-0.15152770-0.42284930-0.8118030040.000077152-0.30591300-0.84750470-1.6248450050-0.002708980-0.00738565-0.01405538-0.022711651-0.000934536-0.01380659-0.03719615-0.071101232-0.010539190-0.09245381-0.23627330-0.441993103-0.034490940-0.25675980-0.64214770-1.190648004-0.074756000-0.52092780-1.29216600-2.388462005-0.129402700-0.88828570-2.20300600-4.07355300Table 3
Expectation values for<r−1>nl: B=0.2,v1=0.1V,v2=0.2V,ℏ=μ=1.
nl<r−1>nlα=0.01Ȧ−1<r−1>nlα=0.02Ȧ−1<r−1>nlα=0.03Ȧ−1<r−1>nlα=0.04Ȧ−100-0.19225140-0.19688090-0.20138910-0.2057764010-0.01142578-0.01191251-0.01239767-0.012881291-0.04285259-0.04902030-0.05517621-0.0613202020-0.04962766-0.05456679-0.05949515-0.064412761-0.02577863-0.03141289-0.03703604-0.042648122-0.01862148-0.02483049-0.03102713-0.0372114430-0.02857951-0.03354061-0.03849154-0.043432301-0.01810582-0.02350037-0.02888420-0.034257352-0.01440590-0.02023623-0.02605494-0.031862033-0.01315972-0.01949603-0.02581969-0.0321307240-0.01947010-0.02444052-0.02940090-0.034351251-0.01401303-0.01927972-0.02453591-0.029781642-0.01188206-0.01748570-0.02307814-0.028659413-0.01069620-0.01662480-0.02254153-0.028446454-0.00997767-0.01620214-0.02241417-0.0286137850-0.01475625-0.01973146-0.02469666-0.029651881-0.01157554-0.01676609-0.02194628-0.027116142-0.01025245-0.01570970-0.02115605-0.026591523-0.00949240-0.01522176-0.02093966-0.026646124-0.00902694-0.01501492-0.02099092-0.026954975-0.19225140-0.19688090-0.02117606-0.02737791Table 4
Expectation values for<T>nl: B=0.2,v1=0.1V,v2=0.2V,ℏ=μ=1.
nl<T>nlα=0.01eV<T>nlα=0.02eV<T>nlα=0.03eV<T>nlα=0.04eV00-0.026806820-0.025402520-0.02391260-0.02233765010-0.015431540-0.013981940-0.01205726-0.0096579981-0.006571382-0.007487332-0.00824097-0.00882454020-0.005631741-0.0033533080.000133090.0048251021-0.004014817-0.004921358-0.00572940-0.0064288032-0.002395157-0.002365251-0.00173894-0.00048491730-0.0020660550.0016844660.007708200.0159995701-0.002916120-0.003925660-0.00491908-0.0058818122-0.001738771-0.001662435-0.001025360.0020262493-0.0010484830.0000196720.002353050.00603390140-0.0000715860.0057053860.015152970.0282610001-0.002372722-0.003544271-0.00480446-0.0061328162-0.001362593-0.001292758-0.000698700.0004514133-0.0007586040.0003541500.027009110.0063576654-0.0021576700.0021543450.006720560.013654220500.0014555760.0097846700.023513930.0426273001-0.002090302-0.003469073-0.00506461-0.0068491022-0.001134154-0.001102620-0.000587910.0004450573-0.0005684970.0005587860.002893010.0065080604-0.0000484470.0023603390.006930530.01382034050.0005069940.0044907060.011886150.023002670Table 5
Expectation values for<p2>nl: B=0.2,v1=0.1V,v2=0.2V,ℏ=μ=1.
nl<p2>nlα=0.01eV/c2<p2>nlα=0.02eV/c2<p2>nlα=0.03eV/c2<p2>nlα=0.04eV/c200-0.053613630-0.050805050-0.04782520-0.0446753110-0.030863080-0.027963880-0.02411452-0.019316001-0.013142760-0.014974660-0.01648195-0.0176490820-0.011263480-0.0067066170.000266180.009650201-0.008029633-0.000984271-0.01145882-0.012857612-0.004790315-0.004730502-0.00347789-0.0009698330-0.0041321090.0033689320.0154164100.031999151-0.005832247-0.007851326-0.009838166-0.011763622-0.003477542-0.003324870-0.0020507330.000405243-0.0020969670.0003934420.0047061080.0120678040-0.0001431730.0114107700.0303059400.056522001-0.004745445-0.007088541-0.000960893-0.012265632-0.002725186-0.002585515-0.0013974110.000902823-0.0015172090.0007083000.0054018210.012715334-0.0004315340.0043086900.0134411400.02730844500.0029111510.0195693400.0470278700.085254611-0.004180604-0.006938146-0.010129220-0.013698202-0.002268308-0.002205244-0.0011758380.000890113-0.0011369900.0011175740.0057860280.013016104-0.0000968940.0047206780.0138610700.0276406750.0010139880.0089814120.0237722900.04600535


### 7.1. Special Cases

(a)
Hellmann potential: substitutingv2=0 into equation (1), then, the potential reduces to Hellmann potential(51)vr=−v1r+Be−αrr.The required energy equation is(52)Enl=ℏ2α2ll+12μ−v1α−ℏ2α28μn+l+1+2μ/ℏ2αB−v1+ll+1n+l+12.The result of equation (52) agrees excellently with Ref. [75].
(b)
Yukawa potential: ifv1=v2=0, then, equation (1) reduces to Yukawa potential(53)vr=Be−αrr.The corresponding energy eigenequation is(54)Enl=ℏ2α2ll+12μ−ℏ2α28μn+l+1+2μB/ℏ2α+ll+1n+l+12.Equation (54) is consistent with result obtain in Ref. [76] in order to prove the mathematical accuracy of our analytical calculation.
(c)
Screened-hyperbolic inversely quadratic potential: substitutingB=v1=0 into equation (1), the potential reduces to screened-hyperbolic inversely quadratic potential(55)vr=v2e−αrcoshαr2.The resulting energy eigenvalue is(56)Enl=ℏ2α2ll+12μ−ℏ2α28μn+12+l+122−2v2μcoshαℏ2+ll+1n+1/2+l+1/22−2v2μcoshα/ℏ22.(d)
Coulomb potential: substitutingα=0 into equation (53), then, the potential reduces to Coulomb potential(57)vr=Br.By substitutingα=0 into equation (54), it gives the corresponding energy eigenvalue for Coulomb’s potential as
(58)Enl=−ℏ2μB22n+l+12.Equation (58) agrees we with result obtain in Ref [76].


## 7.1. Special Cases

(a)
Hellmann potential: substitutingv2=0 into equation (1), then, the potential reduces to Hellmann potential(51)vr=−v1r+Be−αrr.The required energy equation is(52)Enl=ℏ2α2ll+12μ−v1α−ℏ2α28μn+l+1+2μ/ℏ2αB−v1+ll+1n+l+12.The result of equation (52) agrees excellently with Ref. [75].
(b)
Yukawa potential: ifv1=v2=0, then, equation (1) reduces to Yukawa potential(53)vr=Be−αrr.The corresponding energy eigenequation is(54)Enl=ℏ2α2ll+12μ−ℏ2α28μn+l+1+2μB/ℏ2α+ll+1n+l+12.Equation (54) is consistent with result obtain in Ref. [76] in order to prove the mathematical accuracy of our analytical calculation.
(c)
Screened-hyperbolic inversely quadratic potential: substitutingB=v1=0 into equation (1), the potential reduces to screened-hyperbolic inversely quadratic potential(55)vr=v2e−αrcoshαr2.The resulting energy eigenvalue is(56)Enl=ℏ2α2ll+12μ−ℏ2α28μn+12+l+122−2v2μcoshαℏ2+ll+1n+1/2+l+1/22−2v2μcoshα/ℏ22.(d)
Coulomb potential: substitutingα=0 into equation (53), then, the potential reduces to Coulomb potential(57)vr=Br.By substitutingα=0 into equation (54), it gives the corresponding energy eigenvalue for Coulomb’s potential as
(58)Enl=−ℏ2μB22n+l+12.Equation (58) agrees we with result obtain in Ref [76].


## 8. Discussion

Figure1 is the graph of Pekeris approximation against the screening parameter α. The nature of the graph shows that the approximation is suitable for the proposed potential. Variation of the probability density against the internuclear separation at various quantum state for l=0 and l=1, respectively, is shown in Figures 2(a) and 2(b), respectively. In Figure 2(a), the probability density curves produce thermal curves with regular peaks compacted close to the origin with uneven peaks at various internuclear distance. This curve shows that for orbital angular quantum number l=0, there is more concentration of the electron density at the origin for all the quantum state studied. The same situation is also observed for l=1 as shown in Figure 2(b). It can also be observed that at every value of the internuclear distance, the probability density for l=0 is higher than the probability density for l=1. Figure 3 shows variation of the probability density against the internuclear separation at various quantum state for l=0 and l=1 for Hellmann potential presented in Figures 3(a) and 3(b), respectively. More concentration of the electron density is observed at the origin in both cases. It is also seen that the probability density obtained for l=0 is lower than the probability density obtained for l=1. In Figure 4, we presented the variation of the probability density against the internuclear separation at various quantum state for l=0 and l=1 for Yukawa potential as shown in Figures 4(a) and 4(b), respectively. Here, there are more concentration and localization of electron density at the origin for l=0 with uneven distribution of peak curves as shown in Figure 4(a), but this situation is not the same for l=1 as presented in Figure 4(b). The probability density at the second excited state for l=1 remains constant with uniform distribution of the density curves for all values of the internuclear separation.Figure 2
Variation of the probability density plots against the internuclear separation of CPSHEP. (a) Probability density plot for fixedl=0 and (b) probability density plot for fixed l=1.
(a)(b)Figure 3
Variation of the probability density plots against the internuclear separation of Hellmann potential. (a) Probability density plot for fixedl=0 and (b) probability density plot for fixed l=1.
(a)(b)Figure 4
Variation of the probability density plots against the internuclear separation of Yukawa potential. (a) Probability density plot for fixedl=0 and (b) probability density plot for fixed l=1.
(a)(b)In Figure5, the variation of partition function for thermodynamic properties and superstatistics with the temperature parameter β were observed. Here, the partition function increases nonlinearly with β. However, in Figure 5(a), the partition function diverged as β increases, but later converges for the superstatistics as shown in Figure 5(b). It can also be observed that the partition function converges as β becomes positive in the superstatistics in Figure 5(b). In Figure 6, we presented the variation of vibrational mean energy against β for thermodynamic properties and superstatistics, respectively, as shown in Figures 6(a) and 6(b). For the thermal property, the mean energy increases as β goes up for all values of λ as shown in Figure 6(a), and at higher values of β, the mean energy for various λ tends to converge. The superstatistics mean energy rises as the temperature of the system decreases as shown in Figure 6(b). However, at a certain absolute temperature, the superstatistics mean energy increases vertically for various values of the deformed parameter (q). The variation is opposite in Figure 7 which is the variation of the heat capacity against β for different values of λ and deformed parameter q is shown in Figures 7(a) and 7(b), respectively. In each case, the heat capacity decreases monotonically with an increasing β for the nonsuperstatistics (thermodynamic property). At zero value of β, the heat capacity for various λ converged and diverges as β increases gradually. For the superstatistics, the heat capacity for various deformed parameter rises while the temperature cools down. The specific heat capacity has a turning point when β equals -150 as shown in Figure 7(b) while the specific heat capacity for the nonsuperstatistics has a maximum turning point at about 0.15 as shown in Figure 7(a). In Figure 8, the vibrational entropy decreases and diverged while the temperature of the system decreases (β increases) in the nonsuperstatistics as shown in Figure 8(a). This decrease is sharper for negative values of the entropy when β is almost constant as shown in Figure 8(b). The superstatistics entropy varies inversely with β (directly with temperature). This means that when the temperature of the system is raised, the disorderliness of the system also increases for every value of the deformed parameter. The entropy for the superstatistics converged as the temperature parameter tends to zero. In Figure 9, the variation of free energy against the β is seen to be two different steps in the case of a nonsuperstatistics as shown in Figure 9(a). Here, between 0 and 60 values of β, the free energy increases steadily for the various values of λ but beyond this range; the free energy increases sharply at constant β. For the superstatistics, the free energy increases monotonically as the temperature of the system reduces gradually. The free energy is always higher when the deformed parameter is increased as shown in Figure 9(b).Figure 5
(a) Variation of partition function withβ for thermodynamic property and (b) variation of partition function with β for superstatistics for CPSHEP, respectively.
(a)(b)Figure 6
(a) Variation of vibrational mean energy withβ for thermodynamic property and (b) variation of vibrational mean energy with β for superstatistics for CPSHEP, respectively.
(a)(b)Figure 7
(a) Variation of vibrational heat capacity withβ for thermodynamic property and (b) variation of vibrational heat capacity with β for superstatistics for CPSHEP, respectively.
(a)(b)Figure 8
(a) Variation of vibrational entropy withβ for thermodynamic property and (b) variation of vibrational entropy with β for superstatistics for CPSHEP, respectively.
(a)(b)Figure 9
(a) Variation of vibrational free energy withβ for thermodynamic property and (b) variation of vibrational free energy with β for superstatistics for CPSHEP, respectively.
(a)(b)Table1 is the numerical bound state solutions for CPSEHP. The numerical bound state solutions increases with an increase in quantum state, but decreases with an increase in the screening parameter. Tables 2–5 are expectation values for <r−2>nl, <r−1>nl, <T>nl, and <p2>nl, respectively. Here, the numerical values in all cases decrease with an increase in the screening parameter.


## 9. Conclusion

In this work, we apply the parametric Nikiforov-Uvarov method to obtain the bound state solutions of Coulomb plus Screened Exponential Hyperbolic Potential. The resulting energy eigenvalues were presented in a close and compact form. The research work was extended to study thermal properties, superstatistics, and various expectation values. The proposed potential also reduced to Hellmann potential, Yukawa potential, Screened Hyperbolic potential, and Coulomb potential as special cases. The normalized wave function for the mother potential and that of the Hellmann potential are similar, but the normalized wave function of the Yukawa potential seams different. The trend of the thermodynamic and superstatistics curves is in agreement to the results of an existing literature. The results of the thermodynamic properties and superstatistics revealed that the effect of the temperature on the thermodynamic properties and the superstatistics are similar. Finally, this research work has practical applications in physical and chemical sciences.



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