Uncertainty Relation and the Thermal Properties of an Isotropic Harmonic Oscillator (IHO) with the Inverse Quadratic (IQ) Potentials and the Pseudo-Harmonic (PH) with the Inverse Quadratic (IQ) Potentials

: The solutions for a combination of the isotropic harmonic oscillator plus the inversely quadratic potentials and a combination of the pseudo-harmonic with inversely quadratic potentials has not been reported, though the individual potentials have been given attention. This study focuses on the solutions of the combination of the potentials, as stated above using the parametric Nikiforov– Uvarov (PNV) as the traditional technique to obtain the energy equations and their corresponding unnormalized radial wave functions. To deduce the application of these potentials, the expectation values, the uncertainty in the position and momentum, and the thermodynamic properties, such as the mean energy, entropy, heat capacity, and the free mean energy, are also calculated via the partition function. The result shows that the spectra for the PHIQ are higher than the spectra for the IHOIQ. It is also shown that the product of the uncertainties obeyed the Heisenberg uncertainty relation/principle. Finally, the thermal properties of the two potentials exhibit similar behaviours.


Introduction
The reliability of numerical techniques is a fact, but the analytical methods are less time-consuming and less complicated, there are few potential functions that can be admitted via an approximation scheme.The exact solutions of the different quantum potential terms, such as the Manning-Rosen potential [1], Rosen-Morse potential [2], Morse potential [3], Pöschl-Teller potential [4], Coulomb potential [5], Yukawa potential [6], Hellmann potential [7], Harmonic potential [8], and others, in the recent time, have attracted a lot of attention because they contain the necessary information to study quantum models, and they also justify the correctness of the quantum theory.The solutions are valuable results that check and improve the models, as well as the numerical methods formulated to solve the complicated physical systems.The most frequently used traditional techniques are the Nikiforov-Uvarov method [9], asymptotic iteration method [10,11], supersymmetry quantum mechanics [12,13], factorization method [14], 1/N shifted expansion method [15], the supersymmetric WKB method [16], the exact and proper quantization rule [17], the formula method for bound state problems [18], etc.In most cases, the solutions obtained are the eigenvalues and eigenfunctions for both the relativistic and nonrelativistic regimes.It is noted that in early quantum mechanics, there are a limited number of exactly solvable problems.There are other reports on the Schrödinger equation [19,20].Recently, efforts have been devoted to the exactly solvable problems.The solvable problems can be classified as exactly solvable potentials that determine the full spectrum, analytically, for a continuous range of values of the potential parameters.The second solvable problem is the conditionally exactly solvable potentials with solutions that can only be generated for the specific values of the potential parameters.However, the analytic exact solutions of some of these exponential-type potentials (ETPs), such as the Yukawa potential, Hellmann potential, and Frost Musulin potential, are impossible for any = 0 state.Therefore, to obtain the solution to such a system, the use of an approximation scheme to deal with the centrifugal barrier is inevitable.Motivated by the interest in the non-ETPs, the present work examines the influence of the potential parameters on the energy of a system and some expectation values for a combination of the isotropic harmonic oscillator (IHO) and the inversely quadratic (IQ) potential as well as a combination of the pseudo-harmonic (PH) potential with the inversely quadratic (IQ) potential.The combination of the potentials to be studied in this work are given by where ω = 2π f , m is the mass of a string, f is the frequency of the vibration, D 0 is a potential strength, r e is the equilibrium bond separation and D e is the dissociation energy.

Methodology
In this study, the method of the parametric Nikiforov-Uvarov is applied.According to Tezcan and Sever [21,22], a general form of the equation is useful in this method.Using Equation ( 3), the authors give the condition for the energy equation and its corresponding wave function as ( The values of the parametric constants in Equations ( 4) and ( 5) are obtained, as follows:

Bound State Solutions
To solve any quantum system for a given potential function, the Schrödinger equation for a three-dimensional system containing the potential model is given as where E n, is the non-relativistic energy of the system, µ is the reduced mass of a particle, V(r) is the interacting potential, and ψ n, ,m (r) is the total wave function.Setting the total wave function ψ n, ,m (r) = R n, (r)Y ,m (θ, φ)r −1 , the radial Schrödinger equation is obtained as The V cen (r) in Equation ( 8) is a centrifugal potential given by the formula Plugging Equations ( 1) and ( 9) into the radial equation of Equation ( 8) and then, using a transformation y = r 2 , Equation (8) becomes dR n, (y) Comparing Equation (10) with Equation (3), the parameters in Equation ( 6) have their respective values as Substituting the correct parametric constants in Equation (11) into Equations ( 4) and ( 5), respectively, the solutions for the IHOIQ potential are given as R n, (y) = y Using the PHIQ potential given in Equation ( 2), and follow the same procedures to obtain Equation ( 12), we have the energy equation for the PHIQ potential as

Expectation Values
Here, we obtained the expectation values via the Hellmann Feynman theory.According to this theorem [23,24], when the spatial distribution of the electrons has been determined by solving the Schrödinger equation, then, all forces in the system can be calculated through classical electrostatics.If the Hamiltonian H for a particular quantum system is a function of some parameters q, with the eigenvalues and eigenfunctions of H, respectively, given by E n, (q) and R n, (q), theoretically, then, we can write with the effective Hamiltonian of the potential as we can obtain the expectation values.Setting q = D 0 ,q = m, and q = , for H I HOIQ , we have the following expectation values Setting q = µ, and q = D e , for H PH IQ , Oyewumi and Sen [25] have the following expectation values Thermodynamic Properties In this section, we calculate the thermodynamic properties of both potential (1) and potential (2).To do this, first, the energy equation in Equations ( 12) and ( 14), respectively, are written in the form where The partition function is defined as Substituting Equation (23) for Equation (26), the partition function is given as Using Maple version 10.0, the partition function is evaluated as Other thermodynamic properties can be obtained using the partition function.
The vibrational mean energy is given as The vibrational heat capacity is given as The vibrational entropy is given as The vibrational free energy is given as

Discussion
In Table 1, the energy eigenvalues for the two different values of the angular momentum were presented for various quantum states.A linear variation is observed between the quantum state and the energy eigenvalues for the two categories of the potentials.The same trend was also observed between the energy eigenvalue and the angular momentum.The energy eigenvalues of the IHOIQ potential for the ground state and the first excited state with two values of the strength of the IQ potential for the various masses, are presented in Table 2.A linear variation between the mass of the spring and the energy eigenvalues is observed.It has been shown, from the Table 2, that the result of the ground state of the IHO potential with D 0 = 3, is the same as the result of the first excited state of the IHO for the different masses of the spring.The momentum and position expectation values, the uncertainty, and the product of the expectations, as well as the product of the uncertainty of the IHOIQ potential, are presented in Tables 3 and 4. The frequency varies directly with the momentum expectation value, as well as with the uncertainty in the momentum, but inversely with the position expectation value and the uncertainty in the position.The same variation is observed against the mass of the spring in Table 4.The results from Tables 3 and 4 showed that the product of the expectation of the momentum and the position at any frequency or mass is a constant.This was also noticed in the product of the uncertainty.The results of the uncertainty showed that the more precisely a system is measured, the less precisely the other can be measured.This aligned with the Heisenberg uncertainty product given as It is noted that the expectation values in Equations ( 18) and ( 20) have a relation of the form In Tables 5 and 6, the momentum and the position expectation values, the uncertainty and the product of the expectations, as well as the product of the uncertainty of the PHIQ potential, are presented for the various dissociation energy and equilibrium lengths, respectively.The results obtained in Tables 3 and 4 are equally obtained in Table 5.However, the variation of the equilibrium bond length against the expectation values, the uncertainty and their product, are the inverse of what is obtained in Tables 3-5.In all cases, the uncertainty product obeys the standard Heisenberg uncertainty relation.Figures 1-10 showed the thermodynamic properties of the PHIQ and IHOIQ potentials for the various upper bound and temperature parameters.The figures in (a) are the thermal properties for the IHOIQ potential while the figures in (b) are the thermal properties for the IHOIQ potential.The variation of Z(β) against β for the different λ is presented in Figure 1.Z(β) rises as β increases and tends to converge at a higher β.It shows that as the temperature of the system becomes lower, Z(β) for the various λ, compact together.In Figure 2, the variation of U(β) against β for different λ, is shown.U(β) varies inversely with the temperature of the system.At every given temperature, the vibrational U(β) of the IHOIQ potential are higher than the vibrational U(β) of the PHIQ potential.The mean energy tends to converge as the temperature becomes lower.The convergence is more obvious in the IHOIQ potential than in the PHIQ potential at higher temperatures.C(β) and β vary directly with one another, as shown in Figure 3. C(β) at the different λ, converged at the origin for the two potential systems, and diverged as β increased.The divergence becomes greater for the PHIQ potential as β remains on the increase, but for the IHOIQ potential, C(β) converged.Figure 4 presents the vibrational S(β) against β for the various λ.The same behaviour was observed in Figure 1 and is also observed here.Thus, the effect of β on S(β) and Z(β) are the same.Figure 5 shows the variation of the vibrational F(β) against β for the different λ.The vibrational F(β) decreases monotonically as β increases.As the temperature of the system lowers, F(β) becomes smaller and tends to converge.The vibrational Z(λ) against λ for the various β, is shown in Figure 6.As λ increases, Z(λ) also increases.Z(λ) of the two potentials for the different β, are the same.Figure 7 shows the vibrational U(λ) against λ for the various β.U(λ) rises and tends to converge as λ increases.A slight difference is observed in the two potential systems.Figure 8 presents the variation of the vibrational C(λ) against λ for the different β.The effect seen in Figure 7 is also obtained in Figure 8.The variation of the vibrational S(λ) against λ for the different β, is shown in Figure 9.The effect observed in Figure 6 is also observed in Figure 9.The variation of the vibrational F(λ) against λ for the different β, is shown in Figure 10.The vibrational F(λ) decreases exponentially while λ increases linearly.The vibrational F(λ) for the highest β in the IHOIQ potential corresponds to the vibrational F(λ) for the lowest β in the PHIQ potential.
Figure 7 shows the vibrational U(λ) against λ for the various ꞵ.U(λ) rises and tends to converge as λ increases.A slight difference is observed in the two potential systems.Figure 8 presents the variation of the vibrational C(λ) against λ for the different ꞵ.The effect seen in Figure 7 is also obtained in Figure 8.The variation of the vibrational S(λ) against λ for the different ꞵ, is shown in Figure 9.The effect observed in Figure 6 is also observed in Figure 9.The variation of the vibrational F(λ) against λ for the different ꞵ, is shown in Figure 10.The vibrational F(λ) decreases exponentially while λ increases linearly.The vibrational F(λ) for the highest ꞵ in the IHOIQ potential corresponds to the vibrational F(λ) for the lowest ꞵ in the PHIQ potential.

Conclusions
The result of the Schrödinger equation for the IHOIQ (isotropic harmonic oscillator with inversely quadratic) potential and the PHIQ (pseudoharmonic with inversely quadratic) potential has been obtained via a simplified traditional method.It has been shown that the result of the IHOIQ (isotropic harmonic oscillator with inversely quadratic) at the ground state, is the same as the result of the IHO for the first excited state.It was also shown that the spectra obtained for the IHOIQ (isotropic harmonic oscillator with in-

Conclusions
The result of the Schrödinger equation for the IHOIQ (isotropic harmonic oscillator with inversely quadratic) potential and the PHIQ (pseudoharmonic with inversely quadratic) potential has been obtained via a simplified traditional method.It has been shown that the result of the IHOIQ (isotropic harmonic oscillator with inversely quadratic) at the ground state, is the same as the result of the IHO for the first excited state.It was also shown that the spectra obtained for the IHOIQ (isotropic harmonic oscillator with inversely quadratic) are lesser than the spectra obtained for the PHIQ (pseudoharmonic with inversely quadratic).The product of the uncertainty obtained for the two potential models obeyed the Heisenberg uncertainty inequality.Finally, the thermal properties for the two potential models showed similar features.

Figure 1 .
Figure 1.(a,b) Vibrational partition function against the temperature parameter for the IHOIQ potential and the PHIQ potential, respectively.The vibrational partition function increases as the temperature parameter increases for the two interacting potentials.The IHOIQ converges more as the temperature parameter increases.It shows that as the temperature of the system becomes lower, Z(ꞵ) for the various λ, compact together.

Figure 1 .Figure 2 .
Figure 1.(a,b) Vibrational partition function against the temperature parameter for the IHOIQ potential and the PHIQ potential, respectively.The vibrational partition function increases as the temperature parameter increases for the two interacting potentials.The IHOIQ converges more as the temperature parameter increases.It shows that as the temperature of the system becomes lower, Z(β) for the various λ, compact together.Quantum Rep. 2023, 5, FOR PEER REVIEW 1

Figure 2 .
Figure 2. (a,b) Vibrational mean energy against the temperature parameter for the IHOIQ potential and the PHIQ potential, respectively.The vibrational mean energy varies directly with the temperature parameter for both IHOIQ and PHIQ potentials with IHOIQ having the more convergence at higher values of the temperature parameter.At every given temperature, the vibrational U(β) of the IHOIQ potential are higher than the vibrational U(β) of the PHIQ potential.

Figure 2 .Figure 3 .
Figure 2. (a,b)Vibrational mean energy against the temperature parameter for the IHOIQ potential and the PHIQ potential, respectively.The vibrational mean energy varies directly with the temperature parameter for both IHOIQ and PHIQ potentials with IHOIQ having the more convergence at higher values of the temperature parameter.At every given temperature, the vibrational U(ꞵ) of the IHOIQ potential are higher than the vibrational U(ꞵ) of the PHIQ potential.

Figure 3 .
Figure 3. (a,b) Vibrational heat capacity against the temperature parameter for the PH with the IQ potentials and the IHO with the IQ potentials, respectively.The vibrational heat capacity rises as the temperature parameter increases.The heat capacity for PHIQ diverges while that of IHOIQ converges as the temperature parameter rises.The divergence becomes greater for the PHIQ potential as β remains on the increase, but for the IHOIQ potential, C(β) converged.Quantum Rep. 2023, 5, FOR PEER REVIEW 11

Figure 4 .
Figure 4. (a,b) Vibrational entropy against the temperature parameter for the IHOIQ potential and the PHIQ potential, respectively.The vibrational entropy varies directly with the temperature parameter.The entropy at various upper bound converges as the temperature parameter rises.Thus, the effect of ꞵ on S(ꞵ) and Z(ꞵ) are the same.

Figure 4 .
Figure 4. (a,b) Vibrational entropy against the temperature parameter for the IHOIQ potential and the PHIQ potential, respectively.The vibrational entropy varies directly with the temperature parameter.The entropy at various upper bound converges as the temperature parameter rises.Thus, the effect of β on S(β) and Z(β) are the same.

Figure 4 .Figure 5 .
Figure 4. (a,b) Vibrational entropy against the temperature parameter for the IHOIQ potential and the PHIQ potential, respectively.The vibrational entropy varies directly with the temperature parameter.The entropy at various upper bound converges as the temperature parameter rises.Thus, the effect of ꞵ on S(ꞵ) and Z(ꞵ) are the same.

Figure 5 .Figure 6 .
Figure 5. (a,b) Vibrational free energy against the temperature parameter for the IHOIQ potential and the PHIQ potential, respectively.The vibrational free energy decreases monotonically while the temperature parameter increases linearly.Quantum Rep. 2023, 5, FOR PEER REVIEW 12

Figure 6 .
Figure 6.(a,b) Vibrational partition function against the upper bound for the IHOIQ potential and the PHIQ potential, respectively.The partition function for IHOIQ and PHIQ have the same behviour for various temperature parameter.

Figure 6 .Figure 7 .
Figure 6.(a,b) Vibrational partition function against the upper bound for the IHOIQ potential and the PHIQ potential, respectively.The partition function for IHOIQ and PHIQ have the same behviour for various temperature parameter.

Figure 7 .Figure 8 .
Figure 7. (a,b) Vibrational mean energy against the upper bound for the IHOIQ potential and the PHIQ potential, respectively.The mean energy varies directly with the upper bound for different values of the temperature parameter.A slight difference is observed in the two potential systems.Quantum Rep. 2023, 5, FOR PEER REVIEW 13

Figure 8 .
Figure 8. (a,b) Vibrational heat capacity against the upper bound for the IHOIQ potential and the PHIQ potential, respectively.The heat capacity increases with the upper bound for both potentials.Figure 8. (a,b) Vibrational heat capacity against the upper bound for the IHOIQ potential and the PHIQ potential, respectively.The heat capacity increases with the upper bound for both potentials.

Figure 8 .Figure 9 .
Figure 8. (a,b) Vibrational heat capacity against the upper bound for the IHOIQ potential and the PHIQ potential, respectively.The heat capacity increases with the upper bound for both potentials.

Figure 9 .
Figure 9. (a,b) Vibrational entropy against the upper bound for the PH with the IQ potentials and the IHO with the IQ potentials, respectively.The entropy varies directly with the upper bound but converges at higher values of the upper bound.Quantum Rep. 2023, 5, FOR PEER REVIEW 14

Figure 10 .
Figure 10.(a,b) Vibrational free energy against the upper bound for the PH with the IQ potentials and the IHO with the IQ potentials, respectively.The vibrational free energy decreases monotonically as the upper bound increases gradually.The vibrational F(λ) for the highest ꞵ in the IHOIQ potential corresponds to the vibrational F(λ) for the lowest ꞵ in the PHIQ potential.

Figure 10 .
Figure 10.(a,b) Vibrational free energy against the upper bound for the PH with the IQ potentials and the IHO with the IQ potentials, respectively.The vibrational free energy decreases monotonically as the upper bound increases gradually.The vibrational F(λ) for the highest β in the IHOIQ potential corresponds to the vibrational F(λ) for the lowest β in the PHIQ potential.

Table 1 .
Energy eigenvalues of the IHOIQ potential and the PHIQ potential with µ = = m = f = 1 and r e = 0.1.

Table 2 .
Energy eigenvalues of the IHOIQ potential with µ

Table 3 .
Expectation values and uncertainty in the position and the momentum of the IHOIQ potential with µ = = = m = 1 and D 0 = 5 for the various frequencies.

Table 4 .
Expectation values and the uncertainty in the position and momentum of the IHOIQ potential with µ = = = f = 1 and D 0 = 5 for the various masses.

Table 5 .
Expectation values and the uncertainty in the position and momentum at the ground state of the PHIQ potential with µ = = = f = 1 and D 0 = 5 for the various masses.

Table 6 .
Expectation values and the uncertainty in the position and momentum at the ground state of the PHIQ potential with µ = = = f = 1, r e = 0.1 and D e = D 0 = 5 for the various masses.