Fisher Information for a System Composed of a Combination of Similar Potential Models

: The solutions to the radial Schrödinger equation for a pseudoharmonic potential and Kratzer potential have been studied separately in the past. Despite different reports on the Kratzer potential, the fundamental theoretical quantities such as Fisher information have not been reported. In this study, we obtain the solution to the radial Schrödinger equation for the combination of the pseudoharmonic and Kratzer potentials in the presence of a constant-dependent potential, utilizing the concepts and formalism of the supersymmetric and shape invariance approach. The position expectation value and momentum expectation value are calculated employing the Hellmann– Feynman Theory. These expectation values are then used to calculate the Fisher information for both position and momentum spaces in both the absence and presence of the constant-dependent potential. The results obtained revealed that the presence of the constant-dependent potential leads to an increase in the energy eigenvalue, as well as in the position and momentum expectation values. Additionally, the constant-dependent potential increases the Fisher information for both position and momentum spaces. Furthermore, the product of the position expectation value and the momentum expectation value, along with the product of the Fisher information, satisfies both Fisher’s inequality and Cramer–Rao’s inequality.


Introduction
The utility of potential systems in quantum mechanics spans a wide spectrum, encompassing solutions for both bound states and scattering states, as well as various theoretical quantities such as Fisher information, Shannon entropy, variance, Tsallis entropy, Rényi entropy, and information energy.Diverse physical potential models have been employed to derive energy spectra for different systems, resulting in varying outcomes.For instance, when considering the sodium dimer, the energy predictions from the Yukawa potential differs from those generated through the Woods-Saxon potential.Such discrepancies arise due to the distinct physical natures and parameterization of these potential models.In the realm of quantum systems, atomic interactions are often modeled using potential energy functions, which greatly influence the properties exhibited by different systems.These potential functions, particularly those falling within the class of exponential-type potential models, effectively describe diatomic molecules and provide numerical values that align Quantum Rep. 2024, 6 185 with observed data.Many of these energy potential functions have been adapted for suitability by Jia et al. [1] and are commonly explored through appropriate approximation schemes.This exploration has piqued the interest of numerous authors and researchers, particularly in nonrelativistic quantum mechanics, where the ro-vibrational spectra of diverse quantum systems have been extensively studied [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17].For instance, Njoku et al. [18] studied the Hua potential model under the nonrelativistic system.These authors obtained the bound state solutions of the Schrödinger equation for the Hua potential model.The authors numerically examined the effect of the equilibrium bond length on the energy eigenvalue for different quantum states.They further extended their study to the calculation of thermodynamic properties, investigating the effects of the maximum quantum state and the temperature parameter on the partition function and other thermodynamic properties such as mean energy, specific heat capacity, entropy, and free mean energy.Yas , uk et al. [19] studied the Schrödinger equation for a noncentral potential system, employing a combination of radial and angular potential models.Using the method of the separation of variables, these authors calculated the energy equation and the wave function of the system using the Nikiforov-Uvarov method.They obtained special cases of the noncentral potential, such as the Coulomb and Hartmann ring-shaped potentials, and compared the results with those in the existing literature.In ref. [20], Okon et al. studied the solutions of the Schrödinger equation for a combination of the Hulthẻn and an exponential Coulomb-like potential model using the parametric Nikiforov-Uvarov method.They examined the behavior of the energy of the combined potential as well as the subset potential for various quantum states and angular momentum quantum states.The energy of the Hulthẻn potential rises as the quantum state increases.Similarly, the energy rises as the screening parameter increases for the same Hulthẻn potential.Yahya and Issa [21] obtained the solutions of the Schrödinger equation for improved Tietz potential and improved the Rosen-Morse potential model using the methodology of the parametric Nikiforov-Uvarov method.They studied the energy eigenvalue and the wave functions of the two potentials separately.These authors examined the relationship between the energy and the deformed parameter.They also examined the effect of the angular momentum number on the energy of the improved Rosen-Morse potential model at different quantum numbers.
However, there exists another category of potential models tailored toward investigating bound state problems and related quantum systems that are not in the exponential form or P 4, 6,2 numerical values that align with observed data.Many of these energy potential functions have been adapted for suitability by Jia et al. [1] and are commonly explored through appropriate approximation schemes.This exploration has piqued the interest of numerous authors and researchers, particularly in nonrelativistic quantum mechanics, where the ro-vibrational spectra of diverse quantum systems have been extensively studied [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17].For instance, Njoku et al. [18] studied the Hua potential model under the nonrelativistic system.These authors obtained the bound state solutions of the Schrödinger equation for the Hua potential model.The authors numerically examined the effect of the equilibrium bond length on the energy eigenvalue for different quantum states.They further extended their study to the calculation of thermodynamic properties, investigating the effects of the maximum quantum state and the temperature parameter on the partition function and other thermodynamic properties such as mean energy, specific heat capacity, entropy, and free mean energy.Yașuk et al. [19] studied the Schrödinger equation for a noncentral potential system, employing a combination of radial and angular potential models.Using the method of the separation of variables, these authors calculated the energy equation and the wave function of the system using the Nikiforov-Uvarov method.They obtained special cases of the noncentral potential, such as the Coulomb and Hartmann ring-shaped potentials, and compared the results with those in the existing literature.In ref. [20], Okon et al. studied the solutions of the Schrödinger equation for a combination of the Hulthẻn and an exponential Coulomb-like potential model using the parametric Nikiforov-Uvarov method.They examined the behavior of the energy of the combined potential as well as the subset potential for various quantum states and angular momentum quantum states.The energy of the Hulthẻn potential rises as the quantum state increases.Similarly, the energy rises as the screening parameter increases for the same Hulthẻn potential.Yahya and Issa [21] obtained the solutions of the Schrödinger equation for improved Tietz potential and improved the Rosen-Morse potential model using the methodology of the parametric Nikiforov-Uvarov method.They studied the energy eigenvalue and the wave functions of the two potentials separately.These authors examined the relationship between the energy and the deformed parameter.They also examined the effect of the angular momentum number on the energy of the improved Rosen-Morse potential model at different quantum numbers.However, there exists another category of potential models tailored toward investigating bound state problems and related quantum systems that are not in the exponential form or ὂ schl-Teller form.Notably, among these models are the pseudoharmonic potential and the Kratzer potential.The pseudoharmonic potential integrates elements of the harmonic potential model, the inverse potential system, and a constant term, and is thereby capable of reproducing solutions from these three distinct potential models.Unlike many other models, the physical nature of the pseudoharmonic potential discourages the application of approximation schemes.Proposed by Davidson [22], the pseudoharmonic potential was specifically devised to elucidate the rotovibrational states of diatomic molecules.On the other hand, the Kratzer potential serves as a molecular potential used to elucidate molecular structures and atomic interactions.Its widespread application has primarily centered around bound states within the molecular domain.Because of the applications of these potentials, several authors have reported the potentials separately in the field of Physics.Among the reports on the pseudoharmonic potential and the Kratzer potential are the work of Oyewumi and Sen [23], who studied the pure pseudoharmonic potential in the context of bound states and applied their study to diatomic molecules.In ref. [24], Oyewumi et al. studied a Ddimensional system of pseudoharmonic potential and constructed ladder operators.Sever et al. [25] obtained energy spectra for the pseudo-harmonic potential and generated numerical values for some molecules with the first six quantum states.Ikhdair and Sever [26], in their study, calculated the exact polynomial eigensolutions of the Schrödinger equation for the pseudoharmonic potential.In the study, they obtained exact bound-state schl-Teller form.Notably, among these models are the pseudoharmonic potential and the Kratzer potential.The pseudoharmonic potential integrates elements of the harmonic potential model, the inverse potential system, and a constant term, and is thereby capable of reproducing solutions from these three distinct potential models.Unlike many other models, the physical nature of the pseudoharmonic potential discourages the application of approximation schemes.Proposed by Davidson [22], the pseudoharmonic potential was specifically devised to elucidate the roto-vibrational states of diatomic molecules.On the other hand, the Kratzer potential serves as a molecular potential used to elucidate molecular structures and atomic interactions.Its widespread application has primarily centered around bound states within the molecular domain.Because of the applications of these potentials, several authors have reported the potentials separately in the field of Physics.Among the reports on the pseudoharmonic potential and the Kratzer potential are the work of Oyewumi and Sen [23], who studied the pure pseudoharmonic potential in the context of bound states and applied their study to diatomic molecules.In ref. [24], Oyewumi et al. studied a D-dimensional system of pseudoharmonic potential and constructed ladder operators.Sever et al. [25] obtained energy spectra for the pseudo-harmonic potential and generated numerical values for some molecules with the first six quantum states.Ikhdair and Sever [26], in their study, calculated the exact polynomial eigensolutions of the Schrödinger equation for the pseudoharmonic potential.In the study, they obtained exact bound-state energy eigenvalues and the corresponding eigenfunctions analytically.The energy states for several diatomic molecular systems were calculated numerically for various principal and angular quantum numbers.Through a proper transformation, the problem was also solved and made very simple using the known eigensolutions of the anharmonic oscillator potential.Das and Arda [27], in one of the articles studied, deduced the exact analytical solution of the N-dimensional radial Schrödinger equation with pseudoharmonic potential using a Laplace transform approach.The authors examined the variation in the spatial dimensions against the energy of the system.Recently, the exact solutions of the κ-dependent Schrödinger equation with quantum pseudo-harmonic oscillator and its applications to the thermodynamic properties in normal and superstatistics were obtained by Okorie et al. [28].The pseudoharmonic potential is generally used to describe the roto-vibrational states of diatomic molecules, nuclear rotations, and vibrations.The Kratzer potential also received different reports on the nonrelativistic models.Some of the reports on the Kratzer potential include the work of Bayrak et al. [29], who studied the radial Schrödinger equation with the Kratzer potential and obtained the exact analytical solutions to the Kratzer potential using the asymptotic iteration method.They generated numerical values for some molecules using their spectroscopic constants.Despite the extensive reporting on the Kratzer potential, its utilization concerning Fisher information remains underexplored to the best of our understanding.Motivated by the interest in Fisher information for nonexponential-type potentials, the current study aimed to scrutinize the applicability of Fisher information in a scenario involving a combination of the pseudoharmonic potential and the Kratzer potential, alongside a constant-dependent potential.The effects of the constant dependent potential and the potential parameters on the Fisher information will be examined numerically to verify the Cramer-Rao inequality and Fisher inequality.The Fisher information for the combination of these potentials will be obtained using expectation values.The specific form of the potential model under investigation is physically represented as while the Kratzer potential is given as [30] where D e is a dissociation energy, r is the internuclear distance, and r e is an equilibrium bond separation.These two potentials take the form of molecular potential due to their possession of parameters with physical meanings.The combination of Equations ( 1) and ( 2) with a constant-dependent potential is given as The parameter (η) is known as the constant-dependent potential.
The nonrelativistic solutions for Equation (3) will be obtained using a supersymmetric approach.The pseudoharmonic potential and the Kratzer potential have been reported by different authors for some models.

Bound State Solutions
To obtain the solutions of the redial potential system in Equation (3), the radial Schrödinger equation with nonrelativistic energy E n,↕ , reduced Planck constant ℏ, the reduced mass of the molecule µ, and the radial wave function R n,↕ (r) is given as Substituting the radial potential system in Equation (3) into Equation ( 4), we obtain a second-order differential equation of the form where λ = 2µ ℏ 2 .The supersymmetric approach suggests the ground state wave function for the next step.Thus, the ground state wave function is written as where W(r) is referred to as a superpotential function in supersymmetry quantum mechanics.The ground state wave function corresponds to the two partner Hamiltonians given as where In substituting the ground state wave function into Equation ( 5), the resulting solution appears as a differential equation of the form where the superpotential W(r) is proposed as follows: The two terms θ 0 and θ 1 in Equation (10) represent constants in the proposed superpotential, which will soon be determined.The superpotential is proposed based on the interacting potential and the nature of the Riccati equation in Equation ( 9).This will ensure that the property of the left-hand side and the right-hand side of Equation ( 9) are the same.In substituting Equation (10) into Equation ( 9) and conducting subsequent mathematical manipulations and simplifications, the values of the superpotential constants in Equation (10) are obtained as follows: With the aid of Equations ( 7), (8), and (10), the partner potentials in supersymmetry quantum mechanics can be constructed as Equations ( 14) and ( 15) are family potentials and they satisfy the shape invariance condition through the mapping of the form where a 1 is a new set of parameters uniquely determined from the old set a 0 , and R(a 1 ) is a residual term that is independent of the variable r.Since a 1 = a 0 + 1 and by recurrence relation, a n = a 0 + n.From the shape invariance approach, we can write [31- The energy level can be obtained using From Equation (20), we obtain the complete energy eigenvalue equation as

Calculation of Expectation Values
Here, we calculate the position and momentum expectation values using the Hellmann-Feynman Theorem.The Hellmann-Feynman Theorem relates the derivative of the total energy with respect to a parameter to the expectation value of the derivative of the Hamiltonian with respect to the same parameter [34][35][36][37].If the spatial distribution of the electrons is determined by the solution of the Schrödinger equation, then the forces in the system can be calculated using classical electrodynamics.If the Hamiltonian H for a particular system is a function of some parameter v with the eigenvalue and eigenfunctions denoted by E n,↕ (v) and R n,↕ (v), respectively, then we can find the various expectation values provided that the associated normalized eigenfunction R n,↕ (v) is continuous with respect to the parameter v.Then, and The various expectation values can now be calculated.(I): Expectation value r 2 .To obtain the expectation value of r 2 , we set v = D e , and then, (II): Expectation value p 2 .Setting v = µ, we obtain the following expectation value:

Fisher Information
Fisher information is a concept in information theory that measures the amount of information that is available in a given system.In a quantum system, Fisher information is a probabilistic measure of uncertainty in a system.According to Gibilisco and Isola [38], a family of inequalities, related to the uncertainty principle, has been recently proved [39].It is known that Heisenberg and Schrödinger uncertainty principles give lower bounds for the product of variances.Gibilisco et al. [40] proved an uncertainty principle in Schrödinger form where the bound for the product of variances depended on the area spanned by the commutators and with respect to an arbitrary quantum version of the Fisher information.According to refs.[41,42], for quantum mechanical systems, the Fisher information-based uncertainty relation has been used as an alternative to the Heisenberg uncertainty relation (HUR).Therefore, for a system with physical parameters θ and ρ(r, θ) [41,42], where ρ(r, θ) accounts for the details of the probability density.The examination of Fisher information occurs in two distinct parts, namely, Fisher information in the position space and Fisher information in the momentum space.The expressions for Fisher information in both the position and momentum spaces can be written as follows: where ρ(r) is the probability density for the position space, and ρ(r) is the probability density for the momentum space.The implication of Equation ( 28) is that a concentrated density yields a higher quantity, offering a local change in density where the system can be more effectively described from an information-theoretic perspective.In terms of expectation values, the Fisher information for both the position space and the momentum space can thus be expressed as follows [43][44][45][46]] where L is the total angular momentum, and m is the magnetic quantum number.In the absence of the magnetic quantum number m, the terms after the minus sign of Equation ( 29) vanishes, and then, the Fisher information for the position space and the Fisher information for the momentum space in Equation ( 29) can be written in the following form: To check the validity of the product of expectation values as a Cramer-Rao inequality, Dehesa et al. [44], gave the following conditions: These imply that When m = 0, the product of Equation ( 32) can be written as The minimum bound for the Fisher product [43] is given by In using Equations ( 21) and (34), the minimum bound of the product of the position and momentum expectation becomes In substituting Equations ( 25) and ( 26) into Equation (30), the Fisher information for the position space and the momentum space, respectively, becomes

Discussion
Table 1 presents the energy eigenvalues of the combination of the pseudoharmonic potential and the Kratzer potential for various quantum numbers and angular momentums in the absence and presence of the constant-dependent potential for two different values of the dissociation energy.The energy of the combined potentials responds positively to an increase in each of the dissociation energy, quantum number, angular momentum, and constant-dependent potential.This implies that an increase in the dissociation energy, quantum number, angular momentum, or constant-dependent potential increases the energy of the interacting potential system.From Table 1, it is noted that as the angular momentum increases, the energy difference between successive angular momenta increases.In Table 2, the energy of the combination of the pseudoharmonic potential and the Kratzer potential for various quantum numbers and equilibrium bond lengths in the absence and presence of the constant-dependent potential is presented.The numerical values in Table 2 show that the equilibrium bond length varies inversely with the energy of the interacting potential.However, the quantum number and the angular momentum respectively vary directly with the energy of the system.In Table 3, we present the energy eigenvalue of the interacting potential for various dissociation energies and constant-dependent potentials at the ground state and the first excited state.The energy of the combined potential model increases as the dissociation energy and the constant-dependent potential respectively increase.The energy of the system also increases as the quantum number and the angular momentum respectively increase.The numerical values for the Fisher information in position space and momentum space, as well as their product in the absence and presence of the constant-dependent potential, are presented in Tables 4 and 5.In Table 4, the Fisher information in position space and the quantum number vary inversely with each other, but the Fisher information in momentum space varies directly with the quantum number.The Fisher information in both position space and momentum space in the presence of the constant-dependent potential is higher than its counterpart produced in the absence of the constant-dependent potential.The minimum bound for the Fisher product in the absence of constant-dependent potential is 111.4298232, while in the presence of the constant-dependent potential, the minimum bound is 339.29054.These values are above the standard value of 36.Thus, the results in Table 4 satisfy the Fisher inequality given in Equation (34).The variation of the Fisher information shows that a decrease in the Fisher information in position space corresponds to an increase in Fisher information in momentum space.This implies that a diffused density distribution in the configuration space is associated with a localized density distribution in the momentum space.In Table 5, the Fisher information in position space and in momentum space for various values of the equilibrium bond separation is presented in the absence and presence of the constant-dependent potential.In this Table, the Fisher information in position space varies directly with the equilibrium bond separation both in the presence and absence of the constant-dependent potential.However, in the momentum space, the Fisher information varies indirectly with the equilibrium bond separation.The minimum bound for the Fisher product in the absence of constant-dependent potential is 51.4913757, while in the presence of the constant-dependent potential, it is 212.949992.The values of the minimum bound also confirm that the results in Table 5 satisfy the Fisher inequality.The result shows that a diffused density in the configuration space corresponds to the localized density in the momentum space.The variation in the Fisher information against the quantum state in Table 4 is contrary to the variation in Fisher information against the equilibrium bond separation in Table 5 but satisfies the Fisher inequality in both cases.In Table 5, a strongly localized distribution in the momentum space corresponds to a widely delocalized distribution in the position space.The results obtained obey Heisenberg's uncertainty principle.The position and momentum expectation values and their product for various values of the angular momentum number are presented in Table 6 for the absence and presence of the constant-dependent potential.It is noted that as the position space expectation value varies directly with the angular momentum, the momentum space expectation value varies inversely with the angular momentum number.As observed in Tables 4-6, the results with the constant-dependent potential are higher than the results without constant-dependent potential.The product of the expectation values for the presence and absence of the constant-dependent potential is greater than the standard value of 2.25.The minimum bounds for the products of the expectation values in the absence and presence of the constant-dependent potential are 3.9027949 and 12.1497115, respectively.These results satisfy Cramer-Rao's inequality for the product of expectations.Table 7 presents Fisher information for both position space and momentum space and their product in the absence and presence of the constant-dependent potential for various angular momenta.The Fisher information for the position space in both the absence and presence of constant-dependent potential respectively increases as the angular momentum increases.However, the Fisher information for the momentum space in the absence and presence of constant-dependent potential respectively decreases as the angular momentum increases.The product of the Fisher information satisfies the Cramer-Rao inequality but decreases with an increase in the angular momentum.The variation in energy against the quantum number is shown in Figure 1.The energy of the combined pseudoharmonic potential and Kratzer potential varies linearly with the quantum number.Figure 2 shows the variation in the Fisher information against the dissociation energy for various quantum states.The Fisher information for the momentum space decreases for all the quantum states studied as the dissociation energy increases.At D e = 5, Fisher information for the momentum space at different quantum states intercept each other, as shown in Figure 2a.In Figure 2b, the Fisher information for the position space decreases monotonically as the dissociation energy increases.The variation in the Fisher information for the momentum space decreases monotonically as the angular momentum increases, as shown in Figure 3a.At a higher angular momentum, the Fisher information for the momentum space for various quantum states tends to be equal.In Figure 3b, the Fisher information for the position space increases as the angular momentum increases.

Conclusions
The solutions for the combination of the pseudoharmonic potential and Kratzer potential were obtained, and the effects of the constant-dependent potential and the potential parameters on the energy, expectation values, and Fisher information were observed.It was demonstrated that, in the presence of constant-dependent potential, the dissociation energy, quantum number, and angular momentum respectively increase the energy eigenvalue of the system, while the equilibrium bond length reduces the energy.The variation in the quantum number against the Fisher information was found to be contrary to the variation in the equilibrium bond length against the Fisher information.The constant-dependent potential increases the position expectation value but decreases the momentum expectation value.It was discovered that certain parameters increase the Fisher information in position space and decrease Fisher information in momentum space, while some parameters decrease the Fisher information in position space and increase the Fisher information in momentum space.The effect of the potential parameters on the energy and Fisher information in the absence of the constant-dependent potential is the same as in the presence of the constant-dependent potential.The presence of the constant-dependent potential only increases the quantities studied.

Table 4 .
Fisher information for position space and momentum space in the absence and presence of constant-dependent potential for various quantum states and the Fisher product with µ = ℏ = ↕ = 1, r e = 0.5, and D e = 2.5.

Figure 1 . 1 , 5 eD
Figure 1.Variation in energy against the quantum number with

Figure 1 .
Figure 1.Variation in energy against the quantum number with µ = ℏ = 1, r e = 0.25, and D e = 5 for three values of the angular momentum number.

Figure 2 .
Figure 2. Fisher information in momentum ( ) I  (a) and Fisher information in position ( ) I  (b) against the dissociation energy with      and 0.5 e r 

Figure 2 .
Figure 2. Fisher information in momentum I(γ) (a) and Fisher information in position I(ρ) (b) against the dissociation energy with µ = ↕ = ℏ and r e = 0.5.

Figure 2 .Figure 3 .
Figure 2. Fisher information in momentum ( ) I  (a) and Fisher information in position ( ) I  (b) against the dissociation energy with      and 0.5 e r 

Table 1 .
Energy eigenvalue for various quantum states and angular momentum with µ = ℏ = 1 and r e = 0.25 in the absence and presence of constant-dependent potential for two values of the dissociation energy.

Table 2 .
Energy eigenvalue for various quantum states and equilibrium bond length with µ = ℏ = 1 and D e = 5 in the absence and presence of constant-dependent potential for s− wave and ↕− wave.

Table 3 .
Energy eigenvalue for various values dissociation energies and constant-dependent potential with µ = ℏ = 1 and r e = 0.25 at the ground state and first excited state.

Table 5 .
5Fisher information for position space and momentum space at the ground state for various equilibrium bond separations and the Fisher product in the absence and presence of constantdependent potential with µ = ℏ = ↕ = 1 and D e = 2.5.

Table 6 .
Position expectation value and momentum expectation value at the ground state for various angular momentum quantum states and their product in the absence and presence of constantdependent potential with µ = ℏ = 1, r e = 0.5, and D e = 2.5.

Table 7 .
Fisher information for both position space and momentum space at the ground state for various angular momentum quantum states and their product with µ = ℏ = 1, r e = 0.5, and D e = 2.5 in the absence and presence of the constant-dependent potential.