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

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

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18 January 2024

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Abstract

The solutions of the radial Schrödinger equation for a pseudoharmonic potential and Kratzer potentials has been studied separately in the past. The Kratzer potential despite different reports, the basic theoretic quantity such as Fisher information has not been reported. In this study, the solution of the radial Schrödinger equation for the combination of the pseudoharmonic and Kratzer potentials in the presence of a constant-dependent potential is obtained using 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. The expectation values are used to calculate the Fisher information for both the position and momentum spaces in both the absence and presence of the constant-dependent potential. The obtained results 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. The presence of the constant-dependent potential also 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 the Fisher’s inequality and the Cramer-Rao’s inequality.

Keywords:

Subject: Physical Sciences - Mathematical Physics

1. 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, Tsalis entropy, Rẻnyi 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 by 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 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 the thermodynamic properties. The effects of the maximum quantum state as well as the effect of the temperature parameter on the partition function and other thermodynamic properties such as the mean energy, the specific heat capacity, the entropy and the free mean energy were fully obtained. Yașuk et al. [19] studied Schrödinger equation for a non-central potential system. The applied non-central potential model is a combination of both the radial potential and angular potential models. Using the method of separation of variables, these authors calculated the energy equation and the wave function of the system via the Nikiforov-Uvarov method. They obtained the special cases of the non-central potential as the Coulomb and Hartmann ring-shaped potentials and compared the results with those of 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 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 the angular momentum quantum state. 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 Rosen-Morse potential model using the methodology of 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 to investigate bound state problems and related quantum systems that are not in the exponential form or Pὂschl-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, 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 to elucidate molecular structure 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 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 D-dimensional system of pseudoharmonic potential and construct the Ladder operators. Sever et al. [25], obtained energy spectra for the pseudoharmonic potential and generate 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 pseudohamonic potential. In the study, they, exact bound-state energy eigenvalues and the corresponding eigenfunctions are analytically calculated. The energy states for several diatomic molecular systems are calculated numerically for various principal and angular quantum numbers. By a proper transformation, the problem was also solved and make very simple using the known eigensolutions of 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 via Laplace transform approach. The authors examined the variation of the spatial dimensions against the energy of the system. Recently, the exact solutions of κ-dependent Schrödinger equation with quantum pseudoharmonic oscillator and its applications to the thermodynamic properties in normal and superstatistics was 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 in the non-relativistic models. Some of the reports on the Kratzer potential includes 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 by 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 non-exponential-type potentials, the current study aims 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 Fisher information for the combination of these potential, will be obtained using expectation values. The specific form of the potential model under investigation is physically represented as [22].

V_{1} (r) = D_{e} {(\frac{r}{r_{e}} - \frac{r_{e}}{r})}^{2},

(1)

while the Kratzer potential is given as [30]

V_{2} (r) = - D_{e} (\frac{2 r_{e}}{r} - \frac{r_{e}^{2}}{r^{2}}),

(2)

where

D_{e}

is a dissociation energy and

r_{e}

is an equilibrium bond separation. These two potentials fall to the form of molecular potential because they possessed the parameters with physical meanings. The combination of equation (1) and equation (2) with a constant dependent potential is given as

V (r) = [V_{1} (r) + V_{2} (r)] (1 + η) .

(3)

The nonrelativistic solutions for equation (3) will be obtained using supersymmetric approach. The pseudoharmonic potential and the Kratzer potential respectively, has been reported by different authors for some models.

2. Bound State Solutions

To obtain the solutions of the redial potential system in equation (3), the radial Schrödinger equation with non-relativistic energy

E_{n, ℓ},

reduced Planck constant

ℏ,

reduced mass of the molecule

μ,

and the radial wave function

R_{n, ℓ} (r)

is given as

- \frac{ℏ^{2}}{2 μ} \frac{d^{2} R_{n, ℓ} (r)}{d r^{2}} + (V (r) + \frac{ℏ^{2}}{2 μ} \frac{1}{r^{2}}) R_{n, ℓ} (r) = E_{n, ℓ} R_{n, ℓ} (r) .

(4)

Substituting the radial potential system in equation (3) into equation (4), we obtain a second-order differential equation of the form

\frac{d^{2} R_{n, ℓ} (r)}{d r^{2}} + (- λ (E_{n, ℓ} - 2 D_{e} (1 + η)) + \frac{λ D_{e} (1 + η) r^{2}}{r_{e}^{2}} - \frac{2 λ D_{e} r_{e} (1 + η)}{r} + \frac{2 λ D_{e} r_{e}^{2} (1 + η) + ℓ (ℓ + 1)}{r^{2}}) R_{n, ℓ} (r) = 0.

(5)

where

λ = \frac{2 μ}{ℏ^{2}} .

The supersymmetric approach suggests the ground state wave function for the next step. Thus, the ground state wave function is written as

R_{0, ℓ} (r) = e x p (- \int W (r) d r),

(6)

where

W (r)

is referred to as superpotential function in supersymmetry quantum mechanics. The ground state wave function corresponds to the two partner Hamiltonians given as

\begin{array}{l} H_{+} = \hat{A} {\hat{A}}^{†} = - \frac{d^{2}}{d r^{2}} + V_{+} (r) \\ H_{-} = {\hat{A}}^{†} \hat{A} = - \frac{d^{2}}{d r^{2}} - V_{-} (r) \end{array}},

(7)

where

\begin{array}{l} \hat{A} = \frac{d}{d r} - W (r) \\ {\hat{A}}^{†} = - \frac{d}{d r} - W (r) \end{array}} .

(8)

Substituting the ground state wave function into equation (5), the resulting solution appears in a differential equation of the form

W^{2} (r) - \frac{d W (r)}{d r} - λ (E_{0, ℓ} - 2 D_{e} (1 + η)) + \frac{λ D_{e} (1 + η) r^{2}}{r_{e}^{2}} + \frac{2 λ r_{e} D_{e} (r_{e} - r) (1 + η) + ℓ (ℓ + 1)}{r^{2}},

(9)

where the superpotential

W (r)

is proposed as

W (r) = θ_{0} - θ_{1} r^{- 1} .

(10)

The two terms

θ_{0}

and

θ_{1}

in equation (10) are constants in the proposed superpotential that will soon be determined. Substituting equation (10) into equation (9), after some mathematical manipulations and simplifications, the values of the superpotential constants in equation (10) are obtained as

θ_{1} = \frac{- 1 \pm \sqrt{{(1 + 2)}^{2} + 8 λ D_{e} r_{e}^{2} (1 + η)}}{2},

(11)

θ_{0} = λ D_{e} r_{e} (1 + η) θ_{1}^{- 1},

(12)

θ_{0}^{2} = - λ (E_{n, ℓ} + 2 D_{e} r_{e} (1 + η)) - θ_{1} \sqrt{λ^{- 1} D_{e} r_{e}^{- 2} (1 + η)} .

(13)

With the aid of equations (7), (8) and (10), the partner potentials in supersymmetry quantum mechanics can be constructed as

V_{+} (r) = W^{2} (r) + \frac{d W (r)}{d r} = θ_{0}^{2} - \frac{2 θ_{0} θ_{1}}{r} + \frac{θ_{1} (θ_{1} - 1)}{r^{2}},

(14)

V_{-} (r) = W^{2} (r) - \frac{d W (r)}{d r} = θ_{0}^{2} - \frac{2 θ_{0} θ_{1}}{r} + \frac{θ_{1} (θ_{1} + 1)}{r^{2}} .

(15)

Equations (14) and (15) are family potentials and they satisfied the shape invariance condition via mapping of the form

θ_{1} \to θ_{1} + n,

θ_{1} = a_{0} .

It is deduced that

a_{1} = f (a_{0}) = a_{0} + 1,

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 [31,32,33]

R (a_{1}) = V_{+} (a_{0}, r) - V_{-} (a_{1}, r),

(16)

R (a_{2}) = V_{+} (a_{1}, r) - V_{-} (a_{2}, r),

(17)

R (a_{3}) = V_{+} (a_{2}, r) - V_{-} (a_{3}, r),

(18)

R (a_{n}) = V_{+} (a_{n - 1}, r) - V_{-} (a_{n}, r) .

(19)

The energy level can be obtained using

E_{n, ℓ} = \sum_{κ = 1}^{n} R (a_{κ}) = R (a_{1}) + R (a_{2}) + R (a_{3}) .... + R (a_{n}) .

(20)

From equation (20), we obtain the complete energy eigenvalue equation as

E_{n, ℓ} = \frac{2 μ^{2} D_{e}^{2} r_{e}^{2} {(1 + η)}^{2}}{ℏ^{2}} {(n + \frac{1}{2} + \frac{1}{2} Λ)}^{- 2} + \frac{2 ℏ}{μ} \sqrt{\frac{μ D_{e} (1 + η)}{2 r_{e}^{2}}} (2 n + 1 + \frac{1}{2} Λ) - 2 D_{e},

(21)

Λ = \sqrt{{(1 + 2)}^{2} + 8 λ D_{e} r_{e}^{2} (1 + η)} .

(22)

3. Calculation of Expectation Values

Here we calculate the position and momentum expectation values using Hellmann Feynman Theorem. The Hellmann-Feynman Theorem relates the derivative of the total energy with respect to a parameter and to the expectation value of the derivative of the Hamiltonian by 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 parameters

v

with the eigenvalue and eigenfunctions denoted by

E_{n, ℓ} (v)

and

R_{n, ℓ} (v)

respectively, then, we can find the various expectation values provided the associated normalized eigenfunction

R_{n, ℓ} (v)

is continuous with respect to the parameter

v .

Then

\frac{\partial E_{n, ℓ} (v)}{\partial v} = 〈 R_{n, ℓ} (v) | \frac{\partial H (v)}{\partial v} | R_{n, ℓ} (v) 〉 .

(23)

and

H = - \frac{ℏ^{2}}{2 μ} \frac{d^{2}}{d r^{2}} + \frac{ℏ^{2}}{2 μ} \frac{ℓ (ℓ + 1)}{r^{2}} + D_{e} {(\frac{r}{r_{e}} - \frac{r_{e}}{r})}^{2} - D_{e} (\frac{2 r_{e}}{r} - \frac{r_{e}^{2}}{r}) .

(24)

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},

then

〈 r^{2} 〉 = \frac{2 μ}{Λ} (\frac{ℏ Λ_{2} \sqrt{2}}{μ} - \frac{4 μ^{2} D_{e}^{2} r_{e}^{2} {(1 + η)}^{2}}{ℏ^{2} Λ_{0}^{3}}) .

(25)

(II): Expectation value

〈 p^{2} 〉 .

Setting

v = μ,

we have the expectation value

〈 p^{2} 〉 = \frac{4 μ^{3} D_{e}^{2} r_{e}^{2} {(1 + η)}^{2}}{ℏ^{2} Λ_{0}^{2}} (\frac{1}{μ^{2}} - \frac{4 D_{e} r_{e}^{2} (1 + η)}{Λ Λ_{0} ℏ^{2}}) + \frac{\sqrt{2} D_{e} (1 + η)}{μ} (\frac{ℏ^{2} Λ_{1}}{Λ_{2} r_{e}^{2}} + \frac{4 Λ_{2} r_{e}^{2}}{Λ ℏ}) - \frac{ℏ Λ_{1} Λ_{2} \sqrt{2}}{μ^{2}} .

(26)

4. 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. Thus, 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),

[38,39] then,

\int d r ρ (r, θ) = 1,

(27)

where

ρ (r, θ)

accounts for the details of the probability density. The Fisher information is examined in two different parts. These are Fisher information in the position space and Fisher information in the momentum space. The Fisher information for both the position space and momentum space can be written as

\begin{array}{l} I (ρ) = \int_{0}^{\infty} \frac{1}{ρ (r)} {[\frac{d ρ (r)}{d r}]}^{2} d r, \\ I (γ) = \int_{0}^{\infty} \frac{1}{γ (p)} {[\frac{d γ (p)}{d p}]}^{2} d p, \end{array}},

(28)

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 gives a higher quantity that provides a local change in the density where the system can be better described in an information-theoretic manner. In terms of the expectation values, the Fisher information for both the position space and the momentum space can therefore be written as [40,41,42]

\begin{array}{l} I (ρ) = 4 〈 p^{2} 〉 - 2 (2 L + 1) | m | 〈 r^{- 2} 〉, \\ I (γ) = 4 〈 r^{2} 〉 - 2 (2 L + 1) | m | 〈 p^{- 2} 〉 \end{array}},

(29)

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 form

\begin{array}{l} I (ρ) = 4 〈 p^{2} 〉, \\ I (γ) = 4 〈 r^{2} 〉 \end{array}} .

(30)

To check the validity of the product of expectation values as a Cramer-Rao inequality, Dehesa et al. [41], give the following conditions

\begin{array}{l} 〈 p^{2} 〉 \geq {(L + \frac{1}{2})}^{2} 〈 r^{- 2} 〉, \\ 〈 r^{2} 〉 \geq (L + \frac{1}{2}) 〈 p^{- 2} 〉 \end{array}} .

(31)

This implies that

\begin{array}{l} I (ρ) \geq 4 (1 - \frac{2 | m |}{2 L + 1}) 〈 p^{2} 〉, \\ I (γ) \geq 4 (1 - \frac{2 | m |}{2 L + 1}) 〈 r^{2} 〉 \end{array}} .

(32)

When

m = 0,

the product of equation (32) can be written as

I (ρ) I (γ) \geq 16 〈 p^{2} 〉 〈 r^{2} 〉 .

(33)

The minimum bound for the Fisher product [43] is given by

I (ρ) I (γ) \geq 36.

(34)

Using equations (21) and (34), the minimum bound the product of the position and momentum expectation becomes

〈 p^{2} 〉 〈 r^{2} 〉 \geq \frac{9}{4} .

(35)

Substituting equations (25) and (26) into (30), the Fisher information for the position space and the momentum space respectively becomes

I 〈 ρ 〉 = \frac{16 μ^{3} D_{e}^{2} r_{e}^{2} {(1 + η)}^{2}}{ℏ^{2} Λ_{0}^{2}} (\frac{1}{μ^{2}} - \frac{4 D_{e} r_{e}^{2} (1 + η)}{Λ Λ_{0} ℏ^{2}}) + \frac{4 \sqrt{2} D_{e} (1 + η)}{μ} (\frac{ℏ^{2} Λ_{1}}{Λ_{2} r_{e}^{2}} + \frac{4 Λ_{2} r_{e}^{2}}{Λ ℏ}) - \frac{4 \sqrt{2} ℏ Λ_{1} Λ_{2}}{μ^{2}} .

(36)

I 〈 γ 〉 = \frac{8 μ}{Λ} (\frac{ℏ Λ_{2} \sqrt{2}}{μ} - \frac{4 μ^{2} D_{e}^{2} r_{e}^{2} {(1 + η)}^{2}}{ℏ^{2} Λ_{0}^{3}}) .

(37)

5. Discussion

Table 1 presents the energy eigenvalues of the combination of pseudoharmonic potential and the Kratzer potential for various quantum number and angular momentum in the absence and presence of the constant dependent potential for two different values of the dissociation energy. The energy of the combined potentials responded positively to an increase in each of the dissociation energy, quantum number, angular momentum and constant dependent potential. This implies that an increase in any of dissociation energy, quantum number, angular momentum and constant dependent potential, increases the energy of the interacting potential system. From Table 1, it is also noted that as the angular momentum increases, the energy difference between the successive one angular momentum and the next angular momentum also increases. In Table 2, the energy of the combination of the pseudoharmonic potential and the Kratzer potential for various quantum number and equilibrium bond length in the absence and presence of the constant dependent potential is presented. The numerical values in Table 2 showed that the equilibrium bond length varies inversely with the energy of the interacting potential. However, the quantum number and the angular momentum respectively varies directly with the energy of the system. In Table 3. We presented the energy eigenvalue of the interacting potential for various dissociation energy and constant dependent potential at the ground state and the first excited state. The energy of the combined potential model increases as each of the dissociation energy and the constant dependent potential respectively increases. The energy of the system also increases as the quantum number and the angular momentum respectively increases.

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 Table 4 and Table 5. In Table 4, the Fisher information in position space and the quantum number varies inversely with each other but the Fisher information in momentum space varies directly with the quantum number. The Fisher information both in position space and in momentum space in the presence of the constant dependent potential are higher than their 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 result in Table 4 satisfied the Fisher inequality given in equation (34). The variation of the Fisher information showed 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 are 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 is 212.949992. The values of the minimum bound also confirmed that the results in Table 5 satisfied 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 of the Fisher information against the quantum state in Table 4 is contrary to the variation of Fisher information against the equilibrium bond separation in Table 5, but satisfied the Fisher inequality in both cases. In Table 5, a strongly localized distribution in the momentum space corresponds to widely delocalized distribution in the position space. The results obtained obeyed Heisenberg 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 Table 4, Table 5 and Table 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 are greater than the standard value of 2.25. The minimum bound for the product of the expectation values in the absence and presence of the constant dependent potential are 3.9027949 and 12.1497115 respectively. These results satisfied Cramer Rao's inequality for the product of expectation. Table 7 presents Fisher information for both the position space and momentum space and their product in the absence and presence of constant dependent potential for various angular momentum. 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 satisfied the Cramer-Rao inequality but decreases with increase in the angular momentum.

6. Conclusion

The solutions of the combination of pseudoharmonic potential and Kratzer potential was obtained and the effects of the potential parameters on the energy, the expectation values and the Fisher information were all observed. It was shown that the presence of constant dependent potential, the dissociation energy, the quantum number and the angular momentum respectively increases the energy eigenvalue of the system but the equilibrium bond length reduces the energy. It was noticed that the variation of the quantum number against the Fisher information is contrary to the variation of the equilibrium bond length against the Fisher information. The constant dependent potential increases the position expectation value but decreases the momentum expectation value. It is discovered that some 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 of the constant dependent potential. The presence of the constant dependent potential only increases the quantity studied.

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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 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.

$n$	$ℓ$	$η = 0$		$η = 0.5$
$n$	$ℓ$	$D_{e} = 5$	$D_{e} = 10$	$D_{e} = 5$	$D_{e} = 10$
0 1 2 3	0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3	19.1915543 26.8693976 37.5880690 49.3042587 43.8601841 51.8866789 62.7622893 74.5398554 68.9627325 77.0734494 88.0004415 99.8033288 94.1756967 102.316428 113.265083 125.080290	30.2367130 38.6168018 51.8488919 67.2542617 64.5835661 73.5761120 87.2096164 102.806150 99.8304090 109.000745 122.777346 138.455898 135.354381 144.594507 158.434548 174.154043	29.9096370 38.0538010 50.2771052 64.1033929 59.8627152 68.5040080 81.0061110 94.9538017 90.4917350 99.2664182 111.864086 125.862895 121.312779 130.137638 142.776689 156.800703	50.2267069 58.8074820 73.2878939 90.9334004 91.8404299 101.195400 116.302626 134.292068 134.751205 144.358222 159.703767 177.845801 178.109873 187.822519 203.277995 221.497925

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

ℓ -

wave.

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

ℓ -

wave.

$n$	$r_{e}$	$η = 0$		$η = 0.5$
$n$	$r_{e}$	$ℓ = 0$	$ℓ = 1$	$ℓ = 0$	$ℓ = 1$
0 1 2 3 4 5	0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0	12.4202881 9.39252710 8.44361600 7.98103440 24.3346767 15.1059713 12.1681436 10.7362083 36.6586629 21.0620459 16.0437357 13.5932412 49.1359643 27.1474431 20.0143131 16.5207835 61.6833974 33.3080324 24.0475954 19.4986801 74.2674019 39.5153037 28.1239350 22.5134945	14.5803237 9.91579930 8.66857060 8.10491030 26.7141787 15.6968954 12.4202881 10.8734364 39.1157627 21.6877923 16.3126127 13.7395803 51.6272352 27.7929105 20.2940459 16.6735512 64.1920084 33.9654942 24.3346767 19.6561126 76.7857329 40.1804755 28.4161668 22.6743939	21.9178843 18.2993918 17.1549376 36.3343188 25.2162768 21.6715263 19.9405623 16.5947227 51.3010558 32.4241376 26.3605159 23.3994205 66.4978821 39.8011807 31.1657135 26.9411554 81.8076698 47.2832585 36.0520860 30.5450424 97.1793420 54.8339773 40.9968719 34.1964520	24.0760055 18.8139539 17.3764753 16.7170015 38.7409248 25.7961769 21.9178843 20.0746906 53.8062866 33.0409857 26.6233344 23.5421470 69.0498216 40.4404751 31.4398758 27.0902716 84.3845319 47.9369683 36.3343188 30.6990056 99.7706981 55.4973648 41.2850021 34.3541585

Table 3. Energy eigenvalue for various values dissociation energy and constant dependent potential with

μ = ℏ = 1,

and

r_{e} = 0.25,

at the ground state and first excited state.

Table 3. Energy eigenvalue for various values dissociation energy and constant dependent potential with

μ = ℏ = 1,

and

r_{e} = 0.25,

at the ground state and first excited state.

$D_{e}$	$η$	$n = 0$		$n = 1$
$D_{e}$	$η$	$ℓ = 0$	$ℓ = 1$	$ℓ = 0$	$ℓ = 1$
1 2 3 4 5	0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5	4.18168780 8.50000000 12.5793525 16.5379816 20.4202881 24.2487113 6.50000000 14.5379816 22.2487113 29.7921099 37.2291330 44.5908154 8.57935250 20.2487113 31.5215665 42.5908154 53.5303880 64.3780215 10.5379816 25.7921099 40.5908154 55.1551611 69.5701085 83.8790804 12.4202881 31.2291330 49.5303880 67.5701085 85.4430187 103.197567	6.0217369 10.5510171 14.7015198 18.6878788 22.5803237 26.4107479 8.55101710 16.6878788 24.4107479 31.9479971 39.3742236 46.7242693 10.7015198 22.4107479 33.6722999 44.7242693 55.6472139 66.4800000 12.6878788 27.9479971 42.7242693 57.2668299 71.6631972 85.9565223 14.5803237 33.3742236 51.6472139 69.6631972 87.5169397 105.255816	9.72050810 16.2222222 21.9412282 27.2631497 32.3346767 37.2301178 14.2222222 25.2631497 35.2301178 44.6521671 53.7304216 62.5667333 17.9412282 33.2301178 47.2265378 60.5667333 73.4925650 86.1265502 21.2631497 40.6521671 58.5667333 75.7320533 92.4238854 108.781987 24.3346767 47.7304216 69.4925650 90.4238854 110.829779 130.865107	11.6301374 18.4038820 24.2349788 29.6126984 34.7141787 39.6259364 16.4038820 27.6126984 37.6259364 47.0601538 56.1377674 64.9679851 20.2349788 35.6259364 49.6351901 62.9679851 75.8806382 88.5000000 23.6126984 43.0601538 60.9679851 78.1153066 94.7875816 111.126964 26.7141787 50.1377674 71.8806382 92.7875816 113.170292 133.184707

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 = ℓ = 1,

r_{e} = 0.5,

and

D_{e} = 2.5

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 = ℓ = 1,

r_{e} = 0.5,

and

D_{e} = 2.5

$n$	$η = 0$			$η = 0.5$
$n$	$I 〈 ρ 〉$	$I 〈 γ 〉$	$I 〈 ρ 〉 I 〈 γ 〉$	$I 〈 ρ 〉$	$I 〈 γ 〉$	$I 〈 ρ 〉 I 〈 γ 〉$
0 1 2 3 4 5 6 7 8 8 10 11 12 13 14 15	12.9965839 11.8185475 11.2615469 10.9565616 10.7720175 10.6520394 10.5697147 10.5108053 10.4672122 10.4340553 10.4082527 10.3877810 10.3712673 10.3577538 10.3465556 10.3371727	8.5737778 15.2579808 22.1905981 29.2240214 36.3056214 43.4130476 50.5355353 57.6673891 64.8053694 71.9475241 79.0926206 86.2398494 93.3886615 100.538674 107.689612 114.841276	111.4298232 180.3271707 249.9004614 320.1947901 391.0847898 462.4374938 534.1461897 606.1306983 678.3315540 750.7044447 823.2159806 895.8406674 968.5587745 1041.354832 1114.216560 1187.134100	21.5583462 19.6777166 18.7115540 18.1551929 17.8069278 17.5749107 17.4127458 17.2950190 17.2068853 17.1392093 17.0861231 17.0437192 17.0093145 16.9810180 16.9574657 16.9376539	15.7382452 23.7495543 32.1266502 40.6676667 49.2919216 57.9626756 66.6613706 75.3778408 84.1061482 92.8426388 101.584965 110.331565 119.081364 127.833606 136.587746 145.343382	339.2905400 467.3369980 601.1395518 738.3293339 877.7376907 1018.688851 1160.757499 1303.661192 1447.204844 1591.249420 1735.693225 1880.460203 2025.492369 2170.744768 2316.182020 2461.775909

Table 5. Fisher information for position space and momentum space at the ground state for various equilibrium bond separation and the Fisher product in the absence and presence of constant dependent potential with

μ = ℏ = ℓ = 1,

and

D_{e} = 2.5

μ = ℏ = ℓ = 1,

and

D_{e} = 2.5

$r_{e}$	$η = 0$			$η = 0.5$
$r_{e}$	$I 〈 ρ 〉$	$I 〈 γ 〉$	$I 〈 ρ 〉 I 〈 γ 〉$	$I 〈 ρ 〉$	$I 〈 γ 〉$	$I 〈 ρ 〉 I 〈 γ 〉$
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0	6.3749475 12.0910640 14.6231981 16.0247585 16.9078176 17.5129218 17.9525752 18.2860804 18.5475457 18.7579355 18.9308264 19.0753897 19.1980391 19.3033929 19.3948598 19.4750097 19.5458162	6.7680001 6.2814977 6.0423780 5.8983477 5.8018537 5.7326651 5.6806283 5.6400719 5.6075771 5.5809594 5.5587579 5.5399585 5.5238356 5.5098559 5.4976189 5.4868181 5.4772149	43.1456451 75.9499903 88.3588904 94.5195968 98.0966833 100.395716 101.981906 103.134809 104.006792 104.687276 105.231880 105.676868 106.046812 106.358913 106.625548 106.855836 107.056635	13.9837226 20.5118296 23.4211406 25.0442629 26.0739294 26.7834340 27.3012806 27.6955632 28.0056349 28.2557876 28.4618122 28.6344110 28.7810910 28.9072717 29.0169624 29.1131925 29.1982930	13.7463577 13.1590415 12.8653865 12.6881230 12.5694080 12.4843480 12.4204164 12.3706163 12.3307320 12.2980723 12.2708384 12.2477829 12.2280132 12.2108738 12.1958730 12.1826340 12.1708638	192.2252527 269.9160160 301.3220276 317.7646874 327.7338557 334.3737107 339.0932721 342.6111838 345.3299783 347.4917179 349.2502995 350.7080500 351.9355594 352.9830473 353.8871873 354.6753683 355.3684483

Table 6. Position expectation value and momentum expectation value at the ground state for various angular momentum quantum state and their product in the absence and presence of constant dependent potential with

μ = ℏ = 1,

r_{e} = 0.5,

and

D_{e} = 2.5

in the absence and presence of the constant dependent potential.

μ = ℏ = 1,

r_{e} = 0.5,

and

D_{e} = 2.5

in the absence and presence of the constant dependent potential.

$ℓ$	$η = 0$			$η = 0.5$
$ℓ$	$〈 r^{2} 〉$	$〈 p^{2} 〉$	$〈 r^{2} 〉 〈 p^{2} 〉$	$〈 r^{2} 〉$	$〈 p^{2} 〉$	$〈 r^{2} 〉 〈 p^{2} 〉$
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15	2.0753109 2.1434445 2.4796239 3.0345051 3.7149293 4.4660484 5.2591091 6.0785638 6.9155189 7.7645982 8.6223893 9.4866342 10.355785 11.228746 12.104725 12.983131	4.2441101 3.2491460 2.3486134 1.7624127 1.3858907 1.1325692 0.9534294 0.8212049 0.7201144 0.6405721 0.5764852 0.5238228 0.4798244 0.4425416 0.4105633 0.3828440	8.8078482 6.9643639 5.8236779 5.3480505 5.1484858 5.0581087 5.0141894 4.9917465 4.9799647 4.9737848 4.9706794 4.9693156 4.9689582 4.9691872 4.9697553 4.9705140	3.9027949 3.9345613 4.2100767 4.7530480 5.4798280 6.3206480 7.2328358 8.1913537 9.1811818 10.192968 11.220659 12.260199 13.308778 14.364399 15.425605 16.491312	6.5411490 5.3895866 4.1263248 3.1895560 2.5445494 2.0942879 1.7693011 1.5265679 1.3396688 1.1919700 1.0726518 0.9744445 0.8923155 0.8226858 0.7629492 0.7111670	25.5287631 21.2056588 17.3721438 15.1601125 13.9436929 13.2372565 12.7970647 12.5046576 12.2997429 12.1497115 12.0358600 11.9468833 11.8756286 11.8173877 11.7689537 11.7280766

Table 7. Fisher information for both position space momentum space at the ground state for various angular momentum quantum state 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.

Table 7. Fisher information for both position space momentum space at the ground state for various angular momentum quantum state 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.

$ℓ$	$η = 0$			$η = 0.5$
$ℓ$	$I (ρ)$	$I (γ)$	$I (ρ) I (γ)$	$I (ρ)$	$I (γ)$	$I (ρ) I (γ)$
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15	8.30124380 8.57377780 9.91849550 12.1380204 14.8597170 17.8641936 21.0364363 24.3142550 27.6620757 31.0583928 34.4895572 37.9465368 41.4231385 44.9149846 48.4188988 51.9325232	16.976441 12.996584 9.3944536 7.0496510 5.5435626 4.5302766 3.8137178 3.2848197 2.8804576 2.5622883 2.3059406 2.0952913 1.9192976 1.7701664 1.6422531 1.5313761	140.925571 111.429823 93.1788457 85.5688076 82.3757720 80.9297386 80.2270307 79.8679439 79.6794347 79.5805565 79.5308712 79.5090499 79.5033319 79.5069946 79.5160843 79.5282236	15.6111797 15.7382452 16.8403066 19.0121919 21.9193120 25.2825921 28.9313433 32.7654146 36.7247270 40.7718702 44.8826362 49.0407939 53.2351111 57.4575959 61.7024212 65.9652465	26.1645959 21.5583462 16.5052993 12.7582239 10.1781975 8.37715150 7.07720460 6.10627160 5.35867530 4.76788000 4.29060710 3.89777810 3.56926200 3.29074340 3.05179690 2.84466800	408.4602096 339.2905400 277.9543004 242.5618001 223.0990871 211.7961041 204.7530351 200.0745216 196.7958870 194.3953836 192.5737593 191.1501323 190.0100570 189.0782028 188.3032599 187.6492262

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

Abstract

1. Introduction

2. Bound State Solutions

3. Calculation of Expectation Values

4. Fisher Information.

5. Discussion

6. Conclusion

References

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