ANALYTICAL SOLUTIONS TO THE SCHRÖDINGER EQUATION WITH COLLECTIVE POTENTIAL MODELS: APPLICATION TO QUANTUM INFORMATION THEORY

In this study, the energy equation and normalized wave function were obtained by solving the Schrödinger equation analytically utilizing the Eckart-Hellmann potential and the Nikiforov-Uvarov method. Fisher information and Shannon entropy were investigated. Our results showed higher-order characteristic behavior for position and momentum space. Our numerical results showed an increase in the accuracy of the location of the predicted particles occurring in the position space. Also, our results show that the sum of the position and momentum entropies satisfies the lower-bound Berkner, Bialynicki-Birula, and Mycieslki inequality and Fisher information was also satisfied for the different eigenstates. This study's findings have applications in quantum chemistry, atomic and molecular physics, and quantum physics.


Introduction
With various analytical techniques, such as the Nikiforov-Uvarov (NU) method [1][2][3][4][5][6][7][8][9][10], the asymptotic iterative method (AIM) [11], the supersymmetric quantum mechanics method (SUSQM) [12], the Nikiforov-Uvarov functional analysis (NUFA) method [13][14][15][16], the series expansion method [17][18][19][20][21], the WKB approximation [22][23][24], and so on [25], the Schrödinger equation (SE) can be solved for a variety of potentials. Our knowledge of the underlying cause of a quantum system is significantly influenced by the analytical solutions to this equation with a physical potential. This is due to the fact that the eigenvalues and eigenfunctions convey essential information about the quantum system under investigation [26,27]. However, the exact bound state solutions of the SE are possible in some cases [28]. One can solve the SE using appropriate approximation approaches, such as the Pekeris, Greene and Aldrich, and others [29][30][31], to obtain the approximate solutions when the arbitrary angular momentum quantum number is not equal to zero. The eigenvalues and eigenfunctions are of great importance in the study of mass spectra of heavy mesons [32], thermodynamic properties of the system [33], and the quantum theoretic information entropies [34] among others. According to the fundamental principle of information theory put forward by Claude Shannon, the global measures of Shannon entropy and Fisher information are crucial to quantum information-theoretic measures [36]. As a result of its numerous applications in physics and chemistry, scientists have actively investigated Shannon and Fisher entropies in various fields in recent years. The theory of communication is one field in which Shannon and Fisher entropies are applied [37,38]. The theoretical foundation of Fisher information was obtained much earlier [39], but the application was unknown until Sear et al., [40], found a link between Fisher information and the kinetic energy of a quantum system. The significance of the global measure is to investigate the uncertainty associated with the probability distribution [41,42]. The position and momentum spaces of the Shannon entropy have an entropic relation derived by Berkner, Bialynicki-Birula and Mycieslki [43] and expressed as (1 ) r p s s D In    , where D is the spatial dimension. In view of this, many scholars have studied the Shannon and Fisher entropies [44][45][46], for instance, Edet et al., [47] used a class of Yukawa potential to study the global quantum information-theoretic measurements in the presence of magnetic and Aharanov-Bohm (AB) fields. Also, Olendski [48] used the quadratic and inverse quadratic dependencies on the radius to study the Shannon quantum information entropies, Fisher information, and Onicescu energies and complexities in the position and momentum spaces for the azimuthally symmetric two-dimensional nano-ring that is placed into the combination of the transverse uniform magnetic field and the AB flux. For time-dependent harmonic vector potential. Onate et al. [49] found the exact solution to the Feinberg-Horodecki equation. Explicitly, the quantized momentum and its corresponding unnormalized wave functions were obtained. Using the Hellman-Feynman theory, expectation values of time and momentum were used to determine the Fisher information (for time and momentum) and variance (for time and momentum). Also, Under the influence of an improved expression for the Wei potential energy function, Onate et al., [50] obtained an approximate solution of the one-dimensional Klein-Gordon equation. By using specific mappings, it was possible to derive the solution of the SE from that of the Klein-Gordon equation.

, , ,and A A A
A are the strength of the potential,  is the screening parameter and r is inter-particle distance. This paper is organized as follows: In Sect. 2 we solve the Schrödinger equation with the Eckart plus Hellmann potential to obtain the energy equation and normalized wave function. In Sect. 3, the derived eigenfunctions will be used to obtain the numerical computation of the Shannon entropy and Fisher information. In Sect. 3, we present the results and discussion. Conclusions are given in Sect. 4.

Analytical solutions of the Schrödinger equation with Eckart plus Hellmann potential
In this study, we adopt the NU method [1] which is based on solving the second-order differential equation of the hypergeometric type. The details can be found in Appendix A. The Schrödinger equation of a quantum physical system is characterized by a given potential   V r takes the form [72,73] nl E is the energy eigenvalues of the quantum system, l is the angular momentum quantum number,  is the reduced mass of the system,  is the reduced Planck's constant and r is the radial distance from the origin.
Equation (2) cannot be exactly solved using the adopted potential. To deal with the centrifugal barrier, we thus employ an approximation approach suggested by Greene-Aldrich [29]. This approximation is a good approximation to the centrifugal term which is valid for 1, By using the change of variable from to r s , our new coordinate becomes We substitute Eq. (5) into Eq. (4) and after some simplifications; Eq. (6) is gotten as Comparing Eq. (6) with Eq. (A1) we obtain the following parameters where   To find the constant k , the discriminant of the expression under the square root of Eq. (9) must be equal to zero. As such we have that Substituting Eqs. (10) and (11) in Eq. (9) we have Differentiating Eq. (12) gives Substituting Eqs. (11) and (13) into Eq.(A10) gives With ( ) being obtained from Eq.(A7) as Differentiating Eq. (15) yields Substituting Eqs. (16) and (17) into Eq.(A11) and simplifying, yields Equating Eqs. (14) and (18) and substituting Eq. (7) yields the energy eigenvalues equation of the Eckart plus Hellmann potential as 2. We set 2 3 0 A A   and obtain the energy eigenvalues for Eckart potential 3. We set 0     and obtain the energy eigenvalues for Coulomb potential

Wave function
To obtain the corresponding wavefunction, we consider Eqs. (A4) and (A6) and upon substituting Eqs. (8) and (15) and integrating, we get Equation 25 is known as the weight function. By substituting Eqs. (8) and (25) where nl B is normalization constant. Equation (26) is equivalent to where The wave function is given by Using the normalization condition, we obtain the normalization constant as follows Let 0 1 According to Ebomwonyi et al. [74], integral of the form in Eq,(33) can be expresses as x y y n p p x n y n P p dp n x x y n Hence, comparing Eq. (33) with the standard integral of Eq.(34), we obtain the normalization constant as

Shannon Entropy for the Eckart plus Hellmann potential
Entropy is a thermodynamic quantity representing the unavailabity of a systems thermal energy for conversion into mechanical work [53]. The Shannon entropy is defined in position and momentum spaces as [75] EEJP. 4 (2022) Funmilayo Ayedun, Etido P. Inyang, et al and where r S is the position space Shannon entropy, p S is the momentum space Shannon entropy, are the probability densities in the position and momentum spaces, respectively. ( ) p  is the wave function in the momentum coordinate obtained by the Fourier transform of ( ) r  . The Shannon entropic uncertainty relation proposed by Beckner,Bialynicki-Birula and Mycielski(BBM) takes the form [43] (1 In ), where D is the spatial dimension. The probability density's logarithmic functional measure of randomness and uncertainty in a particle's spatial localization is called Shannon entropy. The lower this entropy, the more concentrated is the wave function, the smaller the uncertainty and the higher is the accuracy in predicting the localization of the particle [74].
The corresponding normalized wave function in the momentum space for two low lying states 0,1 n  is obtained as [77]  

Fisher Information theory for the Eckart plus Hellmann potential
We examine the Fisher information in position and momentum spaces. Fisher information is the sole component of the local measure, and is mainly concerned with local changes that occur in probability density [57,78]. Density functional is important for the investigation of Fisher information [79]. It is stated as: Fisher information inequality becomes [80]   2 2 1 9 2 36 1 We solve Eqs. (48) and (49) numerically, which are complicated to solve analytically due to the form of the integral.

RESULTS AND DISCUSSION
In this section, we will discuss our numerical results. For both cases, the screening criterion was set to 0.1 0.9.
   These parameters were selected in order to compare results [57].
Our results were obtained numerically. Fisher information and Shannon entropy give important details about the precision and degree of uncertainty in particle localization predictions. Lower Shannon entropy denotes greater stability, higher localization, and reduced uncertainty. The kinetic energy and Fisher information are related, and more Fisher information indicates greater localization and energy fluctuation. For the various values of  , the numerical results for Shannon entropy and Fisher information are shown in Tables 1 and 2, respectively. The Shannon entropy values show a deceasing order in the position space, which signifies a lower uncertainty and higher accuracy in predicting localization and the stability. This is complimented in the momentum space by an increasing Shannon entropy. For 1 n  , it increased and decreased afterward. This similar behavior is also observed in the momentum EEJP. 4 (2022) Funmilayo Ayedun, Etido P. Inyang, et al spaces. Negative values mean that the Shannon entropy is highly localized [57]. The numerical analysis of Fisher information for 0 n  and 1 n  is shown in Table 2. Similar phenomena are seen in momentum spaces as well and negative values indicate a strongly localized Shannon entropy. Table 2 displays the numerical analysis of Fisher information for 0 n  and 1 n  indicating ground and first excited states respectively. Here, there was a similar pattern of behavior, and the alternative increase and decrease for 1 n  are noticed. The increasing Fisher information observed in these different states implies an increasing localization. In both cases, the Shannon entropy uncertainty relation condition is satisfied as seen in Eq. (40) and Fisher uncertainty relation is satisfied as seen in Eq. (50).

CONCLUSION
In this research, the Schrödinger equation is solved with the Eckart plus Hellmann potential to obtain the energy equation and normalized wave function. We studied the charactertic properties of Shannon entropy and Fisher information for the position and momentum spaces for ground state and first excited state. Our results was presented numerically. We observed a similar behavior for Shannon entropy and Fisher information values. This behavior is related to the probability density distribution's concentration. Our findings showed that several eigenstates had negative values in the position space. This implies a higher localization for the collective potential models. The potential models also show increasing accuracy in predicting particle localization in the position space of Shannon   ORCID IDs Funmilayo Ayedun, https://orcid.org/0000-0001-5421-9305; Etido P. Inyang, https://orcid.org/0000-0002-5031-3297