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Scientific Reports

Information theory and thermodynamic properties for a combined potential model

C. A. Onate, B. B. Deji-Jinadu, O. O. Ajani, Jacob Adebayo Akinpelu, O. Ebomwonyi

Published Jan 21, 2025• 6 citations
Publisher

Abstract

Yukawa potential and Hulthẻn potential are very useful potential models with applications in different areas of physics. The present study examined theoretic measure and thermodynamic properties of energy levels and wave functions for a combination of these two potentials. The effect of screening parameter on Fisher information and Shannon entropy as well as the effect of temperature on the various thermodynamic properties of the combined potentials and its subsets potentials are well studied. The effects of screening parameter on the thermodynamic properties of the combined potentials are also studied. This study shows equivalence of temperature effect on the combined potential and Hulthẻn potential but a different effect on Yukawa potential. Fisher information and the Shannon entropy for both position and momentum spaces as a function of the screening parameter obey Heisenberg principle, satisfy Cramer-Rao inequality and Bialynick-Birula, Mycielski inequality. The Shannon entropy shows the stability of the system for low screening parameter. It is also shown that the strength of Hulthẻn potential outweighed the strength of Yukawa potential in their combination for thermodynamic properties.

DOI
10.1038/s41598-025-86335-x

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Yukawa potential and Hulthẻn potential are very useful potential models with applications in different areas of physics. The present study examined theoretic measure and thermodynamic properties of energy levels and wave functions for a combination of these two potentials. The effect of screening parameter on Fisher information and Shannon entropy as well as the effect of temperature on the various thermodynamic properties of the combined potentials and its subsets potentials are well studied. The effects of screening parameter on the thermodynamic properties of the combined potentials are also studied. This study shows equivalence of temperature effect on the combined potential and Hulthẻn potential but a different effect on Yukawa potential. Fisher information and the Shannon entropy for both position and momentum spaces as a function of the screening parameter obey Heisenberg principle, satisfy Cramer-Rao inequality and Bialynick-Birula, Mycielski inequality. The Shannon entropy shows the stability of the system for low screening parameter. It is also shown that the strength of Hulthẻn potential outweighed the strength of Yukawa potential in their combination for thermodynamic properties.
The approximate solutions of the Schrödinger equation with the central generalized Hulthén and Yukawa potential are investigated within the framework of the functional method. The obtained wave function and the energy levels are used to study the Shannon entropy, Renyi entropy, Fisher information, Shannon Fisher complexity, Shannon power and Fisher-Shannon product in both position and momentum spaces for the ground and first excited states. The theoretic information theories such as Shannon entropies Sr,Sp, Fisher information Ip,Ip and Renyi entropies Rβ as well as the information entropy probability densities ρ(r),ϕ(p) are illustrated graphically. The Bialynicki-Birula – Mycielski (BBM) inequality for the Shannon entropies and the Cramer-Rao inequality for the Fisher information are satisfied for the potential model.
A Schrödinger equation is solved for a cotangent hyperbolic potential using the elegant parametric Nikiforov–Uvarov method. The results obtained are used to calculate the Fisher information for position space and momentum space as well as the Shannon entropy for position space and momentum space at the ground state. The energies of the cotangent potential are observed to be fully bounded for different quantum states. The Fisher information satisfied the Fisher relation known as Cramer–Rao inequality for the Fisher product. Its result also obeyed the Heisenberg principle as its alternative for uncertainty measurement. However, the Shannon entropy only satisfied the Bialynick-Birula, Mycielski inequality for higher values of the screening parameter. Its relation to the Heisenberg principle failed as both the position space and the momentum space are parallel with high entropy squeezing. This potential has not been reported for any system to the best of our understanding.
Abstract The Shannon entropies in both coordinate and momentum spaces for some low-lying eigenstates of the Hulthén potential (HP) are investigated over a wide range of screening parameters where bound states exist. The system wave functions are expanded in terms of multiple groups of Slater-type orbitals and the Hamiltonian is solved by employing the Rayleigh–Ritz variational method. The accuracy of the obtained bound-state eigenenergies, Shannon entropies, and radial mean values are validated by comparing with the analytical solutions of the HP in s-wave states. The Bialynicki–Birula and Mycielski inequalities for Shannon entropies in both the coordinate and momentum spaces as well as their sum are examined in both weak and strong screening situations. It is shown that the variation of Shannon entropy sum is governed by its upper bound for the system in a well-defined bound state. In the critical bound region where the system undergoes a bound-continuum transition, the Shannon entropies and entropy sum differ in states by their orbital angular momenta and their asymptotic behavior can be properly shaped by the rigorous upper and lower bounds in terms of the radial and momentum expectation values ⟨ r ⟩ and ⟨ p ⟩ . We finally extended the calculation to a modified HP (MHP) that has been widely discussed in the literature and compared our results with previous predictions. Benchmark Shannon entropies are provided for further investigation of the information measures of MHPs.
By employing the Nikiforov–Uvarov functional analysis (NUFA) method, we solved the radial Schrödinger equation with the shifted Morse potential model. The analytical expressions of the energy eigenvalues, eigenfunctions and numerical results were determined for selected values of the potential parameters. Variations of different thermodynamic functions with temperature were discussed extensively. Different quantum information-theoretic measures, including Shannon entropy, Fisher information and Fisher–Shannon product of the shifted Morse potential, were investigated numerically and graphically in position and momentum spaces for the ground and the first excited states. The quantum information theories considered satisfied their corresponding inequalities, including Bialynicki–Birula–Mycielski, Stam–Cramer–Rao inequalities and the Fisher–Shannon product relation.
In this research, the radial Schrödinger equation for a newly proposed screened Kratzer–Hellmann potential model was studied via the conventional Nikiforov–Uvarov method. The approximate bound state solution of the Schrödinger equation was obtained using the Greene–Aldrich approximation in addition to the normalized eigenfunction for the new potential model, both analytically and numerically. These results were employed to evaluate the rotational–vibrational partition function and other thermodynamic properties for the screened Kratzer–Hellmann potential. The results obtained have been graphically discussed. Also, the normalized eigenfunction has been used to calculate some information-theoretic measures including Shannon entropy and Fisher information for low-lying states in both position and momentum spaces numerically. The Shannon entropy results obtained agreed with the Bialynicki-Birula and Mycielski inequality, while the Fisher information results obtained agreed with the Stam, Cramér–Rao inequality. Also, alternating increasing and decreasing localization across the screening parameter for both eigenstates was observed.
Abstract In this paper, Shannon entropy and Fisher information is studied for the screened Kratzer potential model and compared with the screened Coulomb in three dimensions. Our results showed similar higher‐order characteristic behavior for position and momentum space. Our numerical results showed that increases in the accuracy of predicting particle location occurred in the position space. Our result shows that the sum of the position and momentum entropies satisfies the lower‐bound Berkner, Bialynicki‐Birula, and Mycieslki inequality. The Stam‐Cramer‐Rao inequalities relation for Fisher information and the expectation values were also satisfied for the different eigenstates.