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Results in Physics

Fisher and Shannon information entropies in 2-dimension for a non-central potential as applied to hydrogen diatomic molecules

Wen‐Li Chen, I. B. Okon, C. A. Onate, Cecilia N. Isonguyo, Sylvester Andrew Ekong, Judith P. Araújo, Abeer E. Aly, A.O. Anna, Eddy Sunday William, Etido Patrick Inyang, K. O. Emeje, Kaushal R. Purohit

Published Jun 3, 2025• 4 citations
Publisher

Abstract

In this paper, Fisher and Shannon information entropies in two-dimensional spaces were analysed for hydrogen diatomic molecules using a non-central exponential-type potential, both in the presence and absence of magnetic and Aharanov-Bohm (AB) flux fields. By solving the two-dimensional Schrödinger equation in the presence of charged Hamiltonian using the parametric Nikiforov-Uvarov (NU) method, we obtained numerical solutions for different potential depths for Fisher and Shannon information entropies. These results adhere to Bialynicki-Birula and Mycielski (BBM) and Fisher-based inequalities. The findings have significant applications in chemical physics, aiding in the analysis of molecular electronic properties, stability and reactivity under varying external influences. Additionally, the study contributes to quantum information by offering insights into the localisation and delocalisation phenomena, with direct applications to quantum computation, entanglement, and communication technologies.

DOI
10.1016/j.rinp.2025.108303

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In this paper, Fisher and Shannon information entropies in two-dimensional spaces were analysed for hydrogen diatomic molecules using a non-central exponential-type potential, both in the presence and absence of magnetic and Aharanov-Bohm (AB) flux fields. By solving the two-dimensional Schrödinger equation in the presence of charged Hamiltonian using the parametric Nikiforov-Uvarov (NU) method, we obtained numerical solutions for different potential depths for Fisher and Shannon information entropies. These results adhere to Bialynicki-Birula and Mycielski (BBM) and Fisher-based inequalities. The findings have significant applications in chemical physics, aiding in the analysis of molecular electronic properties, stability and reactivity under varying external influences. Additionally, the study contributes to quantum information by offering insights into the localisation and delocalisation phenomena, with direct applications to quantum computation, entanglement, and communication technologies.
The Shannon information entropy for scandium fluoride (ScF) and hydrogen (H2) molecules in the presence of magnetic and Aharanov–Bohm (AB) fields with modified Kratzer plus screened Coulomb potential in the position and momentum coordinate is investigated. The wavefunction and probability density are obtained by solving the two-dimensional (2D) Schrödinger wave equation using the Nikiforov–Uvarov functional analysis method and then applied to calculate the Shannon entropy. The graphical behaviour of the 2D wavefunction and probability density is showcased. Our results depict the effects of the external fields on Shannon entropy for ScF and (H2), the negative values of the position Shannon entropies obtained imply higher localisation of the particles and molecules indicating more stability of the system. Another interesting part of this work is that Bialynicki–Birula–Mycielski inequality is tested and proven bounded for both diatomic molecules. A few to mention, this research will enhance the apprehension of the dynamics of quantum molecules and systems subjected to external fields, quantum information signal processing and communication.
In this research article, the influence of magnetic and Aharonov-Bohm flux fields on Shannon entropy with improved Kratzer potential for Scandium fluoride (ScF) and hydrogen (H2) molecules are studied analytically and numerically for three low-lying states. The wave function and probability density which controls the chemical and physical properties of atomic and molecular systems have been obtained by solving the Schrodinger wave equation coupled with charged particle Hamiltonian via the Nikiforov-Uvarov functional analysis method. The total normalized wave function was expressed in terms of the hypergeometric function of Jacobi polynomials and then utilized to evaluate the Shannon entropy for two diatomic molecules in position and momentum spaces. The wave function and electronic densities were graphically analyzed. Our numerical results of Shannon entropies sum as applied to ScF and H2 were verified to obey the Shannon-entropy-based uncertainty relation. Observations from our results also reveal negative values of Shannon entropies in the position coordinates depicting high localization of particles and stability. In Addition, the effects of the external fields on the quantum systems were observed to influence the quantum information-theoretic measure by small decrements in the Shannon entropy values when compared to the case of the absence of the external fields, indicating a further improvement in the localization of particles and system's stability. This work has applications in atomic and molecular dynamic, molecular drug design, nanostructure processes, information technology, amongst others.
Quantum information entropies have been properly studied within the confines of nonrelativistic quantum mechanics ( QM ) by solving Schrödinger wave equation for different considerable potential models to obtained the energy and total normalised wave function which are essential tools for determining various information entropies. However, not much research has been done on evaluation of information entropies within the relativistic quantum mechanics because of its complexities and existence of both multiple and quadratic eigenvalues for fixed quantum state ( n ) and orbital angular quantum number (ℓ)(\ell )(ℓ) . In this work, we investigate Fisher information and Shannon entropy in three dimensional space within the relativistic regime by solving Klein-Gordon ( KG ) equation for equal scalar and vector potential with Schioberg plus Yukawa potential ( SPYP ) using the Parametric Nikiforov-Uvarov ( PNU ) method and compare the results to its nonrelativistic counterpart. It is important to note that the numerical solutions for the three dimensional Fisher information and Shannon entropy obtained for both the relativistic and nonrelativistic wave equations using SPYP adequately satisfies Bialynicki-Birula and Mycieslski (BBM) uncertainty inequality. The radial wave function, radial probability density, momentum space wave function and momentum probability density graphical plots were all obtained for both cases respectively. The numerical computations for Fisher information and Shannon entropy in three dimensional space were carried out for ℓ=0\ell = 0ℓ=0 with different quantum states n=0n= 0n=0 and 1. The investigation of Fisher information and Shannon entropy are relevance in describing localisation and delocalisation mechanism of particles in atomic orbitals within a potential well, relativistic confinement effect and entropy-energy correlation.
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.
This study investigates Fisher and Shannon entropies in one- and three-dimensional systems under the Radial Scalar Power Potential. Using the Nikiforov–Uvarov method combined with the Greene–Aldrich approximation, we derived energy eigenvalues and normalized wavefunctions. The results demonstrate that Shannon and Fisher entropies satisfy fundamental quantum information inequalities, including the Białynicki–Birula–Mycielski and Stam–Cramér–Rao bounds, across different spatial dimensions. Rényi entropy was also analyzed in both position and momentum spaces, revealing its dependence on the screening parameter and highlighting the complementarity in measurement precision between conjugate domains. In particular cases, the Radial Scalar Power Potential reduces to the Kratzer potential, allowing the computation of energy spectra for methylidyne (CH) and nitrogen (N₂) molecules. Energy increases with angular momentum, affecting molecular stability and spectroscopic transitions, while calculated oscillator strengths are in agreement with previous results, thereby validating the Radial Scalar Power Potential model for applications in both quantum information theory and molecular spectroscopy.
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.