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Revista Mexicana de Física

Fisher information and quantum entropies of a 2D system under a pair of non-central scalar and vector potentials

Ahmed Becir, Mustafa Moumni

Published Nov 1, 2025• 0 citations
Publisher

Abstract

In this study, we examine a 2 dimensional system influenced by a non-central potential consisting of a Kratzer potential with a dipole moment, along with a vector potential of the (AB) effect. We explore various information-theoretic measures, including Fisher information, Shannon entropy, Tsallis entropy, and Rényi entropy. Our numerical results show that the Fisher information increases with an increase in dissociation energy and decreases with rising dipole moment, Aharonov–Bohm potential strength, and both the radial and angular quantum numbers. In contrast, the Shannon entropy, the Tsallis entropy, and the Rényi entropy decrease with rising dissociation energy, while they increase with an increase in dipole moment, Aharonov–Bohm potential strength, as well as the principal and angular quantum numbers. These observations collectively indicate that the precision and localization of particles in space are enhanced by the increasing of the dissociation energy and reduced when the dipole moment, Aharonov–Bohm potential strength, and both the radial and angular quantum numbers increase.

DOI
10.31349/revmexfis.71.061301

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BENTHAM SCIENCE PUBLISHERS eBooks•Dec 11, 2013•1 citation
In this study, we examine a 2 dimensional system influenced by a non-central potential consisting of a Kratzer potential with a dipole moment, along with a vector potential of the (AB) effect. We explore various information-theoretic measures, including Fisher information, Shannon entropy, Tsallis entropy, and Rényi entropy. Our numerical results show that the Fisher information increases with an increase in dissociation energy and decreases with rising dipole moment, Aharonov–Bohm potential strength, and both the radial and angular quantum numbers. In contrast, the Shannon entropy, the Tsallis entropy, and the Rényi entropy decrease with rising dissociation energy, while they increase with an increase in dipole moment, Aharonov–Bohm potential strength, as well as the principal and angular quantum numbers. These observations collectively indicate that the precision and localization of particles in space are enhanced by the increasing of the dissociation energy and reduced when the dipole moment, Aharonov–Bohm potential strength, and both the radial and angular quantum numbers increase.
Abstract In this study, we examine a 2 dimensional system influenced by a non-central potential consisting of a Kratzer potential with a dipole moment, along with a vector potential of the (AB) effect. We explore various information-theoretic measures, including Fisher information, Shannon entropy, Tsallis entropy, and Rényi entropy. Our numerical results show that the Fisher information increases with an increase in dissociation energy and decreases with rising dipole moment, Aharonov–Bohm potential strength, and both the radial and angular quantum numbers. In contrast, the Shannon entropy, the Tsallis entropy, and the Rényi entropy decrease with rising dissociation energy, while they increase with an increase in dipole moment, Aharonov–Bohm potential strength, as well as the principal and angular quantum numbers. These observations collectively indicate that the precision and localization of particles in space are enhanced by the increasing of the dissociation energy and reduced when the dipole moment, Aharonov–Bohm potential strength, and both the radial and angular quantum numbers increase.
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.
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.
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.
There has been an interest in studies on informationtheoretic and complexity measures for quantum model systems in the recent years. The information entropy for the trigonometric Rosen-Morse potential, harmonic oscillator, hydrogenic atoms, Morse potential, Yukawa and Hulthen potentials have been investigated. The Kratzer potential plays an important role in atomic and molecular physics and quantum chemistry. Cheng and Dai proposed a new potential consisting of the modified Kratzer potential plus a new proposed ring-shaped potential which has possible applications to ring-shaped organic molecules like cyclic polyenes and benzene. In this work, we shall obtain the Fisher information, Fisher length, Shannon entropy, Shannon length, Renyi entropy, Renyi length, Tsallis entropy and the Fisher-Shannon complexity for the ring-shaped modified Kratzer potential. In this section, we present the definition of the information theoretical measures used in this work which are based on the 1-normalized electron density in the position and momentum space. The Shannon information entropy in the position space is given by Eq. (1). ( ) ( ) log ( )d S ρ ρ ρ = −∫ r r r ... (1)
The Fisher information functional occupies a very special position among the uncertainty measures of quantum systems of any dimensionality. In contrast with (global) entropic measures of Shannon, Renyi or Tsallis types, it contitutes a local measure in the sense that it is sensitive to the oscillatory character of the probability density distribution. Moreover, the Fisher information is closely related to various macroscopic properties of the system described by different density functionals. Here, we review its main analytical properties such as, inequality-based relationships with some radial expectation values and its uncertainty relation. Tighter versions of these inequalities are shown for those systems subject to an arbitrary central potential. Moreover, the Fisher information is computed for D-dimensional hydrogenic systems explicitly in terms of the radial and angular hyperquantum numbers characterizing the quantum states of the system in both position and momentum spaces. Finally, the utility of this quantity is discussed for various physico-chemical processes of recent interest: Abstraction and nucleophilic substitution reactions.