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Science World Journal

FISHER INFORMATION AND QUANTUM LOCALIZATION IN A CALOGERO-LIKE POTENTIAL UNDER VARYING MAGNETIC FIELDS

Rasaq Adekunle Kareem, Maitama A.Y. Hotoro, Oluwaseun Raphael Aderele

Published Sep 29, 2026• 0 citations
OpenAlex

Abstract

This research examines the influence of a particle in a quantum-mechanical system constrained in a time-dependent magnetic field and the strengths of a Calogero-like potential. Analytical expressions for the expectation values of the relevant observables were first derived from the exact energy spectrum using the total confinement frequency, which incorporates both the intrinsic confinement and cyclotron frequencies. These expectation values were subsequently employed to evaluate the Fisher information (FI) in both position space (PS) and momentum space (MS), and to assess the effects of the external magnetic fields and the strength of the Calogero-like potential on quantum localization and uncertainty. The expectation values increase with increasing magnetic field strength, resulting in spatial compression and enhanced localization, while stronger Calogero-like potential strengths mimic centrifugal confinement. The examination of FI for both ground and first-excited states across magnetic quantum numbers highlights that localization sharpens with increasing field strength and potential, with observable divergences between quantum states and magnetic quantum numbers. However, the product of the FI was evaluated as the Cramér-Rao bound, showcasing the interplay between quantum uncertainty and information content. A comparison of the ground and first excited states for two magnetic quantum numbers reveals whether the FI product obeys or violates the Cramér-Rao bound, offering insight into quantum stability and control. These results reinforce the controllability of quantum state localization via external field modulation, with implications for quantum control and confinement in potential applications. To the best of our understanding, this is the first time that the analysis of FI for a quantum system governed by a Calogero-like potential in the presence of a varying magnetic field, specifically using a total confinement frequency, is reported. The results provide new insights into quantum localization, information-theoretic uncertainty, and the controlled manipulation of confined quantum states, with potential applications in quantum control, quantum information processing, and nanoscale confinement technologies.

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This research examines the influence of a particle in a quantum-mechanical system constrained in a time-dependent magnetic field and the strengths of a Calogero-like potential. Analytical expressions for the expectation values of the relevant observables were first derived from the exact energy spectrum using the total confinement frequency, which incorporates both the intrinsic confinement and cyclotron frequencies. These expectation values were subsequently employed to evaluate the Fisher information (FI) in both position space (PS) and momentum space (MS), and to assess the effects of the external magnetic fields and the strength of the Calogero-like potential on quantum localization and uncertainty. The expectation values increase with increasing magnetic field strength, resulting in spatial compression and enhanced localization, while stronger Calogero-like potential strengths mimic centrifugal confinement. The examination of FI for both ground and first-excited states across magnetic quantum numbers highlights that localization sharpens with increasing field strength and potential, with observable divergences between quantum states and magnetic quantum numbers. However, the product of the FI was evaluated as the Cramér-Rao bound, showcasing the interplay between quantum uncertainty and information content. A comparison of the ground and first excited states for two magnetic quantum numbers reveals whether the FI product obeys or violates the Cramér-Rao bound, offering insight into quantum stability and control. These results reinforce the controllability of quantum state localization via external field modulation, with implications for quantum control and confinement in potential applications. To the best of our understanding, this is the first time that the analysis of FI for a quantum system governed by a Calogero-like potential in the presence of a varying magnetic field, specifically using a total confinement frequency, is reported. The results provide new insights into quantum localization, information-theoretic uncertainty, and the controlled manipulation of confined quantum states, with potential applications in quantum control, quantum information processing, and nanoscale confinement technologies.
Fisher information (FI) and Shannon entropy (SP), as fundamental theoretical constructs, play crucial roles in quantum information, offering significant insights into communication systems. This work addresses the radial Schrödinger equation for a cotangent hyperbolic potential, solved using the parametric Nikiforov–Uvarov method. The derived wave functions are utilized to explore FI and SP within a one-dimensional system. The analysis focuses on the influence of the screening parameter and potential depth on these quantities, examining their behavior for the ground state and the first excited state. The study complies with key theoretical principles, including the Cramer–Rao inequality, the Beckner-Bialynicki-Birula-Mycielski (BBM) inequality, and the Heisenberg uncertainty principle. The results shed light on the dynamics of quantum states, emphasizing the relationship between quantum uncertainties and information measures. Significant variations in FI and SP emerge with changes in the screening parameter and potential depth, enhancing the understanding of quantum mechanics in confined systems. The findings also reveal the phenomenon of entropy squeezing, underscoring the complex nature of quantum uncertainties. The model demonstrates realistic behavior, with distinct transition points in SP as a function of potential depth, highlighting unique and intriguing characteristics. These contributions enrich the theoretical framework for investigating quantum systems, providing valuable perspectives on their intricate behaviors.
This work provides analytical study of a two-dimensional quantum harmonic oscillator (HO) coupled with a perpendicular magnetic field (B) in the x-direction and an inverse-square potential in the y-direction. The system’s Hamiltonian is exactly solved via variable separation, to obtain the energy levels and wavefunctions composed of Hermite and associated Laguerre polynomials. Fisher information (FI) and Shannon entropy (SE) for both position space (PS) and momentum space (MS) are fully examined under the magnetic field strength and anisotropic potential parameters. The canonical partition function is computed and thermodynamic quantities such as internal energy, entropy, and Gibbs free energy are evaluated. The result shows that the presence of magnetic field enhances spatial localization and reduces uncertainty in PS but has a revised effect in the MS. In the thermodynamic properties, there is asymmetric thermodynamic responses in the x- and y-directions due to the presence of the magnetic field and the inverse-square potential. This study highlights the deep relationship between quantum information theory and thermodynamics in low-dimensional systems with structured interactions.
We introduce a qqq-deformed Deng-Fan potential (qqqDFP) model that enables controlled modulation of short-range repulsion and long-range attraction while preserving the equilibrium configuration. The model is solved exactly within the framework of the time-independent Schrödinger equation, yielding closed-form expressions for the energy eigenvalues and wave functions in terms of hypergeometric functions. We show that the deformation parameter qqq induces non-uniform spectral shifts and a redistribution of bound states. In particular, for q<1q<1q<1, the system exhibits spectral compression and enhanced spatial localization. In addition, we investigate the system from an information-theoretic perspective using Shannon entropy, Fisher information, and Fisher--Shannon complexity measures in both position and momentum spaces. The results reveal that the deformation parameter governs the redistribution of quantum information, establishing a direct connection between spatial confinement and momentum delocalization in accordance with the Białynicki-Birula and Mycielski entropic uncertainty principle. Stronger deformation pushes the quantum state further from the minimum-uncertainty configuration, increasing the entropic excess above the BBM bound and reducing the information content about complementary observables, even as position-space localization sharpens. The analysis of entropic and Fisher information densities further shows how the deformation reshapes both the local information content and the structural complexity of the quantum states. We show that in the limit q→1q\to 1q→1, the qqqDFP model is reduced to the standard Deng-Fan potential.
Thermomagnetic properties, and its effects on Fisher information entropy with Schioberg plus Manning-Rosen potential are studied using NUFA and SUSYQM methods in the presence of the Greene-Aldrich approximation scheme to the centrifugal term. The wave function obtained was used to study Fisher information both in position and momentum spaces for different quantum states by the gamma function and digamma polynomials. The energy equation obtained in a closed form was used to deduce numerical energy spectra, partition function, and other thermomagnetic properties. The results show that with an application of AB and magnetic fields, the numerical energy eigenvalues for different magnetic quantum spins decrease as the quantum state increases and completely removes the degeneracy of the energy spectra. Also, the numerical computation of Fisher information satisfies Fisher information inequality products, indicating that the particles are more localized in the presence of external fields than in their absence, and the trend shows complete localization of quantum mechanical particles in all quantum states. Our potential reduces to Schioberg and Manning-Rosen potentials as special cases. Our potential reduces to Schioberg and Manning-Rosen potentials as special cases. The energy equations obtained from the NUFA and SUSYQM were the same, demonstrating a high level of mathematical precision.
We investigate the quantum-information properties of a particle confined by the double Morse potential in position and momentum spaces. The quasi-exact solvability of the model gives analytical expressions for the first two bound states, allowing the corresponding probability densities to be analyzed directly. Shannon entropy, Onicescu energy, Fisher information, statistical complexity, and Fisher-Shannon products are evaluated as functions of the parameter AAA, which controls the transition from a well-separated double well to a merged single-well profile. The position distribution is more delocalized and structurally complex when the double-well character is pronounced, whereas the momentum distribution exhibits the complementary trend. As the wells merge, the ground-state Fisher--Shannon product approaches its Gaussian reference value, whereas the excited state retains stronger non-Gaussian structure.
An uncertainty Fisher information relation in quantum mechanics is derived for multidimensional single-particle systems with central potentials. It is based on the concept of Fisher information in the two complementary position and momentum spaces, which is a gradient functional of the corresponding probability distributions. The lower bound of the product of position and momentum Fisher informations is shown to depend on the orbital and magnetic quantum numbers of the physical state and the space dimensionality. Applications to various elementary systems is discussed.