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

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.