Quantum Information Measures in Quartic and Symmetric Potentials using perturbative approach

TL;DR

This paper investigates quantum information measures like Shannon and Fisher information for particles in quartic and symmetric potentials, using perturbation theory to analyze how these measures evolve with potential width and confirming fundamental uncertainty principles.

Contribution

It introduces a perturbative approach to compute quantum information measures in specific potentials and explores their behavior, providing new insights into quantum uncertainty relations.

Findings

Shannon entropy decreases in position space and increases in momentum space with potential width.

Fisher information remains nearly constant for quartic potential, decreases in position space for symmetric potential.

Bialynicki-Birula-Mycielski inequality is validated across cases.

Abstract

We analyze the Shannon and Fisher information measures for systems subjected to quartic and symmetric potential wells. The wave functions are obtained by solving the time-independent Schr\"{o}dinger equation, using aspects of perturbation theory. We examine how the information for various quantum states evolves with changes in the width of the potential well. For both potentials, the Shannon entropy decreases in position space and increases in momentum space as the width increases, maintaining a constant sum of entropies, consistent with Heisenberg's uncertainty principle. The Fisher information measure shows different behaviors for the two potentials: it remains nearly constant for the quartic potential. For the symmetric well potential, the Fisher information decreases in position space and increases in momentum space as localization in position space increases, also consistent with the analogue of Heisenberg's uncertainty principle. Additionally, the Bialynicki-Birula-Mycielski inequality is evaluated across various cases and is confirmed to hold in each instance.