Quantum Information Measures in Quartic and Symmetric Potentials using perturbative approach

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.