Entropy and Information of a harmonic oscillator in a time-varying electric field in 2D and 3D noncommutative spaces

TL;DR

This paper investigates how noncommutative geometry affects the information measures of a harmonic oscillator in a time-varying electric field across 2D and 3D spaces, providing exact solutions and analyzing uncertainty relations.

Contribution

It provides exact solutions for the Schrödinger equation in noncommutative spaces and examines the impact on Fisher information and Shannon entropy, including modifications needed for noncommutative settings.

Findings

Fisher information required modification to satisfy Cramer-Rao inequalities

Noncommutativity alters Fisher information and Shannon entropy in position and momentum

Entropic uncertainty relation remains valid in noncommutative spaces

Abstract

We analyzed the noncommutativity effects on the Fisher information (F_(r,p)) and Shannon entropies (S_(r,p)) of a harmonic oscillator immersed in a time-varying electric field in two and three dimensions. We find the exact solutions of the respective time-dependent Schr\"odinger equation and use them to calculate the Fisher information and the Shannon entropy for the simplest case corresponding the lowest-lying state of each system. While there is no problem in defining the Shannon entropy for noncommutating spaces, the definition of the Fisher information had to be modified to satisfy the Cramer-Rao inequalities. We observe for both systems, how the Fisher information and Shannon entropy in position and momentum change due to the noncommutativity of the space. We verified that the Bialynicki-Birula-Mycielski (BBM) entropic uncertainty relation still holds in the systems considered.