Information-theoretic-based spreading measures of orthogonal polynomials

TL;DR

This paper investigates the spreading properties of orthogonal polynomials used in quantum systems using information-theoretic measures like Fisher information, Rényi, and Shannon entropies, providing new insights beyond standard deviation analysis.

Contribution

It introduces a novel analysis of polynomial spreading in quantum systems through information-theoretic quantities, extending beyond traditional measures.

Findings

Information-theoretic lengths offer new insights into polynomial spreading.

Fisher information and entropy measures capture facets of distribution not seen with standard deviation.

Enhanced understanding of wavefunction localization in quantum systems.

Abstract

The macroscopic properties of a quantum system strongly depend on the spreading of the physical eigenfunctions (wavefunctions) of its Hamiltonian operador over its confined domain. The wavefunctions are often controlled by classical or hypergeometric-type orthogonal polynomials (Hermite, Laguerre and Jacobi). Here we discuss the spreading of these polynomials over its orthogonality interval by means of various information-theoretic quantities which grasp some facets of the polynomial distribution not yet analyzed. We consider the information-theoretic lengths closely related to the Fisher information and R\'enyi and Shannon entropies, which quantify the polynomial spreading far beyond the celebrated standard deviation.