FISHER-SHANNON ENTROPIC UNCERTAINTY RELATIONS AND THEIR POWER-PRODUCTS AS A MEASURE OF ELECTRONIC CORRELATION
By
¹Sudin Singh and ²Aparna Saha
¹Department of Physics, Bolpur College, Bolpur, Birbhum, West Bengal, India-731204
²Department of Physics, Visva-Bharati University, Santiniketan, West Bengal, India-731235
¹Corresponding Author
Email: skyalpha731204@gmail.com, aparnasaha1507@gmail.com
(Received: February 22, 2022; In format: March 19, 2022; Revised: May 19, 2023; Accepted: May 26, 2023)
DOI: https://doi.org/10.58250/jnanabha.2023.53130
Abstract
In this paper, we have presented an analytical model of two electron systems consisting of a many particle correlated wave function with some variational parameters α, λ and µ and used it to quantify the electron-electron correlation described by the wave function containing explicitly r12
(inter atomic distance between two electrons) dependent term. The single particle wave functions and the charge densities have been extracted from the said correlated wave function both for the uncorrelated and correlated systems in coordinate space and its momentum analogs have been obtained by taking the Fourier transform of the coordinate analogs. We have computed and presented the results of the numerical values of the theoretic information entropies of the Shannon entropy, Fisher information entropy, Shannon power and the FisherShannon product. The numerical values are consistently found to satisfy the Beckner, Bialynicki-Birula and Mycielski (BBM ) inequality relation; Stam-Cramer-Rao inequalities or Fisher based uncertainty relation and Fisher-Shannon product relation for the uncorrelated and correlated systems in both the coordinate and momentum spaces.
2020 Mathematical Sciences Classification: 62B10, 94A15, 94A17.
Keywords and Phrases: Coordinate and momentum space; uncorrelated and correlated system; Shannon information entropy; Fisher information entropy; uncertainty relations; Fisher-Shannon product.