Essentials of the Entropic Uncertainty Relations and Measurement of Fisher-Shannon Product of a Free Particle in Three Dimensions

Abstract

The purpose of this article is to calculate the numerical values of the Shannon (S) and Fisher (I) information entropies along with their Shannon power (J) and Fisher-Shannon product (P) for a free particle in three dimensions. In both coordinate and momentum space, the numerical values have been calculated for various values of the cubical box's length (L). Further, the validity of the entropic uncertainty relations such as the Bialynicki-Birula and Myceilski (BBM) inequality relation, Stam-Cramer-Rao inequalities (better known as the Fisher based uncertainty relation) and the Fisher-Shannon product relation have been investigated. The Fisher-Shannon product has been introduced and analyzed, which can be considered as a new correlation measure. The article is basically based on the concept, measurements and on the behaviour of the Shannon entropy, Fisher information entropy, Shannon power and the Fisher-Shannon product. Here, the plane wave solution of the free particle in three dimensions is considered and the wave function is normalized in an arbitrarily large but finite cube. The Fourier transforms of the coordinate space wave function provided the momentum space wave function. The probability densities corresponding to the respective wave functions have been employed to compute the numerical values of the information theoretic quantities such as Shannon entropy, Fisher entropy, Shannon power and the Fisher-Shannon product both in the coordinate and momentum space and the computed values have been presented in the respective tables. Finally, the entropic uncertainty relations investigated have been found consistently satisfying the corresponding inequalities such as Bialynicki-Birula and Mycielski (BBM) inequality, the Stam-Cramer-Rao inequalities and the Fisher-Shannon product relation. The study establishes the validity of the entropic uncertainty relations for the motion of a free particle in three dimensions.