A Theoretical Study on the Information Theoretic Inequalities and Fisher-Shannon Product of a Free Particle

In this article, the plane wave solution for a free particle in three dimensions is considered and the wave function is normalized in an arbitrarily large but finite cube. The momentum space wave function is obtained by taking the Fourier transform of the coordinate space wave function. The probability densities are employed to compute the numerical values of the information theoretic quantities such as Shannon information entropy (S), Fisher information entropy (I), Shannon power (J) and the Fisher–Shannon product (P) both in coordinate and momentum spaces for different values of the length (L) of the cubical box. Numerical values so found satisfy the Beckner, Bialynicki-Birula and Myceilski (BBM) inequality relation; Stam-Cramer-Rao inequalities (better known as the Fisher based uncertainty relation) and Fisher-Shannon product relation. This establishes the validity of the information theoretic inequalities in respect of the motion of a free particle.