Determination of Fisher and Shannon Information for 1D Fractional Quantum Harmonic Oscillator

Abstract

This work uses the Riesz-Feller fractional derivative to examine Fisher information and Shannon entropy in a one-dimensional fractional quantum harmonic oscillator. By computing the fractional derivative of the probability density function, we systematically assess these information-theoretic measures, providing deeper insights into the system’s probabilistic characteristics. In this context, we further explore the effect of the parameter \({\upalpha }\) on the one-dimensional fractional quantum harmonic oscillator and analyze its impact on both Fisher and Shannon parameters. We compute the position and momentum information entropies for low-lying quantum states \((n=0,1,2)\) to gain a clearer understanding of the system’s behavior. Additionally, we investigate key features of Fisher and Shannon densities as well as probability distributions to identify patterns in information distribution. The study also evaluates the validity of the Stam, Cramer-Rao, and Bialynicki–Birula–Mycielski (BBM) inequalities. In particular, we examine whether the BBM inequality holds true in the form \(S_{x}+S_{p}\ge 1+\ln \pi \), in accordance with standard quantum mechanics. Furthermore, we analyze the complexity measure in the context of the 1D Fractional Quantum Harmonic Oscillator, uncovering an increase in disorder within position space as the quantum number n grows.