Shannon and Fisher Entropy for a New Class of Single Hyperbolic Potentials in Fractional Schrödinger Equation

Abstract

We investigate the quantum information entropy for a class of single hyperbolic potentials within the context of the fractional Schrödinger equation (FSE). We find that as the derivative variable $n$ decreases, the position entropy density function $\rho \left(x\right)$ becomes more localized, and its peak heightens. However, there are differences in the degree of localization between the position entropy density functions for each hyperbolic potential, which can be attributed to the varying sizes of the potentials. Conversely, in momentum space, the momentum entropy density function $\rho \left(p\right)$ becomes more delocalized, and its peak lowers as the derivative variable $n$ decreases for both hyperbolic potentials studied. Our analysis also examines the BBM inequality, demonstrating that it is satisfied for different values of the potential depths. Finally, we explore the Fisher entropy and observe that it increases in position space while decreasing in momentum space as the depth of the wells increases. Our findings provide new insights into the behavior of quantum systems governed by hyperbolic potentials within the fractional Schrödinger framework. The observed localization effects in position space, delocalization in momentum space, and the validation of the BBM inequality highlight the role of fractional derivatives in modifying quantum entropy measures. These results deepen our understanding of quantum information entropy in non-local quantum systems. They may have implications for fields such as quantum transport in disordered media, semiconductor physics, and the study of anomalous diffusion processes in quantum mechanics.