Quantum Information Entropy for a Hyperbolic Double Well Potential in the Fractional Schrödinger Equation.

In this study, we investigate the position and momentum Shannon entropy, denoted as Sx and Sp, respectively, in the context of the fractional Schrödinger equation (FSE) for a hyperbolic double well potential (HDWP). We explore various values of the fractional derivative represented by in our analysis. Our findings reveal intriguing behavior concerning the localization properties of the position entropy density, ρs(x), and the momentum entropy density, ρs(p), for low-lying states. Specifically, as the fractional derivative decreases, ρs(x) becomes more localized, whereas ρs(p) becomes more delocalized. Moreover, we observe that as the derivative decreases, the position entropy Sx decreases, while the momentum entropy Sp increases. In particular, the sum of these entropies consistently increases with decreasing fractional derivative . It is noteworthy that, despite the increase in position Shannon entropy Sx and the decrease in momentum Shannon entropy Sp with an increase in the depth of the HDWP, the Beckner-Bialynicki-Birula-Mycielski (BBM) inequality relation remains satisfied. Furthermore, we examine the Fisher entropy and its dependence on the depth of the HDWP and the fractional derivative . Our results indicate that the Fisher entropy increases as the depth of the HDWP is increased and the fractional derivative is decreased.