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 [Formula: see text] and [Formula: see text] , 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 k in our analysis. Our findings reveal intriguing behavior concerning the localization properties of the position entropy density, [Formula: see text] , and the momentum entropy density, [Formula: see text] , for low-lying states. Specifically, as the fractional derivative k decreases, [Formula: see text] becomes more localized, whereas [Formula: see text] becomes more delocalized. Moreover, we observe that as the derivative k decreases, the position entropy [Formula: see text] decreases, while the momentum entropy [Formula: see text] increases. In particular, the sum of these entropies consistently increases with decreasing fractional derivative k. It is noteworthy that, despite the increase in position Shannon entropy [Formula: see text] and the decrease in momentum Shannon entropy [Formula: see text] with an increase in the depth u 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 u of the HDWP and the fractional derivative k. Our results indicate that the Fisher entropy increases as the depth u of the HDWP is increased and the fractional derivative k is decreased.