| Resumen |
In this study, we investigate the position and momentum Shannon entropy, denoted as (Formula presented.) and (Formula presented.), 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 presented.), and the momentum entropy density, (Formula presented.), for low-lying states. Specifically, as the fractional derivative k decreases, (Formula presented.) becomes more localized, whereas (Formula presented.) becomes more delocalized. Moreover, we observe that as the derivative k decreases, the position entropy (Formula presented.) decreases, while the momentum entropy (Formula presented.) 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 presented.) and the decrease in momentum Shannon entropy (Formula presented.) 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. © 2023 by the authors.
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