 In this study, we investigated the position and momentum Shannon entropy, denoted as Sx and Sp, respectively, in the context of the fractional Schrodinger equation, FSE, for a hyperbolic double well potential, HDWP. We explored various values of the fractional derivative represented by K in our analysis. Our findings revealed interesting behavior regarding the localization properties of the position entropy density, rho s, x, and the momentum entropy density, rho s, p, for low lying states. As the fractional derivative K decreased, rho s, x, became more localized, while rho s, p, became more delocalized. Additionally, we observed that as the derivative K decreased, the position Shannon entropy Sx decreased, while the momentum Shannon entropy Sp increased. Notably, despite the increase in position Shannon entropy Sx and the decrease in momentum Shannon entropy Sp with an increase in the depth view of the HDWP, the Bekner-Biolynnicki-Berylomysielski, BBM, inequality relation remained satisfied. Furthermore, we examined the Fischer entropy and its dependence on the depth view of the HDWP and the. This article was authored by our Santana Carrillo, J. M. Velasquez-Pito, Guohua Sun, and others. We are article.tv, links in the description below.