In this work, we investigate the Shannon entropy of four recently proposed hyperbolic potentials through studying position and momentum entropies. Our analysis reveals that the wave functions of the single-well potentials U0,3 exhibit greater localization compared to the double-well potentials U1,2. This difference in localization arises from the depths of the single- and double-well potentials. Specifically, we observe that the position entropy density shows higher localization for the single-well potentials, while their momentum probability density becomes more delocalized. Conversely, the double-well potentials demonstrate the opposite behavior, with position entropy density being less localized and momentum probability density showing increased localization. Notably, our study also involves examining the Bialynicki-Birula and Mycielski (BBM) inequality, where we find that the Shannon entropies still satisfy this inequality for varying depths u¯. An intriguing observation is that the sum of position and momentum entropies increases with the variable u¯ for potentials U1,2,3, while for U0, the sum decreases with u¯. Additionally, the sum of the cases U0 and U3 almost remains constant within the relative value 0.01 as u¯ increases. Our study provides valuable insights into the Shannon entropy behavior for these hyperbolic potentials, shedding light on their localization characteristics and their relation to the potential depths. Finally, we extend our analysis to the Fisher entropy F¯x and find that it increases with the depth u¯ of the potential wells but F¯p decreases with the depth.
Keywords: BBM inequality; Fisher entropy; Shannon entropy; double well potential; hyperbolic potentials; single well potential.