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Quantum Information Entropy on Another Class of Single Hyperbolic Potentials in Fractional Schrödinger Equation

R. Santana Carrillo, David Maya‐Franco, Shi‐Hai Dong, Guo-Hua Sun

Published Jan 1, 2024• 0 citations
Publisher
DOI
10.2139/ssrn.4986986

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In this work we have studied the Shannon information entropy for two hyperbolic single-well potentials in the fractional Schrödinger equation (the fractional derivative number (0<n≤2) by calculating position and momentum entropy. We find that the wave function will move towards the origin as the fractional derivative number n decreases and the position entropy density becomes more severely localized in more fractional system, i.e., for smaller values of n, but the momentum probability density becomes more delocalized. And then we study the Beckner Bialynicki-Birula−Mycieslki (BBM) inequality and notice that the Shannon entropies still satisfy this inequality for different depth u even though this inequality decreases (or increases) gradually as the depth u of the hyperbolic potential U1 (or U2) increases. Finally, we also carry out the Fisher entropy and observe that the Fisher entropy increases as the depth u of the potential wells increases, while the fractional derivative number n decreases.
ABSTRACT In this work, we investigate quantum information entropy for double hyperbolic well potentials within the framework of the fractional Schrödinger equation (FSE). Specifically, we analyze the position and momentum Shannon entropies for two hyperbolic potentials, and , as a function of the fractional derivative order . Our findings reveal that decreasing enhances wave function localization in position space, thereby reducing spatial uncertainty while simultaneously increasing momentum uncertainty. We confirm the validity of the Beckner–Bialynicki‐Birula–Mycielski inequality for both potentials, demonstrating its robustness across different degrees of nonlocality. Furthermore, we explore the behavior of Fisher information, observing that it increases in position space while decreases in momentum space as the well depths grow.