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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 deriv...

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Autores principales: Santana-Carrillo, R., Peto, J. M. Velázquez, Sun, Guo-Hua, Dong, Shi-Hai
Formato: Online Artículo Texto
Lenguaje:English
Publicado: MDPI 2023
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10377981/
https://www.ncbi.nlm.nih.gov/pubmed/37509934
http://dx.doi.org/10.3390/e25070988
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author Santana-Carrillo, R.
Peto, J. M. Velázquez
Sun, Guo-Hua
Dong, Shi-Hai
author_facet Santana-Carrillo, R.
Peto, J. M. Velázquez
Sun, Guo-Hua
Dong, Shi-Hai
author_sort Santana-Carrillo, R.
collection PubMed
description 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.
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spelling pubmed-103779812023-07-29 Quantum Information Entropy for a Hyperbolic Double Well Potential in the Fractional Schrödinger Equation Santana-Carrillo, R. Peto, J. M. Velázquez Sun, Guo-Hua Dong, Shi-Hai Entropy (Basel) Article 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. MDPI 2023-06-28 /pmc/articles/PMC10377981/ /pubmed/37509934 http://dx.doi.org/10.3390/e25070988 Text en © 2023 by the authors. https://creativecommons.org/licenses/by/4.0/Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
spellingShingle Article
Santana-Carrillo, R.
Peto, J. M. Velázquez
Sun, Guo-Hua
Dong, Shi-Hai
Quantum Information Entropy for a Hyperbolic Double Well Potential in the Fractional Schrödinger Equation
title Quantum Information Entropy for a Hyperbolic Double Well Potential in the Fractional Schrödinger Equation
title_full Quantum Information Entropy for a Hyperbolic Double Well Potential in the Fractional Schrödinger Equation
title_fullStr Quantum Information Entropy for a Hyperbolic Double Well Potential in the Fractional Schrödinger Equation
title_full_unstemmed Quantum Information Entropy for a Hyperbolic Double Well Potential in the Fractional Schrödinger Equation
title_short Quantum Information Entropy for a Hyperbolic Double Well Potential in the Fractional Schrödinger Equation
title_sort quantum information entropy for a hyperbolic double well potential in the fractional schrödinger equation
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10377981/
https://www.ncbi.nlm.nih.gov/pubmed/37509934
http://dx.doi.org/10.3390/e25070988
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