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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...
Autores principales: | , , , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
MDPI
2023
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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. |
format | Online Article Text |
id | pubmed-10377981 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2023 |
publisher | MDPI |
record_format | MEDLINE/PubMed |
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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