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Entropy

Quantum Information Entropy for Another Class of New Proposed Hyperbolic Potentials

R. Santana Carrillo, Roberto de J. León‐Montiel, Guo-Hua Sun, Shi‐Hai Dong

Published Sep 5, 2023• 8 citations
Publisher

Abstract

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.

DOI
10.3390/e25091296

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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.
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.
In this study, we investigate the position and momentum Shannon entropy, denoted as Sx and Sp, 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, ρs(x), and the momentum entropy density, ρs(p), for low-lying states. Specifically, as the fractional derivative k decreases, ρs(x) becomes more localized, whereas ρs(p) becomes more delocalized. Moreover, we observe that as the derivative k decreases, the position entropy Sx decreases, while the momentum entropy Sp 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 Sx and the decrease in momentum Shannon entropy Sp 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.
In this work, we study the quantum information entropies for two different types of hyperbolic single potential wells. We first study the behaviors of the moving particle subject to two different hyperbolic potential wells through focusing on their wave functions. The shapes of these hyperbolic potentials are similar, but we notice that their momentum entropy densities change along with the width of each potential and the magnitude of position entropy density decreases when the momentum entropy magnitude increases. On the other hand, we illustrate the behaviors of their position and momentum entropy densities. Finally, we show the variation of position and momentum entropies Sx and Sp with the change of the potential well depth u and verify that their sum still satisfies the BBM inequality relation.
ABSTRACT We investigate the quantum information entropy for a class of single hyperbolic potentials within the context of the fractional Schrödinger equation (FSE). We find that as the derivative variable decreases, the position entropy density function becomes more localized, and its peak heightens. However, there are differences in the degree of localization between the position entropy density functions for each hyperbolic potential, which can be attributed to the varying sizes of the potentials. Conversely, in momentum space, the momentum entropy density function becomes more delocalized, and its peak lowers as the derivative variable decreases for both hyperbolic potentials studied. Our analysis also examines the BBM inequality, demonstrating that it is satisfied for different values of the potential depths. Finally, we explore the Fisher entropy and observe that it increases in position space while decreasing in momentum space as the depth of the wells increases. Our findings provide new insights into the behavior of quantum systems governed by hyperbolic potentials within the fractional Schrödinger framework. The observed localization effects in position space, delocalization in momentum space, and the validation of the BBM inequality highlight the role of fractional derivatives in modifying quantum entropy measures. These results deepen our understanding of quantum information entropy in non‐local quantum systems. They may have implications for fields such as quantum transport in disordered media, semiconductor physics, and the study of anomalous diffusion processes in quantum mechanics.
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.
The Shannon information entropy for the Schrödinger equation with a nonuniform solitonic mass is evaluated for a hyperbolic-type potential. The number of nodes of the wave functions in the transformed space z are broken when recovered to original space x . The position S x and momentum S p information entropies for six low-lying states are calculated. We notice that the S x decreases with the increasing mass barrier width a and becomes negative beyond a particular width a , while the S p first increases with a and then decreases with it. The negative S x exists for the probability densities that are highly localized. We find that the probability density ρ ( x ) for n = 1, 3, 5 are greater than 1 at position x = 0. Some interesting features of the information entropy densities ρ s ( x ) and ρ s ( p ) are demonstrated. The Bialynicki–Birula–Mycielski (BBM) inequality is also tested for these states and found to hold.