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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>x</mi></msub></semantics&g...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-06-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/25/7/988 |
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| author | R. Santana-Carrillo J. M. Velázquez Peto Guo-Hua Sun Shi-Hai Dong |
| author_facet | R. Santana-Carrillo J. M. Velázquez Peto Guo-Hua Sun Shi-Hai Dong |
| author_sort | R. Santana-Carrillo |
| collection | DOAJ |
| description | In this study, we investigate the position and momentum Shannon entropy, denoted as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>x</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>p</mi></msub></semantics></math></inline-formula>, 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 <i>k</i> in our analysis. Our findings reveal intriguing behavior concerning the localization properties of the position entropy density, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ρ</mi><mi>s</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, and the momentum entropy density, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ρ</mi><mi>s</mi></msub><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, for low-lying states. Specifically, as the fractional derivative <i>k</i> decreases, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ρ</mi><mi>s</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> becomes more localized, whereas <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ρ</mi><mi>s</mi></msub><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> becomes more delocalized. Moreover, we observe that as the derivative <i>k</i> decreases, the position entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>x</mi></msub></semantics></math></inline-formula> decreases, while the momentum entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>p</mi></msub></semantics></math></inline-formula> increases. In particular, the sum of these entropies consistently increases with decreasing fractional derivative <i>k</i>. It is noteworthy that, despite the increase in position Shannon entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>x</mi></msub></semantics></math></inline-formula> and the decrease in momentum Shannon entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>p</mi></msub></semantics></math></inline-formula> with an increase in the depth <i>u</i> 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 <i>u</i> of the HDWP and the fractional derivative <i>k</i>. Our results indicate that the Fisher entropy increases as the depth <i>u</i> of the HDWP is increased and the fractional derivative <i>k</i> is decreased. |
| first_indexed | 2024-03-11T01:06:42Z |
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| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-11T01:06:42Z |
| publishDate | 2023-06-01 |
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| spelling | doaj.art-5906cfaa4bd64d3cbd5d36b4aa313f9e2023-11-18T19:13:02ZengMDPI AGEntropy1099-43002023-06-0125798810.3390/e25070988Quantum Information Entropy for a Hyperbolic Double Well Potential in the Fractional Schrödinger EquationR. Santana-Carrillo0J. M. Velázquez Peto1Guo-Hua Sun2Shi-Hai Dong3Centro de Investigación en Computación, Instituto Politécnico Nacional, UPALM, Mexico City 07700, MexicoESIME-Culhuacan, Instituto Politécnico Nacional, Av. Santa Ana 1000, Mexico City 04430, MexicoCentro de Investigación en Computación, Instituto Politécnico Nacional, UPALM, Mexico City 07700, MexicoCentro de Investigación en Computación, Instituto Politécnico Nacional, UPALM, Mexico City 07700, MexicoIn this study, we investigate the position and momentum Shannon entropy, denoted as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>x</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>p</mi></msub></semantics></math></inline-formula>, 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 <i>k</i> in our analysis. Our findings reveal intriguing behavior concerning the localization properties of the position entropy density, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ρ</mi><mi>s</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, and the momentum entropy density, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ρ</mi><mi>s</mi></msub><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, for low-lying states. Specifically, as the fractional derivative <i>k</i> decreases, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ρ</mi><mi>s</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> becomes more localized, whereas <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ρ</mi><mi>s</mi></msub><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> becomes more delocalized. Moreover, we observe that as the derivative <i>k</i> decreases, the position entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>x</mi></msub></semantics></math></inline-formula> decreases, while the momentum entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>p</mi></msub></semantics></math></inline-formula> increases. In particular, the sum of these entropies consistently increases with decreasing fractional derivative <i>k</i>. It is noteworthy that, despite the increase in position Shannon entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>x</mi></msub></semantics></math></inline-formula> and the decrease in momentum Shannon entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>p</mi></msub></semantics></math></inline-formula> with an increase in the depth <i>u</i> 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 <i>u</i> of the HDWP and the fractional derivative <i>k</i>. Our results indicate that the Fisher entropy increases as the depth <i>u</i> of the HDWP is increased and the fractional derivative <i>k</i> is decreased.https://www.mdpi.com/1099-4300/25/7/988hyperbolic double well potentialfractional Schrödinger equationShannon entropyFisher entropy |
| spellingShingle | R. Santana-Carrillo J. M. Velázquez Peto Guo-Hua Sun Shi-Hai Dong Quantum Information Entropy for a Hyperbolic Double Well Potential in the Fractional Schrödinger Equation Entropy hyperbolic double well potential fractional Schrödinger equation Shannon entropy Fisher entropy |
| 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 schrodinger equation |
| topic | hyperbolic double well potential fractional Schrödinger equation Shannon entropy Fisher entropy |
| url | https://www.mdpi.com/1099-4300/25/7/988 |
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