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Riccati and Ermakov Equations in Time-Dependent and Time-Independent Quantum Systems

Riccati and Ermakov Equations in Time-Dependent and Time-Independent Quantum Systems

The time-evolution of the maximum and the width of exact analytic wave packet (WP) solutions of the time-dependent Schrödinger equation (SE) represents the particle and wave aspects, respectively, of the quantum system. The dynamics of the maximum, located at the mean value of position, is governed...

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Main Author: Schuch, D.
Format: Article
Language:English
Published: Інститут математики НАН України 2008
Series:Symmetry, Integrability and Geometry: Methods and Applications
Online Access:http://dspace.nbuv.gov.ua/handle/123456789/149040
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spelling irk-123456789-1490402019-02-20T01:28:52Z Riccati and Ermakov Equations in Time-Dependent and Time-Independent Quantum Systems Schuch, D. The time-evolution of the maximum and the width of exact analytic wave packet (WP) solutions of the time-dependent Schrödinger equation (SE) represents the particle and wave aspects, respectively, of the quantum system. The dynamics of the maximum, located at the mean value of position, is governed by the Newtonian equation of the corresponding classical problem. The width, which is directly proportional to the position uncertainty, obeys a complex nonlinear Riccati equation which can be transformed into a real nonlinear Ermakov equation. The coupled pair of these equations yields a dynamical invariant which plays a key role in our investigation. It can be expressed in terms of a complex variable that linearizes the Riccati equation. This variable also provides the time-dependent parameters that characterize the Green's function, or Feynman kernel, of the corresponding problem. From there, also the relation between the classical and quantum dynamics of the systems can be obtained. Furthermore, the close connection between the Ermakov invariant and the Wigner function will be shown. Factorization of the dynamical invariant allows for comparison with creation/annihilation operators and supersymmetry where the partner potentials fulfil (real) Riccati equations. This provides the link to a nonlinear formulation of time-independent quantum mechanics in terms of an Ermakov equation for the amplitude of the stationary state wave functions combined with a conservation law. Comparison with SUSY and the time-dependent problems concludes our analysis. 2008 Article Riccati and Ermakov Equations in Time-Dependent and Time-Independent Quantum Systems / D. Schuch // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 22 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 37J15; 81Q05; 81Q60; 81S30 http://dspace.nbuv.gov.ua/handle/123456789/149040 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description The time-evolution of the maximum and the width of exact analytic wave packet (WP) solutions of the time-dependent Schrödinger equation (SE) represents the particle and wave aspects, respectively, of the quantum system. The dynamics of the maximum, located at the mean value of position, is governed by the Newtonian equation of the corresponding classical problem. The width, which is directly proportional to the position uncertainty, obeys a complex nonlinear Riccati equation which can be transformed into a real nonlinear Ermakov equation. The coupled pair of these equations yields a dynamical invariant which plays a key role in our investigation. It can be expressed in terms of a complex variable that linearizes the Riccati equation. This variable also provides the time-dependent parameters that characterize the Green's function, or Feynman kernel, of the corresponding problem. From there, also the relation between the classical and quantum dynamics of the systems can be obtained. Furthermore, the close connection between the Ermakov invariant and the Wigner function will be shown. Factorization of the dynamical invariant allows for comparison with creation/annihilation operators and supersymmetry where the partner potentials fulfil (real) Riccati equations. This provides the link to a nonlinear formulation of time-independent quantum mechanics in terms of an Ermakov equation for the amplitude of the stationary state wave functions combined with a conservation law. Comparison with SUSY and the time-dependent problems concludes our analysis.
format Article
author Schuch, D.
spellingShingle Schuch, D.
Riccati and Ermakov Equations in Time-Dependent and Time-Independent Quantum Systems
Symmetry, Integrability and Geometry: Methods and Applications
author_facet Schuch, D.
author_sort Schuch, D.
title Riccati and Ermakov Equations in Time-Dependent and Time-Independent Quantum Systems
title_short Riccati and Ermakov Equations in Time-Dependent and Time-Independent Quantum Systems
title_full Riccati and Ermakov Equations in Time-Dependent and Time-Independent Quantum Systems
title_fullStr Riccati and Ermakov Equations in Time-Dependent and Time-Independent Quantum Systems
title_full_unstemmed Riccati and Ermakov Equations in Time-Dependent and Time-Independent Quantum Systems
title_sort riccati and ermakov equations in time-dependent and time-independent quantum systems
publisher Інститут математики НАН України
publishDate 2008
url http://dspace.nbuv.gov.ua/handle/123456789/149040
citation_txt Riccati and Ermakov Equations in Time-Dependent and Time-Independent Quantum Systems / D. Schuch // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 22 назв. — англ.
series Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv AT schuchd riccatiandermakovequationsintimedependentandtimeindependentquantumsystems
first_indexed 2023-05-20T17:32:03Z
last_indexed 2023-05-20T17:32:03Z
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