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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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2008 |
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Інститут математики НАН України
2008
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/149040 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | 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. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862744169701703680 |
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| author | Schuch, D. |
| author_facet | Schuch, D. |
| 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. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| 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.
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| first_indexed | 2025-12-07T20:34:17Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-149040 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T20:34:17Z |
| publishDate | 2008 |
| publisher | Інститут математики НАН України |
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| spelling | Schuch, D. 2019-02-19T13:12:45Z 2019-02-19T13:12:45Z 2008 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 https://nasplib.isofts.kiev.ua/handle/123456789/149040 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. This paper is a contribution to the Proceedings of the Seventh International Conference “Symmetry in Nonlinear Mathematical Physics” (June 24–30, 2007, Kyiv, Ukraine). The author wishes to thank G. Reinisch for numerous encouraging and inspiring discussions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Riccati and Ermakov Equations in Time-Dependent and Time-Independent Quantum Systems Article published earlier |
| spellingShingle | Riccati and Ermakov Equations in Time-Dependent and Time-Independent Quantum Systems Schuch, D. |
| title | 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_short | 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 |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149040 |
| work_keys_str_mv | AT schuchd riccatiandermakovequationsintimedependentandtimeindependentquantumsystems |