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Application of Lie Systems to Quantum Mechanics: Superposition Rules

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Classical and Quantum Physics

Part of the book series: Springer Proceedings in Physics ((SPPHY,volume 229))

Abstract

We prove that t-dependent Schrödinger equations on finite-dimensional Hilbert spaces determined by t-dependent Hermitian Hamiltonian operators can be described through Lie systems admitting a Vessiot-Guldberg Lie algebra of Kähler vector fields. This result is extended to other related Schrödinger equations, e.g. projective ones, and their properties are studied through Poisson, presymplectic, and Kähler structures. This leads to deriving nonlinear superposition rules for them depending on a lower (or equal) number of solutions than standard linear ones. As an application, we study n-qubit systems and special attention is paid to the one-qubit case.

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Acknowledgements

J. F. Cariñena, J. A. Jover-Galtier and J. Clemente-Gallardo acknowledge partial financial support from MINECO (Spain) grant number MTM2015-64166-C2-1. Research of J. de Lucas was supported under the contract 1100/112000/16. Research of J. Clemente-Gallardo and J. A. Jover-Galtier was financed by projects MICINN Grants FIS2013-46159-C3-2-P. Research of J. A. Jover-Galtier was financed by DGA grant number B100/13 and by “Programa de FPU del Ministerio de Educación, Cultura y Deporte” grant number FPU13/01587.

Author information

Authors and Affiliations

  1. Department of Theoretical Physics, Faculty of Sciences, University of Zaragoza, c. Pedro Cerbuna 12, 50.009, Zaragoza, Spain

    José F. Cariñena & Jesús Clemente-Gallardo

  2. IUI IUMA, University of Zaragoza, c. Pedro Cerbuna 12, 50.009, Zaragoza, Spain

    José F. Cariñena

  3. IUI BIFI, University of Zaragoza, Edificio I+D, c. Mariano Esquillor s/n, 50.018, Zaragoza, Spain

    Jesús Clemente-Gallardo

  4. Centro Universitario de la Defensa de Zaragoza, Academia General Militar, ctra. de Huesca s/n, 50.090, Zaragoza, Spain

    Jorge A. Jover-Galtier

  5. Department of Mathematical Methods in Physics, University of Warsaw, ul. Pasteura 5, 02-093, Warszawa, Poland

    Javier de Lucas

Authors
  1. José F. Cariñena
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  2. Jesús Clemente-Gallardo
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  3. Jorge A. Jover-Galtier
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  4. Javier de Lucas
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Corresponding author

Correspondence to Jesús Clemente-Gallardo .

Editor information

Editors and Affiliations

  1. Università degli Studi di Napoli Federico II, Naples, Italy

    G. Marmo

  2. Institute of Mathematical Sciences (ICMAT), Madrid, Spain

    David Martín de Diego

  3. Universitat Politècnica de Catalunya, Barcelona, Spain

    Miguel Muñoz Lecanda

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Cariñena, J.F., Clemente-Gallardo, J., Jover-Galtier, J.A., de Lucas, J. (2019). Application of Lie Systems to Quantum Mechanics: Superposition Rules. In: Marmo, G., Martín de Diego, D., Muñoz Lecanda, M. (eds) Classical and Quantum Physics. Springer Proceedings in Physics, vol 229. Springer, Cham. https://doi.org/10.1007/978-3-030-24748-5_6

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