Application of Lie Systems to Quantum Mechanics: Superposition Rules

  • José F. Cariñena
  • Jesús Clemente-GallardoEmail author
  • Jorge A. Jover-Galtier
  • Javier de Lucas
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 229)


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.

AMS Classification

34A26 (Primary) 17B81, 34A34, 53Z05 (Secondary) 



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.


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • José F. Cariñena
    • 1
    • 2
  • Jesús Clemente-Gallardo
    • 1
    • 3
    Email author
  • Jorge A. Jover-Galtier
    • 4
  • Javier de Lucas
    • 5
  1. 1.Department of Theoretical Physics, Faculty of SciencesUniversity of ZaragozaZaragozaSpain
  2. 2.IUI IUMA, University of ZaragozaZaragozaSpain
  3. 3.IUI BIFI, University of Zaragoza, Edificio I+DZaragozaSpain
  4. 4.Centro Universitario de la Defensa de Zaragoza, Academia General MilitarZaragozaSpain
  5. 5.Department of Mathematical Methods in PhysicsUniversity of WarsawWarszawaPoland

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