The European Physical Journal Special Topics

, Volume 225, Issue 6, pp 959–976

Fundamental concepts of quantum chaos

  • M. Robnik
Review Session A: Reviews

DOI: 10.1140/epjst/e2016-02649-0

Cite this article as:
Robnik, M. Eur. Phys. J. Spec. Top. (2016) 225: 959. doi:10.1140/epjst/e2016-02649-0
Part of the following topical collections:
  1. Mathematical Modeling of Complex Systems


We review the fundamental concepts of quantum chaos in Hamiltonian systems. The quantum evolution of bound systems does not possess the sensitive dependence on initial conditions, and thus no chaotic behaviour occurs, whereas the study of the stationary solutions of the Schrödinger equation in the quantum phase space (Wigner functions) reveals precise analogy of the structure of the classical phase portrait. We analyze the regular eigenstates associated with invariant tori in the classical phase space, and the chaotic eigenstates associated with the classically chaotic regions, and the corresponding energy spectra. The effects of quantum localization of the chaotic eigenstates are treated phenomenologically, resulting in Brody-like level statistics, which can be found also at very high-lying levels, while the coupling between the regular and the irregular eigenstates due to tunneling, and of the corresponding levels, manifests itself only in low-lying levels.

Copyright information

© EDP Sciences and Springer 2016

Authors and Affiliations

  • M. Robnik
    • 1
  1. 1.CAMTP-Center for Applied Mathematics and Theoretical Physics, University of MariborMariborSlovenia

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