Statistical Properties of Energy Levels of Chaotic Systems: Wigner or Non-Wigner? (original) (raw)

1995, Physical Review Letters

For systems whose classical dynamics is chaotic, it is generally believed that the local statistical properties of the quantum energy levels are well described by random matrix theory. We present here two counterexamples-the hydrogen atom in a magnetic field and the quartic oscillator-which display nearest neighbor statistics strongly different from the usual Wigner distribution. We interpret the results with a simple model using a set of regular states coupled to a set of chaotic states modeled by a random matrix. PACS numbers: 05.45.+b, 05.40.+j Since the pioneering work of Bohigas et al. , it has been numerically checked on a wide variety of systems [2,3] that the local statistical properties of a quantum system, whose classical dynamics is chaotic, are well described by random matrix theory. Especially, the statistical distribution of energy spacing between consecutive levels-also called nearest neighbor spacing (NNS) distribution-has been shown to be in excellent agreement with the spacing distribution between consecutive eigenvalues of random matrices. For spinless systems with time-reversal symmetry, the Hamiltonian is real in a suitable basis and the Gaussian orthogonal ensemble (GOE) of random matrices has to be used. The NNS distribution of this ensemble is very close to the Wigner surmise [2,3]:

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