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arXiv:1801.06389 · Science China Physics Mechanics and Astronomy

Quantum-classical correspondence in integrable systems

Yiqiang Zhao, Biao Wu

Published Apr 23, 2019• 7 citations
arXivHTMLPDF
DOI
10.1007/s11433-018-9348-2

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A general program to show quantum-classical correspondence for bound conservative integrable and chaotic systems is described. The method is applied to integrable systems and the nature of the approach to the classical limit, the cancellation of essential singularities, is demonstrated. The application to chaotic systems requires an understanding of classical Liouville eigenfunctions and a Liouville spectral decomposition, developed herein. General approaches to the construction of these Liouville eigenfunctions and classical spectral projectors in quantum and classical mechanics are discussed and are employed to construct Liouville eigenfunctions for classically chaotic systems. Correspondence for systems whose classical analogs are chaotic is discussed, based on this decomposition, in the following paper [Phys. Rev. A 54, 43 (1996)].
We study a stationary velocity field, defined via the probability current density for an energy eigenstate of a quantum system, and its classical analog determined by the Hamilton-Jacobi equation. For classically integrable systems, the quantum-classical correspondence can be properly established by specifying constants of motion in addition to the Hamiltonian. For classically chaotic systems, the correspondence is broken down; the manifestation of classical chaos in the quantum domain is shown numerically to be the appearance of turbulence in the quantum velocity field.
The long term behavior of non-integrable quantum systems is investigated to find a classical counterpart that describes the statistical properties of dynamical fluctuation of quantum observables. On the basis of the quantum-classical correspondence in the integrable system, we introduce the ensemble of classical trajectories (ECT) , which is defined as a set of weighted classical trajectories whose mean density in phase space is one per Planck cell. In the non-integrable system, due to the rapid breakdown of quantum-classical correspondence of the wavepacket dynamics, the ECT is unable to yield the correct temporal behavior of quantum observables. However, numerical experiments reveal that it gives a good approximation of various statistical properties of the quantum dynamical fluctuation. This result strongly suggests that, as far as the statistical properties are concerned, the dominant part of the quantum dynamical fluctuation is determined by classical dynamics, i.e. a set of classical trajectories. The meaning and interpretation of the ECT in relation to the coherent subsystem is also discussed.