Quantum-classical correspondence via Liouville dynamics. II. Correspondence for chaotic Hamiltonian systems

Abstract

We prove quantum-classical correspondence for bound conservative classically chaotic Hamiltonian systems. In particular, quantum Liouville spectral projection operators and spectral densities, and hence classical dynamics, are shown to approach their classical analogs in the h!0 limit. Correspondence is shown to occur via the elimination of essential singularities. In addition, applications to matrix elements of observables in chaotic systems are discussed. @S1050-2947~96!05212-2# The validity of quantum mechanics as a description of the macroscopic world is contingent upon the reduction of the laws of quantum mechanics to Newton’s laws in the limit where the characteristic actions of a system are large with respect to Planck’s constant @1#. Thus, diagonal and offdiagonal matrix elements must reduce to their classical analogs and quantum dynamics must reproduce the predictions of classical mechanics as h!0. Despite the fundamental importance of quantum-classical correspondence it has only been satisfactorily demonstrated @2‐5# in the very restrictive case of regular systems, i.e., systems that classically possess as many constants of the motion as degrees of freedom. Indeed some authors have suggested that bound quantum systems with a discrete quantum spectrum and a chaotic classical analog may violate the correspondence principle @6#. These doubts about the validity of the correspondence principle for chaotic systems stem from the difficulty of reconciling the quasiperiodic nature of bound state quantum dynamics with the chaotic nature of classical dynamics for the same Hamiltonian. The issue of correspondence for quantum systems whose classical analogs exhibit chaos ~irregular systems! is thus of great interest. Verification of correspondence should be distinguished from the development of semiclassical approximation methods. While semiclassical theories provide a natural starting point for an exploration of the classical limit their existence does not guarantee correspondence. For example, semiclassical theories for regular systems preceded the development of modern quantum mechanics @7#, but an understanding of correspondence for regular systems has only recently been achieved @2,3,5#. By comparison, attempts to develop semiclassical quantization rules for chaotic systems have had some success @8#, whereas the correspondence limit remains largely unexplored @9#. In this paper we demonstrate that the existing semiclassical theories of quantum dynamics for classically chaotic systems are sufficiently well developed to allow us to show that such systems do in fact approach their proper correspondence limits as Planck’s constant approaches zero. This completes the Liouville correspondence program outlined in the preceding paper @5#, and significantly extends the results of our study of quantum maps @10,11#, where we rigorously demonstrated that a nonchaotic quantum map dynamics can completely recover a fully chaotic classical dynamics in the limit h!0. The Liouville picture affords a means of gaining insight into the connections between quantum and classical mechanics @3,12,13#, and is a natural framework for studies of correspondence. As outlined in the preceding paper ~henceforth referred to as paper I !@ 5 #the essential ingredients for Liouville dynamics are eigenstates and eigenvalues of the Liouville operators in both mechanics. In particular, the dynamics is completely characterized by the Liouville eigenfunctions and eigenvalues or the spectral projectors once the class of allowed initial distributions is specified. Here we consider correspondence in chaotic systems from this Liouville perspective.