Classical, semiclassical, and quantum mechanics of a globally chaotic system: Integrability in the adiabatic approximation

Abstract

Articles you may be interested inThe semiclassical regime of the chaotic quantum-classical transition Chaos 15, 033302 (2005); 10. 1063/1.1979227 Comparison of quantum mechanical and semiclassical methods for the determination of transport cross sections and collision integrals AIP Conf.We examine the classical, semiclassical, and quantum mechanics of the Hamiltonian H = ~(p; + p; + x 2 y2). The dynamics of this system are globally chaotic. However, the classical and quantum mechanical problems can be solved analytically by assuming an adiabatic separation of the x and y motion. We construct the canonical transformation to adiabatic action-angle variables and investigate the connection between this integrable approximation and the exact dynamics. In addition, we present a simple semiclassical formula that predicts energy levels in excellent agreement with the exact energy spectrum. The quantum adiabatic potential curves of this system have a very unusual structure-infinitely many curves cross at one point.2328

ħ corrections in semiclassical formulas for smooth chaotic dynamics

“…[27] and a twofold degenerate series EO and OE, where E means even and O means odd, the first two letters referring to the u → −u and v → −v symmetries, the third letter to u ↔ v. Actually, because of the definition of the semiparabolic coordinates (u, v), only eigenvectors invariant under the parity symmetry ψ(−u, −v) = ψ(u, v) correspond to eigenvectors of the 2D hydrogen in a magnetic field, allowing us, in principle, to drop the OE and EO series [4,26]. However, from the semiclassical point of view, one would have to extend all preceding sections to symmetry-projected propagator and Green's function [28], and thus to take into account symmetry properties of the classical Green's function, which is beyond the scope of this paper.…”

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“…It was successful in providing a semiclassical quantization scheme for special integrable dynamical systems, but failed to describe the generic nonintegrable case. Adiabatic invariants play an interesting but minor role in the quantization of chaotic systems [2,3].Since the existence of an adiabatic invariant is the exception rather than the rule, the emergence of a new one quite often teaches us something useful about the system. An example from condensed matter physics is the quantum Hall effect, in which the semiclassical theory is based on two adiabatic invariants: The flux through a cyclotron orbit and the flux enclosed by the orbit center as it slowly drifts along an equipotential [4].…”

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“…If one were now to ask for the most important next step that one could take with the Heisenberg methods, an excellent candidate for an answer would be to produce solutions for a globally chaotic system such as the one studied by Martens et al [46]. It would also be worthwhile to revisit some old ground.…”