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The Journal of Chemical Physics

Classical-quantum correspondence in the distribution dynamics of integrable systems

Charles Jaffé, Paul W. Brumer

Published Mar 1, 1985• 56 citations
Publisher

Abstract

The classical-quantum correspondence is developed in its most natural setting, i.e., that of phase space distributions. Specifically, the Wigner–Weyl representation of the two fundamental quantum densities ‖n≳<n‖ and ‖n≳<m‖, which are eigenstates of the quantum Liouville operator, are shown, for integrable systems, to yield stationary and nonstationary classical Liouville eigenstates in the ℏ→0 limit. The former are uniform distributions on semiclassically quantized tori, whereas the latter are nonuniform distributions on similar tori at intermediate actions. Applications to semiclassical mechanics in action-angle variable, the Heisenberg correspondence for matrix elements and the computation of vibrational intensities are provided.

DOI
10.1063/1.448946

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The classical-quantum correspondence is developed in its most natural setting, i.e., that of phase space distributions. Specifically, the Wigner–Weyl representation of the two fundamental quantum densities ‖n≳<n‖ and ‖n≳<m‖, which are eigenstates of the quantum Liouville operator, are shown, for integrable systems, to yield stationary and nonstationary classical Liouville eigenstates in the ℏ→0 limit. The former are uniform distributions on semiclassically quantized tori, whereas the latter are nonuniform distributions on similar tori at intermediate actions. Applications to semiclassical mechanics in action-angle variable, the Heisenberg correspondence for matrix elements and the computation of vibrational intensities are provided.
We discuss how the language of wave functions (state vectors) and associated non-commuting Hermitian operators naturally emerges from classical mechanics by applying the inverse Wigner-Weyl transform to the phase space probability distribution and observables. In this language, the Schr"odinger equation follows from the Liouville equation, with \hbar ℏ now a free parameter. Classical stationary distributions can be represented as sums over stationary states with discrete (quantized) energies, where these states directly correspond to quantum eigenstates. Interestingly, it is now classical mechanics which allows for apparent negative probabilities to occupy eigenstates, dual to the negative probabilities in Wigner’s quasiprobability distribution. These negative probabilities are shown to disappear when allowing sufficient uncertainty in the classical distributions. We show that this correspondence is particularly pronounced for canonical Gibbs ensembles, where classical eigenstates satisfy an integral eigenvalue equation that reduces to the Schr"odinger equation in a saddle-point approximation controlled by the inverse temperature. We illustrate this correspondence by showing that some paradigmatic examples such as tunneling, band structures, Berry phases, Landau levels, level statistics and quantum eigenstates in chaotic potentials can be reproduced to a surprising precision from a classical Gibbs ensemble, without any reference to quantum mechanics and with all parameters (including \hbar ℏ ) on the order of unity.
After a brief review of the extensive work done on the theory of invariant tori and their quantization, we show that nevertheless an important connection between the quantum and classical theories remains to be exploited. This is the relationship between matrix elements of operators in the energy diagonal representation and Fourier components of the corresponding classical dynamical variables that was the basis for Heisenberg's invention of quantum mechanics. We describe a number of previously unknown or little-known aspects of this relationship, with special emphasis on variational principles and the connection between commutation relations and quantization of action variables. As a single illustration of the utility of these ideas we show that it is possible to obtain approximate solutions to the quantum scheme that are more accurate than the semiclassical approximation with little additional effort compared to the latter.
A new method for the quantization of classical Hamiltonian systems is presented. This method is based upon the correspondence between the Liouville formulation of classical mechanics and the Liouville–von Neumann formulation of quantum mechanics. It does not distinguish between integrable and nonintegrable systems, and consequently, is equally applicable to both types of systems. Further, it treats the indistinguishability of identical particles correctly, and thus, the semiclassical eigenstates have the correct symmetry properties. Application of the method is illustrated by a series of examples. The results are in excellent agreement with quantum mechanics and represent an improvement over results obtained using the uniform semiclassical approximation.
Abstract. We establish the difference between the propagation of semiclassical Wigner functions and classical Liouville propagation. First we re-discuss the semiclassical limit for the propagator of Wigner functions, which on its own leads to their classical propagation. Then, via stationary phase evaluation of the full integral evolution equation, using the semiclassical expressions of Wigner functions, we provide the correct geometrical prescription for their semiclassical propagation. This is determined by the classical trajectories of the tips of the chords defined by the initial semiclassical Wigner function and centered on their arguments, in contrast to the Liouville propagation which is determined by the classical trajectories of the arguments themselves. 1. introduction The Wigner [11] function W is a representation of the quantum density operator ˆρ as a real function on phase space, for a non-relativistic dynamical system with classical-quantum correspondence and n degrees of freedom. Here, all the underlying geometry is euclidean and we can include the Wigner function within the general framework of Weyl representation of operators: Â → A ∈ Ck IR (IR2n), so W(x) = (2π�) −n ρ(x) , where x ≡ (p1,...pn, q1,...qn) denotes a point in phase space. In this context, the quantum evolution of the density operator ˆρt effected by an autonomous external hamiltonian
This paper addresses the general problem of zeroth order representation of resonant dynamics. We investigate the classical, quantum mechanical, and semiclassical transformation properties of two-dimensional isotropic and anisotropic uncoupled harmonic oscillators. The classical and quantal theories are presented in a manner that emphasizes the strong correspondence between the two, and in particular, the SU(2) symmetry exhibited by both the classical and quantum oscillators. The classical canonical transformations relating the action-angle variables appropriate for normal, local, and precessional motion of the isotropic oscillator are derived by explicit calculation of the generating functions. By employing a simple mapping relating the anisotropic and isotropic oscillators, expressions for action-angle variables appropriate for the topology of an arbitrary m:n resonance are determined. The resulting invariant tori are compared with the corresponding quantum mechanical wave functions and phase space densities. The relationship between the classical and quantum mechanical theories is illustrated by determining semiclassical approximations to the unitary transformation matrix elements, which are given in terms of the classical generating functions. Applications to problems of current interest, such as the adiabatic switching method for semiclassical quantization of nonseparable systems, are briefly discussed.
This chapter discusses in some detail the connection between the classical and quantum mechanical equations of motion, largely using examples that are solvable. It recounts how the old quantum theory was formulated by quantizing the action. Choosing irreducible cycles around the tori leads naturally to defining the action and angle variables of motion, and the quantization of action around each cycle. The chapter shows that families of these periodic orbits contribute to the oscillating part of the density of states of an integrable system in the form of a trace formula. It discusses the transition from integrability to chaos in such systems when perturbations are added that couple resonantly to these natural frequencies. The chapter elaborates with a view to develop the semiclassical approximation, and the associated quantization condition. It brings out the similarity between the Schrodinger equation of wave mechanics, and the HJ equation.