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The Journal of Chemical Physics

Application of action-angle variables to the calculation of the density matrix

Kenneth A. Dawson

Published Apr 1, 1985• 0 citations
Publisher

Abstract

Using the analogy of action-angle variables in classical mechanics we introduce the notion of quantum actions in the calculation of the density matrix. The result is an integral representation of the density matrix in its semiclassical limit. This involves only the classical generating functions.
DOI
10.1063/1.448221

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The density of states n(E) is calculated for a bound system whose classical motion is integrable, starting from an expression in terms of the trace of the time-dependent Green function. The novel feature is the use of action-angle variables. This has the advantages that the trace operation reduces to a trivial multiplication and the dependence of n(E) on all classical closed orbits with different topologies appears naturally. The method is contrasted with another, not applicable to integrable systems except in special cases, in which quantization arises from a single closed orbit which is assumed isolated and the trace taken by the method of stationary phase.
Conventional quantum mechanical treatments of many systems have worked with coordinates and momenta that are not canonically conjugate. In this work it is shown how the quantum expressions may be reformulated in terms of the canonical set of action-angle variables, and specific examples of the harmonic oscillator, linear rotor, and triaxial rotor are presented. When expressed in these terms, the quantum mechanics take on a form which can be directly related to analogous results from classical mechanics. In addition, it becomes possible to express the Hamiltonian in the minimum number of coordinates. It is also shown that the common assumption of an exponential form for the overlap of canonical coordinate and momentum eigenstates is false for an asymmetric rotor. This has important implications for the quantization rules applicable to nonseparable systems.
A quantum mechanics is constructed which is patterned on classical Hamilton-Jacobi theory. The dynamical basis of the theory is a quantum Hamilton-Jacobi equation with accompanying physical boundary conditions. Basic objects of the formulation are quantum Hamilton's principal and characteristic functions. As an application of the formalism a theory of quantum action-angle variables is set up. The central feature of this theory is the definition of the quantum action variable which permits the determination of the bound-state energy levels without solving the dynamical equation.
Semiclassical approximations of quantummechanics are known to be invariant with respect to classical canonical transformat ions, even though these are not in general isomorphic to unitary transform- ations in quantum mechanics. However, this rule is not a priori ex- tensive to effects involving t unnelling, since these may be semiclas- sical ly small themselves. An important class of canonical transform - ations b rings the system into action-angle va.riables,which facil i tates the study of tunnell ing near a classical resonance. We verify compu- tational ly that the energy eigenlevels of a resonant system computed in a harmnic oscil l ator basis are in good agreement with the semi- classical values obtained with the use of action-angle variables.
A unified treatment of quantum mechanical oscillatory motion in one dimension is presented in a phase space formalism which is especially adapted to the semiclassical energy domain and closely related to a `naive' quantization of action and angle variables. Characteristics of this scheme are non-classical symmetry properties of the phase space functions representing density operators and observables and the inclusion of half-Bohr orbits besides the familiar Bohr orbits. Over long time intervals the quantum evolution can be well approximated by a Hamiltonian flow along these distinguished classical orbits. The interplay of this reduced classical evolution and the symmetry properties of the phase space functions results in a consistent quantitative description of quantum interference effects which are most clearly seen in the revivals of wavepackets.