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Semiclassical use of Action-Angle Variables in the Presence of Tunnelling

R. Egydio de Carvalho, Alfredo M. Ozorio de Almeida

Published Jan 1, 1988• 0 citations
OpenAlex

Abstract

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.

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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.
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.
Quantum action-angle variables are used to describe and analyze a number of familiar systems. For a given system, the quantum canonical transformation from the old coordinates, e.g., linear or polar, to the new coordinates, action-angle variables, is found by generalizing the corresponding classical transformation using a method based upon the correspondence principle, the Hermiticity and canonical nature of the old coordinates, and the requirement that the Hamiltonian be independent of the quantum angle variable. The bound-state energy levels and other important system properties follow immediately from the canonical transformation. Harmonic oscillators of various dimensions and the three-dimensional angular momentum system are used as illustrations; these illustrations provide interesting alternatives to the usual quantum treatments.
The semiclassical method based on the Einstein-Brillouin-Keller quantization of invariant phase-space tori and the hypothesis of adiabatic invariance of classical action variables is applied in the calculation of highly excited energy levels of two-dimensional coupled oscillator systems. The results obtained are in good agreement with quantum-mechanical calculations even in the regions where the corresponding classical dynamics becomes apparently irregular. General limitations of the method are discussed.
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.
Semiclassical bound state theory is considered from the viewpoint of the phase space representations of quantum mechanics. In the small h/ limit, within a representation defined by the ordering chosen for the noncommuting position and momentum operators, we obtain: (1) equations of motion, which need not be classical and (2) correspondence principles in action-angle variables, regarded as those initial conditions on the equations of motion which dominate in the small h/ limit. The Weyl ordering is seen to generate the classical equations of motion and the normal correspondence result. Possible evaluation of semiclassical bound state energies is discussed by means of the eigenvalue approach to classical mechanics used by Prigogine and co-workers.