Commensurate anisotropic oscillator, SU(2) coherent states and the classical limit

TL;DR

This paper establishes a precise quantum-classical correspondence for the commensurate anisotropic oscillator using SU(2) coherent states, linking quantum states to classical Lissajous orbits through a canonical transformation and SU(2) symmetry.

Contribution

It introduces a novel approach connecting SU(2) coherent states with classical Lissajous orbits via a canonical transformation and a detailed quantum-classical correspondence.

Findings

SU(2) coherent states correspond to ensembles of Lissajous orbits.

A non-bijective canonical transformation maps anisotropic to isotropic oscillators.

Classical limit achieved through expectation values of SU(2) generators.

Abstract

We demonstrate a formally exact quantum-classical correspondence between the stationary coherent states associated with the commensurate anisotropic two-dimensional harmonic oscillator and the classical Lissajous orbits. Our derivation draws upon earlier work of Louck et al [1973 \textit {J. Math. Phys.} \textbf {14} 692] wherein they have provided a non-bijective canonical transformation that maps, within a degenerate eigenspace, the commensurate anisotropic oscillator on to the isotropic oscillator. This mapping leads, in a natural manner, to a Schwinger realization of SU(2) in terms of the canonically transformed creation and annihilation operators. Through the corresponding coherent states built over a degenerate eigenspace, we directly effect the classical limit via the expectation values of the underlying generators. Our work completely accounts for the fact that the SU(2) coherent state in general corresponds to an ensemble of Lissajous orbits.