2005/11/23 by A. J. Bracken, J. G. Wood · 1 citation
Mathematics · Physics and Astronomy · #Classical mechanics #Cold Atom Physics and Bose-Einstein Condensates #Eigenvalues and eigenvectors #Hamiltonian (control theory) #Harmonic oscillator #Hilbert space #Mathematical physics #Mathematics #Method of quantum characteristics #Observable #Operator (biology) #Phase space #Physics #Quantum #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Quantum dynamics #Quantum harmonic oscillator #Quantum mechanics #Quantum process #Semiclassical physics #Time evolution #Wigner distribution function #quant-ph
paper · pdf · doi:10.1103/physreva.73.012104
published as Phys. Rev. A 73(2006), 012104 · Latex2e file. 27 pages, 7 figures. To appear in Phys. Rev. A
arxiv created 2005/11/23 · openalex publication_date 2006/01/10 · arxiv updated 2017/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Quantum mechanics has been formulated in phase space, with the Wigner function as the representative of the quantum density operator, and classical mechanics has been formulated in Hilbert space, with the Groenewold operator as the representative of the classical Liouville density function. Semiclassical approximations to the quantum evolution of the Wigner function have been defined, enabling the quantum evolution to be approached from a classical starting point. Now analogous semiquantum approximations to the classical evolution of the Groenewold operator are defined, enabling the classical evolution to be approached from a quantum starting point. Simple nonlinear systems with one degree of freedom are considered, whose Hamiltonians are polynomials in the Hamiltonian of the simple harmonic oscillator. The behaviour of expectation values of simple observables and of eigenvalues of the Groenewold operator, are calculated numerically and compared for the various semiclassical and semiquantum approximations.