Abstract
Upon revisiting the Hamiltonian structure of classical wavefunctions in Koopman–von Neumann theory, this paper addresses the long-standing problem of formulating a dynamical theory of classical–quantum coupling. The proposed model not only describes the influence of a classical system onto a quantum one, but also the reverse effect—the quantum backreaction. These interactions are described by a new Hamiltonian wave equation overcoming shortcomings of currently employed models. For example, the density matrix of the quantum subsystem is always positive definite. While the Liouville density of the classical subsystem is generally allowed to be unsigned, its sign is shown to be preserved in time for a specific infinite family of hybrid classical–quantum systems. The proposed description is illustrated and compared with previous theories using the exactly solvable model of a degenerate two-level quantum system coupled to a classical harmonic oscillator.
Highlights
Classical–quantum coupling has been an open problem since the rise of quantum mechanics
The proposed classical–quantum wave equation is illustrated on the exactly solvable model of a degenerate two-level quantum system quadratically coupled to a one-dimensional classical harmonic oscillator
The formulation of hybrid classical–quantum dynamics is usually based on fully quantum treatments, in which some kind of factorization ansatz is invoked on the wave function
Summary
Classical–quantum coupling has been an open problem since the rise of quantum mechanics. Despite several efforts [15,16,17,18,19,20,21,22,23], Liealgebraic arguments [11,13] tend to exclude the existence of a closed equation for D possessing a Hamiltonian structure (i.e. comprising the Jacobi identity) Another stream of research on classical–quantum coupling goes back to Sudarshan’s measurement theory [24] of 1976. One of the advantages of Sudarshan’s proposal is that Koopman wavefunctions possess a simple canonical Hamiltonian structure formally equivalent to that underlying Schrödinger’s equation This feature provides a great simplification over the AG approach, which instead is based on density operators and Wigner functions both carrying highly non-canonical. The proposed classical–quantum wave equation is illustrated on the exactly solvable model of a degenerate two-level quantum system quadratically coupled to a one-dimensional classical harmonic oscillator
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