Quantum-Classical Hybrid Systems and Ehrenfest's Theorem

Abstract

The conceptual analysis of quantum mechanics brings to light that a theory inherently consistent with observations should be able to describe both quantum and classical systems, i.e., quantum-classical hybrids. For example, the orthodox interpretation of measurements requires the transient creation of quantum-classical hybrids. Despite its limitations in defining the classical limit, Ehrenfest's theorem makes the simplest contact between quantum and classical mechanics. Here, we generalized the Ehrenfest theorem to bipartite quantum systems. To study quantum-classical hybrids, we employed a formalism based on operator-valued Wigner functions and quantum-classical brackets. We used this approach to derive the form of the Ehrenfest theorem for quantum-classical hybrids. We found that the time variation of the average energy of each component of the bipartite system is equal to the average of the symmetrized quantum dissipated power in both the quantum and the quantum-classical case. We expect that these theoretical results will be useful both to analyze quantum-classical hybrids and to develop self-consistent numerical algorithms for Ehrenfest-type simulations.

Keywords: Ehrenfest’s theorem; open quantum systems; quantum mechanics in phase space; quantum–classical dynamics.

Figure 1

Relation between the quantum and the classical worlds. The classical world is not contained in a larger and more fundamental quantum world. Instead, there is only an overlap between the two worlds. In the picture, the cyan quantum world (Q-world) overlaps with the light orange classical world (C-world), and the green area represents their intersection.