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This paper is concerned with the Wick symbol of time evolving quantum observables. The time dynamics is following either the Heisenberg equation relative to the Schrödinger Hamiltonian, or the time-dependent Hartree–Fock equation. Under very weak assumptions, we prove that the Wick symbol approximatively follows the classical mechanics laws when the semiclassical parameter h tends to zero. For the Heisenberg equation, this is a form of what is commonly called the Ehrenfest theorem. These statements have to be understood in a weaker sense than usual and in return, we do not assume that the Weyl symbol of the initial observable belongs to a class allowing the use of the Egorov theorem. 0. Introduction The aim of this paper is twofold: to prove, under minimal hypotheses, a version of Ehrenfest theorem, and to give a similar result for the time-dependent Hartree– Fock equation (TDHF). We first consider a quantum observable (bounded operator) Ah(t), (t ∈ IR, h > 0), evolving according to the Heisenberg equation associated to the Schrödinger Hamiltonian (see (1.2)). Generally speaking, the Ehrenfest theorem suggests that the average of this observable Ah(t), taken on an h-dependent coherent state, approximatively follows, for small h > 0, the classical mechanics laws [7]. The average of an observable on coherent states is related to its Wick symbol. The first goal of this work is to obtain minimal hypotheses on the initial data Ah(0), in order that the idea conveyed by Ehrenfest is asymptotically verified when h tends to 0. A precise statement of Ehrenfest theorem, using Egorov theorem for the proof, is given in [4, 13] (see also the bibliography therein). It is also noted ([4] and references therein) that the approximation by the classical mechanics laws is not valid uniformly for all time, but up to a maximal time called the Ehrenfest time. These results are obtained assuming that the family of observables (Ah(0)) is a semiclassical pseudodifferential (PDO) operator (see [13] or [17]). Our first goal is therefore to give another precise statement of Ehrenfest theorem, weaker than the one in [13], but in assuming weaker hypotheses on Ah(0) (Theorem 1.1). In particular, we do not assume that Ah(0) is a PDO operator. It is known that an operator in L(IR) is a PDO one in the standard class of Calderón and Vaillancourt, when all its iterated commutators with position and momentum operators are bounded in L(IR) (Beals characterization theorem [2]). In our work, the main assumption is that the single commutators of Ah(0) with position and momentum operators are bounded, without any hypotheses on the iterated commutators. Then we prove that, in some sense, the Received by the editors May 24, 2012.

The classical limit of the Heisenberg and time-dependent Hartree–Fock equations: the Wick symbol of the solution