A practical approach to the description of time evolution via the mean values of a set of a few relevant observables is discussed. The mean values determine, in a self-consistent way, the time propagation of the system. The procedure yields variational formulation, through which closedform equations of motion of Hamiltonian form are derived for the relevant mean values. The approximation can provide an exact description under well-defined conditions. The time evolution is reversible in that the entropy does not increase and that it can be described by a unitary evolution operator. A special case of both practical and formal importance is when the relevant observables form a Lie algebra. The self-consistency conditions can then be explicitly implemented and a symplectic structure can be provided for the reduced phase space. Time displacements (of either the state or the observables) can then be described by a self-consistent Hamiltonian, linear in the generators. An example corresponding to the evolution of a Morse-type oscillator under a time-dependent external perturbation is discussed in detail.

• Received 19 December 1983

DOI:https://doi.org/10.1103/PhysRevA.30.1477