asymptotic properties of time evolution are given by the scattering matrix. A state space with a distinguished propagator is also called a dynamical system. To say time evolution is homogeneous means that for all . In the case of a homogeneous system, the mappings Gt = Ft,0 form a one-parameter group of transformations of X, that is For non-reversible systems, the propagation operators Ft, s are defined whenever t ≥ s and satisfy the propagation identity for any . In the homogeneous case the propagators are exponentials of the Hamiltonian. In quantum mechanics In the Schrödinger picture, the Hamiltonian operator generates the time evolution of quantum states. If is the state of the system at time , then This is the Schrödinger equation. Given the state at some initial time (), if is independent of time, then the unitary time evolution operator is the exponential operator as shown in the equation See also Arrow of time Time translation symmetry Hamiltonian system Propagator Time evolution operator Hamiltonian (control theory) References General references . . . . . Dynamical systems fr:Opérateur d'évolution