Classical mechanics yields a deterministic dynamics only in the case of a closed or isolated system, since an exact prediction of the future or past for a system interacting with an unknown environment is not possible. But even if one is interested in the approximate dynamics of an open classical system it helps to start off with the idealization that the environment and the system of interest are parts of a total system which is isolated, and therefore has a dynamics given by Hamilton’s equations. Similarly, when constructing the intrinsically stochastic dynamics appropriate to quantum theory it is helpful to begin with a closed or isolated system with well-defined boundary conditions in which Schrödinger’s equation has well defined solutions and induces a unitary time development given by a collection of time-evolution operators \(T(t',t)\): if \(\ket{\psi(t)}\) is any solution to Schrödinger’s (time-dependent) equation, then \(\ket{\psi(t')} = T(t',t)\ket{\psi(t)}\). In the case of a time-independent Hamiltonian \(H\) one has \(T(t',t) = \exp[-i(t'-t)H/\hbar]\).