Quantum mechanics of time-dependent systems. Construction of pure states

For time-dependent systems the wavefunction depends explicitly on time and it is not a pure state of the Hamiltonian. We construct operators for which the above wavefunction is a pure state. The method is based on the introduction of conserved quantities $Q$ and the pure states are defined by ${\hat Q}\psi=q\psi$. The conserved quantities are constructed using parametrised mechanics and the Noether theorem.