LETTER TO THE EDITOR: Heisenberg picture approach to the invariants and the exact quantum motions for coupled parametric oscillators

For N-coupled generalized time-dependent oscillators, primary invariants and a generalized invariant are found in terms of classical solutions. Exact quantum motions satisfying the Heisenberg equation of motion are also found. For number states and coherent states of the generalized invariant, the uncertainties in positions and momenta are obtained.