Quantum invariants

We extend the direct approach of Lewis and Leach to quantum theory. For a quantum Hamiltonian $H=\frac{P^2}{2+V(q,t)\mathrm{}}$ we find all quantum invariants which are linear and quadratic in the momentum. The form of the potential for each of these invariants is also determined. For the quadratic case we arrive at the quantum Ermakov system for which exact solutions to the Schrödinger equation have recently been discussed.