Ermakov systems and quantum-mechanical superposition laws

Ermakov systems are pairs of coupled, time-dependent, nonlinear dynamical equations possessing a joint constant of the motion called an Ermakov invariant. The invariant provides a link between the two equations and leads to a superposition law between solutions to the Ermakov pair. Extensive studies of Ermakov systems in classical mechanics have been carried out. Here we present a detailed study of Ermakov systems from a quantum point of view, and prove that the solution to the Schr\"odinger equation for a general Ermakov system can be reduced to the solution of a time-independent Schr\"odinger equation involving the Ermakov invariant. We thereby arrive at a quantum-mechanical superposition law analogous to the classical superposition law.