Dynamics of Quantum Averages and Quantization Condition

The system of ODE, which describes dynamics of quantum averages of operators of coordinate, momentum and second moments in semiclassical approximation is considered (average values are calculated w.r.t. approximate solutions of the evolution Schreodinger equation). It is shown that quantization condition for the Schreodinger operator is related to periodic solutions of the system of ODE. The known Einstein-Brillouin-Keller semiclassical quantization condition follows from the obtained one under additional assumptions.