Derivation and investigation of an averaged quantum equation of motion for a nonlinear oscillator in the field of an harmonic force

Approximate quantum equation for the function $\xi$ is derived from the Schrodinger equation. The function $\xi$ gives the expansion of the oscillator state in the states, which correspond to coherent states in the case of linear oscillator and/or in the quasiclassical case. In the classical limit $h\to 0$ the classical equations of motion are obtained. Quantum effects lead to the diffusion of $\xi$ in the space of parameters which correspond in classical theory to the action-angle variables. The coefficients of diffusion are proportional to $\hbar d^2E/dI^2$ where $E(n)$ are the oscillator terms, $I=\hbar n$ is the action. In the conditions when the classical oscillator has two states allowed for steady motion, the quantum oscillator has due to diffusion only one allowed state, the state with larger energy.