Wave Function of Coupled Quantum-Classical Oscillator

Considering classical environment to be a classical oscillator, we describe the dynamics of a system consisting of a quantum oscillator coupled to classical oscillator. Using the Born-Oppenheimer approximation and the method of adiabatic invariants, we derive the classically effective Hamiltonian describing the complete dynamics of the system and the wave function of quantum part of the system. We also present a new method for investigating the dynamics of the quantum system in classical environment.