For a system composed of a cavity with Kerr medium, a movable mirror (treated as a quantum harmonic oscillator with frequency ωm and a single mode field in the cavity, the time evolution operator and its transformation of the system are given. Time evolution of the system state is obtained with full quantum theory. State of the cavity field at the time of the mirror back to its original state is discussed, and a variety of multicomponent Schr?dinger cat states which are different from those in the published literature are obtained, which is significant for the investigation of the nature and applications of the Schr?dinger cat states.