Abstract: Based on the rotation transformation in the phase space and the technique of integration within an ordered product of operators, the coherent state representations of the rotation operator in quantum phase space, the Fourier transform operator and the parity operator are derived. Then the three-mode rotation operator is introduced.By the three-mode rotation operator, the Hamilton of the three-dimensional coordinate-momentum coupled harmonic oscillators is simply decoupled. Then its exact energy spectrums and eigenstates are obtained.

Key words: rotation operator, coupled harmonic oscillators, energy spectrums, energy eigenstates