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The correspondence between classical and quantum invariants is established.The Ermakov Lewis quantum invariant of the time dependent harmonic oscillatoris translated from the coordinate and momentum operators into amplitude andphase operators. In doing so, Turski's phase operator as well asSusskind-Glogower operators are generalized to the time dependent harmonicoscillator case. A quantum derivation of the Manley-Rowe relations is shown asan example

Amplitude and phase representation of quantum invariants for the time-dependent harmonic oscillator