Quantum Action-Angle Variables for the Harmonic Oscillator

Operators conjugate to the Hamiltonian are constructed explicitly for the quantum harmonic oscillator by two approaches in the space spanned by the eigenstates of $q$ and the eigenstates of $p$. The operators are quantum analogs of a classical angle variable divided by the oscillator frequency. Matrix elements have been evaluated in the coherent state representation. Either conjugate operator can be used to construct an explicitly time-dependent operator invariant. It can be used as the starting point in a new perturbative procedure for constructing invariant operators for nonlinear, nonautonomous Hamiltonians.