A canonical approach to the action-angle formalism in quantum mechanics

In classical mechanics, the action and angle variables (J; Θ) can be found by integrating the momentum p with respect to the coordinate q under the constraint of energy conservation. Because it is not known how to extend Riemann integration to operator functions and variables, the classical method of action-angle formalism cannot be extended to quantum mechanics. We show that by using general quantum canonical transformations, one can transform (p; q) into (J; Θ), by which one of the conjugate variables in the Hamiltonian is eliminated. This algebraic integration by quantum canonical transformations gives not only the operator relations between (p; q) and (J; Θ), but also the eigenfunctions of the Hamiltonian and the eigenfunctions of the phase operator. These results offer a new point of view for the action-angle formalism in quantum mechanics.