Exact solutions to operator differential equations

This paper considers the Heisenberg equations of motion ̇ \.q=-i[q, H], ̇ \.p=-i[p, H], for the quantum-mechanical Hamiltonian $H(p, q)$ having one degree of freedom. It is a commonly held belief that such operator differential equations are intractable. However, a technique is presented here that allows one to obtain exact, closed-form solutions for huge classes of Hamiltonians. This technique, which is a generalization of the classical action-angle-variable methods, allows us to solve, albeit formally and implicitly, the operator differential equations of the anharmonic oscillator whose Hamiltonian is H=p²2+q⁴4.