in which the canonical pair \(\left(q,p\right)\) are Hilbert space operators with commutation relations \(\left[q,p\right]=i\hbar\boldsymbol{1}\)\(\left[q,q\right]=0\), and \(\left[p,p\right]=0\). The term \(\omega(t)\) represents a time dependent angular frequency, \(F\left(t\right)\) stands for a time dependent driven force, and \(G\left(t\right)\) is another time dependent function. These functions are supposed to be at least of class \(C^{2}\). This operator can be seen as a generalization of the Bateman-Caldirola-Kanai (BCK) model for the dissipative harmonic oscillator \cite{Bateman_1931,Caldirola_1941,Kanai_1948}.
Heisenberg's equations for the hamiltonian (\ref{eq:01}) are given by