Further observations
In this work, we showed a procedure for the quantization of the harmonic oscillator with time-dependent frequency, time-dependent driven force, and time-dependent dissipative term. The procedure is based on the construction of the linear invariants of the BCK Hamiltonian (\ref{eq:01}), which turns out to be ladder operators. We also construct the Hilbert space of the system and calculate the wave eigenfunctions.
This approach shows that the fundamental quantities turn out to be the linear invariants. Other attempts of analyzing the quantum oscillator from the dynamical invariant point of view can be found in the literature, most of them are based on the second-order invariant (\ref{27}) as the proper Hamiltonian operator, as the case of \cite{Lewis_1967}. However, the fundamental role of the linear invariants for the quantization of the oscillators can be found in \cite{Malkin_1970,Dodonov_1979}. In fact, the procedure of the ref. \cite{Dodonov_1979} is very close to the one employed here. In the case of the underdamped oscillator, we also report to the refs. \cite{Gitman_2007}, where the authors propose a quantization procedure based on the construction of first-order actions, and also to the ref. \cite{Baldiotti_2011}.
We found that the abstract Hilbert space of the general quadratic oscillator is the same as the simple harmonic oscillator. However, it is not a surprise that the same is not observed with the solutions of the Schrödinger equation, which are also eigenfunctions of the quadratic invariant \(I_Q\). The wave functions \(\psi_n\) are time-dependent and lead, in the general case, to time-dependent expectation values and uncertainty relations for the canonical operators. In the special case of constant parameters, however, the uncertainty relations between \(q\) and \(p\) are time-independent.
We note that the procedure in \cite{Bertin_2012} does not need a Hamiltonian function, but can be implemented from the equation of motion (\ref{eq:02c}). However, some caution would be advised. First, the first-order equations,