The importance of the dynamical invariants of a system should not be underrated. In classical mechanics, the dynamical constants of motion are the variables that allow complete integration of dynamical systems. In classical field theories, symmetries of lagrangian systems are related to continuity equations and time-invariants through the Noether theorem \cite{e2011} In quantum field theory, Casimir invariants of symmetry groups are essential to the understanding of the fundamental particle structure of our universe \cite{Wigner_1939}.
In quantum mechanics, a complete characterization of a quantum system is achieved by the knowledge of a complete set of time-invariant observables, which are also generators of a complete symmetry of the system. The process of quantization, therefore, is accomplished by finding an invariant set of stationary eigenvectors which generates, hopefully, a Hilbert space. Symmetries are linked to invariants, and invariants are linked to the very existence of quantum states, on a very fundamental level.
In time-dependent systems, dynamical invariants play a major role, since the energy is no longer conserved, and sometimes even defined. Particularly, in quantum mechanics, systems with time-dependent Hamiltonians do not have well-defined energy spectra. Even in the case where a complete basis of eigenvectors exists, one cannot be sure that this condition persists in time. When quantization is allowed, the problem of time-dependent hamiltonians can be dealt with by finding a hermitian quadratic invariant, for which the eigenvalue problem is well defined \cite{Lewis_1969}. Time-dependent systems appear in several applications in physics such as ion traps \cite{Paul_1990,Leibfried_2003,Torrontegui_2011}, optical cavities \cite{Johnston_1996}, and to perform algorithms in quantum computation \cite{Sarandy_2011,G_ng_rd__2012}.
There are several methods to calculate dynamical invariants. In the classical case, we have Lutzky's approach \cite{Lutzky_1978,Lutzky_1978a}, which consists of the application of the Noether theorem. Another method is the dynamical algebra approach \cite{Korsch_1979,Kaushal_1981}. Recently, the authors developed a new way to calculate dynamical invariants \cite{Bertin_2012}, which consists of combinations of the equations of motion. These last two methods can be used in both, classical and quantum cases.
In this work, we show how the definition of first-order invariants allows us to approach the quantization of the one-dimension time-dependent, damped, driven harmonic oscillator (TDDDHO). In sec. \ref{832125}, we follow \cite{Bertin_2012} and calculate the linear invariants for the TDDDHO by taking the combinations of the equations of motion. Next, in sec. \ref{965209}, we construct the quadratic invariant and find a Steen-Ermakov-like equation. In sec. \ref{343671}, we perform the quantization of the TDDDHO using the algebra of the first-order invariants. Sec. \ref{745680} presents the coordinate representation in the form of wave eigenfunctions of the quadratic invariant, along with a general expression for the uncertainty relations between the observables \((q,p)\). In section \ref{943594}, we address the problem of the dissipative oscillator with constant parameters and general driven force. Finally, in sec. \ref{549543}, we present our main observations.