A new method for finding the dynamical invariant for a general time-dependent harmonic oscillator is proposed by making use of two linearly independent solutions to the classical equation of motion. It is shown that the dynamical invariant for different gauged Hamiltonians are connected by time-dependent gauge transformations. Therefore, the representation whose bases are the instantaneous eigenstates of the invariant operator is a good representation for the quantum nonautonomous system. In this representation, the wavefunction of the system is gauge covariant and thus any observable physical effect is naturally independent of the choice of gauged Hamiltonians. The exact even and odd coherent states for a time-dependent harmonic oscillator are constructed in terms of these gauge covariant bases. The harmonic oscillator with periodically varying frequency is treated as a demonstration of our general approach.
For a time-dependent harmonic oscillator with an inverse squared singular term, we find the generalized invariant using the Lie algebra of
SU(2) and construct the number-type eigenstates and the coherent states using the spectrum-generating Lie algebra of
SU(1,1). We obtain the evolution operator in both of the Lie algebras. The number-type eigenstates and the coherent states are constructed group-theoretically for both the time-independent and the time-dependent harmonic oscillators with the singular term. It is shown that the squeeze operator transforms unitarily the time-dependent basis of the spectrum-generating Lie algebra of
SU(1,1) for the generalized invariant, and thereby evolves the initial vacuum into a final coherent vacuum.