  We revise the Lewis-Riesenfeld invariant method for solving the quantum
time-dependent harmonic oscillator in light of the Quantum Arnold
Transformation previously introduced and its recent generalization to the
Quantum Arnold-Ermakov-Pinney Transformation. We prove that both methods are
equivalent and show the advantages of the Quantum Arnold-Ermakov-Pinney
transformation over the Lewis-Riesenfeld invariant method. We show that, in the
quantum time-dependent and damped harmonic oscillator, the invariant proposed
by Dodonov & Man'ko is more suitable and provide some examples to illustrate
it, focusing on the damped case.
