  In quantum mechanics, a classical particle is raised to a wave-function,
thereby acquiring many more degrees of freedom. For instance, in the
semi-classical regime, while the position and momentum expectation values
follow the classical trajectory, the uncertainty of a wave-packet can evolve
and beat independently. We use this insight to revisit the dynamics of a 1d
particle in a time-dependent harmonic well. One can solve it by considering
time reparameterizations and the Virasoro group action to map the system to the
harmonic oscillator with constant frequency. We prove that identifying such a
simplifying time variable is naturally solved by quantizing the system and
looking at the evolution of the width of a Gaussian wave-packet. We further
show that the Ermakov-Lewis invariant for the classical evolution in a
time-dependent harmonic potential is actually the quantum uncertainty of a
Gaussian wave-packet. This naturally extends the classical Ermakov-Lewis
invariant to a constant of motion for quantum systems following Schrodinger
equation. We conclude with a discussion of potential applications to quantum
gravity and quantum cosmology.
