  Quantum canonical transformations corresponding to time-dependent
diffeomorphisms of the configuration space are studied. A special class of
these transformations which correspond to time-dependent dilatations is used to
identify a previously unknown class of exactly solvable time-dependent harmonic
oscillators. The Caldirola-Kanai oscillator belongs to this class. For a
general time-dependent harmonic oscillator, it is shown that choosing the
dilatation parameter to satisfy the classical equation of motion, one obtains
the solution of the Schr\"odinger equation. A simple generalization of this
result leads to the reduction of the Schr\"odinger equation to a second order
ordinary differential equation whose special case is the auxiliary equation of
the Lewis-Riesenfeld invariant theory. Time-evolution operator is expressed in
terms of a positive real solution of this equation in a closed form, and the
time-dependent position and momentum operators are calculated.
