  Quantum canonical transformations corresponding to the action of the unitary
operator $e^{i\epsilon(t)\sqrt{f(x)}p\sqrt{f(x)}}$ is studied. It is shown that
for $f(x)=x$, the effect of this transformation is to rescale the position and
momentum operators by $e^{\epsilon(t)}$ and $e^{-\epsilon(t)}$, respectively.
This transformation is shown to lead to the identification of a previously
unknown class of exactly solvable time-dependent harmonic oscillators. It turns
out that the Caldirola-Kanai oscillator whose mass is given by $m=m_0 e^{\gamma
t}$, belongs to this class. It is also shown that for arbitrary $f(x)$, this
canonical transformations map the dynamics of a free particle with constant
mass to that of free particle with a position-dependent mass. In other words,
they lead to a change of the metric of the space.
