On a Class of Quantum Canonical Transformations and the Time-Dependent Harmonic Oscillator

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.