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 √ f(x)p √ 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 and e, 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 = m0e , 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. It is well-known that in quantum mechanics the unitary transformations of the Hilbert space correspond to the canonical transformations of the classical mechanics. Unfortunately, these quantum canonical transformations have not been usually discussed in most textbooks on quantum mechanics. The purpose of this note is to study a class of canonical transformations, namely U := exp [ iǫ(t) 2 {f(x), p} ]