  We characterize point transformations in quantum mechanics from the
mathematical viewpoint. To conclude that the canonical variables given by each
point transformation in quantum mechanics correctly describe the extended point
transformation, we show that they are all selfadjoint operators in
$L^2(\mathbb{R}^n)$ and that the continuous spectrum of each coincides with
$\mathbb{R}$. They are also shown to satisfy the canonical commutation
relations.
