  For a particle moving on a half-line or in an interval the operator $\hat p =
- i \partial_x$ is not self-adjoint and thus does not qualify as the physical
momentum. Consequently canonical quantization based on $\hat p$ fails. Based
upon a new concept for a self-adjoint momentum operator $\hat p_R$, we show
that canonical quantization can indeed be implemented on the half-line and on
an interval. Both the Hamiltonian $\hat H$ and the momentum operator $\hat p_R$
are endowed with self-adjoint extension parameters that characterize the
corresponding domains $D(\hat H)$ and $D(\hat p_R)$ in the Hilbert space. When
one replaces Poisson brackets by commutators, one obtains meaningful results
only if the corresponding operator domains are properly taken into account. The
new concept for the momentum is used to describe the results of momentum
measurements of a quantum mechanical particle that is reflected at impenetrable
boundaries, either at the end of the half-line or at the two ends of an
interval.
