  We address an apparent conflict between the traditional canonical
quantization framework of quantum theory and the spatially restricted quantum
dynamics, when the translation invariance of the otherwise free quantum system
is broken by boundary conditions. By invoking an exemplary case of a particle
in an infinite well, we analyze spectral problems for related, confined and
global, observables. In particular, we show how one can make sense of various
operators pertaining to trapped particles by not ignoring the rest of the real
line (e.g., that space which is never occupied by the particle in question).
