Canonical quantization on the half-line and in an interval based upon an alternative concept for the momentum in a space with boundaries

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