  We construct the exact position representation of a deformed quantum
mechanics which exhibits an intrinsic maximum momentum and use it to study
problems such as a particle in a box and scattering from a step potential,
among others. In particular, we show that unlike usual quantum mechanics, the
present deformed case delays the formation of bound states in a finite
potential well. In the process we also highlight some limitations and pit-falls
of low-momentum or perturbative treatments and thus resolve two puzzles
occurring in the literature.
