  We examine a deformed quantum mechanics in which the commutator between
coordinates and momenta is a function of momenta. The Jacobi identity
constraint on a two-parameter class of such modified commutation relations
(MCR's) shows that they encode an intrinsic maximum momentum; a sub-class of
which also imply a minimum position uncertainty. Maximum momentum causes the
bound state spectrum of the one-dimensional harmonic oscillator to terminate at
finite energy, whereby classical characteristics are observed for the studied
cases. We then use a semi-classical analysis to discuss general concave
potentials in one dimension and isotropic power-law potentials in higher
dimensions. Among other conclusions, we find that in a subset of the studied
MCR's, the leading order energy shifts of bound states are of opposite sign
compared to those obtained using string-theory motivated MCR's, and thus these
two cases are more easily distinguishable in potential experiments.
