  We reconsider the generalization of standard quantum mechanics in which the
position operators do not commute. We argue that the standard formalism found
in the literature leads to theories that do not share the symmetries present in
the corresponding commutative system. We propose a general prescription to
specify a Hamiltonian in the noncommutative theory that preserves the existing
symmetries. We show that it is always possible to choose this Hamiltonian in
such a way that the energy spectrum of the standard and non-commuting theories
are identical, so that experimental differences between the predictions of both
theories are to be found only at the level of the detailed structure of the
energy eigenstates.
