  We tested in the framework of quantum mechanics the consequences of a
noncommutative (NC from now on) coordinates. We restricted ourselves to 3D
rotationally invariant NC configuration spaces with dynamics specified by the
Hamiltonian H = H(kin) + U, H(kin) is an analogue of kinetic energy and U =
U(r) denotes an arbitrary rotationally invariant potential. We introduced the
velocity operator by Heisenberg relation using the commutator of the coordinate
and the Hamiltonian operators. We found that the NC velocity operator possesses
various general, independent of potential, properties: 1) uncertainty relations
indicate an existence of a natural kinetic energy cut-off, 2) vanishing
commutator relations for velocity components, which is non-trivial in the NC
case, 3) modified relation between the velocity operator and H(kin) that
indicates the existence of maximal velocity and confirms the kinetic energy
cut-off, 4) All these results sum up in canonical (general, not depending on a
particular form of the central potential) commutation relations of the
Euclidean group E(4), 5) NC Heisenberg equation for the velocity operator,
relating acceleration to derivatives of the potential.
