  The observable algebra O of SO_q(3)-symmetric quantum mechanics is generated
by the coordinates of momentum and position spaces (which are both isomorphic
to the SO_q(3)-covariant real quantum space R_q^3). Their interrelations are
determined with the quantum group covariant differential calculus. For a
quantum mechanical representation of O on a Hilbert space essential self-
adjointness of specified observables and compatibility of the covariance of the
observable algebra with the action of the unitary continuous corepresent- ation
operator of the compact quantum matrix group SO_q(3) are required. It is shown
that each such quantum mechanical representation extends uniquely to a
self-adjoint representation of O. All these self-adjoint representations are
constructed. As an example an SO_q(3)-invariant Coulomb potential is intro-
duced, the corresponding Hamiltonian proved to be essentially self-adjoint and
its negative eigenvalues calculated with the help of a q-deformed Lenz-vector.
