  Noncommutative quantum mechanics can be considered as a first step in the
construction of quantum field theory on noncommutative spaces of generic form,
when the commutator between coordinates is a function of these coordinates. In
this paper we discuss the mathematical framework of such a theory. The
noncommutativity is treated as an external antisymmetric field satisfying the
Jacoby identity. First, we propose a symplectic realization of a given Poisson
manifold and construct the Darboux coordinates on the obtained symplectic
manifold. Then we define the star product on a Poisson manifold and obtain the
expression for the trace functional. The above ingredients are used to
formulate a nonrelativistic quantum mechanics on noncommutative spaces of
general form. All considered constructions are obtained as a formal series in
the parameter of noncommutativity. In particular, the complete algebra of
commutation relations between coordinates and conjugated momenta is a
deformation of the standard Heisenberg algebra. As examples we consider a free
particle and an isotropic harmonic oscillator on the rotational invariant
noncommutative space.
