  We consider the quantum mechanical equivalence of the Seiberg-Witten map in
the context of the Weyl-Wigner-Groenewold-Moyal phase-space formalism in order
to construct a quantum mechanics over noncommutative Heisenberg algebras. The
formalism is then applied to the exactly soluble Landau and harmonic oscillator
problems in the 2-dimensional noncommutative phase-space plane, in order to
derive their correct energy spectra and corresponding Wigner distributions. We
compare our results with others that have previously appeared in the
literature.
