A new form of Wigner functions on the noncommutative space

Wigner quasi-probability distribution function in phase space is a special (Weyl) representation of the density matrix. It has been very useful in many areas of quantum mechanics. Starting from fundamental principle of the Weyl correspondence, we derive explicit form of the Wigner function (WF) for noncommutative quantum mechanics (NCQM), and prove that it satisfies a generalized *-genvalue equation. We also give some examples to show that the new form of WF indeed has this property.