Exact results for the Wigner transform phase space densities of a   two-dimensional harmonically confined charged quantum gas subjected to a   magnetic field
  Closed form analytical expressions are obtained for the Wigner transform of the Bloch density matrix and for the Wigner phase space density of a two dimensional harmonically trapped charged quantum gas in a uniform magnetic field of arbitrary strength, at zero and nonzero temperatures. An exact analytic expression is also obtained for the autocorrelation function. The strong magnetic field case, where only few Landau levels are occupied, is also examined, and useful approximate expressions for the spatial and momentum densities are given. 
K. Bencheikh and L.M. Nieto