  The quantum mechanical version of a classical model for studying the
orientational degrees of freedom corresponding to a nematic liquid composed of
biaxial molecules is presented. The effective degrees of freedom are described
by operators carrying an SU(3) representation, which allows the explicit
calculation of the partition function in the mean field approximation. The
algebraic consistency conditions are solved numerically and the equilibrium
phases of the system are determined. In particular, the entropy, the specific
heat and the order parameters are presented for different choices of the
constituent biaxial molecules. Our results reproduce the classical calculation
in the limit of high temperatures and high quantum numbers.
