  The non-commutativity of the position and momentum operators is formulated as
an effective potential in classical phase space and expanded as a series of
successive many-body terms, with the pair term being dominant. A non-linear
partial differential equation in temperature and space is given for this. The
linear solution is obtained explicitly, which is valid at high and intermediate
temperatures, or at low densities. An algorithm for solving the full non-linear
problem is given. Symmetrization effects accounting for particle statistics are
also written as a series of effective many-body potentials, of which the pair
term is dominant at terrestrial densities. Casting these quantum functions as
pair-wise additive, temperature-dependent, effective potentials enables the
established techniques of classical statistical mechanics to be applied to
quantum systems. The quantum Ornstein-Zernike equation is given as an example.
