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The quantum states representing classical phase space are given, and theseare used to formulate quantum statistical mechanics as a formally exact doubleperturbation expansion about classical statistical mechanics. One series ofquantum contributions arises from the non-commutativity of the position andmomentum operators. Although the formulation of the quantum states differs, thepresent results for separate averages of position operators and of momentumoperators agree with Wigner (1932) and Kirkwood (1933). The second seriesarises from wave function symmetrization, and is given in terms of $l$-particlepermutation loops in an infinite order re-summation. The series givesanalytically the known exact result for the quantum ideal gas to all orders.The leading correction corrects a correction given by Kirkwood.  The first four quantum corrections to the grand potential are calculated fora Lennard-Jones fluid using the hypernetted chain closure. For helium on liquidbranch isotherms, the corrections range from several times to 1\% of the totalclassical pressure, with the effects of non-commutativity being significantlylarger in magnitude than those of wave function symmetrization. All correctionsare found to be negligible for argon at the densities and temperatures studied.The calculations are computationally trivial as the method avoids having tocompute eigenfunctions, eigenvalues, and numerical symmetrization.

Quantum Statistical Mechanics as an Exact Classical Expansion with Results for Lennard-Jones Helium