Quantum Statistical Mechanics in Classical Phase Space. Test Results for Quantum Harmonic Oscillators

The von Neumann trace form of quantum statistical mechanics is transformed to an integral over classical phase space. Formally exact expressions for the resultant position-momentum commutation function are given. A loop expansion for wave function symmetrization is also given. The method is tested for quantum harmonic oscillators. For both the boson and fermion cases, the grand potential and the average energy obtained by numerical quadrature over classical phase space are shown to agree with the known analytic results. A mean field approximation is given which is suitable for condensed matter, and which allows the quantum statistical mechanics of interacting particles to be obtained in classical phase space.