26 - Quantum Statistics

Abstract

Two types of averaging occur in quantum statistical mechanics, the first for pure quantum mechanical states and the second for a statistical ensemble of pure states. We define and exhibit the properties of density operators and their density matrix representation for both pure and statistical states. For equilibrium states, a statistical density operator depends only on stationary quantum states. We exhibit it in the energy representation for the microcanonical, canonical, and grand canonical ensembles; its use is illustrated for an ideal gas and the harmonic oscillator. Density matrices for spin 1/2 are expressed in terms of a polarization vector and Pauli spin matrices and related to vectors called spinors. Symmetric wave functions for bosons and antisymmetric wave functions for fermions are constructed from single-particle quantum states in terms of occupation numbers by using permutation operators, or Slater determinants for fermions. Weighting factors for states are contrasted for bosons, fermions, and distinguishable classical particles.