This chapter builds upon the previous chapters, applying the method of combining probability theory with Hamiltonian mechanics. To do so, one needs to build a meaningful sample space over states, in this case, quantum states. A substantial part of the chapter discusses how to construct these quantum states out of which one can build a sample space on which to apply a probability measure. Vector states and density operators are introduced and various worked examples are proposed. Once the quantum sample space is identified, the equilibrium quantum statistical mechanics is formulated. The ‘particle in a box’ problem turns out to be analytically intractable, unless we take a certain limit called the semi-classical limit. Heuristics as to what this limit means are proposed. Finally, the von Neumann (quantum) entropy is introduced and analogies with thermodynamics are made. An application to the heat capacity of solids is presented. As complement, the chapter also introduces a classical ‘ring-polymer’ analog of quantum statistical mechanics stating the formal equivalence between a one-particle quantum canonical system and an N-particle classical canonical system.

Unlike classical particles, quantum particles are indistinguishable.Fermions and bosons differ in their quantum statistics, and the consequences of this for their statistical mechanics are explored in the grand canonical ensemble.The Fermi--Dirac and Bose--Einstein distribution functions are derived, and utilized to write thermal averages using the density of states.