8. Quantum Formulation

In this chapter, we present the mathematical formalism used in quantum mechanics first and then derive expressions for canonical and microcanonical partition functions for quantum mechanical systems. This will help you develop familiarity with the basic ideas of quantum mechanics and the bra–ket notation you may encounter when consulting more advanced textbooks on statistical mechanics. The important conclusion of this chapter is that the distinction between classical and quantum mechanical versions of statistical mechanics stems from the explicit expressions for the density of states these mechanics predict and from the manner in which a given system populates the microstates accessible to it. Many optional sections are included to provide explicit derivations of several key results from quantum mechanics we have already used in earlier Chapters, but should probably be omitted upon the first reading. Keeping with our most immediate goals, we will not concern ourselves with experimental findings that forced the radical departure from classical mechanics and the eventual formulation of quantum mechanics in the early twentieth century. Interested readers can find these accounts in earlier chapters of many textbooks on quantum mechanics.