The Configurational Distribution Function in Quantum-Statistical Mechanics

The function FN which gives the relative probability of a configuration N is studied for the case of a quantum-mechanical system to which classical statistics can be applied. By means of a device originally used in quantum electrodynamics FN is obtained in a new form which is very closely related to the form of the analogous classical distribution FCLN. The new form of FN shows clearly how the difference between FN and FCLN is related to the uncertainty principle. As an illustration of the utility of the methods presented here several applications are presented. The first application is a high temperature development of FN as a power series in ℏ/kT which is carried through to the fourth order. The second application is the development of FN as a series of configuration space integrals. A previously published low temperature development of FN is obtained in a simple manner, and it is shown that all higher terms in the development can be written down explicitly by starting from the new form of FN. Finally, several integral equations for FN are presented which give physical insight into its structure, by establishing mathematical analogies with such physical processes as neutron diffusion.