The Configurational Distribution Function in Quantum-Statistical Mechanics

The function F{N} 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 F{N} is obtained in a new form which is very closely related to the form of the analogous classical distribution FCL{N}. The new form of F{N} shows clearly how the difference between F{N} and FCL{N} 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 F{N} as a power series in ℏ/kT which is carried through to the fourth order. The second application is the development of F{N} as a series of configuration space integrals. A previously published low temperature development of F{N} 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 F{N}. Finally, several integral equations for F{N} are presented which give physical insight into its structure, by establishing mathematical analogies with such physical processes as neutron diffusion.