The Pair Distribution Function in Quantum Statistical Mechanics.

Shigeji Fujita

Robert F. O’Connell

William Kunkin, Jerome K. Percus

The method of functional differentiation, used in classical statistical mechanics to obtain approximate integral equations for the pair distribution function, is extended to quantum systems obeying Maxwell-Boltzmann statistics. The grand partition function is written as a path integral, and space-temperature distributions are generated from it by successive functional differentiation. Distributions can be expanded in powers of an external potential via a functional Taylor series. When the series is truncated so that no distributions higher than second order enter and the external potential is specialized to one arising from a fixed particle in the system, coupled integral equations result for the two kinds of pair distributions. The first-order Ursell function calculated from these equations yields the Montroll-Ward ring summation for the grand potential. This approximate Ursell function determines also the Fourier transform of the momentum density (useful in calculating the p = 0 occupation number). Although divergences appear in the low-temperature limit, they can be removed by a simple modification of the independent functional. An extension to Fermi-Dirac and Böse-Einstein statistics is indicated at appropriate places in the text.

Mark Kon

J. H. Irving, Robert W. Zwanzig

This paper is concerned with certain extensions of a formal technique devised by Wigner for handling problems in quantum-statistical mechanics, especially to problems in quantum-mechanical transport processes. The approach is to find the closest possible analogy between classical and quantum-statistical mechanics, so that the extensive work in classical statistical mechanics can be utilized. This analogy is attained with the Wigner distribution function, with which averages of dynamical variables in quantum mechanics may be calculated by integrations in phase space. We will first state some basic properties of distribution functions in classical statistical mechanics, and then state the corresponding properties of the density matrix in quantum mechanics. We will define and discuss the Wigner distribution function, show that it has the desired averaging properties, and obtain the analog of the Liouville equation satisfied by this function. We will derive the analog of the Liouville equation in reduced phase space, and then obtain the equations of hydro-dynamics from quantum-statistical mechanics.

Christian Willy Gruber