The Statistical Mechanical Theory of Transport Processes. V. Quantum Hydrodynamics

Abstract

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.