A unified treatment of dynamics and scattering in classical and quantum statistical mechanics

Abstract

The common formal features of classical and quantum statistical mechanics are investigated at three separate levels: at the level of L 2 spaces of wave-packets on Γ-space, of Liouville spaces B 2 consisting of density operators constructed from such wave-packets, and of phase-space representation spaces P of Γ-distribution functions. It is shown that at the last level the formal similarities become so outstanding that all key quantities in P -spaces, such as Liouville operators, Hamilton functions, position and momentum observables, etc., are represented by expressions which to the zeroth order in ħ coincide in the classical and quantum case, and in some instances coincide completely. Scattering theory on the B 2 Liouville spaces takes on the same formal appearance for classical and quantum statistical mechanics, and to the zeroth order in ħ it coincides in both cases. This makes possible the formulation of a classical approximation to quantum scattering, and of a computational scheme for determining ϱ out from ϱ in for successive orders of ħ.