Quantum Mechanics of Complex Systems, II

Abstract

So far our treatment has been general and schematic. In order actually to calculate the effect of the environment quantitatively, we need an explicit model of the dynamics of the environment and its interaction with the system. At this point one of two situations can arise, corresponding roughly to whether our system is microscopic (and relatively simple) or macroscopic. In the first case, we often have a good a priori knowledge of the Hamiltonian governing the environment and its interaction with the system. This is the case, for example, in small molecules (when for most purposes the total Hamiltonian of the molecule is just the sum of the nuclear and electron kinetic energies and the various Coulomb interactions); it may also be approximately true in other situations of interest in chemical physics or solid-state physics, e.g. in muon diffusion. On the other hand, if we are dealing with the motion of a macroscopic variable, such as the angle made by a pendulum with the vertical, or the relative phase of the Cooper pairs in a Josephson junction, then it is unlikely that we know the details of the microscopic Hamiltonian with any confidence. What is much more likely, and indeed the norm, in this type of case is that we know the effects of the environment on the classical motion of the system, as instantiated for example in the classical friction coefficient. It is then an important question, to which we shall return below, whether a knowledge of such classical effects is sufficient to determine the effects of the environment also on the quantum dynamics of the system.