State Function Formulation of Elementary Relaxation Processes

The approach to thermal equilibrium of a free particle and a harmonic oscillator interacting with a heat bath can be understood from a semiclassical point of view in terms of the time evolution of a quantum mechanical state function appropriate to each system. The state functions are parameterized such that expectation values comply with the predictions of the classical theory of Brownian motion, with the consequence that the Fokker-Planck equation for a free particle in velocity space, the Montroll-Shuler equation for a quantum oscillator in discrete energy space, and their solutions are arrived at by a common procedure. These relaxation processes become visualizable as the progressive development from a coherent, relatively ordered state to that of equilibrium, so that the loss of information as equilibrium is reached and the consequences of random phasing can be traced dynamically in some detail. The unity of approach is reinforced by the demonstration that since both equations represent Markoffian processes, the Montroll-Shuler equation predicts distributions in coordinate space and momentum space which satisfy a Fokker-Planck equation.