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American Journal of Physics

Illustrative calculations using projection techniques

A. Muriel

Published Aug 1, 1977• 4 citations
Publisher

Abstract

Using a pedagogical model introduced by Dresden and Feiock, the quantum mechanical version of the Kac ring model, the following studies are illustrated: (a) solution of the Liouville equation using projection techniques; (b) analysis of the difference between the linear and nonlinear theories; and (c) existence of a kinetic description, with the solution of the Pauli master equation for this example.

DOI
10.1119/1.10777

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Using a pedagogical model introduced by Dresden and Feiock, the quantum mechanical version of the Kac ring model, the following studies are illustrated: (a) solution of the Liouville equation using projection techniques; (b) analysis of the difference between the linear and nonlinear theories; and (c) existence of a kinetic description, with the solution of the Pauli master equation for this example.
Taking the viewpoint that a complete solution of the Liouville equation contains much more information than is necessary in response theory (wherein one is concerned with a particular response), projection techniques are employed to introduce a shortcut in the problem. The treatment supplements and completes the formalism initiated in a previous publication and yields a general equation which is more relevant to response theory in the usual sense. The most natural application is to the theory of transport. The general transport equation is analyzed in the context of electrical conduction. Meaningful similarities to the simple classical Drude model of conduction are exhibited; the relationship of this exact formalism to the Kubo theory of linear response is shown, an exact solution for a step-function stimulus and approximation procedures which go beyond the linear regime are presented, and an expression for the electrical resistivity is explicitly calculated thus making contact with an earlier paper by Kenkre and Dresden. The response equation is also analyzed in the context of electric and magnetic polarization.
Mixed quantum-classical equations of motion are derived for a quantum subsystem of light (mass m) particles coupled to a classical bath of massive (mass M) particles. The equation of motion follows from a partial Wigner transform over the bath degrees of freedom of the Liouville equation for the full quantum system, followed by an expansion in the small parameter μ=(m/M)1/2 in analogy with the theory of Brownian motion. The resulting mixed quantum-classical Liouville equation accounts for the coupled evolution of the subsystem and bath. The quantum subsystem is represented in an adiabatic (or other) basis and the series solution of the Liouville equation leads to a representation of the dynamics in an ensemble of surface-hopping trajectories. A generalized Pauli master equation for the evolution of the diagonal elements of the density matrix is derived by projection operator methods and its structure is analyzed in terms of surface-hopping trajectories.