Chapter 11 - Density Matrix and Product Operator Formalisms

Abstract

This chapter provides a brief description of the density matrix. The term density matrix arises by analogy to classical statistical mechanics, where the state of a system consisting of N molecules moving in a real three-dimensional space is described by the density of points in a 6N-dimensional phase space, which includes three orthogonal spatial coordinates and three conjugate momenta for each of the N particles, thus giving a complete description of the system at a particular time. The density matrix for a spin system includes all the spins and all the spatial coordinates as well. The chapter provides a formula for constructing the density matrix for any system in terms of a set of basis functions and also determines the expectation value of any dynamical variable. The real value of the density matrix approach lies in its ability to describe coherent time-dependent processes, something that could not do with steady-state quantum mechanics. The chapter also presents an expression for the time evolution of the density matrix in terms of the Hamiltonian applicable to the spin system. The product operator formalism is normally applied only to weakly coupled spin systems, where independent operators for I and S are meaningful. The shorthand scheme for representing the evolution of the operators shows the operator at a particular step in the process, then indicates by an arrow the type of transformation and gives the resultant operator. Evolution of a coherence based on a chemical shift is usually indicated by a rotation (superoperator) at the precession frequency multiplied by the period during which the chemical shift acts.