Mixtures and the Density Operator

Abstract

In this chapter we generalize our treatment of two-level systems to include various kinds of damping. Some of these can be incorporated directly into the equations of motion for the probability amplitudes. However two important kinds cannot: upper to lower level decay, and more rapid decay of the electric dipole than the average level decay rate. For these two damping mechanisms, we need a more general description than can be provided by the state vector. Specifically, we need to consider systems for which we do not possess the maximum knowledge allowed by quantum mechanics. In other words, we do not know the state vector of the system, but rather the classical probabilities for having various possible state vectors. Such situations are described by the density operator ρ, which is a sum of projectors ∣ψ _i 〉〈ψ _i ∣ onto the possible state vectors ∣ψ _i 〉, each weighted by a classical probability P _i .