By J. Rammer, H. Smith

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A similar equation may be obtained for the off-diagonal elements of ρ(t). For this see Peier (1972) and Louisell (1973). Eq. 50) contains ρ Rαα (0), which may be taken as a thermodynamic equilibrium state for a large system. This then introduces a temperature as a parameter in the reservoir. 46 Quantum statistical master equation General results of this Pauli equation will be discussed in Chapter 5. The principal applications will be seen in later sections of this book, particularly in the discussion of quantum optics in Chapter 11.

It is at least not exponential. Two time scales exist: one τ B as t → ∞, and the other τ c as t → 0. The lower limit was examined by Horwitz and Marchand (1967). They argue that near t = 0, we may neglect the time integral in Eq. 34). In the remaining term, ρ S (0) is diagonal, and P L P = 0. Thus ρ S (t) near t = 0 is time independent, and there can be no exponential decay. The long-time behavior is difficult to treat and subject to much consideration. Qualitatively, in decay-scattering models, the energy E 0 is bounded from below (E m0 = 0), and a branch cut must appear in the Laplace transform space of the 3 resolvent (Goldberger and Watson, 1964).

Diagonal elements are constant, and the off-diagonal elements oscillate, and ρ nn (t) = n ρ nn (0) = 1. 19) n In the so-called random phase approximation, we replace ρ nn (t) by its average over n − n . Then the oscillations cancel, and ρ¯ nn (t) = ρ nn (0) is time independent. The comments made are also true for any exact diagonal representation, not just the harmonic oscillator being discussed here. We may write the coordinate representation u n (q). 2 Harmonic oscillator 23 whose normalized solution is the Gaussian ω u 0 (q) ≡< q | 0 >= π 1 4 −ωq 2 .