Semiclassical convergence between the classical and quantum is a particular case of convergence for a general operator, found originally by Feingold and Peres [73], and most clearly expressed (as a `sum rule') by Wilkinson [199] (see also [72]). The theoretical statement was that the average squared matrix element of any operator , a `distance' from the diagonal when written in the energy basis, is proportional to the -component of the Fourier transform of the corresponding classical auto-correlation . The typical energy is . Convergence is achieved when averaged over a small region, as shown in Fig. 2.3, and the energy smearing () is larger than . ( is the period of the shortest period orbit). Because of this smearing over a scale , all periodic orbit fluctuations (discussed in the following section) are averaged away.
Looking at (2.48),
and choosing
,
this corresponds in our notation to
If the initial energy distribution (or ) is a smooth function on this smearing scale, then the smoothed would give the resulting quantum dissipation rate, and QCC would follow. This will be the usual assumption about . However, if the system were prepared in a quantum-mechanically narrow distribution (for example a single pure state (2.39) or a Fermi distribution at temperature ), consideration of the unsmeared structure of is required.