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.