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Relation to random wave approximation
The uncorrelated impulse picture of the WNA is an intuitive
estimate for
the classical band profile
.
However, it gives little understanding in the quantum case.
The quantum band profile expresses offdiagonal structure in the
matrix
.
The matrix elements are given in a hardwalled system by Eq.(C.2),
the weighted overlap of eigenstates and on the boundary.
Therefore a first approximation to the quantum band profile might
be reached by assuming that the eigenstates
are random sums of plane waves (Berry's postulate).
Furthermore one could assume no correlations between states .
The number of independent
sized patches on the billiard
surface is
, and these will add in a fashion
because they are uncorrelated.
Over each patch the typical squared eigenfunction normal derivative
is
.
Combining with (2.48) and the fact that in dimensions
the mean level spacing is
gives

(3.13) 
where is the typical value of
.
Note that the predicted band profile is flat (independent of )
because the overlap of random waves on the boundary does not depend strongly
on their wavenumber difference.
Also note that does not appear in this quantum estimate.
The similarity to (3.12) is clear.
Performing the above calculation more carefully with the correct
prefactors (this lengthy result is derived in [46])
gives exactly (3.12).
So, remarkably, a random wave estimate in quantum mechanics is equivalent
to the WNA in classical mechanics, as far as the naive band profile
prediction is concerned.
Figure:
The WNA estimate compared to actual
power spectra for example `special' deformation types:
DI (dilation), TX (translation) and RO (rotation).
See Table 3.2 for definitions.
The WNA fails to predict the vanishing in the small limit.

Figure 3.6:
Similar to Fig. 3.5 except a loglog plot.
This demonstrates the special deformation power laws.
The two dotted lines show
and frequency dependence, for purposes of comparison.
A nonspecial deformation (W2) is also shown to contrast its small
dependence.
Estimation error here is 13% for W2 and RO, 20% for DI and TX.

Table 3.2:
Key to the four `special' deformations in 2D.
The unit vectors and are
in the plane (see Fig. 1),
and is in the
perpendicular direction.
In the case of dilation and rotation
could be made unitless by dividing by a constant length.
key 
description 
deformation field 
DI 
dilation about origin 

TX 
translation 

TY 
translation 

RO 
rotation about origin 


Next: `Special' deformations
Up: The white noise approximation
Previous: The white noise approximation
Alex Barnett
20011003