Relation to random wave approximation

The uncorrelated impulse picture of the WNA is an intuitive estimate for the classical band profile $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$. However, it gives little understanding in the quantum case. The quantum band profile expresses off-diagonal structure in the matrix $(\partial {\mathcal{H}} / \partial x)_{nm}$. The matrix elements are given in a hard-walled system by Eq.(C.2), the weighted overlap of eigenstates $n$ and $m$ 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 $n \neq m$. The number of independent $\lambda_{{\mbox{\tiny B}}}$-sized patches on the billiard surface is $N \sim k^{1-d}{\mathsf{A}}$, and these will add in a $N^{1/2}$ fashion because they are uncorrelated. Over each patch the typical squared eigenfunction normal derivative is $k^2/{\mathsf{V}}$. Combining with (2.48) and the fact that in $d$ dimensions the mean level spacing is $\Delta \sim \hbar k^{2-d} {\mathsf{V}}/m$ gives

$$\tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\omega)\; ... ...{\mbox{\tiny E}}}^3 \frac{1}{{\mathsf{V}}}\, {\mathsf{A}}D^2 ,$$ (3.13)

where $D$ is the typical value of $D{({\mathbf s})}$. Note that the predicted band profile is flat (independent of $\omega$) because the overlap of random waves on the boundary does not depend strongly on their wavenumber difference. Also note that $\hbar$ 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 $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$ 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 $\omega$ limit.

Figure 3.6: Similar to Fig. 3.5 except a log-log plot. This demonstrates the special deformation power laws. The two dotted lines show $\omega^2$ and $\omega^4$ frequency dependence, for purposes of comparison. A non-special deformation (W2) is also shown to contrast its small-$\omega$ 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 ${\mathbf e}_x$ and ${\mathbf e}_y$ are in the plane (see Fig. 1), and ${\mathbf e}_z$ is in the perpendicular direction. In the case of dilation and rotation ${\mathbf D}$ could be made unitless by dividing by a constant length.

key	description	deformation field
DI	dilation about origin	${\mathbf D}({\mathbf r}) = {\mathbf r}$
TX	$x$-translation	${\mathbf D}({\mathbf r}) = {\mathbf e}_x$
TY	$y$-translation	${\mathbf D}({\mathbf r}) = {\mathbf e}_y$
RO	rotation about origin	${\mathbf D}({\mathbf r}) = {\mathbf e}_z \times {\mathbf r}$