Semiclassical connection between quantum and classical band profiles

Semiclassical convergence between the classical and quantum $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$ 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 $\hat{A}$, a `distance' $\hbar \omega$ from the diagonal when written in the energy basis, is proportional to the $\omega$-component of the Fourier transform of the corresponding classical auto-correlation $\langle A(0) A(t) \rangle_{{\mbox{\tiny E}}}$. The typical energy is $E$. Convergence is achieved when averaged over a small region, as shown in Fig. 2.3, and the energy smearing ($M\Delta$) is larger than $\hbar / \tau_1$. ($\tau_1$ is the period of the shortest period orbit). Because of this smearing over a scale $O(\hbar)$, all periodic orbit fluctuations (discussed in the following section) are averaged away.

Looking at (2.48), and choosing $\hat{A} = {\mathcal{F}}$, this corresponds in our notation to

$$\tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\omega)\ \approx \ \tilde{C}_{{\mbox{\tiny E}}}(\omega).$$ (2.57)

This connection is just another statement of the correspondence principle of Shnirelman [177] (also see [91]) that in the semiclassical limit the expectation value of an operator (averaged over many adjacent states) is given by the microcanonical average of the corresponding classical function. In this case, the operator is ${\mathcal{F}}(0){\mathcal{F}}(\tau)$, and its quantum and classical expectations are (2.46) and (2.10) respectively. The operator involves propagation forward in time. The only significant contribution comes from times $\tau \sim t_{{\mbox{\tiny erg}}}$ or less, therefore short-time correspondence of the evolution is an additional requirement for (2.57) to hold.

If the initial energy distribution $\rho(E)$ (or $p_m$) is a smooth function on this $O(\hbar)$ smearing scale, then the smoothed $\tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\omega)$ would give the resulting quantum dissipation rate, and QCC would follow. This will be the usual assumption about $\rho(E)$. 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 $k_{{\mbox{\tiny B}}}T \sim \Delta$), consideration of the unsmeared structure of $\vert\partial {\mathcal{H}} / \partial x_{nm}\vert^2$ is required.