Random Matrix Elements and Eigenfunctions in Chaotic Systems

Details Random Matrix Elements and Eigenfunctions in Chaotic Systems Random Matrix Elements and Eigenfunctions in Chaotic Systems

1997-11-20

arxiv.org/abs/chao-dyn/9711020v1

Phys. Rev. E 57, 7313 (1998)

Physics

Condensed Matter

Mesoscale and Nanoscale Physics

7 pages, no figures, RevTeX

Scientific paper

10.1103/PhysRevE.57.7313

The expected root-mean-square value of a matrix element $A_{\alpha\beta}$ in a classically chaotic system, where $A$ is a smooth, $\hbar$-independent function of the coordinates and momenta, and $\alpha$ and $\beta$ label different energy eigenstates, has been evaluated in the literature in two different ways: by treating the energy eigenfunctions as gaussian random variables and averaging $|A_{\alpha\beta}|^2$ over them; and by relating $|A_{\alpha\beta}|^2$ to the classical time-correlation function of $A$. We show that these two methods give the same answer only if Berry's formula for the spatial correlations in the energy eigenfunctions (which is based on a microcanonical density in phase space) is modified at large separations in a manner which we previously proposed.

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