  This article is concerned with the semi-classical limits of matrix elements
$<F \phi_j, \phi_j>$ of eigenfunctions of the Laplacian $\Delta_g$ of a compact
Riemannian manifold $(M, g)$ with respect to a Fourier integral operator $F$ on
$L^2(M)$. Many results exist for the case where $F$ is a pseudo-differential
operator, but matrix elements of Fourier integral operators involve new
considerations. The limits reflect the extent to which the canonical relation
of $F$ is invariant under the geodesic flow of $(M, g)$. When the canonical
relation is almost nowhere invariant, a density one subsequence of the matrix
elements tends to zero (related results arose first in the study of quantum
ergodic restriction theorems). The limit states are invariant measures on the
canonical relation of $F$ and their invariance properties are explained. The
invariance properties in the case of Hecke operators answers an old question
raised by the author.
