  This undergraduate thesis is concerned with developing the tools of
differential geometry and semiclassical analysis needed to understand the the
quantum ergodicity theorem of Schnirelman (1974), Zelditch (1987), and Colin de
Verdi\`ere (1985) and the quantum unique ergodicity conjecture of Rudnick and
Sarnak (1994). The former states that, on any Riemannian manifold with negative
curvature or ergodic geodesic flow, the eigenfunctions of the Laplace-Beltrami
operator equidistribute in phase space with density 1. Under the same
assumptions, the latter states that the eigenfunctions induce a sequence of
Wigner probability measures on fibers of the Hamiltonian in phase space, and
these measures converge in the weak-* topology to the uniform Liouville
measure. If true, the conjecture implies that such eigenfunctions
equidistribute in the high-eigenvalue limit with no exceptional "scarring"
patterns. This physically means that the finest details of chaotic Hamiltonian
systems can never reflect their quantum-mechanical behaviors, even in the
semiclassical limit.
  The main contribution of this thesis is to contextualize the question of
quantum ergodicity and quantum unique ergodicity in an elementary analytic and
geometric framework. In addition to presenting and summarizing numerous
important proofs, such as Colin de Verdi\`ere's proof of the quantum ergodicity
theorem, we perform graphical simulations of certain billiard flows and
expositorily discuss several themes in the study of quantum chaos.
