Since a classically chaotic system is also ergodic, almost all of its trajectories eventually explore uniformly the entire accessible phase space. Thus, when translating the concept of ergodicity to the quantum realm, it is natural to assume that the eigenstates of the quantum chaotic system would fill the quantum phase space evenly (up to random fluctuations) in the semiclassical limit ℏ ℏ → → 0 . The quantum ergodicity theorems of Shnirelman, Zelditch, and Yves Colin de Verdière proves that the expectation value of an operator converges in the semiclassical limit to the corresponding microcanonical classical average. However, the quantum ergodicity theorem leaves open the possibility of eigenfunctions become sparse with serious holes as ℏ ℏ → → 0 , leaving large but not macroscopic gaps on the energy manifolds in the phase space. In particular, the theorem allows the existence of a subset of macroscopically nonergodic states which on the other hand must approach zero measure, i.e., the contribution of this set goes towards zero percent of all eigenstates when ℏ ℏ → → 0 .