The eigenstate thermalization hypothesis is closely connected to the quantum nature of chaos (see quantum chaos). Furthermore, since a classically chaotic system is also ergodic, almost all of its trajectories eventually explore uniformly the entire accessible phase space, which would imply the eigenstates of the quantum chaotic system fill the quantum phase space evenly (up to random fluctuations) in the semiclassical limit ℏ ℏ → → 0 . In particular, there is a quantum ergodicity theorem showing that the expectation value of an operator converges to the corresponding microcanonical classical average as ℏ ℏ → → 0 . However, the quantum ergodicity theorem leaves open the possibility of non-ergodic states such as quantum scars. In addition to the conventional scarring, there are two other types of quantum scarring, which further illustrate the weak-ergodicity breaking in quantum chaotic systems: perturbation-induced and many-body quantum scars. Since the former arise a combined effect of special nearly-degenerate unperturbed states and the localized nature of the perturbation (potential bums), the scarring can slow down the thermalization process in disordered quantum dots and wells, which is further illustrated by the fact that these quantum scars can be used to propagate quantum wave packets in a disordered nanostructure with high fidelity. On the other hand, the latter form of scarring has been speculated to be the culprit behind the unexpectedly slow thermalization of cold atoms observed experimentally.