Level repulsion: in chaotic (non-integrable) quantum systems, eigenstates avoid having the same energy — small spacings are suppressed. This is the quantum fingerprint of classical chaos.
Bohigas-Giannoni-Schmit conjecture (1984): quantum systems whose classical limit is chaotic have level statistics following Random Matrix Theory (GOE/GUE/GSE ensembles). Integrable systems show Poisson statistics.
Stadium billiard: a rectangle capped by two semicircles (Bunimovich stadium). For a > 0, the classical dynamics is ergodic and chaotic. For a = 0 (rectangle), classical orbits are integrable.