Quantum chaotic systems are physical systems whose classical counterparts exhibit deterministic chaos, while their quantum behavior reflects this through characteristic spectral and eigenstate properties rather than literal trajectory chaos. They are typically defined by Hamiltonians lacking sufficient integrals of motion for integrability and are studied via tools such as random matrix theory, semiclassical analysis, and eigenstate statistics. Hallmarks include Wigner–Dyson level spacing distributions, eigenstate delocalization in appropriate bases, sensitivity of spectral correlations to classical phase-space structures, and signatures linked to the semiclassical limit, exemplified by the Gutzwiller trace formula and correspondence between classical periodic orbits and quantum spectral fluctuations.
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