Nearest-neighbor spacing distributions reveal quantum chaos vs. integrability
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The Bohigas-Giannoni-Schmit conjecture: quantum systems whose classical limit is chaotic have energy level statistics matching the Gaussian Orthogonal Ensemble (GOE) of random matrices. Integrable systems show Poisson statistics (P(s) = e^−s). The transition from level clustering (Poisson) to level repulsion (Wigner surmise: P(s) ≈ (π/2)s·e^(−πs²/4)) diagnoses chaos.