The nearest-neighbor spacing distribution P(s) distinguishes integrable from chaotic quantum systems. Brody's interpolation p(s) = (β+1)A s^β exp(−As^(β+1)) smoothly connects Poisson (β=0) to Wigner-Dyson GOE (β=1).
Random matrix theory predicts that the eigenvalue spacings of chaotic Hamiltonians follow the Wigner surmise P(s) ≈ (πs/2)exp(−πs²/4), while integrable systems show Poisson clustering P(s) = exp(−s). Level repulsion (β=1) prevents eigenvalue degeneracies — quantum chaos has a spectral signature. The Brody parameter β is used as an empirical chaos indicator.