GOE Random Matrix — Quantum Chaos Level Statistics
Wigner-Dyson level repulsion vs Poisson (integrable) spacing statistics
The Gaussian Orthogonal Ensemble (GOE) describes quantum systems whose classical counterpart is chaotic. The hallmark is level repulsion: spacings follow the Wigner surmise P(s) = (π/2)s·exp(−πs²/4), vanishing as s→0. Integrable (Poisson) systems show P(s) = exp(−s) — no repulsion, levels can cluster. This is a fundamental signature of quantum chaos used in nuclear physics, quantum billiards, and condensed matter.