Quantum Chaos & Level Statistics

Energy level spacing distributions: Wigner-Dyson (chaos) vs Poisson (integrable)

Billiard Domain

Trajectories: 5

Stadium billiard: two semicircular caps on a rectangle. Ergodic and mixing — Bunimovich 1979. Every trajectory fills the domain uniformly (except measure-zero integrable ones).

Level Spacing Distribution P(s)

Levels N: 200

Wigner-Dyson (GOE)

Poisson

Simulated

Wigner-Dyson: P(s) = (π/2)s·exp(-πs²/4) — level repulsion at s→0. Poisson: P(s) = e^(-s) — levels cluster. BGS conjecture: quantum chaos ↔ GOE statistics.

Energy Spectrum & Nearest-Neighbor Spacings

Each vertical line is an energy level. Chaotic systems show repulsion (levels avoid each other). Integrable systems allow level crossings (lines cluster).