Quantum Chaos — Stadium Billiard

Eigenstates, scarred wavefunctions, and Wigner-Dyson level statistics in the Bunimovich stadium

Eigenstate info
Mode index—
Approx. kₙ—
ε (elongation)—
−ℏ²Δψ = Eψ, ψ|∂Ω = 0

Circle (ε=0): Poisson stats
Stadium (ε>0): Wigner-Dyson

Scarring (Heller 1984):
eigenstates concentrate
along unstable periodic orbits
The Bunimovich stadium billiard is the archetypal quantum chaos system: a rectangle with semicircular caps. Its classical dynamics is fully chaotic (ergodic, mixing). Quantum mechanically, solving −Δψ = k²ψ with Dirichlet boundary conditions reveals scarred eigenstates — wavefunctions that concentrate along classical periodic orbits (Heller 1984). The level spacing statistics obey Wigner-Dyson distribution P(s) = (πs/2)e^{−πs²/4}, the hallmark of quantum chaos, vs. Poisson P(s) = e^{−s} for integrable systems (circle billiard). This is the quantum signature of classical chaos.