Quantum Chaos: Energy Level Statistics
Wigner-Dyson (GOE/GUE) vs Poisson · Level Repulsion · Spectral Rigidity
Histogram
Wigner surmise (GOE)
Poisson (integrable)
GUE
Ensemble & Parameters
Matrix ensemble
GOE (time-reversal symmetric)
GUE (broken T-reversal)
Poisson (integrable)
Mixed (chaos parameter β)
Matrix size N
100
Chaos parameter β (mixed only)
0.5
Histogram bins
30
Regenerate Sample
Wigner surmise (GOE):
p(s) = (π/2)·s·exp(−πs²/4)
GUE:
p(s) = (32/π²)·s²·exp(−4s²/π)
Poisson (integrable):
p(s) = exp(−s)
Key signatures:
Level repulsion
: p(s→0) ~ sᵝ
β=1 (GOE), β=2 (GUE), β=0 (Poisson)
Spectral rigidity Δ₃:
GOE:
~(1/π²)ln L
Poisson:
~L/15