Diagonal matrix elements ⟨n|O|n⟩ vs energy — smooth for chaotic, fluctuating for integrable
Hilbert dim: 128
Chaotic variance: —
Integrable variance: —
ETH (Deutsch 1991, Srednicki 1994): In a chaotic quantum system, every energy eigenstate
locally looks thermal. Diagonal matrix elements Onn = O(En) vary smoothly with energy —
the eigenstate itself encodes thermodynamics. In integrable systems (many conserved quantities),
eigenstates are highly non-thermal — Onn fluctuates wildly. This explains why isolated
quantum systems thermalize: any initial state at energy E reaches the thermal ensemble expectation ⟨O⟩th.