The diagonal Eigenstate Thermalization Hypothesis

Consider a finite isolated system with a non-integrable Hamiltonian having $N$ degrees of freedom, and work within a single symmetry sector so that the energy eigenstates $|E_i\rangle$ of $\hat{H}$ are non-degenerate. For a large class of operators $\hat{O}$, define

$$O_{ii}=\langle E_i|\hat{O}|E_i\rangle.$$

Let $\langle\hat{O}\rangle_{\mathrm{micro},E_i}$ denote the microcanonical average at energy $E_i$.

Eigenstate Thermalization Hypothesis. The diagonal expectation values satisfy

$$O_{ii}=\langle\hat{O}\rangle_{\mathrm{micro},E_i}+\Delta_i,$$

where $\Delta_i$ has zero mean and $\Delta_i^2$ has magnitude of order

$$\langle\hat{O}^2\rangle_{\mathrm{micro},E_i}\exp(-S(E))\propto \langle\hat{O}^2\rangle_{\mathrm{micro},E_i}\exp(-\operatorname{const.}\times N).$$

This hypothesis asserts that individual energy eigenstates reproduce microcanonical expectation values up to exponentially small fluctuations, providing a mechanism for thermal behavior in isolated non-integrable quantum systems. The source does not state a resolution status for this formulation.