  We prove that any deterministic matrix is approximately the identity in the
eigenbasis of a large random Wigner matrix with very high probability and with
an optimal error inversely proportional to the square root of the dimension.
Our theorem thus rigorously verifies the Eigenstate Thermalization Hypothesis
by Deutsch [Deutsch 1991] for the simplest chaotic quantum system, the Wigner
ensemble. In mathematical terms, we prove the strong form of Quantum Unique
Ergodicity (QUE) with an optimal convergence rate for all eigenvectors
simultaneously, generalizing previous probabilistic QUE results in [Bourgade,
Yau 2017] and [Bourgade, Yau, Yin 2020].
