  After a quench, observables in an integrable system may not relax to the
standard thermal values, but can relax to the ones predicted by the generalized
Gibbs ensemble (GGE) [M. Rigol et al., Phys. Rev. Lett. 98, 050405 (2007)]. The
GGE has been shown to accurately describe observables in various
one-dimensional integrable systems, but the origin of its success is not fully
understood. Here we introduce a microcanonical version of the GGE and provide a
justification of the GGE based on a generalized interpretation of the
eigenstate thermalization hypothesis, which was previously introduced to
explain thermalization of nonintegrable systems. We study relaxation after a
quench of one-dimensional hard-core bosons in an optical lattice. Exact
numerical calculations for up to 10 particles on 50 lattice sites (~10^10
eigenstates) validate our approach.
