  The Quantum Ergodic Conjecture equates the Wigner function for a typical
eigenstate of a classically chaotic Hamiltonian with a delta-function on the
energy shell. This ensures the evaluation of classical ergodic expectations of
simple observables, in agreement with Shnirelman's theorem, but this putative
Wigner function violates several important requirements. Consequently, we
transfer the conjecture to the Fourier transform of the Wigner function, that
is, the chord function. We show that all the relevant consequences of the usual
conjecture require only information contained within a small (Planck) volume
around the origin of the phase space of chords: translations in ordinary phase
space. Loci of complete orthogonality between a given eigenstate and its nearby
translation are quite elusive for the Wigner function, but our local conjecture
stipulates that their pattern should be universal for ergodic eigenstates of
the same Hamiltonian lying within a classically narrow energy range. Our
findings are supported by numerical evidence in a Hamiltonian exhibiting soft
chaos. Heavily scarred eigenstates are remarkable counter-examples of the
ergodic universal pattern.
