Recovering hidden Bloch character: Unfolding electrons, phonons, and slabs

For a quantum state, or classical harmonic normal mode, of a system of spatial periodicity ``R,'' Bloch character is encoded in a wave vector ``K.'' One can ask whether this state has partial Bloch character ``k'' corresponding to a finer scale of periodicity ``r.'' Answering this is called ``unfolding.'' A theorem is proven that yields a mathematically clear prescription for unfolding, by examining translational properties of the state, requiring no ``reference states'' or basis functions with the finer periodicity ($r,k$). A question then arises: How should one assign partial Bloch character to a state of a finite system? A slab, finite in one direction, is used as the example. Perpendicular components kz of the wave vector are not explicitly defined, but may be hidden in the state (and eigenvector |i). A prescription for extracting kz is offered and tested. An idealized silicon (111) surface is used as the example. Slab unfolding reveals surface-localized states and resonances which were not evident from dispersion curves alone.