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 k _z of the wave vector are not explicitly defined, but may be hidden in the state (and eigenvector). A prescription for extracting k_z 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.