  For a quantum state, or classical harmonic normal mode, of a system of
spatial periodicity "R", Bloch character is encoded in a wavevector "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 wavevector are not explicitly defined, but may be hidden
in the state (and eigenvector |i>.) 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.
