Here's a question for all you advanced solid state people out there.  I am studying methods of solid state physics on unusual surfaces, and I find that I do not understand the meaning of what we typically refer to as the "First Brillouin Zone" or the assumptions we make.

     I thoroughly understand Bloch's theorem (several versions of it): for a lattice with periodic potential, certain translation operators (that translate from one direct lattice site to another) commute with the Hamiltonian.  As a result, these two operators have simultaneous eigenstates, which can be uniquely specified by identifying their momentum and energy , so that the simultaneous eigenstates can be written .

     Then we consider the expansion of these simultaneous eigenstates in terms of the momentum eigenstates , and we find that the state only contains certain momentum states - namely those that differ by a reciprocal lattice vector (times ) from .

     These results are what give rise directly to the "usual" form of Bloch's theorem:

, where

The First Brillouin Zone of a lattice seems to arise from indexing these particular subclasses of the momenta states.  It seems that we denote the state with momentum to be "equivalent" to all states with momenta , so that we can freely restrict to the First Brillouin Zone and still extract the same band structure from the Hamiltonian.  What is this equivalence relation we are defining, and how is the restriction to the FBZ valid?  Any help would be greatly appreciated.  Thanks.