Is linear momentum conserved in a system with open boundary conditions? In a one dimensional lattice system with periodic boundary conditions, in which the last and the first site of the lattice are the same site, momentum is conserved modulo a vector of the reciprocal lattice. My question is if for a system with open boundary conditions, in which the single particle wave functions vanish in the two extremes of the chain, momentum or quasi-momentum is conserved. In principle, there is no discrete (nor continuous) translation symmetry, so momentum shouldn't be conserved. However, I wonder if momentum conservation is completely lost or if there are some remains.
 A: Strictly speaking, crystal momentum is not conserved.  For bulk properties of a solid, the consequences of making the approximation that crystal momentum is strictly conserved are negligible.  Of course, for regions near a surface, the lack of periodicity is an essential feature and causes the surface region to have different properties than the bulk.  It's like a layer of a different material on top of the bulk.
"Regular" momentum plays a role in solids only if the effect of the ion lattice can be ignored.  Semiconductor systems can be constructed with thin planar layers in which the electrons behave as a free two-dimensional electron gas.  But even in this case the conserved quantity is different than the $m\vec{v}$ momentum from mechanics.  The particles in solids, what we loosely call "electrons", are actually comprised of a bare electron plus whatever the electron might drag along with it.  The electron can deform the background medium, creating a polarization excitation as it moves.  This quasiparticle is known as a polaron.  A polaron does have a conserved momentum, but it is not equal to the mass of the electron times its velocity.  As above, this momentum is not strictly conserved, and is manifestly not conserved near surfaces, interfaces, defects, etc.
