My book gives a treatment of this problem for crystal vibrations, but I don't really understand it. It says: There is one allowed value of K per volume (2[itex]\pi[/itex]/L)3. But at the same time it has just shown that Kx,Ky,Kz can take values ±2[itex]\pi[/itex]/L which would certainly lead to more combinations of Kx,Ky,Kz within the volume confined by (2[itex]\pi[/itex]/L)3. What am I misunderstanding. Also: applying periodic boundary conditions yields the condition that Kx,Ky,Kz=±n2[itex]\pi[/itex]/L, while fixed ends yielded K=n[itex]\pi[/itex]/L, but my book says the two approaches yield identical results. How is that??