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??
if you read the sentence without pausing, the meaning is slightly different. "There is one allowed value of K per volume , for each polarization and for each branch." so there are multiple K values allowed per volume
No, same combination of kx,ky,kz may correspond to different states with different energies. Look at the acoustic and optic branch in a 1D chain with two types of atoms. This is the simplest example of "branches". Here for each k there are two energies (or frequencies).