# Loss of Momentum Information when Folding Bands into First Brillouin Zone?

I have read these posts in trying to answer my question: $k$-interval for First Brillouin Zone, Band gaps: are they at the centre or at the edge of the Brillouin zone?. I have also tried to find answers in Ashcroft and Mermin, Ibach and Luth, and Kittle to no avail.

In solid state physics, electron bands are usually introduced by deriving the nearly-free-electron dispersion, using Bloch's theorem. From there the professor folds the dispersion into the First Brillouin Zone (FBZ), and says 'voila: bands and gaps'! I understand from a math perspective that for Bloch wave-functions the k vector only adds a phase, and that it has no impact on physical observables. I am having a hard time understanding this physically. Once could say it is easy to unfold the bands back into the original picture in this case, but what about more complicated bands where the originating BZ is not so obvious? My question is: why does folding the energy-momentum dispersion into the FBZ not destroy information about the absolute momentum of states? If it does destroy information, but that fact does not matter, why does it not matter?

A state with well defined momentum outside of the FBZ should be measure-able as such, correct? Then why does the FBZ band representation not limit the band diagram?

• I will shamelessly point to this answer to a closely related question. Note the emphsis there on this sentence: "Both schemes show the same information." There is no information lost. Apr 24, 2020 at 20:37
• Thanks for the link. I'll add it to the post so it's more visual. I think the post helps, but it doesn't fully explain why information isn't lost.
– NJP
Apr 26, 2020 at 20:18
• Actually, it does. In the folded scheme all momenta that lie on any vertical line are the same. If they are the same, nothing is lost. It's like counting modulo 5. I can count fifteen consecutive numbers, and the sixteenth will be 1. If you are talking about the sixteenth you are actually talking about the first. Apr 27, 2020 at 3:49

A Bloch state is labelled by a reciprocal lattice vector $$k$$, which is often sloppily called "momentum" with or without a factor $$\hbar$$. The vector space where $$k$$ lives has symmetry properties that are determined by the structure of the real-space lattice and the space spanned by the lattice basis vectors. Just like translating the wave function by an integer number of lattice sites does not change the wavefunction at all, the same holds true for "translating" the wave function in reciprocal lattice space. This means that the label $$k$$ cannot be unique, just like a site in an infinite and perfectly periodic lattice cannot be either.