Understanding Bloch's Theorem: Does 'n' Label Energy Bands?

[aaaa202]

Bloch's theorem states that the wave functions for electrons in a periodic potential have the form:

ψ_n,k(r) = u_n(r)exp(ik⋅r)

, where u_n has the same periodicity as the potential.
Bloch's theorem is used to calculate energy bands, and my question is:
Does the n in u_n label the different bands of the crystal?

 

[DrDu]

Yes, exactly

 

[aaaa202]

Okay but if remember correctly then a band diagram can be constructed using an empty lattice approximation, where all wave vectors outside the Brillouin zone are mapped back into the Brillouin zone. In this case you get a band diagram, but the different bands need not necessarily belong to different n? or am I wrong? maybe I am mixing up different things.

 

[DrDu]

Try to wrire down the u-n for the empty lattice.