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If I think about Bloch's theorem, it states that

ψ

_{k}(r)=e

^{ik⋅r}u

_{k}(r) where u

_{k}has the periodicity of the lattice. If u is independent of wavevector, well then I'm just multiplying some function by a plane wave. Great. The density, which is a physical observable, then is independent of k and each unit cell has the same density ρ(r).

Within the tight binding approximation, however, one typically applies that phase factor discretely to orbitals inside each unit cell. For a single orbital per unit cell, φ, one might write the wave function as ψ

_{TB}=∑

_{m}e

^{ikRm}φ(r-R

_{m})

This is distinct from multiplying the sum by a continuous plane wave (ψ=e

^{ik⋅R}∑

_{m}φ(r-R

_{m})) as it introduces nodes in the density ρ(r), and makes the density depend on the choice of origin. Both of these satisfy Bloch's theorem, but they seem physically different. I understand if I work through a simple band structure how ψ

_{TB}(k) has some advantages for calculating band structures, but am I getting hung up on something unphysical, wrong, trivial, or subtly relevant?

Thanks for any insight,

James