# Bloch Waves within Tight Binding Approximation

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So I thought I understood something well, and then I went to explain it to someone and it turns out I'm missing something, and I'd appreciate any insight you might have.
If I think about Bloch's theorem, it states that
ψk(r)=eik⋅ruk(r) where uk 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 ψTB=∑m eikRmφ(r-Rm)

This is distinct from multiplying the sum by a continuous plane wave (ψ=eik⋅Rmφ(r-Rm)) 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 ψTB(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