Undergrad Bloch Waves within Tight Binding Approximation

[James J]

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
ψ_k(r)=e^ik⋅ru_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^ikR_mφ(r-R_m)

This is distinct from multiplying the sum by a continuous plane wave (ψ=e^ik⋅R∑_mφ(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

 

[DrDu]

Maybe a simple example helps here: Think of a one dimensional chain of e.g. hydrogen atoms. You can write down the Bloch wavefunctions immediately. Now you can also chose the lattice constant twice as large as the distance of the atoms. Then the basis will contain two atoms. The wavefunction does not change, only the interpretation. You see that the second atom will be out of phase relative to the first one by exp(ika).