I was reading through topocondmat.org, and I am confused about the discussion here (see the section on k.p theory and discretizing continuum models).
To discretize the continuum $k$-space Hamiltonian, they write $k = -i\partial_{x}$ and approximate the derivative as a finite difference then rewrite the Hamiltonian in terms of real-space orbitals. But what I don't understand is why they equate k (crystal momentum) to $-i∂_{x}$ (canonical momentum). Crystal momentum is not the same thing as canonical momentum, and canonical momentum is not a good quantum number for labeling Bloch states, so how does this procedure make any sense? I would think they should replace $k$ with the crystal momentum operator, but I'm not sure what the crystal momentum operator is in real-space. In any case, it clearly cannot be $-i\partial_{x}$, as hitting a Bloch state with this does not yield an eigenvalue of $k$, so how does it make any sense to equate $k$ and $-i\partial_{x}$?
