The first statements says that u_{n,k}(r) is a normalization constant that is independent of position, but I really don’t understand this statement. From what does this follow? There is no elaboration of this in the lecture notes.
Thanks in advance 
And then also, why is the momentum operator zero?
We can think about this in 2 ways, but for both you need the relation \Psi(\vec r)=u(\vec r) e^{i\vec k \cdot \vec r}.
- If we are studying free electrons, the solutions should correspond to plane waves, which are of the form A e^{i\vec k \cdot \vec r}. Here we recognize u(\vec r)=A, a constant term.
- If we are studying free electrons, we have to replace V=0 in the results from subsection 4. This leads us to the same answer, u is a constant.
Now, if u(\vec r) is a constant, momentum will be zero. For this we need to remember that momentum corresponds to a derivative when working in real space.
Something very important 
in this course is to know when and how to use real and momentum space. The right approach can save a lot of time and effort (when building the matrix in problem 3.1 NFEM for example).
Very clear, thanks!!!
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I will see that in a bit 