Understanding Bloch's Theorem: Dependence of u on k

[aaaa202]

As I understand it Blochs theorem says that the solutions to the one electron Schrödinger equation in a periodic potential has the form:

ψ(r) = exp(i k⋅r)u_n(r)
, where u_n(r) has the same periodicity as the lattice and n labels the band number.

Now a detail that confuses me: In a book I am reading (see attached picture) the function u is labeled by two quantum numbers: The band index as well as the wavenumber k. Why does u depend on the wavenumber? Am I making a fuzz out of an unimportant detail?

 

[eljose79]

Thank you for bringing up this question about Bloch's theorem. It is important to pay attention to details when studying quantum mechanics, so your confusion is understandable.

The reason why the function un(r) in Bloch's theorem is labeled by both the band index n and the wavenumber k is because the solutions to the Schrödinger equation in a periodic potential are not unique. In other words, there are multiple possible solutions that satisfy the same periodic boundary conditions.

The band index n represents the different energy levels or bands that electrons can occupy in the periodic potential. Each band has a different energy and can accommodate a certain number of electrons. The wavenumber k, on the other hand, represents the momentum of the electron in the lattice. Different values of k correspond to different states or wavefunctions of the electron.

So, the function un(r) depends on both the band index n and the wavenumber k because it describes a specific state of the electron in the periodic potential. This allows for a more complete description of the electron's behavior in the lattice.

I hope this explanation helps to clarify the role of the wavenumber in Bloch's theorem. Keep up the good work in your studies of quantum mechanics!