I understanding the meaning of Bloch waves

[raeed]

Before reading Bloch theorem i read something to get a feeling to what happens to the energy of electron in a periodic potential, in short what i read said:
Assuming we have a weak periodic potential from -π/a to π/a for example cos(2πx/a), we can write the electron wave function as: α|k>+β|k'>.
my first question is: does this notation mean that the electron is in superposition state of two states, first one being the electron didn't scatter and the second being the electron did scatter?
After that i went on to read Bloch theorem, he stated that the waving function can be written as:
Σu_ke^ikr. and using Fourier series
Σα_k-Ge^i(k-G)r.
my second question is: correct if I'm wrong but since there is no difference between k and k-G in term of energy does that mean that they are the same state? what does the wave function exactly tell us about the electron? for example what does
α_k-G1e^i(k-G₁)r+α_k-G2e^i(k-G₂)r mean?
sorry for the long question I'm just have a hard time trying to connect the dots

 

[ngonyama]

''Since there is no difference between k and k-G in term of energy does that mean that they are the same state?''

In general a point k and the point k-G (or k+G or k+2G etc.) represent different wave functions with different frequencies and different energies, but they all belong to the same irreducible representation of the translation symmetry. They have the same symmetry behavior. This means that you can mix them, like you do with wave functions in chemical bonding. As the energies at k and k+G are generally very different this does not matter much except at the edges of a Brillouin zone. The wave function at k=1/2 and k=1/2-1 = -1/2 have the same energy and so mixing will split them up and cause a gap (like the two 1s states of two adjacent H atoms leading to a bonding and an antibonding sigma orbital). This splitting is what causes semiconductors to have an energy gap.

Another way of saying this is to say that the functions at k, k-G (or k+G or k+2G) are aliases of each other. This is particularly important for digital sampling.