I am studying Bloch's theorem, which can be stated as follows:
The eigenfunctions of the wave equation for a period potential are the product of a plane wave $e^{ik \cdot r}$ times a modulation function $u_{k}(r)$, which has the periodicity of the lattice. In total: $\psi_k (r) = u_k(r)e^{ik\cdot r} $. [Reference: Kittel - Introduction to solid sate physics.]
I have some problems understanding Bloch's theorem in full. Can I view the wavevector $k$ as the actual, physical momentum of the electron, which moves in a periodic potential, i.e., does it define the wavelength via $\lambda = 2\pi/k$? And how does this relate to the fact that all wavevectors can be translated back to the first Brouillon zone?