For each vector of the Bravais lattice we define a translation operator by
for an arbitrary function .
Since the Hamiltonian is periodic, we find
Eq. (4.9) holds for all state functions ; so we can write
Successively applying the translation leads to
Following Eq. (4.10) we can choose the Eigenvectorsystem of to be simultaneously a Eigenvectorsystem of :
Following Eq. (4.12) the Eigenvalues of the translation operator obey
Thus for the Eigenvalues we find
Let’s have a closer look on the primitive translation vectors of the Bravais lattice . For adequate variables we can write
Thus for a general translation vector
we find by successively applying Eq. (4.15)
which may be rewritten as
using Eq. (4.16) and the following definitions:
The above defined vectors
are the basic vectors of the reciprocal lattice.
Summing up, we have shown that for every vector of the real space the Eigenvectors of can be chosen to fulfill the following relation:
This is the definition of the Bloch theorem according to Eq. (4.5). Additional illustrative information you can find in the MaWi II script.
© J. Carstensen (Quantum Mech.)