See also the QCMP lecture about this topic

We now want to remove the free electron approximation and introduce a periodic potential while still working under the Born-Oppenherimer approximation and independent electron model.

Since the potential is periodic (and remembering what we saw about the reciprocal lattice) we can write

with

and

Bloch theorem

The Bloch theorem states that the wave function of a particle in a periodic potential can be expressed as:

Where is the band index (which will often be omitted).

This means that the wave functions are no longer plane waves but are plane waves multiplied by a function which has the same periodicity of the lattice. The Bloch state does NOT have the same periodicity of the lattice.

The same concept can be formulated also as (omitting ).

Consequences

Given a crystal with dimensions which contains primitive cells, the BVK conditions tell us that:

and the Bloch theorem that

By imposing = we need to have the exponential equal to . Since is a vector of the reciprocal lattice it can be written as .

From the relation between the vectors of the direct and reciprocal lattice we get

which requires

and thus

This means that can be written as

Demonstration

todo