Wavefunctions of the crystal - Bloch Theorem

 

 

 

 

The Hamiltonian of a semiconductor crystal has the translation symmetry

 

with being a reciprocal lattice vector.

The Bloch theorem states that the wavefunctions have two “good” quantum numbers, the band index n and a reciprocal vector  such that the wavefunctions of the crystal may be written as:

 

where exhibits the periodicity of the crystal.

It may also be written as:

 

 

Proof of Bloch’s theorem

 

Translation operator

 

 

If R belongs to the Bravais lattice, the operator will commute with the Hamiltionian

 

 

 and therefore

 

 
These two operators form a common set of commuting observable, we can therefore write the wavefunctions using eigenfunctions of both:

 

 

Consid´erons maintenant les propri´et´es de c(R)

 

 et donc

 

 

To simplify somewhat the notation, a_i with i = 1, 2, 3 the basis vectors of the primitive Bravais lattice, and b_i the basis vector of the reciprocal lattice. We can always define (because c(a_i) is always normalized to unity:

 

 

If now R is a translation of the Bravais lattice, it can be written as

 

 then,

 

 
Using the orthogonality relations between the direct and reciprocal basis vectors, written as

 

 
Considering the product between the vector R of the real space and the vector k of the
reciprocal space, written in the basis of the b_i:

 

 and then

 

 

It was in fact the result we had obtained for c(R), that was written as:

 

 
setting naturally x_i = k_i. The wavefunctions may well be written as:

 

 

that is one of the equivalent formulations of the Bloch theorem.