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
If R belongs to the Bravais lattice, the operator will commute with the Hamiltionian
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)
To simplify somewhat the notation, ai with i = 1, 2, 3 the basis vectors of the primitive Bravais lattice, and bi the basis vector of the reciprocal lattice. We can always define (because c(ai) is always normalized to unity:
If now R is a translation of the Bravais lattice, it can be written as
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 bi:
It was in fact the result we had obtained for c(R), that was written as:
setting naturally xi = ki. The wavefunctions may well be written as:
that is one of the equivalent formulations of the Bloch theorem.