Set of supplementary notes.

4.2. BLOCH’S THEOREM FROM DISCRETE TRANSLATIONAL SYMMETRY 57
By taking out a factor e iqr , this can be rewritten as
ψ q (r) = e iqr ∑ G
c q−G e −iGr = e iqr u j,q (r)
where the function u j,q (r) is built from periodic function e −iGr , and must therefore have
the same periodicity as the lattice.
This is Bloch’s theorem:
Eigenstates of the one-electron Hamiltonian can be chosen to be a plane wave
multiplied by a function with the periodicity of the Bravais lattice.
4.2 Bloch’s theorem from discrete translational symmetry
Another way of thinking about Bloch’s theorem is to consider what happens to an eigenstate
of the Hamiltonian (kinetic energy plus periodic lattice potential), if it is translated in space.
An arbitrary translation operation will not necessarily produce an eigenstate, because the new
state, generated by this translation, may not match the lattice potential correctly. If the
translation operation is matched to the lattice, however, so if it is a translation by a Bravais
lattice vector, then the resulting state is also an eigenstate of the Hamiltonian. The reason
for this lies in the connection between symmetry and quantum mechanics, which is discussed
in quantum mechanics courses: if an operator (such as the Hamiltonian) is unchanged under
a change of coordinate system (i.e. a symmetry operation such as translation, rotation, etc.),
then applying a symmetry operation on the eigenstate of such an operator produces another
eigenstate of the operator, with the same eigenvalue as the original one.
Either the two eigenstates produced by the symmetry operation are actually the same and
differ only by a complex prefactor, or they are different, in which case we are dealing with a
set of degenerate eigenstates.
In the first case, it is clear that the original eigenstate is also an eigenstate of the symmetry
operation. In the second case, it can be shown that we can always choose from the subspace of
degenerate eigenstates a set of eigenstates which are also eigenstates of the symmetry operation.
Lattices are symmetric under discrete translation of the coordinate system by lattice vectors.
Accordingly, the eigenstates of the Hamiltonian can be chosen to be eigenstates of the discrete
lattice translation operation. This is the underlying origin of Bloch’s theorem, which we will
now explore in more detail.
4.2.1 Symmetry in quantum mechanics – applied to the lattice
Consider a symmetry operator ˆT, e.g. translation
〈 〉
r|ˆTψ = ψ(r + a).
If the Hamiltonian Ĥ commutes with the symmetry operator ˆT, then this implies that ˆT
maps one eigenstate of the Hamiltonian Ĥ onto another eigenstate of Ĥ with the same energy: