Set of supplementary notes.

4.2. BLOCH’S THEOREM FROM DISCRETE TRANSLATIONAL SYMMETRY 59
Ĥψ(r) =
[ ˆp
2
2m + V (r) ]
ψ(r) = Eψ(r) ,
where V (r + R) = V (r) for ∀ R in a Bravais lattice.
can be chosen such that
ψ (n)
k (r) = eik·r u nk (r) , where u (n)
k
(r + R) = u(n)
k
(r) (4.6)
Or, equivalently:
ψ (n)
k
(r + R) = eik·R ψ (n) (r) (4.7)
k
Here, n is called the band index. It is necessary, because there may be several distinct
eigenstates of Ĥ with the same symmetry label k. The band index distinguishes between these.
Note that whereas the potential is periodic, the wavefunction ψ(r) is not. It is formed
from multiplying a plane wave state with a periodic function, which has the same translational
symmetry as the lattice.
The two forms of Bloch’s theorem can be shown to be equivalent – each implies the other.
For instance, applying ˆT R to the product e ik·r u nk (r) in (4.6) will produce the phaseshift e ik·R
required by (4.7). Conversely, substituting a product e ik·r u nk (r) (where u(r) can be any function,
not necessarily periodic) into (4.7) will produce (4.6) and demonstrate that u(r) indeed
has to be periodic.
The Bloch states (plane wave × periodic function) are similar to eigenstates of free electrons
(just plane waves), but the choice of periodic function gives additional freedom in labelling
states. Note, for instance that for any reciprocal lattice vector g, e igr is periodic with same
periodicity as the Bravais lattice, which follows from the definition of the reciprocal lattice
vectors g. This can be used to relabel a Bloch state k with a new wavevector k − g by
introducing a different periodic function u (n) = e igr u (m) :
[ ]
ψ (m)
k
(r) = eikr u (m)
k (r) = eikr e −igr e igr u (m)
k (r) = e i(k−g)r u (n)
k−g
(r) = ψ(n)
k−g (r)
In this case, exactly the same function ψ(r) can be labelled by wavevector k, if the periodic
function in the Bloch state is u (m)
k
(r), or by wavevector k − g, if the corresponding periodic
function is u (n)
k−g (r) = eigr u (m)
k
(r). This implies that for every state labelled with a k− vector
outside the first Brillouin zone, we can find an identical state which can be labelled with a vector
q = k − g inside the first Brillouin zone. From this, we conclude:
Any quantity that depends on the wavefunction,
in particular energy, is periodic in wavevector space.