Set of supplementary notes.

Chapter 4
Electrons in a periodic potential
Our modelling of electrons in solids – both in terms of the classical Drude model and the
quantum mechanical Sommerfeld model – has so far ignored the presence of the electrostatic
potential caused by the positively charged ions. In crystalline lattices, the spatial dependence
of this potential has the same symmetry as the lattice, and this greatly simplifies the problem.
In particular, the potential is subject to discrete translational symmetry
4.1 Schrödinger equation in a periodic potential
We consider first a formal treatment in terms of a complete set of basis functions, namely
the set of all plane wave states which satisfy the periodic boundary conditions. The results
from this treatment can be used to obtain Bloch’s theorem, which is one of the cornerstones
of electronic structure in solids. Next, we will approach Bloch’s theorem from a more abstract
but also more elegant direction, which uses the translational symmetry of the lattice directly.
ˆp2
We are looking for solutions to Ĥ |ψ〉 = ( + V ) |ψ〉 = E |ψ〉, where V (r) is periodic.
2m
Because V (r) has the same periodicity as the lattice, it can be Fourier-expanded. We define
its Fourier components at reciprocal lattice vectors G as
V G = 1 ∫
∫
d 3 r e −iG·r 1
V (r) =
d 3 r e −iG·r V (r) , (4.1)
Vol.
Vol. per cell
unit cell
and conversely expand the spatial dependence of the potential as
V (r) = ∑ G
V G e iG·r . (4.2)
Since the potential is real, V ∗ G = V −G. The Fourier component for G = 0, V G = V 0 is the
average of the potential, which we set to zero. If it were not zero, this would simply add a
constant to all the energy values obtained in the following.
We build the eigenstate |ψ〉 from the plane wave states |k〉, defined such that 〈r|k〉 = e ikr .
These form a complete set of basis vectors for ’well-behaved’ functions, as is shown in functional
analysis (for example, the completeness of this set of functions is the reason why Fourier
transforms work).
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