On the Summation Ranges of Wavevectors
Tianhao Zhang∗
March 23, 2025
Abstract
The summation of wavevectors in formulas appearing in solid state physics are
explored, and the summation ranges are summarized.
Contents
1 Where do these wavevectors come from?
1.1 Wavevectors of plane wave solutions . . . . . . . . . . . . . . . . . . . . .
1.2 Quantum number characterizing the translational symmetry of a periodic
potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
2 Useful theorems
2.1 Summation identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Fourier expansion for periodic functions . . . . . . . . . . . . . . . . . . .
2
2
2
3 Where do summations over k come from?
3.1 Fourier expansion for periodic functions . . . . . . . . . . . . . . . . . . .
3.2 Constructing tight-bonding Blöch functions/lattice Fourier transform . . .
4
4
5
Reference
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1
2
Where do these wavevectors come from?
Let’s consider 3-dimensional systems with periodic boundary condition.
1.1
Wavevectors of plane wave solutions
In Sommerfeld’s free electron theory, the energy eigenfunctions that satisfies the timeℏ2
independent Schrödinger equation − 2m
∇2 ψ(r) = εψ(r) are planewaves
1
ψk (r) = √ eik·r , k ∈ R3 .
V
In this free electron case, the periodic boundary condition is
ψ(x + L, y, z) = ψ(x, y, z),
ψ(x, y + L, z) = ψ(x, y, z),
ψ(x, y, z + L) = ψ(x, y, z).
∗
School of Physics, Peking University
1
Invoking the periodic boundary condition, the wavevector of a plane wave takes values
only on a discrete set
2π
ki =
mi , mi ∈ Z, i = 1, 2, 3.
L
1.2
Quantum number characterizing the translational symmetry
of a periodic potential
Blöch’s theorem says that for a particle in a periodic
whose periodicity is de Ppotential
3
1
scribed by the direct lattice DL ≡ Z{a1 , a2 , a3 } :=
, i.e.
i=1 ni ai ni ∈ Z, i = 1, 2, 3
V (r + R) = V (r), ∀R ∈ DL, the energy eigenfunctions may be chosen as simultaneous
eigenfunctions of all the translations TR , R ∈ DL
ψk (r) = eik·r uk (r), k ∈ R3 ,
where uk has the same periodicity as the potential, that is, uk (r + R) = uk (r), ∀R ∈ DL.
Taking the lattice structure into account, the periodic boundary condition is now
ψ(r + Ni ai ) = ψ(r), i = 1, 2, 3,
(1)
where Ni is the number of sites in the i-th direction. (The total site number is N =
N1 N2 N3 .) Invoking the periodic boundary condition, the quantum number k takes values
only on a discrete set
3
X
mi
bi , mi ∈ Z.
k=
N
i
i=1
Note that only those k’s in BZ are independent physical degrees of freedom.
2
Useful theorems
2.1 Summation identities
See Figure 1. When k in Eq. (F.1) in Figure 1 is not restricted in BZ but allowed to take
on any value consistent with the periodic boundary condition, a more general result hold
X
X
eik·R = N
δk,G .
(2)
R∈DL
2.2
G∈RL
Fourier expansion for periodic functions
Convention 2.1 (Fourier transform). The Fourier transform of a well-behaved function
f (r) and its inverse transformation are defined as
Z
1
fk =
d3 re−ik·r f (r),
V V
Z
d3 k −ik·r
f (r) = V
e
fk .
(2π)3
1
The reciprocal lattice of DL is defined as
RL := G ∈ R3 eiG·R = 1, ∀R ∈ DL
where bi ’s are defined by ai · bj = 2πδij .
2
= Z{b1 , b2 , b3 },
Figure 1: Appendix F of Aschroft and Mermin [1]
Lemma 2.2. If f (r) is a well-behaved function with periodicity described by the direct
lattice DL ≡ Z{a1 , a2 , a3 }, then its Fourier compoment fk is zero unless k = G for some
G ∈ RL, i.e.
X
δk,G fG .
fk =
G∈RL
Proof. By definition 2.1,
Z
1
fk =
d3 re−ik·r f (r)
V V
Z
1 X
=
d3 re−ik·(r+R) f (r + R)
V R∈DL cell
Z
1 X −ik·R
=
d3 re−ik·r f (r)
e
V R∈DL
cell
Z
X
N
=
d3 re−iG·r f (r)
δk,G
V cell
G∈RL
X
=
δk,G fG ,
G∈RL
where in the third equality the periodicity of f (r) has been used, Rthe fourth equality
is Rbecause of Eq. (2), and the final equality follows from fG = V1 V d3 re−iG·r f (r) =
1
V
.
d3 re−iG·r f (r) where Ω = N
Ω cell
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Theorem 2.3 (Fourier expansion of a periodic function). If f (r) is a well-behaved function with periodicity described by the direct lattice DL ≡ Z{a1 , a2 , a3 }, then in the thermodynamic limit L → ∞ where
Z
X
d3 k
=V
,
3
some range (2π)
k∈some range
f (r) can be expanded as
X
f (r) =
where fG = Ω1
eiG·r fG ,
G∈RL
R
cell
d3 re−iG·r f (r).
3
Where do summations over k come from?
3.1
Fourier expansion for periodic functions
These summations are over the whole RL.
Example 3.1. Expand the periodic potential V (r), which satisfies V (r+R) = V (r), ∀R ∈
DL.
By theorem 2.3,
X
eiG·r VG .
V (r) =
G∈RL
Example 3.2. Expand the Blöch function ψk (r) = eik·r uk (r), where uk satisfies uk (r +
R) = uk (r), ∀R ∈ DL.
By theorem 2.3,
X
uk (r) =
e−iG·r ck−G
G∈RL
ik·r
⇒ ψk (r) = e uk (r)
X
=
ei(k−G)·r ck−G .
G∈RL
A much more confusing example is the following:
Example 3.3. Expand the wavefunction ψ(r) that satisfies the periodic boundary condition Eq. (1).
˜ ≡ Z{ã1 , ã2 , ã3 }
Clearly this ψ(r) has a periodicity described by the direct lattice DL
ai
where ãi = Ni , thus
X
ψ(r) =
eik·r ck ,
P3
˜
k∈RL
P3
i
˜ mi ∈ Z.
where k = i=1 mi b̃i = i=1 m
b ∈ RL,
Ni i
If we write k = q − G, q ∈ BZ, G ∈ RL in the above summation, then
X
ψ(r) =
eik·r ck
˜
k∈RL
=
X X
ei(q−G)·r cq−G
q∈BZ G∈RL
≡
X
c̃q ψq (r),
q∈BZ
4
P
where ψq (r) = G∈RL ei(q−G)·r c̃q−G , q ∈ BZ is a Blöch function, and c̃q c̃q−G = cq−G .
This is just the mathematical formulation that any wavefunction that satisfies the periodic
boundary condition Eq. (1) can be expressed as a physical superposition of the Blöch
functions, i.e., the Blöch functions form a complete basis of the Hilbert space —— this is
true because the Blöch functions are eigenfunctions of the Hamiltonian. Here “physical”
emphasizes that this summation is only over BZ but not the whole RL.
3.2
Constructing tight-bonding Blöch functions/lattice Fourier
transform
These summations are physical, i.e. are over BZ but not the whole RL.
In tight-bonding model, one chooses one atomic orbital2 on each atom, assuming
that these single-electron states are orthonormal and complete. Being aware of Blöch’s
theorem, one constructs Blöch states in terms of these orbitals as
Convention 3.4. “Fourier transform”]
1 X ik·R
|ψk ⟩ = √
e
|ψR ⟩ , k ∈ BZ.
N R∈DL
This formula is orally referred to as a “Fourier transform”, although it actually isn’t.
The reason for choosing k to be in BZ is that only those k’s in BZ are physically
independent. Actually, the Blöch states have a periodicity described by RL,
1 X ik·R
e
|ψR ⟩ = |ψk ⟩ ,
|ψk+G ⟩ = √
N R∈DL
so we have already included all possible states although k is restricted to BZ.
Another advantage for restricting k to BZ is that in this case, the “inverse Fourier transform” is simple (using Eq. (F.4) in Figure. 1)
1 X −ik·R
|ψR ⟩ = √
e
|ψk ⟩ .
N k∈BZ
References
[1] N. W. Ashcroft and N. D. Mermin, “Solid state physics,” Cengage Learning, 1976.
2
Maybe several orbitals, but there is no essential difference, so let’s consider only monatomic solids.
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