---
title: Bloch's Theorem and Energy Bands
module: Band Theory
moduleNumber: 6
lessonNumber: 1
order: 601
summary: >
  An electron in a periodic potential has stationary states that are plane
  waves modulated by a lattice-periodic envelope. This lesson proves Bloch's
  theorem two ways, defines crystal momentum and the band index, counts the
  allowed wavevectors from Born–von Kármán boundary conditions, and sets up the
  extended, reduced, and repeated-zone descriptions of a band.
topics: [Band Theory]
draft: false
sources:
  - book: Ashcroft & Mermin
    ref: "Ch. 8 — Electron Levels in a Periodic Potential: Bloch's Theorem"
  - book: Kittel
    ref: "Ch. 7 — Energy Bands"
  - book: Simon
    ref: "Ch. 11 — Electrons in a Periodic Potential"
---

The [free-electron gas](/condensed-matter/free-electron-fermi-gas/sommerfeld-model-and-heat-capacity)
treats a metal's conduction electrons as a Fermi gas in a flat box, ignoring
the ions entirely. That model reproduces the electronic heat capacity and the
order of magnitude of the conductivity, but it cannot explain why some solids
conduct and others insulate, why a few metals carry current as though the
charge carriers were positive, or why the mean free path can reach centimetres
at low temperature. Every one of these failures traces to the same omission:
the periodic potential of the ion cores. This module restores that potential.
The single structural fact that organizes the whole subject is that the
electron sees a potential with the periodicity of the lattice,

$$
U(\vec r + \vec R) = U(\vec r) \qquad \text{for every lattice vector } \vec R,
$$

and the eigenstates of a Hamiltonian with that symmetry have a rigidly
constrained form. Bloch's theorem states that form.

## The periodic potential

A crystal is a [Bravais lattice](/condensed-matter/crystal-structure/bravais-lattices-and-crystal-systems)
of points $\vec R = n_1\vec a_1 + n_2\vec a_2 + n_3\vec a_3$ with a basis of
ions attached to each point. An electron moving through it feels the summed
Coulomb attraction of all the ion cores (and, in a mean-field sense, the
averaged repulsion of the other electrons). Because the ions sit on a
periodic array, the resulting one-electron potential inherits that period:
translating by any $\vec R$ maps the crystal onto itself and leaves $U$
unchanged. The one-electron Hamiltonian is

$$
H = -\frac{\hbar^2}{2m}\nabla^2 + U(\vec r), \qquad U(\vec r + \vec R) = U(\vec r).
$$

The potential is not weak in any absolute sense; near an ion it plunges to
tens of electron-volts. What matters is only its periodicity. Two limiting
approximations bracket every real band structure: the
[nearly-free-electron model](/condensed-matter/band-theory/nearly-free-electron-model),
which treats the periodic modulation as a weak perturbation on plane waves,
and the [tight-binding model](/condensed-matter/band-theory/tight-binding-method),
which starts from isolated-atom orbitals and lets them hybridize. Bloch's
theorem holds exactly regardless of the potential's strength, so it governs
both limits.

$$
% caption: The crystal potential U(x) along a line of ions repeats with the
% lattice period a; deep wells sit at the ion cores and the potential rises
% between them. Only the periodicity, not the depth, enters Bloch's theorem.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-0.3,0) -- (8.4,0) node[below] {position};
  \draw[->, black] (0,-2.6) -- (0,0.7) node[left] {$U$};
  % periodic well train, period 2
  \draw[acc, very thick, domain=0.2:8.0, samples=200, variable=\x]
    plot ({\x},{-2.2*exp(-((\x-1)*(\x-1))/0.08) - 2.2*exp(-((\x-3)*(\x-3))/0.08)
      - 2.2*exp(-((\x-5)*(\x-5))/0.08) - 2.2*exp(-((\x-7)*(\x-7))/0.08)});
  \foreach \x in {1,3,5,7}{ \fill[black] (\x,0) circle (2pt); }
  \draw[black, <->] (1,0.28) -- (3,0.28) node[midway, above] {$a$};
\end{tikzpicture}
$$

## Bloch's theorem

> **Theorem (Bloch's theorem).** The stationary states of a one-electron
> Hamiltonian with a periodic potential can be chosen so that each is a plane
> wave times a function with the periodicity of the lattice,
> $$
> \psi_{n\vec k}(\vec r) = e^{i\vec k\cdot\vec r}\,u_{n\vec k}(\vec r),
> \qquad u_{n\vec k}(\vec r + \vec R) = u_{n\vec k}(\vec r),
> $$
> for every lattice vector $\vec R$. Equivalently, the state satisfies
> $\psi_{n\vec k}(\vec r + \vec R) = e^{i\vec k\cdot\vec R}\,\psi_{n\vec k}(\vec r)$.

The label $\vec k$ is a wavevector; $n$ is the **band index**, distinguishing
the discrete solutions found at fixed $\vec k$. A state of this form is a
**Bloch wave**: a running plane wave $e^{i\vec k\cdot\vec r}$ whose amplitude
is not constant but modulated by the lattice-periodic envelope $u_{n\vec k}$,
which repeats the internal structure of the potential inside every cell.

$$
% caption: A Bloch wave (bottom) is the product of a plane wave of wavevector k
% (top) and a lattice-periodic envelope u (middle) that repeats every cell;
% the product oscillates on the plane-wave scale while its detail recurs cell
% to cell.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % plane wave (top)
  \begin{scope}[yshift=4.0cm]
    \draw[black] (0,0) -- (8.2,0);
    \draw[acc, thick, domain=0:8, samples=200, variable=\x]
      plot ({\x},{0.55*cos(180*\x/1.0)});
    \node[anchor=west, black] at (8.3,0) {plane wave};
  \end{scope}
  % periodic envelope (middle)
  \begin{scope}[yshift=2.0cm]
    \draw[black] (0,0) -- (8.2,0);
    \draw[acc, thick, domain=0:8, samples=240, variable=\x]
      plot ({\x},{0.45 + 0.30*cos(360*\x/1.0)});
    \node[anchor=west, black] at (8.3,0) {envelope $u$};
    \foreach \x in {0.5,1.5,2.5,3.5,4.5,5.5,6.5,7.5}{ \fill[black] (\x,0) circle (1.2pt); }
  \end{scope}
  % product (bottom)
  \begin{scope}[yshift=0cm]
    \draw[black] (0,0) -- (8.2,0);
    \draw[acc, very thick, domain=0:8, samples=300, variable=\x]
      plot ({\x},{(0.45 + 0.30*cos(360*\x/1.0))*cos(180*\x/1.0)});
    \node[anchor=west, black] at (8.3,0) {Bloch wave};
  \end{scope}
\end{tikzpicture}
$$

### Proof from translation operators

Define the **lattice translation operator** $T_{\vec R}$ that shifts a function
by a lattice vector, $T_{\vec R}f(\vec r) = f(\vec r + \vec R)$. Because $U$ is
periodic and $\nabla^2$ is translation-invariant, $T_{\vec R}$ commutes with the
Hamiltonian, $[T_{\vec R}, H] = 0$. Successive translations add, so the
translations commute among themselves, $T_{\vec R}T_{\vec R'} = T_{\vec R + \vec R'}
= T_{\vec R'}T_{\vec R}$.

> **Proof.** A set of mutually commuting operators that also commute with $H$
> has a common eigenbasis, so each stationary state can be chosen an
> eigenstate of every $T_{\vec R}$:
> $$
> T_{\vec R}\psi = c(\vec R)\,\psi.
> $$
> The multiplication law of the translations forces the eigenvalues to
> multiply, $c(\vec R)c(\vec R') = c(\vec R + \vec R')$. A function on the
> lattice obeying $c(\vec R + \vec R') = c(\vec R)c(\vec R')$ is an exponential:
> writing $c(\vec a_i) = e^{2\pi i x_i}$ for the three primitive vectors and
> using the additivity gives $c(\vec R) = e^{i\vec k\cdot\vec R}$ with
> $\vec k = x_1\vec b_1 + x_2\vec b_2 + x_3\vec b_3$ expressed in the
> [reciprocal basis](/condensed-matter/crystal-structure/reciprocal-lattice-and-brillouin-zones)
> $\vec b_i$. Normalizability requires the $x_i$ real, so $\lvert c(\vec R)\rvert = 1$.
> Then $\psi(\vec r + \vec R) = T_{\vec R}\psi(\vec r) = e^{i\vec k\cdot\vec R}\psi(\vec r)$,
> which is the second form of the theorem. Setting $u(\vec r) = e^{-i\vec k\cdot\vec r}\psi(\vec r)$
> and checking $u(\vec r + \vec R) = e^{-i\vec k\cdot\vec r}e^{-i\vec k\cdot\vec R}\psi(\vec r + \vec R)
> = e^{-i\vec k\cdot\vec r}\psi(\vec r) = u(\vec r)$ recovers the product form. $\;\blacksquare$

The physical content is that the crystal's discrete translation symmetry, like
any symmetry, labels its eigenstates by the eigenvalue of the symmetry
operation. Here the eigenvalue is the phase $e^{i\vec k\cdot\vec R}$ picked up
on translating by $\vec R$, and $\vec k$ is the label. Two wavevectors differing
by a reciprocal-lattice vector $\vec G$ give identical phases, since
$e^{i\vec G\cdot\vec R} = 1$ by definition of the reciprocal lattice. So $\vec k$
and $\vec k + \vec G$ label the same translational symmetry, and every distinct
label lies in one [primitive cell of the reciprocal lattice](/condensed-matter/crystal-structure/reciprocal-lattice-and-brillouin-zones)
— the **first Brillouin zone**.

### Proof from the Fourier expansion

The second proof is constructive and produces the equation that the
nearly-free-electron model solves. Expand the periodic potential in reciprocal
-lattice plane waves, the only ones with the lattice period,

$$
U(\vec r) = \sum_{\vec G} U_{\vec G}\, e^{i\vec G\cdot\vec r},
$$

and expand any wavefunction obeying Born–von Kármán boundary conditions in the
allowed plane waves,

$$
\psi(\vec r) = \sum_{\vec q} c_{\vec q}\, e^{i\vec q\cdot\vec r}.
$$

Substituting both into $H\psi = E\psi$ and matching the coefficient of each
$e^{i\vec q\cdot\vec r}$ gives the **central equation**,

$$
\left(\frac{\hbar^2 q^2}{2m} - E\right) c_{\vec q} + \sum_{\vec G} U_{\vec G}\, c_{\vec q - \vec G} = 0.
$$

The potential only couples $c_{\vec q}$ to coefficients $c_{\vec q - \vec G}$
whose wavevector differs by a reciprocal-lattice vector. A given $\vec q$ is
therefore linked only to the set $\{\vec q - \vec G\}$, and never to
wavevectors outside that family. Fix one representative $\vec k$ in the first
zone; the central equation is a closed linear system in the coefficients
$\{c_{\vec k - \vec G}\}$. Its solution is a wavefunction built entirely from
plane waves $\vec k - \vec G$,

$$
\psi_{\vec k}(\vec r) = \sum_{\vec G} c_{\vec k - \vec G}\, e^{i(\vec k - \vec G)\cdot\vec r}
= e^{i\vec k\cdot\vec r}\underbrace{\sum_{\vec G} c_{\vec k - \vec G}\, e^{-i\vec G\cdot\vec r}}_{u_{\vec k}(\vec r)},
$$

and the bracketed sum is periodic because each $e^{-i\vec G\cdot\vec r}$ is.
This reproduces Bloch's theorem and shows that at fixed $\vec k$ the central
equation has a discrete ladder of solutions — one eigenvalue $E_n(\vec k)$ per
band index $n$.

$$
% caption: The central equation couples the plane wave k only to those
% displaced by reciprocal-lattice vectors G; the potential component U_G links
% k to k minus G, and the closed chain of couplings defines one Bloch state.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-0.4,0) -- (9.0,0) node[below] {wavevector};
  \foreach \x/\lab in {1/{k minus 2G}, 3/{k minus G}, 5/{k}, 7/{k plus G}, 8.6/{k plus 2G}}{
    \fill[acc] (\x,0) circle (2.6pt);
    \node[black, anchor=south, font=\scriptsize] at (\x,0.12) {$\lab$};
  }
  % coupling arcs
  \draw[acc, thick, ->] (5,-0.12) to[bend right=40] (3,-0.12);
  \draw[acc, thick, ->] (5,-0.12) to[bend left=40] (7,-0.12);
  \draw[black, thick, ->] (3,-0.55) to[bend right=35] (1,-0.55);
  \draw[black, thick, ->] (7,-0.55) to[bend left=35] (8.6,-0.55);
  \node[acc, anchor=north] at (4,-0.62) {$U_{G}$ coupling};
\end{tikzpicture}
$$

## Crystal momentum

The quantity $\hbar\vec k$ is the **crystal momentum**. It is not the electron's
true momentum. A Bloch state is not an eigenstate of the momentum operator
$-i\hbar\nabla$, because acting with $\nabla$ on $e^{i\vec k\cdot\vec r}u_{n\vec k}$
also differentiates the envelope:

$$
-i\hbar\nabla\,\psi_{n\vec k} = \hbar\vec k\,\psi_{n\vec k} - i\hbar\,e^{i\vec k\cdot\vec r}\nabla u_{n\vec k},
$$

and the second term does not vanish. A Bloch state is a superposition of the
plane-wave momenta $\hbar(\vec k - \vec G)$, weighted by the $c_{\vec k - \vec G}$.
What $\hbar\vec k$ does capture is the state's transformation under lattice
translations, and it is that quantity which is conserved when Bloch electrons
scatter off one another or off phonons. In such a process the total crystal
momentum is conserved only up to a reciprocal-lattice vector,

$$
\vec k_1 + \vec k_2 = \vec k_3 + \vec k_4 + \vec G,
$$

the same selection rule that governs
[phonon collisions](/condensed-matter/lattice-dynamics/phonons-quantization-and-dos):
processes with $\vec G = 0$ are **normal**, those with $\vec G \neq 0$ are
**Umklapp**. The distinction between true momentum (conserved absolutely, from
continuous translation symmetry) and crystal momentum (conserved mod $\vec G$,
from discrete translation symmetry) is the recurring theme of transport in
crystals.

## The band index and $E_n(\vec k)$

At each $\vec k$ the eigenvalue problem is that of the periodic Hamiltonian
restricted to Bloch waves of that wavevector: substitute
$\psi = e^{i\vec k\cdot\vec r}u(\vec r)$ into $H\psi = E\psi$ to get an
eigenvalue equation for $u$ on a single primitive cell,

$$
\left[\frac{\hbar^2}{2m}\left(-i\nabla + \vec k\right)^2 + U(\vec r)\right] u_{n\vec k}(\vec r) = E_n(\vec k)\, u_{n\vec k}(\vec r),
$$

with periodic boundary conditions on the cell. A Hermitian eigenvalue problem
on a finite region has a discrete, bounded-below spectrum, so for each $\vec k$
the energies form a discrete ladder labeled $n = 1, 2, 3, \dots$. As $\vec k$
varies continuously through the first zone, each $E_n(\vec k)$ traces out a
continuous sheet — an **energy band**. The set of all bands is the crystal's
band structure. Because $\vec k$ and $\vec k + \vec G$ label the same state,
each band is periodic in reciprocal space,

$$
E_n(\vec k + \vec G) = E_n(\vec k),
$$

so the entire spectrum is determined by its values on the first Brillouin zone.

> **Definition (Energy band and band gap).** An **energy band** is the
> continuous function $E_n(\vec k)$ obtained by following the $n$-th eigenvalue
> of the periodic Hamiltonian across the first Brillouin zone. A **band gap** is
> a range of energies containing no $E_n(\vec k)$ for any $n$ or $\vec k$: no
> Bloch state exists at those energies, so an electron cannot propagate with
> such an energy.

## Born–von Kármán boundary conditions and the count of states

To count states one must make the crystal finite without introducing a surface.
The **Born–von Kármán** (periodic) boundary condition wraps the crystal onto
itself after $N_i$ cells along each primitive direction,

$$
\psi(\vec r + N_i\vec a_i) = \psi(\vec r), \qquad i = 1, 2, 3,
$$

with $N = N_1 N_2 N_3$ the total number of primitive cells. Applying the Bloch
relation $N_i$ times gives $e^{i N_i\vec k\cdot\vec a_i} = 1$, so the allowed
wavevectors are quantized,

$$
\vec k = \sum_{i=1}^{3} \frac{m_i}{N_i}\,\vec b_i, \qquad m_i \in \mathbb{Z}.
$$

These points form a fine uniform mesh in reciprocal space. The volume per
allowed $\vec k$ is $(2\pi)^3/V$ with $V = N\,v_{\text{cell}}$ the crystal
volume — identical to the free-electron count, because the density of allowed
wavevectors depends only on the box size, not on the potential. Counting how
many fall inside one Brillouin zone (of volume $(2\pi)^3/v_{\text{cell}}$) gives

$$
\frac{(2\pi)^3/v_{\text{cell}}}{(2\pi)^3/V} = \frac{V}{v_{\text{cell}}} = N.
$$

Each band contains exactly $N$ distinct wavevectors — one per primitive cell in
the crystal. Including the two spin states, a band holds $2N$ electrons. This
integer count is the hinge of the metal–insulator distinction developed in the
[Fermi-surface lesson](/condensed-matter/band-theory/fermi-surfaces-and-semiclassical-dynamics):
a crystal with an even number of electrons per cell can exactly fill an integer
number of bands, and a filled band carries no current.

$$
% caption: Born–von Kármán conditions quantize k to a uniform mesh; exactly N
% allowed points (N the number of crystal cells) lie in the first Brillouin
% zone, here a square zone in two dimensions.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % zone boundary square
  \draw[acc, very thick] (-2.4,-2.4) rectangle (2.4,2.4);
  % mesh of allowed k-points
  \foreach \i in {-2,-1.6,-1.2,-0.8,-0.4,0,0.4,0.8,1.2,1.6,2}{
    \foreach \j in {-2,-1.6,-1.2,-0.8,-0.4,0,0.4,0.8,1.2,1.6,2}{
      \fill[black] (\i,\j) circle (0.9pt);
    }
  }
  \fill[acc] (0,0) circle (2pt);
  \node[acc, anchor=south west, font=\scriptsize] at (0.08,0.05) {$k=0$};
  % reciprocal axes
  \draw[black, ->] (-2.9,0) -- (3.1,0) node[below] {$k_x$};
  \draw[black, ->] (0,-2.9) -- (0,3.0) node[left] {$k_y$};
  \node[acc, anchor=south] at (0,2.5) {Brillouin zone};
  \draw[black, <->] (-2.4,-2.75) -- (-2.0,-2.75) node[midway, below, font=\scriptsize] {$\frac{b}{N}$};
\end{tikzpicture}
$$

## Zone schemes

The periodicity $E_n(\vec k + \vec G) = E_n(\vec k)$ means one band structure
admits three equivalent drawings. In the **extended-zone scheme** each band is
placed in a different Brillouin zone, so the curve marches outward through
successive zones and, for a weak potential, resembles the free-electron
parabola $E = \hbar^2 k^2/2m$ with small gaps at the zone edges. In the
**reduced-zone scheme** every band is translated by the appropriate $\vec G$
back into the first zone, stacking the branches into a set of curves indexed by
$n$ — the standard band-structure plot. In the **repeated-zone scheme** each
reduced band is copied into every zone, making the full periodicity manifest and
letting one follow a Fermi surface across zone boundaries without bookkeeping.
The three carry identical information; the choice is a matter of convenience.

$$
% caption: Three portrayals of the same band structure. Extended: successive
% bands occupy successive zones, tracing the folded free-electron parabola.
% Reduced: every branch folded into the first zone. Repeated: the reduced bands
% copied periodically in k.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % ---- extended (left) ----
  \begin{scope}[xshift=0cm]
    \draw[black, ->] (-2.1,0) -- (2.3,0) node[below] {$k$};
    \draw[black, ->] (0,0) -- (0,3.2) node[left] {$E$};
    \draw[black, dashed] (1,0) -- (1,3.1);
    \draw[black, dashed] (-1,0) -- (-1,3.1);
    \draw[black, dashed] (2,0) -- (2,3.1);
    \draw[black, dashed] (-2,0) -- (-2,3.1);
    % parabola pieces
    \draw[acc, very thick, domain=-1:1, samples=60, variable=\x] plot ({\x},{0.9*\x*\x});
    \draw[acc, very thick, domain=1:2, samples=60, variable=\x] plot ({\x},{0.9*\x*\x+0.35});
    \draw[acc, very thick, domain=-2:-1, samples=60, variable=\x] plot ({\x},{0.9*\x*\x+0.35});
    \node[black, anchor=south west, font=\scriptsize] at (0.2,3.2) {extended};
  \end{scope}
  % ---- reduced (middle) ----
  \begin{scope}[xshift=6.0cm]
    \draw[black, ->] (-1.4,0) -- (1.6,0) node[below] {$k$};
    \draw[black, ->] (0,0) -- (0,3.2) node[left] {$E$};
    \draw[black, dashed] (1,0) -- (1,3.1);
    \draw[black, dashed] (-1,0) -- (-1,3.1);
    \draw[acc, very thick, domain=-1:1, samples=60, variable=\x] plot ({\x},{0.9*\x*\x});
    \draw[acc, very thick, domain=-1:1, samples=60, variable=\x] plot ({\x},{1.25+0.9*(1-abs(\x))*(1-abs(\x))});
    \node[black, anchor=south west, font=\scriptsize] at (0.2,3.2) {reduced};
  \end{scope}
  % ---- repeated (right) ----
  \begin{scope}[xshift=11.4cm]
    \draw[black, ->] (-1.6,0) -- (2.2,0) node[below] {$k$};
    \draw[black, ->] (0,0) -- (0,3.2) node[left] {$E$};
    \foreach \s in {-1,0,1}{
      \draw[acc, very thick, domain=-1:1, samples=40, variable=\x] plot ({\x+2*\s},{0.9*\x*\x});
    }
    \node[black, anchor=south west, font=\scriptsize] at (0.2,3.2) {repeated};
  \end{scope}
\end{tikzpicture}
$$

The vocabulary of this lesson — Bloch waves, crystal momentum, the band index,
the count of $2N$ states per band, the zone schemes — is the coordinate system
in which every later result is stated. The
[nearly-free-electron model](/condensed-matter/band-theory/nearly-free-electron-model)
solves the central equation for a weak potential and shows how the free-electron
parabola breaks into bands with gaps of size $2\lvert U_{\vec G}\rvert$ at the
zone boundaries; the [tight-binding model](/condensed-matter/band-theory/tight-binding-method)
builds the same bands from the opposite limit of localized orbitals.

[^am]: Ashcroft & Mermin, Ch. 8.
[^kittel]: Kittel, Ch. 7.
[^simon]: Simon, Ch. 11.