Bloch's Theorem

Definition

Put a single electron into the potential of a perfect crystal — a potential that repeats itself exactly under every lattice translation R, so that V(r + R) = V(r). You might expect the wavefunction to be a messy, lattice-specific thing. Bloch's theorem says it is astonishingly simple: every energy eigenstate is a plane wave multiplied by a function that shares the lattice's own periodicity.

ψ_nk(r) = e^(ik·r) · u_nk(r),   with   u_nk(r + R) = u_nk(r)

Read it as two layers. The factor e^(ik·r) is a smooth, long-wavelength envelope — exactly the free-electron plane wave — that carries the macroscopic phase across the whole crystal. The factor u_nk(r) is a cell-periodic part that wiggles violently on the atomic scale, piling up charge near the ions and thinning out in between. The integer n is the band index; the vector k is the Bloch wavevector, living in reciprocal space.

An equivalent, often more useful statement: translating a Bloch state by a lattice vector multiplies it by a pure phase.

ψ_nk(r + R) = e^(ik·R) · ψ_nk(r)

Nothing about the probability density |ψ|² changes from one unit cell to the next — only the phase advances. The electron is genuinely spread over the entire crystal, not bound to one atom.

How it works — symmetry forces the form

The mechanism is pure symmetry. Define the translation operator T_R that shifts everything by a lattice vector R. Because the Hamiltonian contains V(r) and V is periodic, H commutes with every T_R, and all the T_R commute with each other. Commuting operators share eigenstates, so we can label each energy eigenstate by the eigenvalues of the translations.

A translation operator is unitary, so its eigenvalues are pure phases. Consistency of T_{R1} T_{R2} = T_{R1+R2} forces those phases to take the form e^(ik·R) for some vector k. That is the whole content of Bloch's theorem: an energy eigenstate must satisfy ψ(r + R) = e^(ik·R) ψ(r), and writing ψ = e^(ik·r) u(r) makes u automatically lattice-periodic. The discrete translational symmetry of the crystal does all the work — no detailed knowledge of V is needed for the form, only for the energies.

Now substitute the Bloch form back into the Schrödinger equation. For each fixed k you get an eigenvalue problem for u_nk inside a single unit cell, with periodic boundary conditions. A finite cell with periodic boundaries yields a discrete ladder of solutions — that is where the band index n comes from. Let k roam over the Brillouin zone and each discrete level E_n(k) sweeps out a continuous curve: a band. The infinite crystal's continuum of energies is thereby reorganized into a countable stack of bands.

Crystal momentum, and why it lives on a torus

The label ħk is the crystal momentum. It looks like ordinary momentum and behaves like it in many equations, but it is not the mechanical momentum: the operator −iħ∇ acting on a Bloch state does not return ħk, because the periodic factor u_nk(r) carries momentum of its own.

The deeper subtlety: a continuous translation symmetry would conserve true momentum exactly (Noether's theorem). The crystal only has discrete translation symmetry, so the conserved label is weaker — it is defined only up to a reciprocal lattice vector G, because e^(i(k+G)·R) = e^(ik·R) for every lattice vector R. States at k and k + G are literally the same state. Reciprocal space is effectively a torus, and we conventionally fold everything into the first Brillouin zone.

The physical payoff appears in scattering. When an electron absorbs a phonon or photon of wavevector q,

k_final = k_initial + q + G,   for some reciprocal lattice vector G

The lattice as a whole can soak up a momentum kick of ħG. When G = 0 the process is "normal"; when G ≠ 0 it is an Umklapp ("flip-over") process, and Umklapp scattering is what gives a clean metal a finite thermal and electrical resistance at room temperature. Crystal momentum is conserved — but only modulo the reciprocal lattice.

A worked example — the nearly free electron and a real gap

Take a 1D crystal of lattice spacing a = 3.0 Å (0.30 nm), and treat the weak periodic potential as a perturbation on free electrons. The reciprocal lattice vectors are G = 2πm/a. Free-electron energies are E = ħ²k²/(2m_e), and they cross at the zone boundary k = ±π/a, where the forward wave e^(ikx) and the back-scattered wave e^(i(k−G)x) become degenerate.

The periodic potential mixes those two degenerate plane waves. Degenerate perturbation theory gives two standing-wave combinations:

• cos(πx/a) — charge piled on the ion cores, lower potential energy.
• sin(πx/a) — charge piled between the ions, higher potential energy.

They split apart by an energy gap equal to twice the relevant Fourier component of the potential:

E_gap = 2 |V_G|

Plug in numbers. The free-electron energy at the first zone boundary is

E(π/a) = ħ²/(2m_e) · (π/a)²
       = (1.055e-34)² / (2 · 9.11e-31) · (π / 3.0e-10)²
       ≈ 4.2 eV

If the potential's first Fourier component is |V_G| ≈ 0.5 eV, the gap opens to E_gap ≈ 1.0 eV — squarely in the semiconductor range, comparable to silicon's 1.12 eV. The lower band, from 0 up to roughly 4.2 − 0.5 ≈ 3.7 eV, is allowed; the window from about 3.7 eV to 4.7 eV is forbidden; the next allowed band begins above it. A single ~0.5 eV ripple in the potential carved a 1 eV hole out of the energy spectrum.

Counting states seals the physics. A crystal of N = 10²³ unit cells gives each band exactly 2N = 2×10²³ states. If this material contributes one electron per cell, the lower band is half-full — a metal. Two electrons per cell exactly fill the lower band, leaving the upper band empty across a 1 eV gap — a semiconductor. The same band structure, different electron count, completely different material.

Variants and regimes

Regime / model	What it assumes	Bloch picture	Where it fits
Free electron (V → 0)	No potential at all	u_nk → constant; ħk = true momentum	Alkali-metal sea, sanity check
Nearly free electron	Weak periodic potential	Small gaps 2|V_G| at zone boundaries	Simple metals (Na, Al)
Tight binding (LCAO)	Strong, localized atomic potentials	Narrow bands from overlapping orbitals	d-band metals, graphene
Kronig–Penney	1D periodic square wells	Exactly solvable bands + gaps	Teaching model, superlattices
Wannier functions	Localized basis instead of k-space	Bloch sums Fourier-transformed to real space	Correlated electrons, polarization
k·p method	Expand around a band extremum	Effective mass m* from band curvature	Semiconductor device design
Magnetic field (Hofstadter)	Add a B-field, broken simple periodicity	Magnetic Bloch states, fractal spectrum	Quantum Hall, moiré lattices

Group velocity and effective mass

A Bloch state of definite k has no group velocity issue — but a wavepacket built around k moves with a well-defined velocity equal to the slope of the band:

v_g = (1/ħ) ∂E_n(k)/∂k

Apply an external force F (an electric field, say). The semiclassical equations of motion read:

ħ dk/dt = F        (force changes crystal momentum)
a = (1/ħ²) (∂²E/∂k²) F   ⇒   m* = ħ² / (∂²E/∂k²)

The effective mass m* is set entirely by the curvature of the band. A sharply curved band → small m*, a light fast electron; a flat band → large m*, a heavy slow one. In GaAs the conduction-band effective mass is only m* ≈ 0.067 m_e, which is half the reason GaAs transistors switch fast. And near the top of a band, where E(k) curves downward, m* is negative — the cleanest formal reason to invent holes, vacancies that behave like positive charges with positive mass.

Common pitfalls and misconceptions

• "ħk is the electron's momentum." It is crystal momentum, a symmetry label conserved only mod ħG. The true momentum expectation value generally differs because u_nk carries momentum. They coincide only in the free-electron limit.
• "The electron is localized on an atom." A Bloch state is delocalized across the entire crystal; |ψ|² is identical in every unit cell. Localization on atoms is the Wannier picture, a Fourier transform of the Bloch states, not the same object.
• "Gaps need a strong potential." Even an infinitesimal periodic potential opens gaps at every zone boundary — the gap is 2|V_G|, proportional to the relevant Fourier component, not to the total potential depth.
• "More electrons always means more conduction." An exactly filled band carries no current no matter how many electrons it holds — every state's velocity is cancelled by its partner at −k. Conduction needs a partly filled band, which is why insulators exist.
• "Bloch's theorem requires a specific potential." The plane-wave-times-periodic form follows from translational symmetry alone. Only the band energies E_n(k) depend on the details of V(r).
• "It only works in 1D." Everything generalizes: k and R are vectors, the Brillouin zone is a 3D polyhedron, and bands are surfaces E_n(k) over that zone. The 1D pictures are just easy to draw.

Applications

• Semiconductors and transistors. The entire band-gap engineering of silicon, GaAs, and modern heterostructures rests on Bloch's E_n(k). Doping shifts the Fermi level within bands; the gap sets the on/off ratio.
• Metals vs. insulators. Band filling — partly filled (metal) versus fully filled with a gap above (insulator/semiconductor) — is the textbook explanation of why some solids conduct and others don't.
• Optoelectronics. An electron dropping across the gap emits a photon of energy ≈ E_gap; LEDs and laser diodes are gap engineering. Direct vs. indirect gaps (whether the band extrema share the same k) decide whether a material can emit light efficiently — silicon's indirect gap is why it makes poor LEDs.
• Photonic and acoustic crystals. The same theorem applies to electromagnetic and sound waves in periodic media, producing photonic and phononic band gaps — frequencies that simply cannot propagate.
• Topological materials. Modern topological insulators are classified by global properties of the Bloch wavefunctions u_nk across the Brillouin zone (Berry phase, Chern numbers).
• Density functional theory. First-principles materials calculations solve the Kohn–Sham equations on a grid of Bloch k-points, computing band structures from scratch for novel compounds and batteries.

Derivation and performance analysis

Why is the Bloch picture so computationally powerful? Because it shrinks an intractable problem to a tiny one. Without symmetry, an electron in a crystal of N ~ 10²³ atoms is a Schrödinger problem over the whole macroscopic solid. Bloch's theorem replaces it with a problem on a single unit cell, parametrized by a continuous label k that you can sample on a modest grid.

Concretely, a practical band-structure calculation expands u_nk in a plane-wave basis of size G_max and diagonalizes a Hamiltonian matrix at each k-point. If you keep M plane waves, each diagonalization is O(M³); you do it at, say, a 12×12×12 grid of k-points — a few thousand small matrices rather than one matrix of dimension 10²³. That is the difference between "impossible" and "runs on a laptop overnight."

The folding is exact, not an approximation: because states repeat with period G in reciprocal space, sampling the first Brillouin zone captures everything. Symmetry of the crystal (point group) reduces the work further to the irreducible wedge of the zone — often a factor of 8 to 48 fewer k-points. The numbers worth keeping:

Quantity	Symbol / value	Note
Bloch wavefunction	ψ_nk(r) = e^(ik·r) u_nk(r)	Plane wave × cell-periodic part
Cell periodicity	u_nk(r + R) = u_nk(r)	Same period as the lattice
Crystal momentum	ħk, conserved mod ħG	Not mechanical momentum
States per band	2N	N cells × 2 spins
Gap size (NFE)	E_gap = 2|V_G|	Twice the Fourier component
Group velocity	v = (1/ħ) ∂E/∂k	Slope of the band
Effective mass	m* = ħ² / (∂²E/∂k²)	Inverse band curvature
Worked gap example	≈ 1.0 eV at a = 3.0 Å	From |V_G| ≈ 0.5 eV

Bloch's theorem is one of those rare results where a symmetry argument that fits on a napkin reorganizes an entire field. From "discrete translational invariance" alone you get plane-wave-times-periodic states, crystal momentum, the Brillouin zone, bands and gaps — and from there metals, insulators, semiconductors, LEDs, and the transistor in the device you're reading this on.