Bloch's Theorem

Introduction

The free-electron model ignores the crystal's periodic ionic potential. An actual electron sees a landscape repeated from cell to cell. Solving Schrödinger's equation independently for every atom in a macroscopic sample would be hopeless, but periodicity supplies a powerful simplification. Bloch's theorem says an electronic energy eigenstate can be chosen as a plane-wave-like phase multiplied by a function that repeats with the lattice. This form is the foundation for energy bands E n(k), Brillouin zones and much of modern materials modelling.

Core explanation

For an ideal single-electron Hamiltonian with V(r + R) = V(r) for every lattice vector R, the Hamiltonian commutes with each lattice translation. Its eigenstates may therefore be chosen to acquire only a phase under translation: ψ nk(r + R) = exp(i k·R)ψ nk(r) . Equivalently, ψ nk(r) = exp(i k·r)u nk(r) with u nk(r + R) = u nk(r). The index n labels distinct bands at the same wavevector k. The periodic factor captures the modulation caused by the ions; the exponential captures the phase progression from cell to cell. The MIT quantum-physics lecture notes derive this relation using translation operators.

The probability density of one Bloch state is periodic: ψ nk(r + R) ² = ψ nk(r) ² because the phase has unit magnitude. That does not mean the electron is confined to one cell. A perfect-crystal Bloch state extends across the periodic region while displaying a repeated density pattern. A useful analogy is a travelling wave whose amplitude is rippled by a repeating grating. The function u is not generally constant, so a Bloch wave is not simply a free plane wave.

Two k labels differing by reciprocal vector G give the same phase under every lattice translation because exp(iG·R) = 1. Hence k is defined modulo a reciprocal-lattice vector, and one can choose representative k values within a first Brillouin zone. This is a bookkeeping equivalence, not an assertion that a crystal electron has a uniquely measured free-space momentum ħk. The periodic lattice can exchange ordinary momentum with the electron by reciprocal-lattice increments, while crystal momentum labels symmetry and is conserved modulo G in suitable interactions.

Energy depends on both band index and k. If V were perfectly constant, E(k) would be a free-electron parabola. Periodic potential couples waves whose wavevectors differ by G and can open gaps near zone boundaries. The slope dE/dk governs group velocity of a wavepacket, and local curvature helps define effective mass. Thus Bloch's theorem does not by itself say whether a material is a metal: electron filling and band gaps still matter. MIT materials lecture notes on periodic potentials show how translation symmetry leads to the Bloch form and a crystal-momentum label.

The theorem applies exactly to an ideal periodic single-particle problem. A real sample has surfaces, impurities, vacancies, disorder and electron interactions. Bloch states are still an excellent bulk reference for many crystals, but a local defect can create a localised state, and a surface can support modes that need boundary conditions absent from an infinite-crystal calculation. An amorphous solid lacks exact long-range translation symmetry, so no globally exact Bloch k labels its states. In interacting crystals, quasiparticle band descriptions can survive approximately, but strongly correlated systems may require a more careful many-body treatment.

Step-by-step reasoning

1. Check that the potential repeats under a lattice translation R. 2. Apply the translation operator to an energy eigenstate. 3. Assign its unit-magnitude translation eigenvalue exp(i k·R). 4. Factor the state into exp(i k·r) times a lattice-periodic u nk. 5. Restrict equivalent k labels to one Brillouin zone and then calculate E n(k).

Visual explanation

Draw a row of equal wells representing ionic sites. Over them draw a smooth plane-wave phase whose crest shifts from cell to cell, then draw a smaller repeating ripple in amplitude. Their product is the Bloch wave. Beneath, draw each cell with the same probability-density shape even though the complex wavefunction changes phase. A reciprocal-vector arrow connects equivalent k labels at opposite edges of a chosen zone.

Real-world analogy

A patterned carpet repeats every metre. One can walk along it with a steady stride while the appearance underfoot changes within each metre in the same way. The walking rhythm resembles the overall phase factor; the repeating carpet pattern resembles u. A boundary or torn patch interrupts the ideal repetition even though the pattern is still useful over most of the room.

Real-world example

In a silicon crystal, calculated valence and conduction bands are indexed by k across the first Brillouin zone. Their extrema occur at different k values, helping explain why ordinary silicon's optical emission is less efficient than that of a direct-gap material. Bloch's theorem is what makes that k-resolved comparison meaningful for the bulk crystal; a nanocrystal or disordered film needs extra attention to confinement and broken symmetry.

Why?

Why does ψ ² repeat even when ψ itself gains a phase? Multiplication by exp(i k·R) changes the complex phase but not the modulus. The measured probability density of that one ideal Bloch state is therefore identical in translation-equivalent cells.

Common misconception

"Bloch's theorem says an electron moves freely without sensing atoms." It says the periodic Hamiltonian permits a special form of eigenstate. The periodic part u and band energies encode substantial effects of the atoms, including gaps and directional dispersion.

Worked example

Question: In a one-dimensional lattice with period a = 0.40 nm, take k = π/a. What phase does ψ acquire after translating by one cell? What happens to its probability density?

Reasoning: Bloch's relation gives ψ(x+a) = exp(ika)ψ(x). With k = π/a, ka = π and exp(iπ) = −1. The complex wavefunction changes sign from one cell to the next. Its density is ψ(x+a) ² = −ψ(x) ² = ψ(x) ², so a sign change does not imply an alternation of measured probability density. Adding 2π/a to k gives the same translation phase.

Answer: The phase factor is −1, while the probability density repeats unchanged.

Quick check

1. Is a Bloch function simply a plane wave when the lattice potential is nonconstant? Answer: No. Its plane-wave phase is multiplied by a lattice-periodic factor that encodes the periodic potential.

Exam focus

Write both Bloch forms and identify which part is periodic. Explain why k is equivalent modulo G and why ψ ² is periodic. State that a crystal momentum label is not automatically the same as free-particle mechanical momentum.

Advanced insight

Bloch functions have a phase freedom at each k: multiplying a state by exp[iφ n(k)] leaves its energy and density unchanged. How this phase changes through k-space underlies Berry connections and modern descriptions of polarization and topological bands. The elementary theorem supplies the states, while these later concepts examine their geometry across k.

Summary

For a periodic potential, electronic states can be chosen as ψ nk(r) = exp(i k·r)u nk(r), with u periodic on the lattice. Translation changes a state by a phase, so its density repeats. k is defined modulo reciprocal vectors and labels energy bands; real defects and boundaries limit the exact idealisation.

Practice questions

1. What property of V(r) allows Bloch's theorem? Answer: It repeats under every lattice translation, V(r+R)=V(r). 2. Which part of ψ nk is periodic, exp(i k·r) or u nk(r)? Answer: The factor u nk(r) repeats with the crystal lattice. 3. Why can k and k+G represent equivalent translation phases? Answer: exp(iG·R)=1 for every lattice translation R. 4. Give one case where an exact bulk Bloch label is inappropriate. Answer: A strongly disordered amorphous solid, a localised defect state or a boundary mode not described by a purely infinite periodic model.