Does crystal momentum obey ordinary momentum conservation?

Crystal momentum is conserved only up to a reciprocal lattice vector, because the lattice can absorb momentum in quantized amounts; it labels Bloch states and governs selection rules but is not the true mechanical momentum of the electron.

Why does Bloch's theorem imply bands rather than a continuum?

For each crystal momentum the periodic Schrödinger problem has a discrete ladder of solutions indexed by band number; letting the momentum vary across the zone sweeps each level into a continuous band, with energy ranges in between that no state occupies, the gaps.

Bloch's Theorem and Energy Bands

Bloch's theorem states that the wavefunction of an electron in a periodic lattice is a plane wave times a lattice-periodic function, which immediately organizes the allowed energies into bands.

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Definition

Bloch's theorem asserts that the energy eigenstates of an electron in a periodic potential have the form of a plane wave modulated by a function with the periodicity of the lattice; the eigenvalues, as a function of the crystal momentum, form continuous energy bands separated by forbidden gaps.

Scope

This topic proves and interprets Bloch's theorem: the periodic potential forces eigenstates to be Bloch waves labeled by a crystal momentum and a band index, the spectrum splits into energy bands separated by gaps, and the bands can be displayed in the extended, reduced, or repeated zone schemes. It covers the meaning of crystal momentum, the group velocity of Bloch electrons, and the count of states per band. It is the foundation on which the model approximations and Fermi-surface topics build.

Core questions

Why does lattice periodicity force the electronic wavefunctions to take the Bloch form?
What is crystal momentum, and how does it differ from ordinary momentum?
How does the band index together with crystal momentum label every electronic state?
Why are there exactly as many states in a band as there are primitive cells in the crystal?

Key concepts

Bloch wavefunction and lattice-periodic part
Crystal momentum and band index
Energy bands and band gaps
Extended, reduced, and repeated zone schemes
Group velocity of a Bloch electron

Key theories

Bloch's theorem: For a single electron in a periodic potential, the eigenstates are products of a plane wave and a periodic function, so each is indexed by a crystal momentum in the Brillouin zone and a discrete band index, yielding a band-structured spectrum.

Clinical relevance

Bloch's theorem is the cornerstone of solid-state physics: it explains why electrons move ballistically through a perfect crystal, defines the band structure used to classify conductors and insulators, and underlies essentially every electronic-structure calculation.

History

Felix Bloch proved the theorem in his 1928 doctoral work (published 1929), supervised by Heisenberg, resolving why electrons are not strongly scattered by the dense lattice of ions; the result generalizes Floquet's earlier one-dimensional theory of periodic differential equations.

Key figures

Felix Bloch
Gaston Floquet
Rudolf Peierls

Seminal works

bloch1929
ashcroft1976

Frequently asked questions

Does crystal momentum obey ordinary momentum conservation?: Crystal momentum is conserved only up to a reciprocal lattice vector, because the lattice can absorb momentum in quantized amounts; it labels Bloch states and governs selection rules but is not the true mechanical momentum of the electron.
Why does Bloch's theorem imply bands rather than a continuum?: For each crystal momentum the periodic Schrödinger problem has a discrete ladder of solutions indexed by band number; letting the momentum vary across the zone sweeps each level into a continuous band, with energy ranges in between that no state occupies, the gaps.