含义Meaning
周期势场中电子的本征函数是平面波乘以与晶格同周期的函数。能量依赖波矢 k 形成能带，是固体能带理论的基础（1928）。
In a periodic potential electron eigenfunctions are plane waves times a lattice-periodic function; energies depend on wavevector k and form bands — the foundation of band theory (1928).
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(\mathbf k\) | 晶体波矢（第一布里渊区内）crystal wavevector (first Brillouin zone) | m⁻¹ |
| \(n\) | 能带指标band index | 1 |
| \(\mathbf R\) | 晶格平移矢量lattice translation | m |
推导Derivation
哈密顿量与平移算符 \(\hat T_\mathbf R\) 对易，可取共同本征函数：\(\hat T_\mathbf R\psi = \lambda_\mathbf R\psi\)。
H commutes with lattice translations, so eigenfunctions are shared.
平移群可交换且 \(|\lambda| = 1\)（周期性边界），所以 \(\lambda_\mathbf R = e^{i\mathbf k\cdot\mathbf R}\)。
Translations commute and \(|\lambda| = 1\), so \(\lambda_\mathbf R = e^{i\mathbf k\cdot\mathbf R}\).
令 \(u = e^{-i\mathbf k\cdot\mathbf r}\psi\)，即证 u 具有晶格周期性。
Then \(u = e^{-i\mathbf k\cdot\mathbf r}\psi\) is lattice-periodic.
一维原子链（紧束缚 / 休克尔）：\(E(k) = \alpha + 2\beta\cos ka\)，带宽 \(4|\beta|\)；化学上即无限长多烯的 π 能带。
For a 1D chain \(E(k) = \alpha + 2\beta\cos ka\), bandwidth 4|β| — the π band of an infinite polyene.
适用条件与局限Assumptions & limitations
假设完美无限晶体；表面、缺陷、无序破坏平移对称。电子相关强的材料（莫特绝缘体）不能只用单电子能带描述。
Assumes a perfect infinite crystal; strongly correlated materials (Mott insulators) defy single-electron bands.