## Crystal momentum, effective mass, group velocity we call $\hbar k$ the crystal momentum, but it is not the momentum of electron, and [[Bloch's theorem and Born-von Karman boundary condition#^73ff80|Bloch's function]] is not eigenfunction of momentum operator. But we can get expectation of momentum operator for a Bloch wave as $\hbar k +\hbar \sum_{G}| c_{G}|^2G$ and the group velocity $v(k)=\braket{\Psi_k|\frac{\hat{p}}{m}|\Psi_k}=\frac{1}{\hbar} \frac{\mathrm{d}E(k)}{\mathrm{d}k}$ effective mass $\frac{1}{m^*} = \frac{1}{\hbar^2} \frac{\mathrm{d}^2E(k)}{\mathrm{d}k^2}$ A more restrict form of effective mass can be deduced from $k\cdot p$ theory, which is shown in [[Band structure near band extrema, k.p theory|band structure near band extrema: k.p theory]] chapter.