k·p perturbation theory ($H_{\mathbf{k}}=H_0+\hslash \mathbf{k}\cdot \mathbf{p}/m_0+{\hslash }^2{k}^2/(2m_0)$)

For a local periodic scalar potential, writing a Bloch wave as $e^{i\mathbf{k}\cdot \mathbf{r}}{u}_{n\mathbf{k}}$ gives the displayed cell-periodic Hamiltonian before additional spin-orbit terms. Momentum matrix elements mix zone-centre bands when k is varied. Perturbatively eliminating remote bands changes the effective mass; a multiband retained subspace describes degeneracy, nonparabolicity and coupled hole bands. Spin-orbit coupling and the chosen reference wavevector must be included consistently.