Bloch Hamiltonian

The Bloch Hamiltonian acts on the cell-periodic part of a Bloch state at fixed wavevector. If ${\psi }_{n\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot \mathbf{r}}{u}_{n\mathbf{k}}(\mathbf{r})$, defineFor a scalar periodic potential, its kinetic term is $(-i\hslash \nabla +\hslash \mathbf{k}{)}^2/(2m)$ on cell-periodic functions.

Solving this family of cell problems produces electronic band structure. In a tight-binding description, the Bloch Hamiltonian is a matrix on the orbitals within a primitive cell, not the full infinite real-space matrix. Basis choices can change its entries without changing the physical band energies. The distinction between the full wavefunction and its periodic factor also matters when computing matrix elements and geometric phases.

For a real scalar lattice-periodic potential, let $|n⟩$ be a nondegenerate zone-centre periodic eigenstate with energy $E_n$. Conjugation by $e^{i\mathbf{k}\cdot \mathbf{r}}$ gives $H(\mathbf{k})=H(0)+\hslash \mathbf{k}\cdot \mathbf{p}/m+{\hslash }^2{k}^2/(2m)$. The zone-centre state can be chosen real, so $⟨n|\mathbf{p}|n⟩=0$. Second-order nondegenerate perturbation theory then givesTime-reversal symmetry makes this isolated spinless band even in $\mathbf{k}$. In a direction $\mathbf{e}$, the curvature givesA degeneracy requires a matrix treatment before extracting curvatures; a nonlocal potential or spin-orbit terms can also alter the simple momentum vertex.