= Bloch Hamiltonian
{c}
The Bloch Hamiltonian acts on the cell-periodic part of a <Bloch state> at fixed <wavevector>. If $\psi_{n\mathbf k}(\mathbf r)=e^{i\mathbf k\cdot\mathbf r}u_{n\mathbf k}(\mathbf r)$, define
$$
H(\mathbf k)=e^{-i\mathbf k\cdot\mathbf r}He^{i\mathbf k\cdot\mathbf r},\qquad H(\mathbf k)u_{n\mathbf k}=E_n(\mathbf k)u_{n\mathbf k}.
$$
For a scalar periodic potential, its kinetic term is $(-i\hbar\nabla+\hbar\mathbf k)^2/(2m)$ 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.
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