Bloch bands and reciprocal space

Consider a time-independent single-particle Hamiltonian with a potential that repeats after a lattice vector. It commutes with the corresponding translation, so energy eigenstates can be chosen to diagonalize translation as well. In one dimension, Bloch’s theorem writes the wavefunction as a plane-wave phase times a cell-periodic function. The periodic factor contains the structure within each unit cell. It is generally not constant, so replacing every Bloch state by a free plane wave throws away physical information about the crystal.

The theorem assumes exact periodicity for the Hamiltonian under study. A finite surface, a defect, or disorder breaks that symmetry; Bloch states can still provide a useful expansion basis without becoming exact eigenstates. Interactions introduce another distinction: a noninteracting band Hamiltonian may describe a reference problem, while observable electronic excitations in an interacting solid require a many-body analysis. Bloch’s theorem concerns translations, not the Bloch sphere of a two-level state.

Adding a reciprocal vector $G=2\pi m/a$ to $k$ leaves the translation eigenvalue unchanged because $e^{iGa}=1$. A corresponding change in the periodic factor represents the same Bloch wavefunction. Therefore, choose one Brillouin zone as a set of inequivalent crystal momenta, conventionally a half-open interval of width $2\pi/a$. Its endpoints are equivalent, not two extra independent states. With $L$ cells and periodic boundary conditions, there are exactly $L$ allowed momenta per band and per spin, matching the number of retained localized orbitals.

Fourier transforming a chain with one orbital per cell turns nearest-neighbor hopping into a diagonal energy function. The two directions of a link contribute complex-conjugate phases; their sum produces a cosine. The energy variation across the Brillouin zone is the bandwidth, which measures the kinetic scale within this model. Changing the sign convention for hopping moves the location of the band minimum, so state the Hamiltonian before interpreting a sketch. The number of bands follows the number of orbitals per unit cell, rather than the number of drawn zone repetitions.

For a narrow wavepacket that stays in one isolated band under weak, slowly varying perturbations, the group velocity is the energy slope divided by the reduced Planck constant. A band extremum therefore has zero group velocity even when the crystal-momentum label is nonzero in a chosen zone convention. Local band curvature defines an effective mass for approximately quadratic motion; that mass need not equal the vacuum electron mass and can change sign near a band maximum. These are controlled single-band statements rather than a complete treatment of acceleration, interband transitions, or scattering.

Filling is essential. A completely filled, isolated noninteracting band has canceling velocity contributions in equilibrium and cannot acquire an ordinary intraband current through infinitesimal redistribution at zero temperature. A partially filled band permits low-energy rearrangements; conductivity also requires lifetimes, disorder, interactions, and the measurement setup. In the wave lab, a cell-to-cell phase analogy can be explored by changing relative phase between two classical waves. It illustrates interference only; it does not compute the cosine band, reciprocal labels, electronic occupations, or a Fermi level.