A radical anion has negative net electric charge and an unpaired electron.

A radical cation has positive net electric charge and an unpaired electron.

Light initiates methane chlorination by photodissociation of $\mathrm{C}{\mathrm{l}}_2$ through homolysis. The elementary bookkeeping isHere $h\nu$ denotes the photon energy. The first propagation step replaces a chlorine radical with a methyl radical; the second regenerates a chlorine radical. Adding the two propagation equations cancels both free radicals and gives $\mathrm{C}{\mathrm{H}}_4+\mathrm{C}{\mathrm{l}}_2⟶\mathrm{C}{\mathrm{H}}_3\mathrm{Cl}+\mathrm{HCl}$: chloromethane and hydrogen chloride are made while the chain carrier is regenerated.^[ref]

For example, recombination $2{\mathrm{C}{\mathrm{H}}_3}^{\bullet }⟶{\mathrm{C}}_2{\mathrm{H}}_6$ of methyl radicals makes ethane and a covalent bond from two unpaired electrons, removing two carriers. Recombination of two ${\mathrm{Cl}}^{\bullet }$ also terminates carriers; in the gas phase, stabilization of the product requires disposal of released energy, for example to a third collision partner. The figure omits that partner. Termination competes with propagation, and further chlorination can consume chloromethane, so this cycle alone does not predict a pure product or a measured reaction rate.^[ref]

If a Hamiltonian changes sufficiently slowly while an eigenstate remains separated from the rest of the spectrum by a nonzero gap, evolution keeps the state in the corresponding instantaneous eigenspace up to a dynamical and geometric phase. A degenerate isolated eigenspace can undergo a unitary holonomy within itself.

Shor's factoring algorithm uses quantum period finding to determine a modular order and then extracts factors classically with greatest common divisors.

Bloch's theorem describes energy eigenstates of a Quantum Hamiltonian invariant under Bravais lattice translations. They can be chosen in the formTranslation by a lattice vector $\mathbf{R}$ therefore multiplies the state by $e^{i\mathbf{k}\cdot \mathbf{R}}$. The wavevector $\mathbf{k}$ labels crystal momentum and is equivalent modulo a reciprocal lattice vector.

This converts a problem over an entire ideal crystal into a family of cell-periodic problems, organized by the Bloch Hamiltonian. Their energy eigenvalues form energy bands over a Brillouin zone. The periodic function carries the microscopic structure missing from a bare plane wave. Boundaries, defects and disorder break exact translation invariance, so the theorem's states serve as a basis or approximation there rather than automatic exact eigenstates.

The BB84 protocol encodes random classical bits in independently chosen computational and Hadamard bases. Sender and receiver publicly compare bases and retain only positions where their bases agree; disturbance of a test sample detects eavesdropping.

Grover search rotates a uniform state toward a marked subspace and finds one of $M$ marked entries using order $\sqrt{N/M}$ oracle calls.