  We consider a free hydrogen atom composed of a spin-1/2 nucleus and a
spin-1/2 electron in the standard model of non-relativistic QED. We study the
Pauli-Fierz Hamiltonian associated with this system at a fixed total momentum.
For small enough values of the fine-structure constant, we prove that the
ground state is unique. This result reflects the hyperfine structure of the
hydrogen atom ground state.
