Theory [ edit ] An illustration of the precession of a spin wave with a wavelength that is eleven times the lattice constant about an applied magnetic field. The projection of the magnetization of the same spin wave along the chain direction as a function of distance along the spin chain. In 1 + 1, 2 + 1 and 3 + 1 dimensions this equation admits several integrable and non-integrable extensions like the Landau-Lifshitz equation, the Ishimori equation and so on. For a ferromagnet J> 0 and the ground state of the Hamiltonian | 0 ⟩ {\displaystyle |0\rangle } is that in which all spins are aligned parallel with the field H. That | 0 ⟩ {\displaystyle |0\rangle } is an eigenstate of H {\displaystyle {\mathcal {H}}} can be verified by rewriting it in terms of the spin-raising and spin-lowering operators given by:

The simplest way of understanding spin waves is to consider the Hamiltonian H {\displaystyle {\mathcal {H}}} for the Heisenberg ferromagnet: H = − 1 2 J ∑ i , j S i ⋅ S j − g μ B ∑ i H ⋅ S i {\displaystyle {\mathcal {H}}=-{\frac {1}{2}}J\sum _{i,j}\mathbf {S} _{i}\cdot \mathbf {S} _{j}-g\mu _{\rm {B}}\sum _{i}\mathbf {H} \cdot \mathbf {S} _{i}} H = − 1 2 J ∑ i , j S i z S j z − g μ B H ∑ i S i z − 1 4 J ∑ i , j ( S i + S j − + S i − S j + ) {\displaystyle {\mathcal {H}}=-{\frac {1}{2}}J\sum _{i,j}S_{i} where J is the exchange energy, the operators S represent the spins at Bravais lattice points, g is the Landé g-factor, μ B is the Bohr magneton and H is the internal field which includes the external field plus any "molecular" field. Note that in the classical continuum case and in 1 + 1 dimensions the Heisenberg ferromagnet equation has the form