The simplest way of understanding spin waves is to consider the Hamiltonian  for the Heisenberg ferromagnet:
where  is the exchange energy, the operators  represent the spins at Bravais lattice points,  is the Landé -factor,  is the Bohr magneton and  is the internal field which includes the external field plus any "molecular" field. Note that in the classical continuum case and in  dimensions the Heisenberg ferromagnet equation has the form
In  and  dimensions this equation admits several integrable and non-integrable extensions like the Landau-Lifshitz equation, the Ishimori equation and so on. For a ferromagnet  and the ground state of the Hamiltonian  is that in which all spins are aligned parallel with the field . That  is an eigenstate of  can be verified by rewriting it in terms of the spin-raising and spin-lowering operators given by:
resulting in
where  has been taken as the direction of the magnetic field. The spin-lowering operator  annihilates the state with minimum projection of spin along the -axis, while the spin-raising operator  annihilates the ground state with maximum spin projection along the -axis. Since
for the maximally aligned state, we find
where N is the total number of Bravais lattice sites. The proposition that the ground state is an eigenstate of the Hamiltonian is confirmed.
One might guess that the first excited state of the Hamiltonian has one randomly selected spin at position  rotated so that
but in fact this arrangement of spins is not an eigenstate. The reason is that such a state is transformed by the spin raising and lowering operators. The operator  will increase the -projection of the spin at position  back to its low-energy orientation, but the operator  will lower the -projection of the spin at position . The combined effect of the two operators is therefore to propagate the rotated spin to a new position, which is a hint that the correct eigenstate is a spin wave, namely a superposition of states with one reduced spin. The exchange energy penalty associated with changing the orientation of one spin is reduced by spreading the disturbance over a long wavelength. The degree of misorientation of any two near-neighbor spins is thereby minimized. From this explanation one can see why the Ising model magnet with discrete symmetry has no spin waves: the notion of spreading a disturbance in the spin lattice over a long wavelength makes no sense when spins have only two possible orientations. The existence of low-energy excitations is related to the fact that in the absence of an external field, the spin system has an infinite number of degenerate ground states with infinitesimally different spin orientations. The existence of these ground states can be seen from the fact that the state  does not have the full rotational symmetry of the Hamiltonian , a phenomenon which is called spontaneous symmetry breaking.