Derivation of the Hamiltonian of the Heisenberg model

[genloz]

Homework Statement

Show that the Hamiltonian of the Heisenberg model can be written as:
$$H=\sum^{N}_{k=1}[H_{z}(k)+H_{f}(k)]$$
where
$$H_{z}(k)\equivS^{z}(k)S^{z}(k+1)$$
$$H_{f}(k)\equiv(1/2)[S^{+}(k)S^{-}(k+1)+S^{-}(k)S^{+}(k+1)]$$

Homework Equations

As above

The Attempt at a Solution

I read through this page: http://phycomp.technion.ac.il/~riki/H2_molecule.html
but I still don't really understand.

 

[genloz]

Sorry that second equation should be
Hz(k)=S^z(k)S^z(k+1)