Eigenstates of coupled Angular Momentum SO I have a hamiltonian:
$$H=\alpha J_1\cdot J_2$$
And I am asked to find the eigenstates and eigenvalues of this Hamiltonian in terms of products of the eigenstates of the z components of the individual spins.
The wording of this question is kind of confusing me.  I could solve it by rewriting it in terms of $(J_1+J_2)^2,J_1^2,J_2^2$ and then find the eigenvalues, but that is not in terms of the z components.  How should I go about doing this?  Thanks
 A: The question is essentially just asking you to perform Clebsch-Gordan (CG) decomposition, namely a change of basis on the Hilbert space $\mathcal H_{j_1}\otimes\mathcal H_{j_2}$ of the composite system of spins.
Recall that for each of the Hilbert spaces $\mathcal H_{j_1}$ and $\mathcal H_{j_1}$ there exist orthonormal bases of eigenvectors of $\mathbf J_1^2, J_1^z$ and $\mathbf J_2^2, J_2^z$ respectively, and these bases are 
\begin{align}
  \text{for $\mathcal H_{j_1}$}&:\qquad  \{|j_1, m_1\rangle\,|\,m_1=-j_1,-j_1+1,\dots,j_1-1, j_1\} \\
\text{for $\mathcal H_{j_2}$}&:\qquad  \{|j_2, m_2\rangle\,|\,m_2=-j_2,-j_2+1,\dots,j_2-1, j_2\}
\end{align}
It follows that an orthonormal basis of eigenvectors for the composite Hilbert space $\mathcal H_{j_1}\otimes\mathcal H_{j_2}$ consists of the set of all tensor products of these basis elements;
\begin{align}
  \text{for $\mathcal H_{j_1}\otimes\mathcal H_{j_2}$}:\qquad  \{|j_1, m_1\rangle|j_2,m_2\rangle\,|\,-j_1\leq m_1\leq j_1, -j_2\leq m_2\leq j_2\}  \tag{$\star$}
\end{align}
This is what being referred to as

"...products of eigenstates of the z components of individual spins."

So once you've found the eigenstates of the Hamiltonian, which you will presumably write in terms of the basis
\begin{align}
  \{|j,m,j_1,j_2\rangle\},
\end{align}
 you just need to decompose them into the tensor product basis in $(\star)$ above.
