I'm trying to solve this 3 part question.
1)Suppose we have two 1/2 spin particles. The total spin vector operator for the pair is $\hat{S} = \hat{s_1} +\hat{s_2}$. Write down a representation of the product eigenbasis of $\hat{\sigma_z}\otimes \hat{\sigma_z}$ for their Hilbert space $\mathbb{C}^2 \otimes \mathbb{C}^2$. Show that the $\hat{\sigma_z}\otimes \hat{\sigma_z}$ product basis states are eigenstates of $\hat{S_z}$ and state their eigenvalues.
2)The Hamiltonian of the two spins is
$$ \hat{H} = J\hat{s_1} . \hat{s_2}$$
Where J the coupling strength. Show that $[\hat{H},\hat{S_z}] = 0$.
3)Take the tensor product of the Pauli matrices to show that the corresponding matrix representation of $\hat{H}$ in the $\hat{\sigma_z}\otimes \hat{\sigma_z}$ product basis is $$\hat{H} = \frac{{\hbar}^2J}{4} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 2 & 0 \\ 0 & 2 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} $$
1)I know that
$$\hat{\sigma_z}\otimes \hat{\sigma_z} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} $$ But how can we use that as the eigenbasis. And for that matter how would it look like in any pauli direct product matrix representation. Is it just the vectors shown on each of the columns/rows of this matrix?
2)Given $\hat{S_z}$ I believe I can compute this part.
3)Again given the previous parts maybe I can do it?