kakarukeys

An easy (not easy) 1st year undergraduate project:

Energy eigenvalues of a single spin-1/2 system which Hamiltonian is given by

H = - k S^z

are -/+ (1/2 k).

I got that using the spin operator S^z = 1/2 Sigma^z (let h_bar = 1), Hamiltonian is in diagonalized form:

( -1/2 k 0
0 1/2k )

So, Eigenvectors are given by (1, 0), (0, 1), spin up and spin down respectively.

How do I compute the energy eigenvalues, eigenvectors of an double identical spin-1/2 system which Hamiltonian is given by

H = - J vec(S1) dot vec(S2) - k S1^z - k S2^z

J is a const. > 0

After going some references, Griffiths, Sakurai, Merzbacher, etc,
I have no idea how to begin,

I have the following problems in mind:
(1) if |u u> represents both spins up, (1/sqrt 2)(|u d> +/- |d u>) represent one spin up, |d d> represent both spins down. |u u> and |d d> should be eigenstates of the system (because they are ground states), for |u d>, |d u> states I am not sure.

(2) I can't use same spin matrices for S1^z, S2^z. But there is only one S^z matrix namely, 1/2 Sigma^z. Could S1^z be a tensor product of 1/2 Sigma^z with an identity matrix, and S2^2 the other way round? If it is so, I have no idea how to do the maths!

(3) Do vec(S1) and vec(S2) commute?

Anyone could give me some hints?