Finding energy eigenvalues with perturbation

[boudreaux]

Homework Statement

There are two spin-1 particles $$S_1,S_2$$

Relevant Equations

$$S_1^2 = S_2^2 = 1(1+1)\hbar^2 = 2\hbar^2$$

there's a strong magnetic field in the Z direction so

$$H_0 = -B*(S_1 +S_2)$$ (all z components here)

I need to fin the eigen-energies and eigenstates of this hamiltonian. Then add a small perturbation

$$H_1 = -J S_1 S_2$$ (still z components) and find the energy levels again.

I know the basis I should use is |m_1,m_2> and that each m can be 1,0,-1 but how do I get the eigenvalues from this?

 

[nrqed]

Homework Statement:: There are two spin-1 particles $$S_1,S_2$$
Relevant Equations:: $$S_1^2 = S_2^2 = 1(1+1)\hbar^2 = 2\hbar^2$$

there's a strong magnetic field in the Z direction so

$$H_0 = -B*(S_1 +S_2)$$ (all z components here)

I need to fin the eigen-energies and eigenstates of this hamiltonian. Then add a small perturbation

$$H_1 = -J S_1 S_2$$ (still z components) and find the energy levels again.

I know the basis I should use is |m_1,m_2> and that each m can be 1,0,-1 but how do I get the eigenvalues from this?

You really mean $H_1 = - J (S_1)_z (S_2)_z$ and not $- J \vec{S_1} \cdot \vec{S_2}$, right?
Then to get the energies, just apply that operator to all the states $| m_1 m_2 \rangle$. These are eigenstates of the $S_z$ operators.