Solving Energy Levels of 2-Spin 1/2 System

[ghostflame]

Homework Statement

Find the energy levels of a 2-spin 1/2 system with spinoperators S1 and S2 in an external magnetic field. The hamiltonian is of the form,

H= A ( 1-$$\frac{2S_{1}}{h}$$ . $$\frac{2 S_{2}}{h}$$ )+ $$\frac{\mu B}{h}$$(S$$_{1,z}$$+S$$_{2,z}$$)

The h is a h-bar, constants A, B, and S1 and S2 the spin operators

Homework Equations

I have to solve the equation H l$$\psi$$> = El$$\psi$$>

The Attempt at a Solution

The spin system is has a basis, l$$\uparrow\uparrow>,\left| \uparrow\downarrow>,\left|\downarrow\uparrow>,\left|\downarrow\downarrow>$$

so any $$\left| \psi$$> is a linear combination of the basis above, but i don't know how i can calculate the eigenvalues of the above equation. I have a feeling i have to use the Pauli matrices but iam not sure. Anyone has an idea? It should be a 3 level system...

 

[ghostflame]

I know the matrices S1 and S2 commute, is also know that S1,z + S2,z = S,z

couldn't that help?

 

[gabbagabbahey]

Why not try expanding your wavefunction into the spin basis and then using that to calculate $$H|\psi\rangle$$? Under what circumstances is your result a constant multiple of $$|\psi\rangle$$?