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?