I start saying I'm Italian, so my English is not very well and I will probably make many grammar mistakes ( forgive me for that)...
I have to find eigenvalues and eigenvectors of this Hamiltonian described by two particles of spin $1 \over 2 $:
$\hat{H} = A (\hat S_{1y} - \hat S_{2y})^2$, where $A$ is a real constant.
All the problems I solved till now presented Hamiltonian with explicit ( or easy implicit) $\hat J_z $ and $\hat J^2$ dependency.
I think I can say that a basis of total Hilbert space $\hat V_1 \times \hat V_2$ is :
$|1/2 ,1/2\rangle|1/2 ,1/2\rangle,\ |1/2 ,1/2\rangle|1/2 ,-1/2\rangle,\ |1/2 ,-1/2\rangle|1/2 ,1/2\rangle,\ |1/2 ,-1/2\rangle|1/2 ,-1/2\rangle$.
The text of the problem also remembers me the expression of annihilator/construction operators $\hat J_\pm$ .
How can I solve this kind of problem without finding the form of both matrices? does it exist an easier way? Thanks in advance, I'm new to Quantum Mechanics.