I found this problem, and so far I am stumped. I was wondering if anyone wanted to solve it with me, or help me calculate eigenvectors, or just give insight on my questions.
Consider a system of two spin-1/2 particles interacting through the Hamitonian $$H = A(S_x^2 − S_y^2) + BS_z^2,$$ where $A$ and $B$ are constants and $S_x$, $S_y$ and $S_z$ are the three components of the total spin of the system. Find the energy spectrum and the corresponding eigenvectors.
It's been awhile since I did QM, and I know you can't have simultaneous eigenvectors of more than one measurement of spin along an axis, but how does that relate to this problem where there Sx^2 and Sy^2 are in the hamiltonian? I've been trying to work out 4x4 matrices, but I can't find good eigenvectors, but even then I am confused how applying a pauli matrix (say for x or y) transforms the eigenvector to another eigenvector, which I think is another example of not being able to define simultaneous eigenvectors etc. How would one start this problem? I will break it down to S^2 and Sz terms, but there will be remaining terms, will they just not contribute to the energy spectrum? If someone much more knowledgable than me could solve this problem easily and post it, that would clarify so much to me. For now, I'll try to read Feynman's lecture on the subject.