For given Hamiltonian, is spin conserved?

[Leicester Fantasy]

A system consisting of two spins is described by the Hamiltonian (b>0)
H = aσ₁ ⋅ σ₂ + b(σ_1z - σ_2z)
where a and b are constants.
(a) Is the total spin S = ½ (σ₁ + σ₂) conserved? Which components of S, if any, are conserved?
(b) Find the eigenvalues of H and the corresponding eigenstates in terms of the eigenstates of the total spin S.

 

[blue_leaf77]

Welcome in PF!
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As for your current problem, can you please show us your own initial attempt?

 

[Leicester Fantasy]

Thank you for your attention sir. I'm first time PH, so I made a mistake sorry.
I can't understand how do I know the spins of the system if there is given a Hamiltoninan.
In (a), I think that there's no term for time, so the Hamiltoninan does not change along the time. But I don't know the how to solve this problem.
In (b), should I use some spinors? How do I express the eigenstates? matrix, vector, or ket notation? would you give me a example, please?

 

[blue_leaf77]

For (a), you are actually asked to calculate ##[H,S_x]##, ##[H,S_y]##, and ##[H,S_z]##. Then find which of them vanish, which ones do not. Here ##S_i## for ##i=x,y,z## are the component of the total spin.