For given Hamiltonian, is spin conserved?

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Poster has been reminded to use the HH Template and show their work
A system consisting of two spins is described by the Hamiltonian (b>0)
H = aσ1 ⋅ σ2 + b(σ1z - σ2z)
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.
 
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Welcome in PF!
Next time, please follow the given template if you post under the homework section.

As for your current problem, can you please show us your own initial attempt?
 
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?
 
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.
 

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