# For given Hamiltonian, is spin conserved?

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A system consisting of two spins is described by the Hamiltonian (b>0)
H = aσ1 ⋅ σ2 + b1z - 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.

blue_leaf77
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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