What are the eigenstates of the anti-ferromagnetic dimer Hamiltonian?

[opaka]

Homework Statement

The hamiltonian of a simple anti-ferromagnetic dimer is given by

H=JS(1)\bulletS(2)-μB(S_z(1)+S_z(2))

find the eigenvalues and eigenvectors of H.

Homework Equations

The Attempt at a Solution

The professor gave the hint that the eigenstates are of S²=(S(1)+S(2))², S(1)², S(2)², and S_z. So I know I should have four eigenvalues. but I still have no Idea how to get this into a form that I recognize as being able to get eigenvalues from. (a matrix, a DiffEQ, etc.)

Please help

 

[vela]

The hint is to help you to deal with the $\vec{S}_1\cdot\vec{S}_2$ term. Rewrite that term in terms of $\vec{S}^2$, $\vec{S}_1^2$, and $\vec{S}_2^2$.

 

[opaka]

When I do that, and apply the spin operators, S² ket (S,S_z)=s(s+1) ket (s,s_z) and S_z ket (S,S_z) = s_zket (s,s_z)(sorry, couldn't find the ket symbol in latex)
I get
H = J/2 (s(s+1) - s₁(s₁+1)-s₂(s₂+1))-μB(s_1z+s_2z)

Is this correct?

 

[vela]

Yes. Now you can calculate what H does to simultaneous eigenstates of $\vec{S}^2$, $S_z$, $\vec{S}_1^2$, and $\vec{S}_2^2$. Recall that these are exactly the states that you got from adding angular momenta.

 

[opaka]

I get four answers : J/4 + μB, J/4 -μB, J/4 and - 3J/4. Is this right? These look like the singlet and triplet state energies, but with an added B term.

 

[vela]

Yeah, that looks right.

 

[opaka]

Thanks so much Vela! you've been a wonderful help.