# Eigenstates of Hamiltonian

1. Feb 12, 2013

### opaka

1. The problem statement, all variables and given/known data
The hamiltonian of a simple anti-ferromagnetic dimer is given by

H=JS(1)$\bullet$S(2)-μB(Sz(1)+Sz(2))

find the eigenvalues and eigenvectors of H.
2. Relevant equations

3. 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 Sz. 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.)

Last edited: Feb 12, 2013
2. Feb 12, 2013

### vela

Staff Emeritus
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 $\vec{S}_2^2$.

3. Feb 12, 2013

### opaka

When I do that, and apply the spin operators, S2 ket (S,Sz)=s(s+1) ket (s,sz) and Sz ket (S,Sz) = szket (s,sz)(sorry, couldn't find the ket symbol in latex)
I get
H = J/2 (s(s+1) - s1(s1+1)-s2(s2+1))-μB(s1z+s2z)

Is this correct?

4. Feb 12, 2013

### vela

Staff Emeritus
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.

5. Feb 12, 2013

### 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.

6. Feb 12, 2013

### vela

Staff Emeritus
Yeah, that looks right.

7. Feb 12, 2013

### opaka

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