Eigenvalues of Hamiltonian operator

[Juli]

Homework Statement

Consider a system with two spin-1 particles, which is described by the Hamiltonian operator

$H = \lambda \vec{S}_1 \cdot \vec{S}_2$

with $\lambda \in \mathbb{R}$.

1. Express H in terms of the total spin $\vec{S} = \vec{S}_1 + \vec{S}_2$.

2. What eigenvalues does H have and how are these degenerate?

Relevant Equations

$\vec{S}^2 = S(S+1)\hbar^2$
$\vec{S_1}^2 = S_1(S_1+1)\hbar^2$
$\vec{S_2}^2 = S_2(S_2+1)\hbar^2$

Hello, I try to solve this problem, and I think a) wasn't too hard, I have the following solution:

$H = \lambda (\frac{\vec{S^2-(\vec{S_1}^2+\vec{S_2}^2)}{2})$.
I struggle with 2. I find it very abstract. When I have H as a matrix I know how to calculate eigenvalues, but I don't know how to proceed with this general approach.

I tried to go somewhere with the eigenvalues of S, but I didn't get far...

Can someone help me solve this?

 

[kuruman]

Expand the right-hand side of the operator $~S^2=(\vec S_1+\vec S_2)\cdot(\vec S_1+\vec S_2)~$ and solve for the operator $\vec S_1\cdot \vec S_2$. Its eigenvalues are the eigenvalues of the operator on the right-hand side.

 

[Juli]

I did the first task and got this $H = \lambda (\frac{\vec{S^2-(\vec{S_1}^2+\vec{S_2}^2)}{2})$
But I don't know how to get the eigenvalues of the operator on the right-hand side. Are they the eigenvalues of the individual spins?