- #1
Juli
- 21
- 5
- 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?
##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?