# Homework Help: Eigenvalue for a Hamiltonian

1. Nov 23, 2011

### yuanyuan5220

1. The problem statement, all variables and given/known data
I am solving a Hamiltonian including a term $$(x\cdot S)^2$$

2. Relevant equations
The Hamiltonian is like this form:

H=L\cdot S+(x\cdot S)^2

where L is angular momentum operator and S is spin operator. The eigenvalue for $$L^2 , S^2$$ are $$l(l+1), s(s+1)$$

3. The attempt at a solution

If the Hamiltonian only has the first term, it is just spin orbital coupling and it is easy to solve. The total J=L+S, L2 and S2 are quantum number. However, when we consider the second term $$(x\cdot S)^2$$, it becomes much harder. The total J is still a quantum number. We have $$[(x\cdot S)^2, J]=0$$. However, $$[(x\cdot S)^2,L^2]≠0$$
The L is no long a quantum number anymore.

Anybody have ideas on how to solve this Hamiltonian?

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