Spin Commutation Relations

[Diracobama2181]

Homework Statement

Consider a tetrahedron with four spin (1/2) particles, one at each of the vertices. Suppose the Hamiltonian is given by $$H=\sum_{i\neq q}S_iS_j$$. Show that all three components of the total spin $$J =\sum_{i}S_i$$ of the system commutes with $$H$$.

Relevant Equations

$$S^2=S_{1}^2+S_{2}^2+S_{3}^2+S_{4}^2+2\sum_{i\neq q}S_iS_j$$

$$H$$ can be rewritten as $$H=\frac{1}{2}(S^2-S_{1}^2-S_{2}^2-S_{3}^2-S_{4}^2)$$. Let's focus on the x component, $$J^x=\sum_{i}S_i^x$$. Now $$S_1^x$$ commutes with $$S^2_1, S^2_2, S^2_3, S^2_4$$, but does it commute with $$S^2$$? If not, what is the exact relation between $$S^2$$ and $$S_1^x$$?

 

[Diracobama2181]

Nevermind, figured it out. $$S_1^x$$ does not commute with $$S^2$$. However, $$S^2=J^2$$, so $$J^2$$ commutes with $$H$$,which I believe implies each component of $$J$$ also commutes.