# Interaction of Two Spin 1/2 Particles

1. Dec 13, 2015

### -Newton-

Hi buddies.

I recently finished my quantum mechanics course, however, I would like to know the solution of this exercise because i couldn´t solve it on my last exam, and i would like to take this doubt off.

An operator $F$ describing the interaction of two spin $\frac{1}{2}$ particles has the form:
$F=c+d {\sigma}_{1}\cdot{\sigma}_{2}$
where $c$ and $d$ are constants, ${\sigma}_{1}$ and ${\sigma}_{2}$are Pauli matrices of the spin.
Prove that $F$ , $j^2$ and ${j}_{z}$ can be meassure simultaneusly.
Where $j$ is the total angular momentum; also you must consider that
${\sigma}_{1}\neq{\sigma}_{2}$.

I had the idea to check that operators $F$ with $j^2$ and $F$ with ${j}_{z}$ Commute to conclude that the observable can be measured simultaneously. But I'm not sure if that's okay, and i don't know how to do it because ${\sigma}_{1}\neq{\sigma}_{2}$.

The individual spin operator is proportional to the corresponding Pauli matrix, therefore you can write for $F$, $F = c +d'\mathbf{j}_1 \cdot \mathbf{j}_2$ where $d'$ is another constant. Then consider $j^2 = (\mathbf{j}_1 + \mathbf{j}_2)^2$, from this pull the resulting $\mathbf{j}_1 \cdot \mathbf{j}_2$ to one side alone and plug in to $F$.