Interaction of Two Spin 1/2 Particles

[-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}##.

I'll appreciate your help.

 

[blue_leaf77]

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.

That's the right direction, because if those observables are commute to each other, a measurement of one observable will not alter the measurement result from the previously measured observable. Therefore the three observables can be measured in a single series of measurement without the need to repeat the procedure for measurement of each obseravble.
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 into ##F##.