Showing that S^2 and Sz Commute with the System Hamiltonian

[bon]

Homework Statement

A system with two spins of magnitude 1/2 have spin operators S1 and S2 and total spin S = S1 + S2

B is a B-field in the z direction (0,0,B)

The Hamiltonian for the system is given by H = m S1 . S2 + c B.S where m,c are constants.


By writing the Hamiltonian in terms of S, show that S^2 and Sz commute with the hamiltonian.

Homework Equations

The Attempt at a Solution

So i know that S^2 = S1 ^2 + S2 ^2 + 2S1.S2

so S1.S2 can be written as 1/2 (S^2 - S1 ^2 - S2 ^2)

But how do i simplify this?

My guess is that I can replace each of S1^2 and S2 ^2 with 3/4..but is this right? why is it justified? if not, what do i do?

Thanks

 

[Mindscrape]

You don't need to simplify. You just have to show that [H,S^2]=[S^2,H] and same for Sz. You should really post this in advanced physics.