Commutator trouble

[Ylle]

1. The problem statement, all variables and given/known data

Hi...
I'm having something about the Interaction/Dirac picture.
The equation of motion, for an observable A that doesn't depend on time in the Schrödinger picture, is given by:

\[i\hbar \frac{d{{A}_{I}}}{dt}=\left[ {{A}_{I}},{{H}_{0}} \right]\]
where:

\[{{\hat{H}}_{0}}=\hbar \omega {{\hat{a}}^{\dagger }}\hat{a}+\frac{\hbar {{\omega }_{0}}{{{\hat{\sigma }}}_{z}}}{2}\]

From this I have to commutate with \[{{\hat{\sigma }}_{+}}\], \[{{\hat{\sigma }}_{-}}\] and \[{{\hat{\sigma }}_{z}}\], where \[{{\hat{\sigma }}_{z}}\] is the last of the Pauli matrices, and \[{{\hat{\sigma }}_{\pm }}=\frac{\left( {{{\hat{\sigma }}}_{x}}\pm i{{{\hat{\sigma }}}_{y}} \right)}{2}\].


2. Relevant equations

?

3. The attempt at a solution
Is it just as always ? By inserting, and then just take the normal commutator, and get:


\begin{align}
& \left[ {{\sigma }_{z}},{{H}_{0}} \right]=...=0 \\
& \left[ {{\sigma }_{+}},{{H}_{0}} \right]=...=-\hbar {{\omega }_{0}}\left[ \begin{matrix}
0 & 1 \\
0 & 0 \\
\end{matrix} \right] \\
& \left[ {{\sigma }_{-}},{{H}_{0}} \right]=...=\hbar {{\omega }_{0}}\left[ \begin{matrix}
0 & 0 \\
1 & 0 \\
\end{matrix} \right] \\
\end{align}


Or am I way off ?
I'm kinda stuck, so a hint would be helpfull :)

Thanks in advance.


Regards

[nickjer]

Are you supposed to answer it in terms of matrices. I haven't checked your answer but it can be simply done just by knowing the commutation relations between the pauli matrices.

[Ylle]

I think so...
There is given a hint that I should look how to commutate the spin matrices, where fx. [Sx, Sy] = ihSz (i = complex number, h = h-bar) - if I remember correctly.

[nickjer]

You can use that commutation you just wrote out, along with the permutations of it to solve the problem in terms of spin matrices alone.