Solving Commutator Trouble with Interaction/Dirac Picture

[Ylle]

Homework Statement

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 $\[{{\hat{\sigma }}_{\pm }}=\frac{\left( {{{\hat{\sigma }}}_{x}}\pm i{{{\hat{\sigma }}}_{y}} \right)}{2}\]$.

Homework Equations

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The Attempt at a Solution

Is it just as always ? By inserting, and then just take the normal commutator, and get:

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

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. [S_x, S_y] = ihS_z (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.