# Homework Help: Commutator trouble

1. May 25, 2010

### 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 SchrÃ¶dinger 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

2. May 25, 2010

### 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.

3. May 26, 2010

### 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.

4. May 26, 2010

### 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.

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