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

?

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. [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.

 

[paranormal]

,

As a scientist, it is important to approach problems with a logical and analytical mindset. In this case, it seems like you are on the right track by using the commutator to solve for the equations of motion. However, it would be helpful if you provided more context or background information about the problem you are trying to solve. This will allow for a more thorough and accurate response. Additionally, it may be useful to provide any relevant equations or attempts at solving the problem so far. Overall, it is important to provide as much information as possible in order to receive the most helpful and accurate response.