Solving for the Time-Dependent Vector in QM Density Matrix

[Spinny]

We have a spin state described by a time-dependent density matrix

\rho(t) = \frac{1}{2}\left(\mathbf{1}+\mathbf{r}(t)\cdot \mathbf{\sigma} \right)

Initial condition for the motion is \mathbf{r} = \mathbf{r}_0 at t = 0. We are then asked to give a general expression for \rho(t) in terms of the time evolution (TE) operator, and use that to find the time-dependent vector \mathbf{r}(t).

The density matrix expression in terms of the TE operator i got to be

\rho(t) = \mathcal{U}(t)\rho_0 [\mathcal{U}(t)]^{\dagger}

where \rho_0 = \rho(t = 0). Now that I'm going to find the time-dependent vector \mathbf{r}(t) I'm having a bit more trouble. I've started with the equation

\rho_0 = \frac{1}{2}\left(\mathbf{1}+\mathbf{r}_0\cdot \mathbf{\sigma} \right)

let the TE operator and it's adjoint operate on it from the left and right respectively. That has left me with the relation

\mathcal{U}(t) \mathbf{r}_0 \cdot \mathbf{\sigma}[\mathcal{U}(t)]^{\dagger} = \mathbf{r}(t)\cdot \mathbf{\sigma}

Furthermore, the Hamiltonian for the system is

H = \frac{1}{2}\hbar \omega_c \sigma_z

So, assuming that I'm on track so far, does anyone have any suggestions as to how I may proceed next?

What I tried was to first use Euler's formula to write out the TE operator and its adjoint. Then I expanded the cosine and the sine part separately and found that the cosine part only contains even powers of the exponent, thus making all \sigma_z become unity. For the sine part, which contains only odd powers of the exponent, all powers of \sigma_z are equal to the matrix itself.

The problem became when I put all this into the relation I need to solve, as it gave many parts containing \sigma_z, \mathbf{\sigma} and/or \mathbf{r}_0 multiplied in different orders, and I'm not really sure how to handle that.

So, what I need to know is if I'm on the right track, or maybe I'm ignoring something or perhaps there's an easier way to do this that I should look into. Suggestions are appreciated.

 

[vanesch]

What I tried was to first use Euler's formula to write out the TE operator and its adjoint.

In the basis where \sigma_z is diagonal, I would think that the time evolution operator is easily found, because the exponentiation applies directly to the eigenvalues. The TE operator would be diagonal in that basis too.
Just a suggestion, I didn't work it out myself.

 

[Spinny]

Of course! Since \sigma_z is diagonal, when it's the exponential, you just get a the same matrix with each diagonal element exponentiated!

Now I finally got an expression for the time dependent vector! It remains, however, to be seen if it's the correct expression.

Anyway, thanks for helping me on my way, vanesch. I'll probably be back with more questions though.