We have a spin state described by a time-dependent density matrix(adsbygoogle = window.adsbygoogle || []).push({});

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

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

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

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

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

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

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

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

Furthermore, the Hamiltonian for the system is

[tex]H = \frac{1}{2}\hbar \omega_c \sigma_z[/tex]

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 [tex]\sigma_z[/tex] become unity. For the sine part, which contains only odd powers of the exponent, all powers of [tex]\sigma_z[/tex] 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 [tex]\sigma_z[/tex], [tex]\mathbf{\sigma}[/tex] and/or [tex]\mathbf{r}_0[/tex] 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.

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# QM Density Matrix Question

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