Spinny
Oct14-05, 06:31 PM
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.
\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.