Determine which operator has the following form when expressed in matrix spinor notation

Question

I am new to spinor notation and I came across the following matrix:

What operator has the following form when expressed in matrix spinor notation;

$[[W]]=\frac{\hbar \xi}{2} \begin{pmatrix}\hat L_z & \hat L_x-i\hat L_y\\ \hat L_x +i\hat L_y&-\hat L_z\end{pmatrix}$.

Any help with this is appreciated.

Answer 1

Your operator is $$ \frac{\hbar \xi }{2} \sum_{i=1}^3 \sigma_i \otimes \hat L_i , $$ which some write as $$ \xi~ {\mathbf S}\cdot {\mathbf L}~. $$ Is this what you have in mind?