Spin vector representation for 1/2 spin

Question

I know that, for 1/2 spin systems, the projection of the spin vector along one of the base's axis can be represented using Pauli's matrices as $\hat{S}_i = \frac{\hbar}{2}\sigma_i$.

While studying spin-orbit coupling I've seen the actual spin vector being written as $\vec{S}=\frac{\hbar}{2} \vec{\sigma}$. This expression confuses me since I don't understand what $\vec{\sigma}$ and $\vec{S}$ actually are: vectors with operators as coordinates?

Moreover, the Hamiltonian of the SO interaction is proportional to $\vec{B} \cdot \vec{S}$. Although I understand the physical meaning (potential energy of a magnetic dipole in a magnetic field), I don't know how to evaluate the dot product: if $\vec{S}$ is really a vector of operators, should I multiply each operator for the corresponding component of $\vec{B}$?

And if that is the case, what is the physical meaning of a vector with operators as coordinates?

Answer 1

The spin vector $\vec{S}$ is indeed a vector of operators acting on your spin system, made up of the two $|{\uparrow}\rangle$ and $|{\downarrow}\rangle$ spin states. In other words, each component of the spin vector is a pauli matrix, i.e., $$ S_x = \frac{\hbar}{2} \sigma_x $$ $$ S_y = \frac{\hbar}{2} \sigma_y $$ $$ S_z = \frac{\hbar}{2} \sigma_z $$

When evaluating the $\vec{B} \cdot \vec{S}$ product, you can then use the dot product in its regular form, such that $$ \begin{eqnarray} \vec{B} \cdot \vec{S} &=& B_x S_x + B_y S_y + B_z S_z \\ &=& \frac{\hbar}{2} \left( B_x \sigma_x + B_y \sigma_y + B_z \sigma_z\right) \end{eqnarray} $$ which is a 2 by 2 matrix with the coordinates of $\vec{B}$ inside, depending on the orientation of $\vec{B}$.