# Spin vector representation for 1/2 spin

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?

• $\vec{S}$ is a vector where each component is a matrix. Commented Jan 15, 2021 at 13:56
• It is called the "Pauli vector", checkable in Wikipedia. Commented Jan 15, 2021 at 14:50
• Commented Jan 15, 2021 at 15:02

## 1 Answer

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 $$\vec{B}$$.