(quantum-mechanics:angular-momentum:spin)=
# Spin

## Properties

In analogy with the spatial angular momentum, and using Cartesian coordinates

$$\begin{aligned}
  \left[ S_a, S_b \right] & = i \hbar \, \varepsilon_{abc} S_c \\
  \left[ S_a, S^2 \right] & = 0 \ . \\
\end{aligned}$$

As the operators $S_z$ and $S^2$ commute, they share the same eigenvectors $| s, m_s \rangle$ (all the eigvecs?) and their eigenproblems read

$$\begin{aligned}
  S_z | s, m_s \rangle & = \hbar m_s | s, m_s \rangle \\
  S^2 | s, m_s \rangle & = \hbar^2 s ( s + 1 ) | s, m_s \rangle \ , 
\end{aligned}$$

The value of $s$ depends on the system of interest: for an electron $s = \frac{1}{2}$. The value of the spin projection quantum number $m_s$ belongs to $m_s \in \{ -s, -s+1, \dots, s-1, s \}$. For an electron, $m_s \in \left\{ - \frac{1}{2}, \frac{1}{2} \right\}$.