16.2. Spin

16.2.1. Properties

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

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

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

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

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, \(m_s \in \left\{ - \frac{1}{2}, \frac{1}{2} \right\}\).