Spin

Phase 3: Advanced Quantum Systems

Syllabus Goal

Spin concepts and Addition of Angular Momentum.

Intrinsic Angular Momentum

Spin is an intrinsic property, not spatial rotation. Electron spin $s = 1/2$. $$ S_z = \pm \frac{\hbar}{2} $$

Pauli Matrices

$$ \sigma_x = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}, \quad \sigma_y = \begin{pmatrix} 0 & -i \ i & 0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix} $$

Addition of Angular Momentum

Total $J = L + S$. This coupling leads to spectral fine structure. Clebsch-Gordan coefficients relate the coupled basis $|J, M\rangle$ to the uncoupled basis $|m_\ell, m_s\rangle$.