Part V: Spin & Angular Momentum
Intrinsic angular momentum (spin), Pauli matrices, general angular momentum theory, addition of angular momenta, and the rotation group—essential for understanding atomic structure and particle physics.
50+ pages | 6 chapters
1. Spin-1/2 Systems (9 pages)
Stern-Gerlach experiment, spin states, spinors
Key concepts: $|\uparrow\rangle, |\downarrow\rangle$, spin operator $\hat{S} = \frac{\hbar}{2}\vec{\sigma}$
2. Pauli Matrices & Spinors (7 pages)
Pauli matrices, eigenstates, spin precession
$\sigma_x, \sigma_y, \sigma_z$, anticommutation: $\{\sigma_i, \sigma_j\} = 2\delta_{ij}$
3. General Angular Momentum (10 pages)
Commutation relations, ladder operators, Casimir operators
$[\hat{J}_i, \hat{J}_j] = i\hbar\epsilon_{ijk}\hat{J}_k$, eigenvalues $j(j+1)\hbar^2$, $m\hbar$
4. Addition of Angular Momenta (10 pages)
Composite systems, total angular momentum, triplet/singlet states
$\hat{\vec{J}} = \hat{\vec{J}}_1 \otimes \mathbb{I} + \mathbb{I} \otimes \hat{\vec{J}}_2$, $|j_1-j_2| \leq j \leq j_1+j_2$
5. Clebsch-Gordan Coefficients (8 pages)
CG coefficients, tables, selection rules
$|j_1,m_1\rangle \otimes |j_2,m_2\rangle = \sum_{j,m} \langle j_1,m_1;j_2,m_2|j,m\rangle |j,m\rangle$
6. Rotation Group SO(3) & SU(2) (6 pages)
Lie groups, representations, spinor representations
Rotation operator: $\hat{R}(\theta,\hat{n}) = e^{-i\theta\hat{n}\cdot\hat{\vec{J}}/\hbar}$, SU(2) double cover