Introduction: The Transformation Coefficients
Clebsch-Gordan coefficients (CG coefficients) relate the uncoupled and coupled bases for angular momentum. They're essential for calculating transition probabilities and selection rules.
Definition
\[|j, m\rangle = \sum_{m_1, m_2} C^{j m}_{j_1 m_1; j_2 m_2} |j_1, m_1; j_2, m_2\rangle\]Notation: \(C^{j m}_{j_1 m_1; j_2 m_2} = \langle j_1, m_1; j_2, m_2 | j, m\rangle\)
Selection Rule
CG coefficients are zero unless \(m = m_1 + m_2\). This is conservation of the z-component.
Example: Two Spin-1/2 Particles
Triplet (j = 1):
- \(|1, 1\rangle = |\uparrow\uparrow\rangle\)
- \(|1, 0\rangle = \frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle + |\downarrow\uparrow\rangle)\)
- \(|1, -1\rangle = |\downarrow\downarrow\rangle\)
Singlet (j = 0):
- \(|0, 0\rangle = \frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle - |\downarrow\uparrow\rangle)\)
The Quantum Connection
CG coefficients appear in atomic physics (fine structure), nuclear physics (nuclear shell model), and particle physics (quark model). Tables of CG coefficients are standard references, and they embody the symmetry properties of rotations.