5. Clebsch-Gordan Coefficients
Clebsch-Gordan (CG) coefficients are the overlap coefficients between uncoupled and coupled angular momentum bases. They are fundamental to calculating matrix elements in quantum systems with multiple angular momenta.
Definition
The Clebsch-Gordan coefficient is defined as:
Alternative notation (Condon-Shortley):
The coupled state is then:
Inverse relation (due to orthonormality):
Selection Rules and Properties
CG coefficients are nonzero only when:
1. Triangle inequality:
2. Projection condition:
3. Parity under permutation:
4. Orthonormality:
5. Completeness:
6. Reality (with phase convention):
Special Cases and Formulas
Maximum $m$ state ($j = j_1 + j_2$, $m = j_1 + j_2$):
Minimum $j$ with maximum $m$ ($j = |j_1 - j_2|$, $m = |j_1 - j_2|$):
Coupling with $j_2 = 1/2$:
Coupling with $j_2 = 1$ (vector coupling):
Example: Two Spin-1/2 ($j_1 = j_2 = 1/2$)
Triplet states ($j = 1$):
Singlet state ($j = 0$):
Note the minus sign in singlet (antisymmetric combination)
Tabulated Values
Example table for $j_1 = 1, j_2 = 1/2$:
| $m_1$ | $m_2$ | $j = 3/2$ | $j = 1/2$ |
|---|---|---|---|
| 1 | 1/2 | 1 ($m=3/2$) | — |
| 1 | -1/2 | $\sqrt{2/3}$ ($m=1/2$) | $\sqrt{1/3}$ ($m=1/2$) |
| 0 | 1/2 | $\sqrt{1/3}$ ($m=1/2$) | $-\sqrt{2/3}$ ($m=1/2$) |
| 0 | -1/2 | $\sqrt{1/3}$ ($m=-1/2$) | $\sqrt{2/3}$ ($m=-1/2$) |
| -1 | 1/2 | $\sqrt{2/3}$ ($m=-1/2$) | $-\sqrt{1/3}$ ($m=-1/2$) |
| -1 | -1/2 | 1 ($m=-3/2$) | — |
Full tables available in standard references (e.g., Edmonds, Varshalovich)
3j Symbols (Wigner 3j)
Alternative notation with better symmetry properties:
Properties:
- Invariant under even permutations of columns
- Pick up sign $(-1)^{j_1+j_2+j_3}$ under odd permutations
- Vanishes unless $m_1 + m_2 + m_3 = 0$ and triangle inequality satisfied
- Symmetric role for all three angular momenta
Useful for products of three angular momentum states
Applications in Matrix Elements
Wigner-Eckart theorem: Matrix element of tensor operator factorizes:
Separates geometric (CG coefficient) from dynamical (reduced matrix element) factors
Example: Electric dipole transition
Dipole operator $\vec{d}$ is vector ($k=1$), so:
CG coefficient enforces selection rules $\Delta l = \pm 1$, $\Delta m = 0, \pm 1$
Intensity ratios: Ratios of transition probabilities determined purely by CG coefficients (geometry), independent of atomic details!
Computational Methods
1. Recursion relations:
Use ladder operators $\hat{J}_\pm$ to relate different $m$ values
2. Racah formula:
Explicit closed-form expression (complicated sum over factorials)
Useful for symbolic computation but unwieldy for hand calculation
3. Software implementations:
- Python:
scipy.special.clebsch_gordan - Mathematica:
ClebschGordan[...] - Symbolic: WignerSymbols.jl (Julia), sympy (Python)
Physical Examples
1. Hydrogen fine structure:
Coupling $\vec{L}$ and $\vec{S}$ to get $\vec{J}$ splits energy levels
Example: 2p splits into $2p_{1/2}$ and $2p_{3/2}$
2. Nuclear magnetic resonance (NMR):
Coupling nuclear spin $\vec{I}$ with electron spin determines hyperfine structure
CG coefficients determine intensity of spectral lines
3. Molecular spectroscopy:
Electronic, vibrational, and rotational angular momenta couple
Selection rules from CG coefficients predict allowed transitions
4. Particle physics:
Combining quark spins and orbital angular momenta in hadrons
Decay amplitudes involve CG coefficients (e.g., $\Delta \to N + \pi$)
Historical Note: Alfred Clebsch and Paul Gordan developed these coefficients in the 1860s in the context of algebraic invariant theory, long before quantum mechanics! Their work on binary forms was reinterpreted by Wigner and others in the 1930s as the mathematics of angular momentum coupling. This is a beautiful example of pure mathematics finding unexpected physical applications decades later.