📖 Background & Definitions
When two angular momenta j₁ and j₂ are combined,
the coupled eigenstates |J,M⟩ of the total angular momentum J = j₁ + j₂
are linear combinations of the uncoupled product basis |j₁,m₁⟩⊗|j₂,m₂⟩.
The expansion coefficients are the Clebsch–Gordan (CG) coefficients:
|J, M⟩ = Σ_{m₁,m₂} ⟨j₁m₁; j₂m₂ | J M⟩ |j₁,m₁⟩|j₂,m₂⟩
Selection rules — coefficient is zero unless:
- M = m₁ + m₂ (z-component conservation)
- |j₁ − j₂| ≤ J ≤ j₁ + j₂ (triangle rule)
- |mᵢ| ≤ jᵢ for i = 1, 2
Allowed J values and dimension check
J ∈ { |j₁−j₂|, |j₁−j₂|+1, …, j₁+j₂ }
Σ_J (2J+1) = (2j₁+1)(2j₂+1) [dimension check]
Example: ½ ⊗ ½ → J=0 (dim 1) ⊕ J=1 (dim 3) → 1+3 = 4 = 2·2 ✓
Orthogonality and symmetry relations
Orthogonality:
Σ_{m₁m₂} ⟨j₁m₁;j₂m₂|JM⟩ ⟨j₁m₁;j₂m₂|J'M'⟩ = δ_{JJ'} δ_{MM'}
Σ_{JM} ⟨j₁m₁;j₂m₂|JM⟩ ⟨j₁m₁';j₂m₂'|JM⟩ = δ_{m₁m₁'} δ_{m₂m₂'}
Exchange symmetry:
⟨j₁m₁;j₂m₂|JM⟩ = (−1)^{j₁+j₂−J} ⟨j₂m₂;j₁m₁|JM⟩
Time-reversal:
⟨j₁m₁;j₂m₂|JM⟩ = (−1)^{j₁+j₂−J} ⟨j₁−m₁;j₂−m₂|J−M⟩
Compute ⟨j₁m₁; j₂m₂ | JM⟩
M is fixed automatically as m₁ + m₂ — the only non-zero case. Results expressed as exact ±√(p/q).
First angular momentum
Second angular momentum
Total angular momentum
M = —
Adjust inputs above to compute
CG Table for fixed j₁, j₂, J
Rows = m₁, columns = m₂. Cell value: ⟨j₁m₁; j₂m₂ | J, m₁+m₂⟩. Keep j ≤ 4 for fast computation.
Wigner–Eckart Theorem
Factorisation of tensor operator matrix elements into geometry (CG) and physics (reduced matrix element).
The Wigner–Eckart theorem factorises any matrix element of a rank-k spherical tensor operator
T(k)q into a geometric part (a CG coefficient, carrying all m-dependence)
and a reduced matrix element ⟨α′j′ ‖ T(k) ‖ αj⟩ that is independent of m, m′, q:
⟨α′, j′, m′ | T(k)_q | α, j, m⟩
= ⟨j, m; k, q | j′, m′⟩ × ⟨α′j′ ‖ T(k) ‖ αj⟩ / √(2j′+1)
Here α, α′ label all other (non-angular) quantum numbers.
The reduced matrix element encodes the physics; the CG coefficient is purely geometric.
Once the reduced matrix element ⟨α′j′ ‖ T(k) ‖ αj⟩ is known (from one measurement or
calculation), all (2j+1)(2j′+1) individual matrix elements
⟨j′m′|T(k)q|jm⟩ are determined. This is enormously powerful for selection rules,
hyperfine matrix elements, and optical transition strengths.
Selection rules (from the CG coefficient)
- m′ = m + q (z-projection conservation)
- |j − k| ≤ j′ ≤ j + k (triangle rule — now involving operator rank k)
- Matrix element is zero unless the above are satisfied, regardless of the physics
Common cases in AMO physics
Electric dipole (E1), k = 1
Δj = 0, ±1 (no 0→0)
Δm = 0 (π polarisation)
Δm = ±1 (σ± polarisation)
Magnetic dipole (M1), k = 1
Same Δj, Δm rules as E1
But Δl = 0 (no parity change)
Electric quadrupole (E2), k = 2
Δj = 0, ±1, ±2 (no 0→0, ½→½)
Δm = 0, ±1, ±2
Hyperfine coupling
All ⟨F,m_F | T | F′,m_F′⟩ matrix elements
reduce to a single reduced matrix element
⟨F ‖ T ‖ F′⟩ — making hyperfine calculations
tractable via the WE theorem.
Relation to Wigner 3j symbols
⟨j₁m₁; j₂m₂ | J M⟩ = (−1)^{j₁−j₂+M} √(2J+1) × ⎛ j₁ j₂ J ⎞
⎝ m₁ m₂ −M ⎠
3j symbols are more symmetric: invariant under even column permutations,
acquire phase (−1)^{j₁+j₂+J} under odd permutations or flipping all m-signs.
In Python/sympy: sympy.physics.wigner.wigner_3j(j1, j2, j3, m1, m2, m3)