Math 2.0

Quantum theory Symmetries, angular momentum and identical particles Theorem

Wigner–Eckart theorem

Builds on Angular momentum operatorClebsch–Gordan coefficientsIrreducible representationSU(2)3-j symbol

Statement

Let ℋ\mathcal{H} be a Hilbert space with an angular momentum operator J=(J1,J2,J3)J = (J_1, J_2, J_3), and put J±=J1±iJ2J_\pm = J_1 \pm iJ_2. Let ∣αjm⟩\lvert \alpha j m \rangle be an orthonormal family of standard angular momentum states, where α\alpha is an extra label. This means J2∣αjm⟩=ℏ2j(j+1)∣αjm⟩J^2 \lvert \alpha j m \rangle = \hbar^2 j(j+1) \lvert \alpha j m \rangle and J3∣αjm⟩=ℏm∣αjm⟩J_3 \lvert \alpha j m \rangle = \hbar m \lvert \alpha j m \rangle. It also means that J±J_\pm acts with the Condon–Shortley convention. Let k∈{0,12,1,…}k \in \set{0, \frac12, 1, \ldots}, and let TqT_q, q=−k,…,kq = -k, \ldots, k, be operators forming an irreducible tensor operator of rank kk: [J3,Tq]=ℏq Tq,[J±,Tq]=ℏk(k+1)−q(q±1)  Tq±1.[J_3, T_q] = \hbar q\, T_q, \qquad [J_\pm, T_q] = \hbar \sqrt{k(k+1) - q(q \pm 1)}\; T_{q \pm 1}. Then for all α,j,α′,j′\alpha, j, \alpha', j' there is a number ⟨α′j′∥T∥αj⟩\langle \alpha' j' \Vert T \Vert \alpha j \rangle with the following property. It does not depend on mm, m′m' or qq, and ⟨α′j′m′∣Tq∣αjm⟩=⟨jm,kq∣j′m′⟩ ⟨α′j′∥T∥αj⟩\langle \alpha' j' m' \rvert T_q \lvert \alpha j m \rangle = \langle j m, k q \mid j' m' \rangle \, \langle \alpha' j' \Vert T \Vert \alpha j \rangle for all m,m′,qm, m', q, where ⟨jm,kq∣j′m′⟩\langle j m, k q \mid j' m' \rangle is a Clebsch–Gordan coefficient.

Remarks

The relations say that Tq∣αjm⟩T_q \lvert \alpha j m \rangle transforms under rotations like the vector ∣kq⟩⊗∣jm⟩\lvert k q \rangle \otimes \lvert j m \rangle of Vk⊗VjV_k \otimes V_j, where VjV_j is the irreducible representation of SU(2) of dimension 2j+12j + 1. So the matrix elements define an intertwining map from Vk⊗VjV_k \otimes V_j to Vj′V_{j'}. By Schur's lemma such maps form a space of dimension at most one, spanned by the Clebsch–Gordan projection. The reduced matrix element ⟨α′j′∥T∥αj⟩\langle \alpha' j' \Vert T \Vert \alpha j \rangle is the scalar factor; conventions differ by a factor such as 2j′+1\sqrt{2j' + 1}.

Example: a vector operator, such as the position xx or the electric dipole operator, gives a tensor of rank k=1k = 1 with T0=x3T_0 = x_3 and T±1=∓12(x1±ix2)T_{\pm 1} = \mp \frac{1}{\sqrt2}(x_1 \pm i x_2). The theorem gives the dipole selection rules: the matrix element vanishes unless m′=m+qm' = m + q and ∣j−1∣≤j′≤j+1\abs{j - 1} \le j' \le j + 1, with j=j′=0j = j' = 0 excluded.

Going deeper

For a student

The theorem is Schur's lemma applied to the rotation group. It yields selection rules, since a Clebsch–Gordan coefficient vanishes unless m′=m+qm' = m + q and j′j' lies between ∣j−k∣\abs{j - k} and j+kj + k. It also gives the ratios of intensities of spectral lines within a multiplet and the projection theorem for vector operators. It is often written with the 3-j symbol in place of the Clebsch–Gordan coefficient.

Reading path

To understand this page from the ground up, read these 32 pages in order. The foundations come first.

Show the path as a graph
3-j symbol, concept, no evidence level3-j symbolconceptAngular momentum operator, concept, no evidence levelAngular momentum oper…conceptClebsch–Gordan coefficients, concept, no evidence levelClebsch–Gordan coeffi…conceptCross product, concept, no evidence levelCross productconceptDeterminant, concept, no evidence levelDeterminantconceptGroup, concept, no evidence levelGroupconceptIdentity matrix, object, no evidence levelIdentity matrixobjectIrreducible representation, concept, no evidence levelIrreducible represent…conceptLie group, concept, no evidence levelLie groupconceptMatrix, definition, no evidence levelMatrixdefinitionReal projective space, object, no evidence levelReal projective spaceobjectVector space, definition, no evidence levelVector spacedefinitionRing, definition, no evidence levelRingdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionDivision ring, definition, no evidence levelDivision ringdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionOrthogonal group, object, no evidence levelOrthogonal groupobjectUnitary matrix, definition, no evidence levelUnitary matrixdefinitionThe quaternions, example, no evidence levelThe quaternionsexampleDeterminant, definition, no evidence levelDeterminantdefinitionUnitary group, object, no evidence levelUnitary groupobjectSpecial orthogonal group, object, no evidence levelSpecial orthogonal gr…objectSpecial unitary group SU(n), example, no evidence levelSpecial unitary group…exampleRotation group SO(3), example, no evidence levelRotation group SO(3)exampleSU(2), example, no evidence levelSU(2)exampleWigner–Eckart theorem, statement, no evidence levelWigner–Eckart theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. 3-j symbol concept
  2. Angular momentum operator concept
  3. Clebsch–Gordan coefficients concept
  4. Cross product concept
  5. Determinant concept
  6. Group concept
  7. Identity matrix object
  8. Irreducible representation concept
  9. Lie group concept
  10. Matrix definition
  11. Real projective space object
  12. Vector space definition
  13. Ring definition
  14. Square matrix definition
  15. Transpose definition
  16. Linear map definition
  17. Unit of a ring definition
  18. Conjugate transpose definition
  19. Multilinear map definition
  20. Orthogonal matrix definition
  21. Division ring definition
  22. Alternating multilinear map definition
  23. Orthogonal group object
  24. Unitary matrix definition
  25. The quaternions example
  26. Determinant definition
  27. Unitary group object
  28. Special orthogonal group object
  29. Special unitary group SU(n) example
  30. Rotation group SO(3) example
  31. SU(2) example
  32. Wigner–Eckart theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Wigner%E2%80%93Eckart_theorem
About this page
TagRZV2Z7XCJ5
URIm2://RZV2Z7XCJ5
Kindstatement, theorem
WritingW0 stub
MSC81R05, 22E70, 20C35
StatusThis copy of the site has no build data. Run m2 status RZV2Z7XCJ5.
AboutAngular momentum operator, Clebsch–Gordan coefficients
UsesAngular momentum operator, Clebsch–Gordan coefficients, Irreducible representation, SU(2), 3-j symbol
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Changed3 November 2026

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