Wigner 3j symbols
E674699
angular momentum coupling coefficient
mathematical object
special function
tensor coupling coefficient
Wigner 3j symbols are mathematical coefficients used in quantum mechanics and angular momentum theory to describe the coupling and recoupling of three angular momenta with well-defined symmetry properties.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Wigner 3j symbols canonical | 2 |
| Wigner 6j symbols | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7593279 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Wigner 3j symbols Context triple: [Condon–Shortley phase, relatedTo, Wigner 3j symbols]
-
A.
Clebsch–Gordan coefficients
Clebsch–Gordan coefficients are numerical factors in quantum mechanics and representation theory that describe how to combine two angular momenta (or group representations) into a single resultant one.
-
B.
Wigner–Eckart theorem
The Wigner–Eckart theorem is a fundamental result in quantum mechanics that factorizes matrix elements of tensor operators into a reduced matrix element and a purely geometric part given by Clebsch–Gordan coefficients, greatly simplifying angular momentum calculations.
-
C.
Racah algebra
Racah algebra is a mathematical structure in representation theory and quantum mechanics that encodes the symmetries and coupling properties of angular momenta, particularly through Racah coefficients and related special functions.
-
D.
Jacobi polynomials
Jacobi polynomials are a family of classical orthogonal polynomials depending on two parameters, widely used in approximation theory, numerical analysis, and solutions of differential equations.
-
E.
Brillouin–Wigner perturbation theory
Brillouin–Wigner perturbation theory is a formulation of quantum mechanical perturbation theory that uses an energy-dependent effective Hamiltonian to obtain improved approximations to eigenvalues and eigenstates.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Wigner 3j symbols Target entity description: Wigner 3j symbols are mathematical coefficients used in quantum mechanics and angular momentum theory to describe the coupling and recoupling of three angular momenta with well-defined symmetry properties.
-
A.
Clebsch–Gordan coefficients
Clebsch–Gordan coefficients are numerical factors in quantum mechanics and representation theory that describe how to combine two angular momenta (or group representations) into a single resultant one.
-
B.
Wigner–Eckart theorem
The Wigner–Eckart theorem is a fundamental result in quantum mechanics that factorizes matrix elements of tensor operators into a reduced matrix element and a purely geometric part given by Clebsch–Gordan coefficients, greatly simplifying angular momentum calculations.
-
C.
Racah algebra
Racah algebra is a mathematical structure in representation theory and quantum mechanics that encodes the symmetries and coupling properties of angular momenta, particularly through Racah coefficients and related special functions.
-
D.
Jacobi polynomials
Jacobi polynomials are a family of classical orthogonal polynomials depending on two parameters, widely used in approximation theory, numerical analysis, and solutions of differential equations.
-
E.
Brillouin–Wigner perturbation theory
Brillouin–Wigner perturbation theory is a formulation of quantum mechanical perturbation theory that uses an energy-dependent effective Hamiltonian to obtain improved approximations to eigenvalues and eigenstates.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
angular momentum coupling coefficient
ⓘ
mathematical object ⓘ special function ⓘ tensor coupling coefficient ⓘ |
| appearsIn |
Gaunt coefficients
NERFINISHED
ⓘ
addition of spherical harmonics ⓘ evaluation of matrix elements of tensor operators ⓘ multipole expansion of interactions ⓘ spherical tensor operator theory ⓘ |
| belongsTo |
representation theory of SU(2)
ⓘ
theory of angular momentum in quantum mechanics ⓘ |
| definedOver | half-integer and integer angular momentum quantum numbers ⓘ |
| expressibleInTermsOf | Clebsch–Gordan coefficients NERFINISHED ⓘ |
| hasAlternativeFormulation |
group-theoretical coupling coefficients of SU(2)
ⓘ
integrals of three spherical harmonics ⓘ |
| hasComputationalUse |
numerical evaluation of transition amplitudes
ⓘ
symbolic manipulation in computer algebra systems ⓘ tabulation of angular momentum coupling coefficients ⓘ |
| hasNotation | ( j1 j2 j3 ; m1 m2 m3 ) with 3x2 array brackets ⓘ |
| hasProperty |
completeness relations
ⓘ
orthogonality relations ⓘ real values in standard phase convention ⓘ vanish when m quantum numbers do not sum to zero ⓘ vanish when triangle inequalities are violated ⓘ well-defined phase convention ⓘ |
| hasSelectionRule |
j1 j2 j3 satisfy angular momentum addition rules
ⓘ
m1 plus m2 plus m3 equals zero ⓘ parity condition j1 plus j2 plus j3 integer ⓘ triangle condition for j1 j2 j3 ⓘ |
| hasSymmetryProperty |
invariance under even permutations of columns
ⓘ
phase factor under column permutations ⓘ sign change under odd permutations of columns ⓘ |
| namedAfter | Eugene Wigner NERFINISHED ⓘ |
| relatedTo |
Clebsch–Gordan coefficients
NERFINISHED
ⓘ
Racah coefficients NERFINISHED ⓘ Wigner 6j symbols ⓘ Wigner 9j symbols NERFINISHED ⓘ |
| standardReference |
Angular Momentum in Quantum Mechanics by A. R. Edmonds
NERFINISHED
ⓘ
Quantum Theory of Angular Momentum by D. A. Varshalovich et al. NERFINISHED ⓘ |
| usedFor |
coupling of three angular momenta
ⓘ
evaluation of Clebsch–Gordan coefficient symmetries ⓘ recoupling of angular momenta ⓘ |
| usedIn |
angular momentum theory
ⓘ
atomic physics ⓘ molecular spectroscopy ⓘ nuclear physics ⓘ quantum chemistry ⓘ quantum mechanics ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Wigner 3j symbols Description of subject: Wigner 3j symbols are mathematical coefficients used in quantum mechanics and angular momentum theory to describe the coupling and recoupling of three angular momenta with well-defined symmetry properties.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Wigner 6j symbols