Clebsch–Gordan coefficients
E262449
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Clebsch–Gordan coefficients canonical | 5 |
| Clebsch–Gordan constraints | 1 |
| Clebsch–Gordan decomposition | 1 |
| Wigner 3-j symbols | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2408487 — 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: Clebsch–Gordan coefficients Context triple: [Alfred Clebsch, notableWork, Clebsch–Gordan coefficients]
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A.
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.
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B.
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.
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C.
Russell–Saunders coupling
Russell–Saunders coupling is an atomic physics scheme that describes how individual electron orbital and spin angular momenta combine to determine the total angular momentum of an atom, especially in light atoms.
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D.
Weyl character formula
The Weyl character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible finite-dimensional representations of semisimple Lie algebras and Lie groups.
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E.
Gell-Mann matrices
Gell-Mann matrices are a set of eight 3×3 traceless Hermitian matrices that serve as the generators of the SU(3) Lie algebra in quantum chromodynamics and other areas of particle physics.
- 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: Clebsch–Gordan coefficients Target entity description: 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.
-
A.
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.
-
B.
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.
-
C.
Russell–Saunders coupling
Russell–Saunders coupling is an atomic physics scheme that describes how individual electron orbital and spin angular momenta combine to determine the total angular momentum of an atom, especially in light atoms.
-
D.
Weyl character formula
The Weyl character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible finite-dimensional representations of semisimple Lie algebras and Lie groups.
-
E.
Gell-Mann matrices
Gell-Mann matrices are a set of eight 3×3 traceless Hermitian matrices that serve as the generators of the SU(3) Lie algebra in quantum chromodynamics and other areas of particle physics.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
quantum mechanics concept ⓘ representation theory concept ⓘ |
| appearsIn |
addition of spin-1/2 systems
ⓘ
coupling of orbital and spin angular momentum in atoms ⓘ quantum theory of angular momentum ⓘ |
| canBeExpressedAs | square root factors times factorials and sums ⓘ |
| canBeTabulated | finite tables for small j values ⓘ |
| computedBy |
computer algebra systems
ⓘ
explicit closed-form formulas ⓘ recursive relations ⓘ |
| definedFor |
magnetic quantum numbers m1 and m2
ⓘ
pairs of angular momentum quantum numbers j1 and j2 ⓘ resultant angular momentum quantum number J ⓘ resultant magnetic quantum number M ⓘ |
| field |
angular momentum theory
ⓘ
group theory ⓘ quantum mechanics ⓘ representation theory ⓘ |
| hasProperty |
depend on phase conventions
ⓘ
real in standard phase conventions ⓘ symmetric up to phase factors under interchange of j1 and j2 ⓘ vanish when selection rules are not satisfied ⓘ |
| namedAfter |
Alfred Clebsch
ⓘ
Paul Gordan ⓘ |
| relatedTo |
rotation group SO(3)
ⓘ
surface form:
SO(3) Lie group
rotation group SU(2) ⓘ
surface form:
SU(2) Lie group
Wigner 3j symbols ⓘ Wigner 3j symbols ⓘ
surface form:
Wigner 6j symbols
Wigner–Eckart theorem ⓘ spherical harmonics ⓘ |
| satisfies |
completeness relations
ⓘ
orthogonality relations ⓘ selection rule absolute value of j1 minus j2 lessOrEqual J lessOrEqual j1 plus j2 ⓘ selection rule m1 plus m2 equals M ⓘ triangle inequality for angular momenta ⓘ |
| usedFor |
adding quantum angular momenta
ⓘ
calculations in atomic physics ⓘ calculations in nuclear physics ⓘ calculations in particle physics ⓘ changing basis between coupled and uncoupled angular momentum states ⓘ combining two angular momenta ⓘ computing selection rules in spectroscopy ⓘ constructing total angular momentum eigenstates ⓘ coupling spin and orbital angular momentum ⓘ decomposing SO(3) representations ⓘ decomposing SU(2) representations ⓘ decomposing tensor products of representations ⓘ evaluating matrix elements of tensor operators ⓘ spectroscopic term coupling ⓘ |
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: Clebsch–Gordan coefficients Description of subject: 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.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Clebsch–Gordan decomposition
this entity surface form:
Wigner 3-j symbols
this entity surface form:
Clebsch–Gordan constraints