Wigner–Eckart theorem
E98261
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Wigner–Eckart theorem canonical | 7 |
| Wigner–Eckart factorization | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T818200 — 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–Eckart theorem Context triple: [Eugene Wigner, knownFor, Wigner–Eckart theorem]
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A.
Wick’s theorem
Wick’s theorem is a fundamental result in quantum field theory that expresses time-ordered products of field operators as sums of normal-ordered products with all possible contractions, forming the basis for deriving Feynman rules and diagrammatic expansions.
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B.
Gell-Mann–Nishijima formula
The Gell-Mann–Nishijima formula is a key relation in particle physics that connects a particle’s electric charge to its isospin and hypercharge, helping classify hadrons within the quark model.
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C.
Feynman–Hellmann theorem
The Feynman–Hellmann theorem is a result in quantum mechanics that relates the derivative of an energy eigenvalue with respect to a parameter in the Hamiltonian to the expectation value of the corresponding derivative of the Hamiltonian.
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D.
Gell-Mann–Low theorem
The Gell-Mann–Low theorem is a fundamental result in quantum field theory that rigorously connects interacting quantum fields to free fields via the adiabatic switching-on of interactions, underpinning the use of perturbation theory and the Dyson series.
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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–Eckart theorem Target entity description: 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.
-
A.
Wick’s theorem
Wick’s theorem is a fundamental result in quantum field theory that expresses time-ordered products of field operators as sums of normal-ordered products with all possible contractions, forming the basis for deriving Feynman rules and diagrammatic expansions.
-
B.
Gell-Mann–Nishijima formula
The Gell-Mann–Nishijima formula is a key relation in particle physics that connects a particle’s electric charge to its isospin and hypercharge, helping classify hadrons within the quark model.
-
C.
Feynman–Hellmann theorem
The Feynman–Hellmann theorem is a result in quantum mechanics that relates the derivative of an energy eigenvalue with respect to a parameter in the Hamiltonian to the expectation value of the corresponding derivative of the Hamiltonian.
-
D.
Gell-Mann–Low theorem
The Gell-Mann–Low theorem is a fundamental result in quantum field theory that rigorously connects interacting quantum fields to free fields via the adiabatic switching-on of interactions, underpinning the use of perturbation theory and the Dyson series.
-
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 |
result in angular momentum theory
ⓘ
theorem in quantum mechanics ⓘ |
| appliesTo |
angular momentum eigenstates
ⓘ
irreducible tensor operators ⓘ tensor operators ⓘ |
| assumes |
conservation of total angular momentum
ⓘ
tensor operator transforms irreducibly under rotations ⓘ |
| basedOn |
representation theory of SU(2)
ⓘ
rotational symmetry ⓘ |
| coreConcept | factorization of matrix elements ⓘ |
| dynamicPartGivenBy | reduced matrix element ⓘ |
| dynamicPartIndependentOf | magnetic quantum numbers ⓘ |
| field |
mathematical physics
ⓘ
quantum mechanics ⓘ theoretical physics ⓘ |
| generalizationOf | selection rules from rotational invariance ⓘ |
| geometricPartDependsOn | angular momentum quantum numbers ⓘ |
| geometricPartGivenBy | Clebsch–Gordan coefficients ⓘ |
| hasProperty |
independent of magnetic quantum numbers in reduced matrix element
ⓘ
separates dynamics from geometry ⓘ simplifies angular momentum calculations ⓘ uses selection rules from angular momentum algebra ⓘ |
| holdsFor | discrete angular momentum spectra ⓘ |
| holdsIn | Hilbert space of angular momentum eigenstates ⓘ |
| involves |
Clebsch–Gordan coefficients
ⓘ
Wigner–Eckart theorem self-linksurface differs ⓘ
surface form:
Wigner–Eckart factorization
angular momentum coupling ⓘ reduced matrix element ⓘ rotation group SU(2) ⓘ spherical tensor operators ⓘ |
| mathematicalFormulationUses |
group representation theory
ⓘ
spherical harmonics ⓘ |
| namedAfter |
Carl Eckart
ⓘ
Eugene Wigner ⓘ |
| relatedTo |
Clebsch–Gordan coefficients
ⓘ
surface form:
Clebsch–Gordan decomposition
Racah algebra ⓘ Clebsch–Gordan coefficients ⓘ
surface form:
Wigner 3-j symbols
Racah algebra ⓘ
surface form:
Wigner 6-j symbols
Wigner 9-j symbols ⓘ |
| statesThat | matrix elements of irreducible tensor operators factorize into a reduced matrix element and a purely geometric factor ⓘ |
| usedFor |
atomic spectroscopy calculations
ⓘ
deriving selection rules for transitions ⓘ evaluation of electromagnetic transition matrix elements ⓘ molecular spectroscopy ⓘ nuclear structure calculations ⓘ |
| usedIn |
many-body quantum systems with angular momentum coupling
ⓘ
quantum scattering theory ⓘ quantum theory of radiation ⓘ |
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–Eckart theorem Description of subject: 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.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Wigner–Eckart factorization