Wigner–Eckart theorem

E98261

result in angular momentum theory theorem in quantum mechanics

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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All labels observed (2)

Label	Occurrences
Wigner–Eckart theorem canonical	7
Wigner–Eckart factorization	1

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 ⓘ

Referenced by (8)

Full triples — surface form annotated when it differs from this entity's canonical label.

Eugene Wigner → knownFor → Wigner–Eckart theorem ⓘ

Wigner–Eckart theorem → involves → Wigner–Eckart theorem self-linksurface differs ⓘ

this entity surface form: Wigner–Eckart factorization

Wigner Jenő Pál → knownFor → Wigner–Eckart theorem ⓘ

Condon–Shortley phase → relatedTo → Wigner–Eckart theorem ⓘ

Clebsch–Gordan coefficients → relatedTo → Wigner–Eckart theorem ⓘ

Carl Eckart → knownFor → Wigner–Eckart theorem ⓘ

Carl Eckart → coDeveloperOf → Wigner–Eckart theorem ⓘ

Carl Eckart → hasNotableTheoremNamedAfter → Wigner–Eckart theorem ⓘ