Clebsch–Gordan coefficients

E262449

mathematical concept quantum mechanics concept representation theory concept

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.

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All labels observed (4)

Label	Occurrences
Clebsch–Gordan coefficients canonical	5
Clebsch–Gordan constraints	1
Clebsch–Gordan decomposition	1
Wigner 3-j symbols	1

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 ⓘ

Referenced by (8)

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

Alfred Clebsch → notableWork → Clebsch–Gordan coefficients ⓘ

Alfred Clebsch → notableConcept → Clebsch–Gordan coefficients ⓘ

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

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

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

this entity surface form: Clebsch–Gordan decomposition

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

this entity surface form: Wigner 3-j symbols

Condon–Shortley phase → relatedTo → Clebsch–Gordan coefficients ⓘ

Penrose spin networks → vertexCondition → Clebsch–Gordan coefficients ⓘ

this entity surface form: Clebsch–Gordan constraints