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.
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
NERFINISHED
→
|
| 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
NERFINISHED
→
Wigner–Eckart factorization NERFINISHED → angular momentum coupling → reduced matrix element → rotation group SU(2) NERFINISHED → spherical tensor operators → |
| mathematicalFormulationUses |
group representation theory
→
spherical harmonics → |
| namedAfter |
Carl Eckart
NERFINISHED
→
Eugene Wigner NERFINISHED → |
| relatedTo |
Clebsch–Gordan decomposition
NERFINISHED
→
Racah algebra NERFINISHED → Wigner 3-j symbols NERFINISHED → Wigner 6-j symbols NERFINISHED → 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 (1)
| Subject (surface form when different) | Predicate |
|---|---|
|
Eugene Wigner
→
|
knownFor |