Condon–Shortley phase

E179791

The Condon–Shortley phase is a sign convention introduced in quantum mechanics to standardize the definition of spherical harmonics and angular momentum coupling coefficients.

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Condon–Shortley phase canonical 1

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Predicate Object
instanceOf concept in quantum mechanics
phase convention
sign convention
appearsIn angular momentum theory monographs
quantum mechanics textbooks
spectroscopic notation for atomic states
tables of angular momentum coefficients
appliesTo associated Legendre polynomials P_l^m(x)
orbital angular momentum eigenstates
spherical harmonics Y_l^m(θ,φ)
total angular momentum eigenstates
assumes fixed phase choice for |l,m⟩ basis
conventionFor magnetic quantum number m dependence
phase of Clebsch–Gordan coefficients
phase of |l,m⟩ basis states
defines relative sign between positive and negative m components
sign factor (-1)^m in associated Legendre functions
sign factor (-1)^m in spherical harmonics
documentedIn Theory of Atomic Spectra (1935)
surface form: The Theory of Atomic Spectra by Condon and Shortley
ensures consistency between different angular momentum coupling schemes
orthonormality of spherical harmonics with standard formulas
simple transformation under complex conjugation of spherical harmonics
standard parity properties of spherical harmonics
field angular momentum theory
atomic physics
quantum mechanics
hasEffect affects relative phases but not physical observables
changes sign of some coefficients if omitted
historicalContext introduced in early 20th century atomic spectroscopy
motivatedBy need for consistent angular momentum algebra
need for uniform sign conventions in spectroscopy tables
namedAfter Edward Condon
surface form: Edward U. Condon

G. H. Shortley
surface form: George H. Shortley
relatedTo Clebsch–Gordan coefficients
Wigner 3j symbols
Wigner–Eckart theorem
representation theory of SO(3)
representation theory of SU(2)
spherical tensor operators
standardizes overall sign of angular momentum coupling coefficients
overall sign of spherical harmonics
relative phases of magnetic quantum number states
usedIn angular momentum coupling coefficients
definition of Clebsch–Gordan coefficients
definition of Wigner 3j symbols
definition of Wigner 6j symbols
definition of Wigner 9j symbols
definition of associated Legendre functions
definition of spherical harmonics

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

Edward Condon knownFor Condon–Shortley phase