ClebschGordan

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

where

Specific values

Specialized values

Nonphysical cases

Fixed (Subscript[j, 1]), (Subscript[j, 2]), (Subscript[m, 1]), (Subscript[m, 2])

Fixed (Subscript[j, 1]), (Subscript[j, 2]), j, (Subscript[m, 2])

Fixed (Subscript[j, 1]), j, (Subscript[m, 1])

Fixed (Subscript[j, 1]), (Subscript[j, 2]), j

Fixed (Subscript[j, 1]), (Subscript[j, 2])

Fixed (Subscript[j, 1]), (Subscript[m, 1])

General characteristics

Domain and analyticity

⟨(Subscript[j, 1]) (Subscript[j, 2]) (Subscript[m, 1]) (Subscript[m, 2]) | (Subscript[j, 1]) (Subscript[j, 2]) j m⟩ is a six-argument function. The condition 𝒫𝒽𝓎𝓈𝒾𝒸𝒶ℓ𝒬 restricts the arguments to integers or half-integers (interpretated as quantum-mechanical spin quantum numbers) that fulfill certain inequalities. Without the condition 𝒫𝒽𝓎𝓈𝒾𝒸𝒶ℓ𝒬, and the prefactor ensuring (Subscript[m, 1])+(Subscript[m, 2])==m, ⟨(Subscript[j, 1]) (Subscript[j, 2]) (Subscript[m, 1]) (Subscript[m, 2]) | (Subscript[j, 1]) (Subscript[j, 2]) j m⟩ would be an analytic function in all six arguments.

Symmetries and periodicities

Reflection symmetry

Series representations

Generalized power series

Other series representations

Series of binomial coefficients

Integral representations

On the real axis

Of the direct function

Multiple integral representations

For the direct function itself

Involving the direct function

Integral representations of negative integer order

Generating functions

Identities

Recurrence identities

Consecutive neighbors

Functional identities

General relations

Arguments changing by 1/2

Arguments (Subscript[j, 1]), (Subscript[j, 2]), j changing by 1

Arguments (Subscript[j, 1]), (Subscript[j, 2]), (Subscript[m, 1]), (Subscript[m, 2]) changing by 1

Arguments (Subscript[j, 2]), j, (Subscript[m, 2]), m changing by 1

Arguments (Subscript[m, 1]), (Subscript[m, 2]), m equal to zero

Involving two Clebsch Gordan coefficients

Involving three Clebsch Gordan coefficients

Summation

Finite summation

Involving one Clebsch Gordan coefficient

Involving two Clebsch Gordan coefficients

Involving three Clebsch Gordan coefficients

Involving four Clebsch Gordan coefficients

Involving two Clebsch Gordan coefficients and one 6j symbol

Representations through more general functions

Through hypergeometric functions

Involving HypergeometricPFQRegularized

Involving HypergeometricPFQ

Through Meijer G

Classical cases

Through other functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With related functions

Theorems

Wigner-Eckhart  theorem

The matrix element of an irreducible tensor operator in an angular momentum basis is the product of an angular momentum dependent factor containing the Clebsch-Gordan coefficients and a term which is rotationally invariant and independent of all projection quantum numbers (Subscript[m, 1]),(Subscript[m, 2]),m .

History

A. Clebsch (1872); P. Gordan (1875); H. Weyl (1928); E.P. Wigner (1928, 1931).

References

L.C. Biedenharn and J.D. Louck, Angular Momentum in Quantum Physics, Addison-Wesley, Reading, 1981.

L.C. Biedenharn and J.D. Louck, The Racah-Wigner Algebra in Quantum Theory, Addison-Wesley, Reading, 1981.

M.E. Rose, Elementary Theory of Angular Momentum, Dover, New York, 1995.

D.A. Varshalovich, A.N. Moskalev and V.K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, Singapore, 1988.

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