ThreeJSymbol

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

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[j, 3])}, {(Subscript[m, 1]), (Subscript[m, 2]), (Subscript[m, 3])}}) ) is a six-argument function. The condition 𝒫𝒽𝓎𝓈𝒾𝒸𝒶ℓ𝒬 restricts the arguments to integers or half-integers (interpreted as quantum-mechanical spin quantum numbers) that fulfill certain inequalities. Without the condition 𝒫𝒽𝓎𝓈𝒾𝒸𝒶ℓ𝒬, and the prefactor ensuring (Subscript[m, 1])+(Subscript[m, 2])+(Subscript[m, 3])==0, ( ({{(Subscript[j, 1]), (Subscript[j, 2]), (Subscript[j, 3])}, {(Subscript[m, 1]), (Subscript[m, 2]), (Subscript[m, 3])}}) ) would be an analytic function in all six arguments over (ℂ^6).

Symmetries and periodicities

Permutation symmetry

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

Involving two 3j symbols

Involving three 3j symbols

Summation

Finite summation

Involving one 3j symbol

Involving two 3j symbols

Involving three 3j symbols

Involving four 3j symbols

Involving five 3j symbols

Involving six 3j symbols

Representations through more general functions

Through hypergeometric functions

Involving HypergeometricPFQRegularized

Involving HypergeometricPFQ

Through Meijer G

Classical cases for the direct function itself

Representations through equivalent functions

With related functions

Theorems

Representation as a magic square

For a physically realizable 3j -symbol ( ({{(Subscript[j, 1]), (Subscript[j, 2]), (Subscript[j, 3])}, {(Subscript[m, 1]), (Subscript[m, 2]), (Subscript[m, 3])}}) ) the following 3×3 matrix is a magic square, all rows and columns summing up to (Subscript[j, 1])+(Subscript[j, 2])+(Subscript[j, 3]):

(({{-(Subscript[j, 1])+(Subscript[j, 2])+(Subscript[j, 3]), (Subscript[j, 1])-(Subscript[j, 2])+(Subscript[j, 3]), (Subscript[j, 1])+(Subscript[j, 2])-(Subscript[j, 3])}, {(Subscript[j, 1])-(Subscript[m, 1]), (Subscript[j, 2])-(Subscript[m, 2]), (Subscript[j, 3])-(Subscript[m, 3])}, {(Subscript[j, 1])+(Subscript[m, 1]), (Subscript[j, 2])+(Subscript[m, 2]), (Subscript[j, 3])+(Subscript[m, 3])}}))

The quantum chemistry Gaunt coefficients

The quantum chemistry Gaunt coefficients

a(m,n,μ,ν,p)==(((2 p+1) (p-m-μ))/(2(p+m+μ))) (∫_(-1)^1)(Subscript[P, n]^m)(x) (Subscript[P, ν]^μ)(x) (Subscript[P, p]^(m+μ))(x)dx

can be expressed in 3j symbols.

History

– E. P. Wigner (1931, 1940)
– G. Racah (1942)

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.

Copyright