# ThreeJSymbol

ThreeJSymbol[{j1,m1},{j2,m2},{j3,m3}]

gives the values of the Wigner 3 symbol.

# Details

• The 3 symbols vanish except when and the satisfy a triangle inequality.
• The parameters of ThreeJSymbol can be integers, halfintegers or symbolic expressions.
• The 3 symbols and ClebschGordan coefficients in the Wolfram Language satisfy the relation .

# Examples

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## Basic Examples(2)

Evaluate a 3 symbol with integer arguments:

Use symbolic arguments to obtain exact symbolic answers:

## Scope(2)

ThreeJSymbol works with integer and halfinteger arguments:

## Applications(3)

Plot 3 symbol coefficients as a function of and :

Apply angular momentum operators to spherical harmonics:

Evaluate Gaunt coefficients:

Evaluate as an explicit triple integral of spherical harmonics:

## Properties & Relations(4)

Evaluate the completely symbolic case of ThreeJSymbol:

Demonstrate sum orthogonality:

Convert a 3 symbol to a Regge R symbol:

Check the symmetry of the Regge R symbol:

Compare with the sum definition:

## Possible Issues(1)

A message is issued and the result 0 returned in the case :

Wolfram Research (1991), ThreeJSymbol, Wolfram Language function, https://reference.wolfram.com/language/ref/ThreeJSymbol.html.

#### Text

Wolfram Research (1991), ThreeJSymbol, Wolfram Language function, https://reference.wolfram.com/language/ref/ThreeJSymbol.html.

#### CMS

Wolfram Language. 1991. "ThreeJSymbol." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ThreeJSymbol.html.

#### APA

Wolfram Language. (1991). ThreeJSymbol. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ThreeJSymbol.html

#### BibTeX

@misc{reference.wolfram_2024_threejsymbol, author="Wolfram Research", title="{ThreeJSymbol}", year="1991", howpublished="\url{https://reference.wolfram.com/language/ref/ThreeJSymbol.html}", note=[Accessed: 17-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_threejsymbol, organization={Wolfram Research}, title={ThreeJSymbol}, year={1991}, url={https://reference.wolfram.com/language/ref/ThreeJSymbol.html}, note=[Accessed: 17-September-2024 ]}