gives the ClebschGordan coefficient for the decomposition of in terms of .


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


open allclose all

Basic Examples  (2)

Evaluate a ClebschGordan coefficient with integer arguments:

Use symbolic arguments to obtain exact symbolic answers:

Scope  (2)

ClebschGordan works with integer and halfinteger arguments:

For symbolic input ClebschGordan evaluates to ThreeJSymbol:

Applications  (3)

Plot ClebschGordan coefficients as a function of and :

Decompose a spherical harmonic into a sum of products of two spherical harmonics:

Apply angular momentum operators to spherical harmonics:

Properties & Relations  (2)

Evaluate the completely symbolic case of ClebschGordan:

Demonstrate sum orthogonality:

Possible Issues  (1)

A message is issued and the result of 0 is returned when :

Neat Examples  (1)

Visualize contours constructed from ClebschGordan coefficients:

Wolfram Research (1991), ClebschGordan, Wolfram Language function,


Wolfram Research (1991), ClebschGordan, Wolfram Language function,


Wolfram Language. 1991. "ClebschGordan." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (1991). ClebschGordan. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_clebschgordan, author="Wolfram Research", title="{ClebschGordan}", year="1991", howpublished="\url{}", note=[Accessed: 21-May-2024 ]}


@online{reference.wolfram_2024_clebschgordan, organization={Wolfram Research}, title={ClebschGordan}, year={1991}, url={}, note=[Accessed: 21-May-2024 ]}