gives the Wigner D-function .


gives the Wigner D-function .


gives the Wigner D-function .


  • The Wigner D-function gives the matrix element of a rotation operator parametrized by Euler angles in a dimensional unitary representation of a rotation group when parameters , , are physical, i.e. all integers or half-integers such that .
  • For unphysical parameters, WignerD is defined by an analytic continuation.
  • The Wolfram Language uses phase conventions where .


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Basic Examples  (1)

Scope  (1)

Evaluate numerically for physical parameters:

Generalizations & Extensions  (1)

Evaluate numerically for unphysical parameters:

Applications  (1)

Construct a rotation matrix for a spin-1/2 representation:

Check unitarity:

Build a 3D vector from spinors:

Spinor basis translates into coordinate basis:

Coordinate transformation induced by unitary transformation on spinors:

Construct the resulting rotation matrix directly using Euler's angles:

Properties & Relations  (4)

For vanishing parameter m1, WignerD reduces to SphericalHarmonicY:

Matrix elements of the Wigner D matrix satisfy certain symmetry relations:

WignerD functions form an orthogonal basis on the group:

The integral is equal to :

The product of two WignerD functions can be expanded in terms of WignerD functions using ClebschGordan coefficients:

Wolfram Research (2010), WignerD, Wolfram Language function, https://reference.wolfram.com/language/ref/WignerD.html.


Wolfram Research (2010), WignerD, Wolfram Language function, https://reference.wolfram.com/language/ref/WignerD.html.


Wolfram Language. 2010. "WignerD." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/WignerD.html.


Wolfram Language. (2010). WignerD. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WignerD.html


@misc{reference.wolfram_2023_wignerd, author="Wolfram Research", title="{WignerD}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/WignerD.html}", note=[Accessed: 10-December-2023 ]}


@online{reference.wolfram_2023_wignerd, organization={Wolfram Research}, title={WignerD}, year={2010}, url={https://reference.wolfram.com/language/ref/WignerD.html}, note=[Accessed: 10-December-2023 ]}