This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# WignerD

 WignerD gives the Wigner D-function . WignerDgives the Wigner D-function . WignerDgives 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.
• Mathematica uses phase conventions where .
 Out[1]=
 Out[2]=
 Scope   (1)
Evaluate numerically for physical parameters:
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:
For vanishing parameter , 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:
New in 8