WignerD
Details
- 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
![TemplateBox[{j, {m, _, 1}, {m, _, 2}, psi, theta, phi}, WignerD]=exp(ⅈ m_1 psi+ⅈ m_2phi) TemplateBox[{j, {m, _, 1}, {m, _, 2}, 0, theta, 0}, WignerD]](Files/WignerD.en/10.png "TemplateBox[{j, {m, _, 1}, {m, _, 2}, psi, theta, phi}, WignerD]=exp(ⅈ m_1 psi+ⅈ m_2phi) TemplateBox[{j, {m, _, 1}, {m, _, 2}, 0, theta, 0}, WignerD]")
.
Examples
open all close allBasic Examples (2)
Scope (4)
Evaluate numerically for physical parameters:
WignerD[{1, 0, 1}, 20.2 Degree, 30 Degree, 11.0 Degree]Wigner D-function 
when 
with 
and 
implicitly zero:
WignerD[{1 / 2, 1 / 2, -1 / 2}, θ]Wigner D-function 
when 
with only 
implicitly zero:
WignerD[{1 / 2, 1 / 2, 1 / 2}, θ, 0]Wigner D-function 
when 
with 
and 
implicitly zero:
WignerD[{3 / 2, 1 / 2, -1 / 2}, θ]Wigner D-function 
when 
with only 
implicitly zero:
WignerD[{3 / 2, 3 / 2, -1 / 2}, θ, ϕ]Wigner D-function 
when 
with 
and 
implicitly zero:
WignerD[{2, 2, 0}, θ]
WignerD[{2, 1, -1}, ψ, θ, ϕ]Generalizations & Extensions (1)
Applications (1)
Construct a rotation matrix for a spin-1/2 representation:
U = Table[WignerD[{1 / 2, m1, m2}, ψ, θ, ϕ], {m1, -1 / 2, 1 / 2}, {m2, -1 / 2, 1 / 2}]Simplify[ConjugateTranspose[U].U, {θ, ϕ, ψ}∈Reals]Build a 3D vector from spinors:
SpinorToVector[spi_] := Table[Conjugate[spi].PauliMatrix[i].spi, {i, 3}]Spinor basis translates into the coordinate basis:
{SpinorToVector[{1, 1} / Sqrt[2]], SpinorToVector[{-I, 1} / Sqrt[2]], SpinorToVector[{1, 0}]}Coordinate transformation induced by unitary transformation on spinors:
rm = Assuming[{θ, ϕ, ψ}∈Reals,
{SpinorToVector[U.{1, 1} / Sqrt[2]], SpinorToVector[U.{-I, 1} / Sqrt[2]], SpinorToVector[U.{1, 0}]}//Transpose//FullSimplify]Construct the resulting rotation matrix directly using Euler's angles:
rm == EulerMatrix[{ψ, θ, ϕ}]Properties & Relations (4)
For vanishing parameter m1, WignerD reduces to SphericalHarmonicY:
Table[WignerD[{ℓ, 0, m}, θ, ϕ] == Sqrt[(4 π/2ℓ + 1)]SphericalHarmonicY[ℓ, m, θ, ϕ], {ℓ, 0, 3}, {m, -ℓ, ℓ}]//Simplify//TableFormMatrix elements of the Wigner D-matrix satisfy certain symmetry relations:
MatrixForm /@ Table[WignerD[{j, m1, m2}, ψ, θ, ϕ] == (-1)^m1 - m2Conjugate[WignerD[{j, -m1, -m2}, ψ, θ, ϕ]], {j, 1 / 2, 3 / 2, 1 / 2}, {m1, -j, j}, {m2, -j, j}]//Simplify[#, 0 < θ < π && 0 < ϕ < 4π && 0 < ψ < 4π]&MatrixForm /@ Table[WignerD[{j, m1, m2}, ψ, θ, ϕ] == (-1)^m1 - m2WignerD[{j, m2, m1}, ϕ, θ, ψ], {j, 1 / 2, 3 / 2, 1 / 2}, {m1, -j, j}, {m2, -j, j}]//SimplifyWignerD functions form an orthogonal basis on the 
group:
With[{j1 = 1 / 2, j2 = 1 / 2},
Table[Integrate[Conjugate[WignerD[{j1, m11, m12}, ψ, θ, ϕ]]WignerD[{j2, m21, m22}, ψ, θ, ϕ]Sin[θ], {θ, 0, Pi}, {ψ, 0, 4Pi}, {ϕ, 0, 4Pi}], {m11, -j1, j1}, {m21, -j2, j2}, {m12, -j1, j1}, {m22, -j2, j2}]//MatrixForm]![2/(2 j_1+1) (4 pi)^2TemplateBox[{{{j, _, 1}, ,, {j, _, 2}}}, KroneckerDeltaSeq]TemplateBox[{{{m, _, {(, 11, )}}, ,, {m, _, {(, 21, )}}}}, KroneckerDeltaSeq]TemplateBox[{{{m, _, {(, 12, )}}, ,, {m, _, {(, 22, )}}}}, KroneckerDeltaSeq]](Files/WignerD.en/32.png "2/(2 j_1+1) (4 pi)^2TemplateBox[{{{j, _, 1}, ,, {j, _, 2}}}, KroneckerDeltaSeq]TemplateBox[{{{m, _, {(, 11, )}}, ,, {m, _, {(, 21, )}}}}, KroneckerDeltaSeq]TemplateBox[{{{m, _, {(, 12, )}}, ,, {m, _, {(, 22, )}}}}, KroneckerDeltaSeq]"){kind=small}
With[{j1 = 1 / 2, j2 = 1 / 2},
Table[2 / (2j1 + 1)(4Pi) ^ 2 KroneckerDelta[m11, m21]KroneckerDelta[m12, m22]KroneckerDelta[j1, j2], {m11, -j1, j1}, {m21, -j2, j2}, {m12, -j1, j1}, {m22, -j2, j2}]//MatrixForm]The product of two WignerD functions can be expanded in terms of WignerD functions using ClebschGordan coefficients:
With[{l1 = 3 / 2, m11 = 1 / 2, m12 = -1 / 2, l2 = 1, m21 = -1, m22 = 1}, WignerD[{l1, m11, m12}, ψ, θ, ϕ]WignerD[{l2, m21, m22}, ψ, θ, ϕ] == Underoverscript[∑, j = Abs[l1 - l2], l1 + l2]ClebschGordan[{l1, m11}, {l2, m21}, {j, m11 + m21}] ClebschGordan[{l1, m12}, {l2, m22}, {j, m12 + m22}] WignerD[{j, m11 + m21, m12 + m22}, ψ, θ, ϕ]]//SimplifyRelated Links
MathWorldText
Wolfram Research (2010), WignerD, Wolfram Language function, https://reference.wolfram.com/language/ref/WignerD.html.
CMS
Wolfram Language. 2010. "WignerD." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/WignerD.html.
APA
Wolfram Language. (2010). WignerD. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WignerD.html