Theorie
The Wigner-D functions are special functions on the rotation group \(SO(3)\).
In terms of Matthies (ZYZ-convention) Euler angles \({\bf R} = {\bf R}(\alpha,\beta,\gamma)\) the \(L_2\)-normalized Wigner-D function of degree \(n\) and orders \(k,l \in \{-n,\dots,n\}\) is defined by
\[ D_n^{k,l}({\bf R}) = \sqrt{2n+1} \, \mathrm e^{-\mathrm i k\gamma} \mathrm d_n^{k,l}(\cos\beta) \,e^{-\mathrm i l\alpha} \]
where \(d_n^{k,l}\), denote the real valued Wigner-d functions, which are defined in terms of Jacobi polynomial \(P_s^{a,b}\) by
\[ d_n^{k,l}(x) = (-1)^{\nu} \binom{2n-s}{s+a}^{\frac12} \binom{s+b}{b}^{-\frac12} \left(\frac{1-x}{2}\right)^{\frac{a}{2}} \left(\frac{1+x}{2}\right)^{\frac{b}2} P_s^{a,b}(x)\]
using the constants \(a =|k-l|\), \(b =|k+l|\), \(s = n - \max\{|k|,|l|\}\) and
\[ \nu = \begin{cases} \min\{0,k\}+\min\{0,l\} & \text{if } l \geq k,\\ \min\{0,k\}+\min\{0,l\} + k+l & \text{otherwise}. \end{cases}\]
The differential representation of the Wigner-d functions reads as
\[ d_n^{k,l}(x) = (-1)^{n-l+\min\{0,k\}+\min\{0,l\}} \frac1{2^n} \left( \frac{(n+l)!}{(n+k)!(n-k)!(n-l)!} \right)^{1/2} (1-x)^{\frac{k-l}2} (1+x)^{-\frac{k+l}2} \frac{d^{n-l}}{dx^{n-l}}((1-x)^{n-k}(1+x)^{n+k})\]
The above definition of the Wigner-D functions in MTEX is slightly different to other well known definitions from literature, they are defined compatible to the spherical harmonics which form an orthonormal basis on the 2-sphere and therefore are use to build S2FunHarmonics.
The differences of the Wigner-D functions to common definitions are: * In the definition of \(D_n^{k,l}\) the indices \(k,l\) are changed in the \(e^{i\cdot}\) functions. * \(D_n^{k,l}\) are normalized by the constant \(\sqrt{2n+1}\). * The sign of \(d_n^{k,l}\) is multiplied with \(\min\{0,k\}+\min\{0,l\}\).
Common definitions can be found in
D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii. Quantum theory of angular momentum. Irreducible tensors, spherical harmonics, vector coupling coefficients, 3nj symbols. World Scientific Publishing Co.,Inc., Teaneck, NJ, 1988. isbn: 9971-50-107-4. doi: 10.1142/0270.
In MTEX the Wigner-D and Wigner-d functions are available through the command Wigner_D
% the Wigner-d function of degree 1
beta = 0.5;
d = Wigner_D(1,beta)
% the Wigner-D function of degree 1
R = rotation.rand;
D = sqrt(3) * Wigner_D(1,R)d =
0.9388 -0.3390 -0.0612
0.3390 0.8776 0.3390
-0.0612 -0.3390 0.9388
D =
0.0447 - 0.0040i -0.1195 + 0.3705i 1.2748 + 1.1051i
0.1524 + 0.3582i -1.6422 + 0.0000i 0.1524 - 0.3582i
1.2748 - 1.1051i -0.1195 - 0.3705i 0.0447 + 0.0040iHere the orders \(k\), \(l\) work as row and column indices.
Series Expansion
The Wigner-D functions form an orthonormal basis in \(L_2(SO(3))\). Hence, we can describe functions on the rotation group \(SO(3)\) by there harmonic representation using the class SO3FunHarmonic.
Hence we define the Wigner-D function \(D_1^{1,-1}\) by
D = SO3FunHarmonic([0;0;0;1])
D.eval(R)D = SO3FunHarmonic (xyz → xyz)
antipodal: true
isReal: false
bandwidth: 1
weight: 0
ans =
1.2748 - 1.1051iVarious normalization for the Wigner-D functions are common in the literature.
Here we define the \(L_2\)-norm by
\[ \| f \|_2 = \left(\frac1{8\pi^2}\,\int_{SO(3)} \lvert f( {\bf R}) \rvert^2 \,\mathrm d {\bf R} \right)^{1/2} \]
such that the norm of the constant function \(f=1\) is \(1\). Take a look on the section Integration of SO3Fun's.
Using that definition the Wigner-D functions in MTEX are normalized, i.e. \(\| D_n^{k,l} \|_2 = 1\) for all \(n,k,l\).
norm(D)ans =
1Some important formulas for Wigner-D functions
The Wigner-D functions are the matrix elements of the representations \(D_n \colon SO(3) \to \mathbb C^{(2n+1)\times(2n+1)}\) on \(SO(3)\). Since representations are group homomorphisms, we have \(D_n( {\bf R} \, {\bf Q} ) = \frac1{\sqrt{2n+1}} \, D_n( {\bf Q} ) \, D_n( {\bf R} ).\) Hence we get
\[ D_n^{k,l}( {\bf R} \, {\bf Q} ) = \frac1{\sqrt{2n+1}} \sum_{j=-n}^n D_n^{k,j}( {\bf Q} )\,D_n^{j,l}( {\bf R} ). \]
Some symmetry properties of Wigner-D functions yields
\[ D_n^{k,l}( {\bf R} ) = \overline{D_n^{l,k}( {\bf R}^{-1} )}. \]
Symmetry properties of Wigner-d functions
The Wigner-d functions by construction fulfill a lot of symmetry properties. Some important are
\[ d_n^{k,l}(x) = d_n^{-k,-l}(x) = (-1)^{k+l}\, d_n^{l,k}(x) = (-1)^{k+l}\, d_n^{-l,-k}(x)\]
\[ d_n^{k,l}(x) = (-1)^{n+k+l}\,d_n^{-k,l}(-x) = (-1)^{n+k+l}\,d_n^{k,-l}(-x) \]
\[d_n^{k,l}(\cos\beta) = (-1)^{k+l}\,d_n^{k,l}(\cos(-\beta))\]