= Wigner D-matrix
{c}
{title2=$D^{(j)}_{mm'}(\phi,\theta,\psi)$}
{wiki}
For active <rotations> generated by dimensional <angular momentum operators>, define
$$
U=e^{-i\phi J_3/\hbar}e^{-i\theta J_2/\hbar}e^{-i\psi J_3/\hbar}.
$$
The <irreducible spin representation> <matrix> is
$$
D^{(j)}_{mm'}(\phi,\theta,\psi)
=e^{-im\phi}d^{(j)}_{mm'}(\theta)e^{-im'\psi},
\qquad
d^{(j)}_{mm'}(\theta)=\langle j,m|e^{-i\theta J_2/\hbar}|j,m'\rangle.
$$
The factors of $\hbar$ are required because $J_3|j,m\rangle=\hbar m|j,m\rangle$. One may equivalently use dimensionless generators $j_i=J_i/\hbar$.
For <spin one-half>, the middle <matrix> is
$$
d^{(1/2)}(\theta)=
\begin{pmatrix}
\cos(\theta/2)&-\sin(\theta/2)\\
\sin(\theta/2)&\cos(\theta/2)
\end{pmatrix}.
$$
A full $2\pi$ <rotation> acts as $(-1)^{2j}I$. Only integer-<spin> representations descend to the three-dimensional <special orthogonal group>; half-integer representations distinguish the two lifts of the same spatial <rotation>.
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