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Angular momentum operator

Spin

Irreducible spin representation

{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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Wigner D-matrix ( $D_{m m^{'}}^{(j)} (ϕ, θ, ψ)$ )

Wigner small-d matrix ( $d_{m^{'} m}^{j} (θ) = ⟨ j, m^{'} ∣ e^{- i J_{y} θ} ∣ j, m ⟩$ )

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Wigner D-matrix (Dmm′(j)​(ϕ,θ,ψ))

Wigner small-d matrix (dm′mj​(θ)=⟨j,m′∣e−iJy​θ∣j,m⟩)