Wigner D-matrix

Remarks

Since $J_z$ is diagonal, $D^j_{m' m}(\alpha, \beta, \gamma) = e^{-i m' \alpha}\, d^j_{m' m}(\beta)\, e^{-i m \gamma}$. The entries of $d^j(\beta)$ are real, and Wigner's formula is $$d^j_{m' m}(\beta) = \sum_{s} \frac{(-1)^{m' - m + s} \sqrt{(j+m')!\,(j-m')!\,(j+m)!\,(j-m)!}}{(j+m-s)!\, s!\, (m'-m+s)!\, (j-m'-s)!} \left(\cos\tfrac{\beta}{2}\right)^{2j+m-m'-2s} \left(\sin\tfrac{\beta}{2}\right)^{m'-m+2s},$$ summed over the integers $s$ for which all factorials have nonnegative arguments.

Example: for $j = \frac12$, $J_y = \frac12 \sigma_y$ with the Pauli matrix $\sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$, and $\sigma_y^2 = I$. So $e^{-i \beta J_y} = \cos\frac{\beta}{2}\, I - i \sin\frac{\beta}{2}\, \sigma_y$, which gives $d^{1/2}_{\frac12 \frac12} = \cos\frac{\beta}{2}$ and $d^{1/2}_{\frac12, -\frac12} = -\sin\frac{\beta}{2}$. Wigner's formula with $m' = \frac12$, $m = -\frac12$ has only the term $s = 0$, and it also gives $-\sin\frac{\beta}{2}$.

Non-example: the matrix $\begin{pmatrix} \cos\frac{\beta}{2} & \sin\frac{\beta}{2} \\ \sin\frac{\beta}{2} & \cos\frac{\beta}{2} \end{pmatrix}$ is not a D-matrix, since it is not unitary for $\sin \beta \ne 0$, while every $D^j$ is unitary.

For integer $j = \ell$, $d^\ell_{00}(\beta) = P_\ell(\cos\beta)$, the Legendre polynomial. Also $\overline{D^\ell_{m 0}(\alpha, \beta, \gamma)} = \sqrt{\frac{4\pi}{2\ell+1}}\, Y_\ell^m(\beta, \alpha)$, where $Y_\ell^m$ is a spherical harmonic with polar angle $\beta$ and azimuth $\alpha$. The bra-ket notation $\langle u \rvert A \lvert v \rangle$ means the inner product of $u$ with $A v$.