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Angular momentum operator

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Irreducible spin representation

{c}
{title2=$D^{(j)}_{mm'}(\phi,\theta,\psi)$}
{wiki}

For active rotations with $U=e^{-i\phi J_3}e^{-i\theta J_2}e^{-i\psi J_3}$, the <irreducible spin representation> <matrix> is $D^{(j)}_{mm'}=e^{-im\phi}d^{(j)}_{mm'}(\theta)e^{-im'\psi}$, where $d^{(j)}_{mm'}=\langle j,m|e^{-i\theta J_2}|j,m'\rangle$. Its spin-one-half middle <matrix> is $\begin{pmatrix}\cos(\theta/2)&-\sin(\theta/2)\\\sin(\theta/2)&\cos(\theta/2)\end{pmatrix}$. A $2\pi$ rotation acts by $(-1)^{2j}$; only integer-spin representations descend to the <special orthogonal group> in three <dimensions>.

Wigner D-matrix

Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 45 / 3 / Solution

Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 43 / 2 / Solution

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

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Wigner D-matrix (Dmm′(j)​(ϕ,θ,ψ))

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

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