= Solution
In the ordered <orthonormal basis> $(|\tfrac12,\tfrac12\rangle,|\tfrac12,-\tfrac12\rangle)$, the <spin one-half> <angular momentum operators> are $J_i=(\hbar/2)\sigma_i$, with <Pauli matrices>
$$
\boxed{J_1=\frac\hbar2\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad
J_2=\frac\hbar2\begin{pmatrix}0&-i\\i&0\end{pmatrix},\qquad
J_3=\frac\hbar2\begin{pmatrix}1&0\\0&-1\end{pmatrix}.}
$$
The <ladder operators> are
$$
\boxed{J_+=\hbar\begin{pmatrix}0&1\\0&0\end{pmatrix},\qquad
J_-=\hbar\begin{pmatrix}0&0\\1&0\end{pmatrix}.}
$$
Explicit multiplication gives $J_+J_-=\hbar^2\operatorname{diag}(1,0)$ and $J_-J_+=\hbar^2\operatorname{diag}(0,1)$. Their <commutator> is therefore
$$
\boxed{[J_+,J_-]=\hbar^2\begin{pmatrix}1&0\\0&-1\end{pmatrix}=2\hbar J_3.}
$$
Back to article page
