= Spin-one angular momentum matrices
{title2=$J^2=2\hbar^2I$}
= Spin-one matrices
{synonym}
In the ordered <angular momentum eigenstate> basis $(|1,1\rangle,|1,0\rangle,|1,-1\rangle)$, with positive raising coefficients,
$$
J_1=\frac\hbar{\sqrt2}\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix},\quad
J_2=\frac\hbar{\sqrt2}\begin{pmatrix}0&-i&0\\i&0&-i\\0&i&0\end{pmatrix},\quad
J_3=\hbar\operatorname{diag}(1,0,-1).
$$
To derive them, $[J_3,J_+]=\hbar J_+$ restricts $J_+$ to its two adjacent superdiagonal entries. Write these real entries as $A,B\geq0$. Since $J_-=J_+^\dagger$, the relation $[J_+,J_-]=2\hbar J_3$ reads $\operatorname{diag}(A^2,B^2-A^2,-B^2)=2\hbar^2\operatorname{diag}(1,0,-1)$, forcing $A=B=\sqrt2\hbar$. Adding and subtracting the <ladder operators> gives the displayed <matrices>. Direct multiplication verifies the <angular momentum commutation relations> and $J^2=2\hbar^2I$.
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