The $\ell=1$ Angular Momentum Operators*

We will use states of definite $L_z$, the $Y_{1m}$.

$$\begin{eqnarray*} & \langle \ell m'\vert L_z\vert\ell m\rangle =m\hbar \delta_{m'm} \\ & L_z=\hbar\left(\matrix{1&0&0\cr 0&0&0\cr 0&0&-1}\right) \end{eqnarray*}$$

$$\begin{eqnarray*} \langle \ell m'\vert L_\pm\vert\ell m\rangle &=&\sqrt{\ell (\... ...hbar\left(\matrix{0&0&0\cr \sqrt{2}&0&0\cr 0&\sqrt{2}&0}\right) \end{eqnarray*}$$

$$\begin{eqnarray*} L_x&=&{1\over 2}(L_+ + L_-)={\hbar\over\sqrt{2}}\left(\matrix... ...r\over\sqrt{2}i}\left(\matrix{0&1&0\cr -1&0&1\cr 0&-1&0}\right) \end{eqnarray*}$$

What is the dimension of the matrices for $\ell =2$?
Dimension 5. Derive the matrix operators for $\ell =2$.
Just do it.