Angular momentum ladder operator
= Angular momentum ladder operator
{title2=$L_\pm$}
The operators $L_\pm=L_x\pm iL_y$ satisfy
$$
[L_z,L_\pm]=\pm\hbar L_\pm,
\qquad
[L^2,L_\pm]=0.
$$
They therefore change the <magnetic quantum number> $m$ by $\pm1$ while preserving the total angular-momentum quantum number $\ell$.