Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 2B / Section B / 35 / a

The operator ${\hat{J}}_z$ describes the $z$ component of angular momentum, whereas ${\hat{J}}^2$ describes its total squared magnitude. For a spherical harmonic $Y_{JM}$ the corresponding eigenvalues areThus the magnitude is $\hslash \sqrt{J(J+1)}$, not $J\hslash$. For ordinary orbital rotation $J=0,1,2,\dots$ and $M=-J,-J+1,\dots ,J$. The simultaneous values are possible because these two operators commute.