We have also $$ |\hat{J}|=\hbar\sqrt{J(J+1)}, $$ with the the following degeneracy $$ M_J=-J, -J+1, \dots, J-1, J. $$ With a given value of \( L \) and \( S \) we can then determine the possible values of \( J \) by studying the \( z \) component of \( \hat{J} \). It is given by $$ \hat{J}_z=\hat{L}_z+\hat{S}_z. $$ The operators \( \hat{L}_z \) and \( \hat{S}_z \) have the quantum numbers \( L_z=M_L\hbar \) and \( S_z=M_S\hbar \), respectively, meaning that $$ M_J\hbar=M_L\hbar +M_S\hbar, $$ or $$ M_J=M_L +M_S. $$ Since the max value of \( M_L \) is \( L \) and for \( M_S \) is \( S \) we obtain $$ (M_J)_{\mathrm{maks}}=L+S. $$