Single-particle and two-particle quantum numbers, brief review on angular momenta etc
Using \( \hat{J}=\hat{L}+\hat{S} \) we get
$$
|\hat{J}| \ge |\hat{L}|-|\hat{S}|,
$$
or
$$
|\hat{J}|=\hbar\sqrt{J(J+1)}\ge |\hbar\sqrt{L(L+1)}-
\hbar\sqrt{S(S+1)}|.
$$