# How is multiplicity given by 2S+1?

Suppose there are two electrons in an atom with $s_1 = \frac{1}{2}$, $l_1 = 1$ and $s_2 = \frac{1}{2}$, $l_2 = 1$. Hence the total $S$ (of the atom) may be +1 or 0. And total $L$ is either $+2$, $+1$ or $0$.

Now If we consider

\begin{align} S=1,L=2 &\to 2S+1=3; J=3,2,1\\ S=0,L=2 &\to 2S+1=1; J=2\\ S=1,L=1 &\to 2S+1=3; J=2,1,0\\ S=0,L=1 &\to 2S+1=1; J=1\\ S=1,L=0 &\to 2S+1=3; J=1\end{align}

But the last one does not show that $2S+1$ is multiplicity as it has only one $J$ value. Where am I making a mistake?

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The formula $2S+1$ for total multiplicity has nothing to do with spin-orbit coupling, so you should forget about $L$. If you know that a system has spin $S$, then it can have $S_z = -S,-S+1,\ldots,S-1,S$, so you have $2S+1$ possibilities. –  Vibert Feb 5 '13 at 20:08
Hi user22180 - I removed the extra question at the end of your post, as we like to limit each post to contain only one question. You can always post it separately. –  David Z Feb 5 '13 at 20:54
@Vibert that could be the basis of an answer. –  David Z Feb 5 '13 at 20:54
@Vibert , Then why is it called by multiplicity?And how is it different from degeneracy in the ABSENCE of magnetic field? –  user22180 Feb 7 '13 at 21:54
@user22180: it's not different at all. The following statement is always true: a system of spin/angular momentum $J$ has multiplicity $2J+1.$ (It is really a maths statement: the dimension of the $SU(2)$ representation $J$ is $2J+1$.) In practice, you need to think which spin you use. For a single electron, it is clearly the electron spin $S$, in a system coupled with orbital angular momentum $L$ you can choose a basis labelled by the total spin $J = L+S.$ –  Vibert Feb 8 '13 at 11:54