# What are $S$ and $L$?

In a helium atom, the total spin of the electrons is 0, suggesting that the total spin quantum number S is the sum of the ms quantum numbers (1/2-1/2=0). However, many sources say that the total spin quantum number S is the sum of the s quantum numbers, so in helium S should be 1 (s is always 1/2 for electrons).

In much the same way, I've seen some sources say that L is the sum of the ml quantum numbers and others say that L is the sum of the l quantum numbers.

So, which is correct?

Both $$L$$ and $$S$$ are quantum angular momenta.

Angular momentum is different in quantum mechanics (QM). Almost everything is different in QM.

In QM, angular momentum is complete if you give two numbers: the "angular momentum" and its "third component" $$M$$. So you give the pairs $$|l,\ m_l\rangle$$ and $$|s, \ m_s\rangle$$

So, any angular momentum is defined with those two numbers. But those numbers behave very differently.

The third component is additive. You add $$m$$'s without problem:

Total $$m$$ = $$m_1+m_2$$

On the other hand, $$l$$'s behave very differently. When we add two angular momenta, the total angular momentum (let's call it $$J$$), is different: it can go from $$|L-S|$$ to $$L+S$$

And the thing is that both things are called "sum", that's the reason of the confusion.

When you add two angular momenta (vectors) $$\vec{L}+\vec{S}=\vec{J}$$ That's okay, but this is the same as saying

$$|l,\ m_l\rangle \ + \ |s,\ m_s\rangle \ = \ |j,\ m_j\rangle$$

And the total $$m$$ adds normally: $$m_j=m_l+m_s$$, okay.

But $$j$$ is different. $$j$$ can go from $$|l-s|$$ to $$|l+s|$$ in unit steps.

In all angular momenta, as you know, $$m$$ can go from $$-l$$ to $$+l$$. And also $$m_s$$ from $$-s$$ to $$+s$$; and $$m_j$$ from $$-j$$ to $$+j$$.

So, if you have $$s=½$$, you can have $$m_s=\pm ½$$

But if you add two spins, $$s_1+s_2$$, the resulting $$j$$ can be 0 or 1.

For $$j=0$$, $$m_j$$ can only be 0 as well. But for $$j=1$$, $$m$$ can have three values: $$m_j={-1,0,1}$$.

So, "summing two spins" can mean that total spin is "J=1" but "M=0".

• Brilliant, thank you very much. – Ali Chopping Jan 9 at 12:13
• while in general this is an excellent answer, may I object to the line $|l, m_l\rangle + |s, m_s⟩ = |j, m_j\rangle$? This makes it seem as if $l$ and $s$ are referring to the same part of Hilbert space, and that $|j, m_j\rangle$ is something like a superposition of them, while this is not correct. The correct formatting will be $|l, m_l \rangle \otimes |s, m_s\rangle = |j, m_j\rangle$ – yu-v Jan 9 at 13:53