# Angular Momentum Addition Theorem - Sanity Check

Looking back at my quantum mechanics notes, the angular momentum addition theorem is listed as:

$j=j_1+j_2,j_1+j_2-1, ..., |j_1-j_2|$ (Using conventional notation)

, but I'm a little unsure how to interpret the introduction of the modulus operation ($|...|$) and couldn't easily find any examples.

I'm assuming you apply the modulus to any expression which would otherwise yield a negative value for $j$?

I'd appreciate a nod from someone in the know :-).

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## 1 Answer

This is an absolute value. So if $j_1>j_2$ then you get $j_1-j_2$ as the lower bound for $j$, otherwise $j_2-j_1$.

For this to really make sense, you need to know that the sequence eventually gets to $|j_1-j_2|$. For convenience I will take $j_1>j_2$ without any loss of generality. Then we want to show that the difference between $j_1+j_2$ and $j_1-j_2$ is a nonnegative integer. But the difference is just $2j_2$ which is clearly a nonnegative integer, since $j_2$ is a nonnegative integer or half-integer.

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Hey thanks, are you saying that if $j_1$ is always greater than $j_2$, then there's no need for the mod? – TCTopCat Apr 24 '12 at 23:34
Yes. If you pick $j_1$ to be the larger of the two then it's unnecessary. – Logan M Apr 24 '12 at 23:35
Thanks, it's all coming back to me now. I seem to have a habit of over-complicating things ... – TCTopCat Apr 24 '12 at 23:45