Suppose you want to add 2 spin-1 particles.(adsbygoogle = window.adsbygoogle || []).push({});

I understand you can get the j=1 triplet by $$e_{ijk}|j\rangle |k\rangle $$ where i, j, k run from -1, 0 , 1.

The idea is that levi-civita symbol is a tensor under SO(3) rotations, so the contraction with the $$|j\rangle |k\rangle$$ tensor gives a vector under rotation, which is the triplet.

However, isn't kronecker delta also a tensor under rotation SO(3)? So why can't you get the singlet from:

$$\delta_{jk} |j\rangle |k\rangle = |1\rangle |1\rangle +|0\rangle |0\rangle+|-1\rangle |-1\rangle $$

Looking at the textbooks, the middle |00> term is with a negative sign.

Also, is there a group theoretic way to get the quintuplet?

Thanks.

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# Adding angular momenta using levi-civta symbol

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