Why does spin add in quantum mechanics?

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Xian
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Why is it that spin adds? I mean sure, the classical analogy tells us that angular momentum is additive, but that I feel is a cop out. What inside of the formalism for quantum mechanics tells us that we should be adding spin of two particles?
 

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It doesn't. You can add spin 1 and spin 1 and get spin 1. Or 3 and 4 and get 2. Or...
 

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