I thought given that electrons have spin $\pm \frac{1}{2}$, the values would be $\frac{1}{2}+\frac{1}{2}+\frac{1}{2}$, $\frac{1}{2}+\frac{1}{2}-\frac{1}{2}$, $\frac{1}{2}-\frac{1}{2}-\frac{1}{2}$ and $-\frac{1}{2}-\frac{1}{2}-\frac{1}{2}$, i.e. $\frac{3}{2}$, $\frac{1}{2}$, $-\frac{1}{2}$ and $-\frac{3}{2}$. I have been told this is not correct. What am I doing wrong?
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1$\begingroup$ Does this answer your question? Total spin of system of two spin-$1/2$ particles $\endgroup$– Carl WitthoftCommented Mar 4, 2021 at 14:47
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$\begingroup$ of possible related interest, quantummechanics.ucsd.edu/ph130a/130_notes/node312.html $\endgroup$– Carl WitthoftCommented Mar 4, 2021 at 14:48
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Total spin is zero or positive. You are confusing $S$ and $m_S$.
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2$\begingroup$ This could be improved by discussing why this is so. A single $e^-$ can have spin $-\frac{1}{2}$ which sure "looks" negative! $\endgroup$ Commented Mar 4, 2021 at 14:44
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$\begingroup$ @CarlWitthoft No, it can't. You, too, are confusing $S$ and $m_S$. $\endgroup$– my2ctsCommented Mar 4, 2021 at 15:18
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2$\begingroup$ that was my point: toss in some text to explain those two terms and how they are related. $\endgroup$ Commented Mar 4, 2021 at 15:49
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$\begingroup$ Why should I? All you have to do is read the first section of the wikipedia article, or any other article on spin. en.wikipedia.org/wiki/Spin_(physics)#Quantum_number $\endgroup$– my2ctsCommented Mar 4, 2021 at 17:43