Suppose we have two spin-1/2 particles with no orbital angular momentum. We choose to work with the eigenbasis of total angular momentum $S^2$ and $S_z$, which gives us the triplet and the singlet states:
$$ \begin{align}
(s=1, \rm triplet, \rm symmetric)
\begin{cases}
&|11\rangle=|\uparrow \uparrow\rangle \\ &|10\rangle=\frac{1}{\sqrt{2}}\left(|\uparrow \downarrow\rangle + |\downarrow \uparrow \rangle\right) \\ &|1-1\rangle=|\downarrow \downarrow\rangle
\end{cases} \\ \\
(s=0, \rm singlet, \rm antisymmetric)
\begin{cases}
&|00\rangle=\frac{1}{\sqrt{2}}\left(|\uparrow \downarrow\rangle – |\downarrow \uparrow \rangle\right)
\end{cases}
\end{align} $$
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Both the triplet and the singlet states have integer-valued total spins. This suggests the composite system of two spin-1/2 particles behaves bosonically. Although the triplet state respects this by being totally symmetric, the singlet state is totally antisymmetric. Given there are no other parts to the wavefunction for us to antisymmetraize, we are stuck with a totally antisymmetric state describing $s=0$, which is for bosons. What am I missing here that gives rise to this contradiction?
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Are the four listed states above always allowed? Or, does it depend on whether the two spin-1/2 particles are identical or distinguishable?
I'm thinking that if they are distinguishable, then all four states are allowed, keeping in mind my confusion described in Question 1 (that is, I think the system should behave bosonically, but the singlet state is antisymmetric).
If the particles are identical, then I cannot tell them apart, and as far as I can tell, I have a composite system of two fermions, and I know the composite state must be antisymmetric. Therefore, only the singlet state would be allowed.
Best Answer
There is no contradiction. A particle's spin is not its only attribute. A two-fermion state must be antisymmetric with respect to the exchange of all of their attributes, not just spin. If the state is symmetric with respect to the exchange of their spins, then it is antisymmetric with respect to the excahge of their other attributes (such as location or momentum, not shown in the OP), and conversely.
The four states shown are allowed whether the particles are distinguishable or "identical" (same species). For the identical-particle case, consider the two electrons in orthohelium and parahelium (https://en.wikipedia.org/wiki/Helium_atom). For the non-identical-particle case, consider the various possible states of a hydrogen atom, taking into account the parallel or antiparallel configuration of the electron/nucleon spin (http://www.feynmanlectures.caltech.edu/III_12.html).