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Two spin-1/2 particles either are part of a spin-1 triplet or a spin-0 singlet. The singlet is antisymmetric but bosons need to be symmetric wave functions.

So does the spatial part of the wave function need to be antisymmetric in the singlet and symmetric in the triplet? What if we're only considering the spin state, how can we have both symmetric and antisymmetric spins? Please answer both questions. Thanks

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The spatial part of the wave function is antisymmetric for the triplet and symmetric for the singlet. You can check this by considering the He atom. Its ground state is a singlet with two electrons in its 1s orbital. Clearly a doubly occupied 1s orbital is symmetric.

The two fermion wave function must be antisymmetric under fermion exchange. Since the spin part of the triplet is symmetric, its spatial part must be antisymmetric. The spin part of the singlet must be antisymmetric, in order to be orthogonal to the singlet, so the singlet spatial part must be symmetric.

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