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I am new to this forum! I write here hoping someone can help me.

I have found a statement in my quantum mechanics book that I really don't understand.

We have a system of two electrons. If both are in a singlet state, the orbital part of the wavefunction of the system must be symmetric. Otherwise, if they are both in a triplet state, it must be antisymmetric.

In my previous studies I have seen that a singlet state is antisymmetric, and that a triplet one is symmetric. I know that in this case we have a system of two particles, but how do you explain that?

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The singlet and triplet labels apply to pairs of electrons, or more generally any spin-half fermions.

For these types of particles, the wavefunction must be antisymmetric under particle exchange. Each particle has a spin and orbital part. To swap the particles, we must swap both the spin and orbital parts. This means if one is symmetric, the other must be antisymmetric to make the whole swap antisymmetric. In other words, a swap induces either $+1$ or $-1$. We want the overall value to be $-1$ so the individual ones must be one of each.

In your case where you see the singlet as antisymmetric and the triplet as symmetric, this refers to the spin part. As a result, the orbital part must be the other way around.

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If both are in a singlet state

They can't be "both" in a singlet state: only the system of these two electrons can, not the electrons individually.

In my previous studies I have seen that a singlet state is antisymmetric, and that a triplet one is symmetric.

This is true, but for the spinor part of the wavefunction. The orbital is still symmetric in a singlet state and antisymmetric in a triplet state. Remember that the final spin-orbital must be antisymmetric, so spatial and spin parts must have different exchange parity.

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