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In terms of statistical physics I thought the microcanonical partition function can be interpreted as summing over all possible quantum numbers. Neglecting indistinguishability in the case of two electrons this should result in the following possible states:
$$\left|\uparrow\uparrow\right\rangle,\left|\uparrow\uparrow\right.\rangle,\left|\downarrow\downarrow\right.\rangle,\left|\downarrow\downarrow\right.\rangle,\left|\uparrow\downarrow\right.\rangle,\left|\downarrow\uparrow\right.\rangle$$ Now introducing the factor $$\frac{1}{N!}$$ to account for the indistinguishability with respect to the exchange of the electrons results with $N=2$ in $3$ total possible states. However, the Spin-1/2-system can be realized in $4$ states, the singlett and the triplett ones. I am not convinced at all with my argumentation but can´t find my mistake, either. I am happy if you could help me!

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I think there are still two states in your list, with A up B down and B up A down. Similarly there are also A down B up and B down A up.

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