Symmetry of fermion wavefunction

Question

I'm studying identical particles and I'm thinking about something related to fermions. I have always heard that more than two electrons can't occupy the same energy level, because their spins are opposite and a third electron would have its spin oriented in the same direction of one of the other two. Now I'm studying the total spin, and actually pairs of electrons (spin $1/2$) have 4 ways of rearranging themselves $\left( |S, M\rangle\right)$: $$|1, 1\rangle = \uparrow \uparrow ,$$ $$|1, 0\rangle = \frac{1}{\sqrt{2}} \left(\uparrow \downarrow + \downarrow \uparrow \right),$$ $$|1, -1\rangle = \downarrow \downarrow,$$ $$|0, 0\rangle = \frac{1}{\sqrt{2}} \left(\uparrow \downarrow - \downarrow \uparrow \right).$$ The first 3 spin wavefunctions ($S = 1$) are symmetric, while the fourth one ($S = 0$) is antisymmetric. So, because of the symmetrization postulate, the total wavefunction of a fermion must be antisymmetric. Bearing this in mind, not only do we need for the electrons' spins to be opposite but they also have to be following the $|0, 0\rangle$ wavefuntion in order to be able to be in the same energy level, right?

Answer 1

The condition is on the overall wavefunction of a fermion to be antisymmetric. In case of two electron systems the overall wavefunction can be written as a product of the spatial part and the spin part.

This means that the $|0,0\rangle$ must have a symmetric space part (both electrons in the same orbital for example) and for the other three, the space part must be antisymmetric.

So yes. For them to be in the same level, the spin must compulsorily must be in the $|0,0\rangle$ state.