[Physics] Why is Helium ground state a singlet

Question

I see that in the 1$s$ orbital the electrons have to have different spins, still there is two possible states,

• a singlet state with an anti-symmetric superposition of spins and symmetric spatial part
• a triplet state with symmetric superposition of spins and anti-symmetric spatial part

So, the question reduces to WHY the first one? I have read somewhere else that a ground state has to have a symmetric spatial wave function always but without further reasoning why this should be necessarily true.

Answer 1

Think of the spatial part alone. The unique combined state where both are in the 1s state is symmetric.

For the total state to be antisymmetric we must have that the spin part is antisymmetric (because the full state space is the tensor product of the spatial and the spin part). This is the lowest possible energy state.