Consider two non-interacting Fermions (half-integer spin) confined in a 'box'. Construct the anti-symmetric wavefunctions and compare the corresponding ground-state energies of the two systems; one with particles of identical spin and the other with particles of opposite spin.
Now, the wavefunction with opposite spin is incredibly easy, One can just duplicate the spin wavefunction for helium, right?
What confuses me is the "ground state" energy of a wavefunction with two fermions of identical spin. Isn't that impossible? Wouldn't this look like non-excited helium with the same spin electrons in the 1S orbital (which can't be done)?