Solution (source code)

= Solution

The <Pauli principle> requires the complete <wavefunction> of identical <fermions> to be antisymmetric under exchanging the full coordinates, including <spin>:
$$
\boxed{\Psi(x_2,x_1)=-\Psi(x_1,x_2),\qquad x_i=(\mathbf r_i,s_i).}
$$
It is the total spatial-and-spin state that changes sign; the spatial factor alone need not be antisymmetric.