The quantum mechanical wave function of a given state with quantum numbers \( \lambda \) (encompassing all quantum numbers needed to specify the system), ignoring time, is $$ \Psi_{\lambda}=\Psi_{\lambda}(x_1,x_2,\dots,x_A), $$ with \( x_i=(\boldsymbol{r}_i,\sigma_i,\tau_i) \) and the projections of \( \sigma_i \) and \( \tau_i \) take the values \( \{-1/2,+1/2\} \). We will hereafter always refer to \( \Psi_{\lambda} \) as the exact wave function, and if the ground state is not degenerate we label it as $$ \Psi_0=\Psi_0(x_1,x_2,\dots,x_A). $$