Our Hamiltonian is invariant under the permutation (interchange) of two particles. Since we deal with fermions however, the total wave function is antisymmetric. Let \( \hat{P} \) be an operator which interchanges two particles. Due to the symmetries we have ascribed to our Hamiltonian, this operator commutes with the total Hamiltonian, $$ [\hat{H},\hat{P}] = 0, $$ meaning that \( \Psi_{\lambda}(x_1, x_2, \dots , x_A) \) is an eigenfunction of \( \hat{P} \) as well, that is $$ \hat{P}_{ij}\Psi_{\lambda}(x_1, x_2, \dots,x_i,\dots,x_j,\dots,x_A)= \beta\Psi_{\lambda}(x_1, x_2, \dots,x_j,\dots,x_i,\dots,x_A), $$ where \( \beta \) is the eigenvalue of \( \hat{P} \). We have introduced the suffix \( ij \) in order to indicate that we permute particles \( i \) and \( j \). The Pauli principle tells us that the total wave function for a system of fermions has to be antisymmetric, resulting in the eigenvalue \( \beta = -1 \).