## Definitions and notations

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 $$\hat{P}_{ij}\Psi_{\lambda}(x_1, x_2, \dots,x_i,\dots,x_j,\dots,x_A)= \beta\Psi_{\lambda}(x_1, x_2, \dots,x_i,\dots,x_j,\dots,x_A),$$ 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 $$\beta = -1$$.