Consider the Slater determinant $$ \Phi_{\lambda}^{AS}(x_{1}x_{2}\dots x_{N};\alpha_{1}\alpha_{2}\dots\alpha_{N}) =\frac{1}{\sqrt{N!}}\sum_{p}(-)^{p}P\prod_{i=1}^{N}\psi_{\alpha_{i}}(x_{i}). $$ where \( P \) is an operator which permutes the coordinates of two particles. We have assumed here that the number of particles is the same as the number of available single-particle states, represented by the greek letters \( \alpha_{1}\alpha_{2}\dots\alpha_{N} \).
a) Write out \( \Phi^{AS} \) for \( N=3 \).
b) Show that $$ \int dx_{1}dx_{2}\dots dx_{N}\left\vert \Phi_{\lambda}^{AS}(x_{1}x_{2}\dots x_{N};\alpha_{1}\alpha_{2}\dots\alpha_{N}) \right\vert^{2} = 1. $$
c) Define a general onebody operator \( \hat{F} = \sum_{i}^N\hat{f}(x_{i}) \) and a general twobody operator \( \hat{G}=\sum_{i>j}^N\hat{g}(x_{i},x_{j}) \) with \( g \) being invariant under the interchange of the coordinates of particles \( i \) and \( j \). Calculate the matrix elements for a two-particle Slater determinant $$ \langle\Phi_{\alpha_{1}\alpha_{2}}^{AS}|\hat{F}|\Phi_{\alpha_{1}\alpha_{2}}^{AS}\rangle, $$ and $$ \langle\Phi_{\alpha_{1}\alpha_{2}}^{AS}|\hat{G}|\Phi_{\alpha_{1}\alpha_{2}}^{AS}\rangle. $$ Explain the short-hand notation for the Slater determinant. Which properties do you expect these operators to have in addition to an eventual permutation symmetry?