This exercise serves to convince you about the relation between two different single-particle bases, where one could be our new Hartree-Fock basis and the other a harmonic oscillator basis.

Consider a Slater determinant built up of single-particle orbitals \( \psi_{\lambda} \), with \( \lambda = 1,2,\dots,A \). The unitary transformation $$ \psi_a = \sum_{\lambda} C_{a\lambda}\phi_{\lambda}, $$ brings us into the new basis. The new basis has quantum numbers \( a=1,2,\dots,A \). Show that the new basis is orthonormal. Show that the new Slater determinant constructed from the new single-particle wave functions can be written as the determinant based on the previous basis and the determinant of the matrix \( C \). Show that the old and the new Slater determinants are equal up to a complex constant with absolute value unity. (Hint, \( C \) is a unitary matrix).

Starting with the second quantization representation of the Slater determinant $$ \Phi_{0}=\prod_{i=1}^{n}a_{\alpha_{i}}^{\dagger}|0\rangle, $$ use Wick's theorem to compute the normalization integral \( \langle\Phi_{0}|\Phi_{0}\rangle \).