Exercise 1: Hartree-Fock Slater determinant

Consider a Slater determinant built up of orthogonal 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 \( a=1,2,\dots,A \).

a) Show that the new basis is orthogonal.

b) 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 \( C \).

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.