## 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.

Hint: $$C$$ is a unitary matrix.