### D.55 Why the Fock operator is Hermitian

To verify that the Fock operator is Hermitian, first note that is Hermitian since it is an Hamiltonian. Next if you form the inner product , the first term in , the Coulomb term, can be taken to the other side since it is just a real function. The second term, the exchange one, produces the inner product,

and if you take the operator to the other side, you get

and writing out these inner products as six-di­men­sion­al spatial integrals and sums over spin, you see that they are the same.