it follows that the expectation value changes sign under inversion. We used here the fact that I , being a symmetry operator, is unitary : I − − 1 = I ∗ ∗ and by definition the Hermitian adjoint I ∗ ∗ may be moved from bra to ket and then becomes I ∗ ∗ ∗ ∗ = I . Since the only quantity that is equal to minus itself is the zero, the expectation value vanishes,