Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 4 / 58 / a / iv / Solution

The inference is most useful when the contributing normalized amplitudes are large. Positivity gives a precise illustration: let $f_h=F(h)/F(0)$ and regard $\rho (x)/F(0)$ as a probability density over a period. From $⟨{\cos }^2(2\pi hx)⟩\ge ⟨\cos (2\pi hx){⟩}^2$,Thus $|f_h|>1/\sqrt{2}$ forces $F(2h)>0$. If $F(h)$ is weak, this bound allows either sign; a small $F(2h)$ also contributes too little to the density sketch to impose a persuasive sign constraint. Large unnormalized intensity alone should not be confused with a large amplitude relative to $F(0)$.