Past exam of the theoretical physics II course of the University of Cambridge / 2014 / Paper 1 / 5 / d

Binomial factorial moments give $⟨N_1(N_1-1)⟩=N(N-1)p^2$, its mode-zero counterpart $N(N-1)(1-p{)}^2$, and $⟨N_0{N}_1⟩=N(N-1)p(1-p)$. Consequently the exact projected expectation isSince $p(1-p)={\sin }^2\chi /4$ and $n=N/(2\pi )$,The finite-$N$ correction is simply the replacement $N\mapsto N-1$ in the interaction term. Number correlations must be retained to obtain that correction.