I am reading section 8.5.1 of http://f3.tiera.ru/2/P_Physics/PS_Solid%20state/Giuliani%20G.,%20Vignale%20G.%20Quantum%20theory%20of%20the%20electron%20liquid%20%28CUP,%202005%29%28ISBN%200521821126%29%28799s%29_PS_.pdf [Broken] (page 442 of the book, page 465 of the pdf). The author claims the discontinuous jump in the occupation number is of size Z, where [itex]Z = |\langle 0,N+1|a^\dagger |0,N\rangle|^2 = |\langle 0,N-1|a |0,N\rangle|^2[/itex] I.e. Z is the probability that the system will be in an N+1 (N-1) ground state after an electron is added (removed). They say this follows from equation 8.123 but I don't find it obvious how. Is there an easy logical path from equation 8.123 to the above relation?