Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 81 / 2 / a / Solution

Write $b_k=⟨b_k⟩+\delta b_k$. The mean-field approximation drops the quadratic product of the fluctuations:For a Hermitian pairing interaction, the two linear terms are Hermitian conjugates. Defining the BCS anomalous average and pairing field by ${\Delta }_k=-{\sum }_{k^{\prime }}{V}_{k,k^{\prime }}⟨b_{k^{\prime }}⟩$, and writing ${\xi }_k={ϵ}_k-\mu$, givesThe final constant corrects the double counting in the reduced BCS pairing Hamiltonian.