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

The occupation state is an eigenstate of the projected Quantum Hamiltonian. Since $a_j^{†}{a}_j^{†}{a}_j{a}_j=N_j(N_j-1)$,Equivalently, the bracket is $N(N-1)+2N_0{N}_1$, where $N=N_0+N_1$; there is no self-interaction for a single particle.