where N i is the number of particles in state | ϕ ϕ i ⟩ ⟩ . The above equality can be proven by noting that a ( ϕ ϕ i ) | ϕ ϕ 1 , ϕ ϕ 2 , ⋯ ⋯ , ϕ ϕ i − − 1 , ϕ ϕ i , ϕ ϕ i + 1 , ⋯ ⋯ , ϕ ϕ n ⟩ ⟩ ν ν = N i | ϕ ϕ 1 , ϕ ϕ 2 , ⋯ ⋯ , ϕ ϕ i − − 1 , ϕ ϕ i + 1 , ⋯ ⋯ , ϕ ϕ n ⟩ ⟩ ν ν a † † ( ϕ ϕ i ) | ϕ ϕ 1 , ϕ ϕ 2 , ⋯ ⋯ , ϕ ϕ i − − 1 , ϕ ϕ i + 1 , ⋯ ⋯ , ϕ ϕ n ⟩ ⟩ ν ν = N i | ϕ ϕ 1 , ϕ ϕ 2 , ⋯ ⋯ , ϕ ϕ i − − 1 , ϕ ϕ i , ϕ ϕ i + 1 , ⋯ ⋯ , ϕ ϕ n ⟩ ⟩ ν ν then N i ^ ^ | Ψ Ψ ⟩ ⟩ ν ν = a † † ( ϕ ϕ i ) a ( ϕ ϕ i ) | ϕ ϕ 1 , ϕ ϕ 2 , ⋯ ⋯ , ϕ ϕ i − − 1 , ϕ ϕ i , ϕ ϕ i + 1 , ⋯ ⋯ , ϕ ϕ n ⟩ ν ν = N i a † † ( ϕ ϕ i ) | ϕ ϕ 1 , ϕ ϕ 2 , ⋯ ⋯ , ϕ ϕ i − − 1 , ϕ ϕ i + 1 , ⋯ ⋯ , ϕ ϕ n ⟩ ν ν = N i N i | ϕ ϕ 1 , ϕ ϕ 2 , ⋯ ⋯ , ϕ ϕ i − − 1 , ϕ ϕ i , ϕ ϕ i + 1 , ⋯ ⋯ , ϕ ϕ n ⟩ ν ν = N i | Ψ Ψ ⟩ ⟩ ν ν