Swapping the labels in the first term then gives 1 = 1 Z ∑ ∑ α α , α α ′ ( e − − β β E α α ′ − − ζ ζ e − − β β E α α ) | ⟨ ⟨ α α ∣ ∣ ψ ψ k † † ∣ ∣ α α ′ ⟩ ⟩ | 2 , which is exactly the result of the integration of ρ.