The sum rule can be proved by considering the expectation value of the commutator, 1 = 1 Z ∑ ∑ α α ⟨ ⟨ α α ∣ ∣ e − − β β ( H − − μ μ N ) [ ψ ψ k , ψ ψ k † † ] − − ζ ζ ∣ ∣ α α ⟩ ⟩ , and then inserting a complete set of eigenstates into both terms of the commutator: 1 = 1 Z ∑ ∑ α α , α α ′ e − − β β E α α ( ⟨ ⟨ α α ∣ ∣ ψ ψ k ∣ ∣ α α ′ ⟩ ⟩ ⟨ ⟨ α α ′ ∣ ∣ ψ ψ k † † ∣ ∣ α α ⟩ ⟩ − − ζ ζ ⟨ ⟨ α α ∣ ∣ ψ ψ k † † ∣ ∣ α α ′ ⟩ ⟩ ⟨ ⟨ α α ′ ∣ ∣ ψ ψ k ∣ ∣ α α ⟩ ⟩ ) .