If the particular single-particle states that are chosen are 'single-particle energy eigenstates', i.e. [ H − − μ μ N , ψ ψ α α † † ] = ξ ξ α α ψ ψ α α † † , then for | n ⟩ ⟩ an eigenstate: ( H − − μ μ N ) ∣ ∣ n ⟩ ⟩ = E n ∣ ∣ n ⟩ ⟩ , so is ψ ψ α α ∣ ∣ n ⟩ ⟩ : ( H − − μ μ N ) ψ ψ α α ∣ ∣ n ⟩ ⟩ = ( E n − − ξ ξ α α ) ψ ψ α α ∣ ∣ n ⟩ ⟩ , and so is ψ ψ α α † † ∣ ∣ n ⟩ ⟩ : ( H − − μ μ N ) ψ ψ α α † † ∣ ∣ n ⟩ ⟩ = ( E n + ξ ξ α α ) ψ ψ α α † † ∣ ∣ n ⟩ ⟩ .