Assume | ψ ψ 2 ⟩ ⟩ = T ^ ^ ( x ) | ψ ψ ⟩ ⟩ as stated above. ⟨ ⟨ ψ ψ 2 | r ^ ^ | ψ ψ 2 ⟩ ⟩ = ( ⟨ ⟨ ψ ψ | ( T ^ ^ ( x ) ) † † ) r ^ ^ ( T ^ ^ ( x ) | ψ ψ ⟩ ⟩ ) = ⟨ ⟨ ψ ψ | r ^ ^ | ψ ψ ⟩ ⟩ + x using the normalization condition ⟨ ⟨ ψ ψ | ψ ψ ⟩ ⟩ = 1 , and the commutator result proven in a previous section.