This function must be real, because both P and − iD are Hermitian, ( P + i D ) | x ⟩ ⟩ = f ( x ) | x ⟩ ⟩ , rotating each state | x ⟩ ⟩ by a phase f (x), that is, redefining the phase of the wavefunction: ψ ψ ( x ) → → e − − i f ( x ) ψ ψ ( x ) . The operator iD is redefined by an amount: i D → → i D + f ( X ) , which means that, in the rotated basis, P is equal to − iD.