The following discussion uses the bra–ket notation. One may write ψ ψ ( x ) = ⟨ ⟨ x | ψ ψ ⟩ ⟩ = ∫ ∫ d p ⟨ ⟨ x | p ⟩ ⟩ ⟨ ⟨ p | ψ ψ ⟩ ⟩ = ∫ ∫ d p e i x p / ℏ ℏ ψ ψ ~ ~ ( p ) 2 π π ℏ ℏ , so the tilde represents the Fourier transform, in converting from coordinate space to momentum space. It then holds that p ^ ^ = ∫ ∫ d p | p ⟩ ⟩ p ⟨ ⟨ p | = − − i ℏ ℏ ∫ ∫ d x | x ⟩ ⟩ d d x ⟨ ⟨ x | , that is, the momentum acting in coordinate space corresponds to spatial frequency, ⟨ ⟨ x | p ^ ^ | ψ ψ ⟩ ⟩ = − − i ℏ ℏ d d x ψ ψ ( x ) .