In the general case, which includes mixed states, it is the Wigner transform of the density matrix : W ( x , p ) = 1 π π ℏ ℏ ∫ ∫ − − ∞ ∞ ∞ ∞ ⟨ ⟨ x − − y | ρ ρ ^ ^ | x + y ⟩ ⟩ e 2 i p y / ℏ ℏ d y , where ⟨ x | ψ ⟩ = ψ (x). This Wigner transformation (or map) is the inverse of the Weyl transform, which maps phase-space functions to Hilbert-space operators, in Weyl quantization.