The quantity ⟨ ⟨ x , p | Ψ Ψ ( t ) ⟩ ⟩ is the probability amplitude for a classical particle to be at point x with momentum p at time t . According to the axioms above, the probability density is given by ρ ρ ( x , p ; t ) = | ⟨ ⟨ x , p | Ψ Ψ ( t ) ⟩ ⟩ | 2 . Utilizing the identity ∂ ∂ ∂ ∂ t ρ ρ ( x , p ; t ) = ⟨ ⟨ Ψ Ψ ( t ) | x , p ⟩ ⟩ ∂ ∂ ∂ ∂ t ⟨ ⟨ x , p | Ψ Ψ ( t ) ⟩ ⟩ + ⟨ ⟨ x , p | Ψ Ψ ( t ) ⟩ ⟩ ( ∂ ∂ ∂ ∂ t ⟨ ⟨ x , p | Ψ Ψ ( t ) ⟩ ⟩ ) ∗ ∗ as well as (KvN dynamical eq in xp), we recover the classical Liouville equation