In classical statistical mechanics, the probability density (with respect to Liouville measure) obeys the Liouville equation i ∂ ∂ ∂ ∂ t ρ ρ ( x , p , t ) = L ^ ^ ρ ρ ( x , p , t ) with the self-adjoint Liouvillian L ^ ^ = − − i ∂ ∂ H ( x , p ) ∂ ∂ p ∂ ∂ ∂ ∂ x + i ∂ ∂ H ( x , p ) ∂ ∂ x ∂ ∂ ∂ ∂ p , where H ( x , p ) denotes the classical Hamiltonian (i.e. the Liouvillian is i times the Hamiltonian vector field considered as a first order differential operator). The same dynamical equation is postulated for the KvN wavefunction i ∂ ∂ ∂ ∂ t ψ ψ ( x , p , t ) = L ^ ^ ψ ψ ( x , p , t ) , thus ∂ ∂ ∂ ∂ t ψ ψ ( x , p , t ) = [ − − ∂ ∂ H ( x , p ) ∂ ∂ p ∂ ∂ ∂ ∂ x + ∂ ∂ H ( x , p ) ∂ ∂ x ∂ ∂ ∂ ∂ p ] ψ ψ ( x , p , t ) , and for its complex conjugate ∂ ∂ ∂ ∂ t ψ ψ ∗ ∗ ( x , p , t ) = [ − − ∂ ∂ H ( x , p ) ∂ ∂ p ∂ ∂ ∂ ∂ x + ∂ ∂ H ( x , p ) ∂ ∂ x ∂ ∂ ∂ ∂ p ] ψ ψ ∗ ∗ ( x , p , t ) . From ρ ρ ( x , p , t ) = ψ ψ ∗ ∗ ( x , p , t ) ψ ψ ( x , p , t ) follows using the product rule that ∂ ∂ ∂ ∂ t ρ ρ ( x , p , t ) = [ − − ∂ ∂ H ( x , p ) ∂ ∂ p ∂ ∂ ∂ ∂ x + ∂ ∂ H ( x , p ) ∂ ∂ x ∂ ∂ ∂ ∂ p ] ρ ρ ( x , p , t ) which proves that probability density dynamics can be recovered from the KvN wavefunction.