where the Liouville operator L is defined using the commutator L = 1 ℏ ℏ [ H , ⋅ ⋅ ] in the quantum case and using the Poisson bracket L = − − i { H , ⋅ ⋅ } in the classical case. We assume here that the Hamiltonian does not have explicit time-dependence. The derivation can also be generalized towards time-dependent Hamiltonians. This equation is formally solved by