If the Hamiltonian has explicit time-dependence (for example, if the quantum system interacts with an applied external electric field that varies in time), it will usually be advantageous to include the explicitly time-dependent terms with H 1 , S , leaving H 0 , S time-independent. We proceed assuming that this is the case. If there is a context in which it makes sense to have H 0 , S be time-dependent, then one can proceed by replacing e ± ± i H 0 , S t / ℏ ℏ by the corresponding time-evolution operator in the definitions below.