We first consider the case where the final states are discrete. The expansion of a state in the perturbed system at a time t is | ψ ψ ( t ) ⟩ ⟩ = ∑ ∑ n a n ( t ) e − − i E n t / ℏ ℏ | n ⟩ ⟩ . The coefficients a n (t) are yet unknown functions of time yielding the probability amplitudes in the Dirac picture. This state obeys the time-dependent Schrödinger equation: