Substituting into the time-dependent Schrödinger equation H | ψ ψ ( t ) ⟩ ⟩ = i ℏ ℏ ∂ ∂ ∂ ∂ t | ψ ψ ( t ) ⟩ ⟩ and premultiplying by ⟨ ⟨ i | produces d a i ( t ) d t = − − i ∫ ∫ C d ε ε Ω Ω i ε ε e − − i ( ω ω − − ω ω i ) t a ε ε ( t ) , where Ω Ω i ε ε = ⟨ ⟨ i | H ′ | ε ε ⟩ ⟩ / ℏ ℏ , and premultiplying by ⟨ ⟨ ε ε ′ | produces d a ε ε ( t ) d t = − − i Ω Ω ε ε i e i ( ω ω − − ω ω i ) t a i ( t ) . We made use of the normalisation ⟨ ⟨ ε ε ′ | ε ε ⟩ ⟩ = δ δ ( ε ε ′ − − ε ε ) . Integrating the latter and substituting into the former, d a i ( t ) d t = − − ∫ ∫ C d ε ε Ω Ω i ε ε Ω Ω ε ε i ∫ ∫ 0 t d t ′ e − − i ( ω ω − − ω ω i ) ( t − − t ′ ) a i ( t ′ ) . It can be seen here that d a i / d t at time t depends on a i at all earlier times t ′ , i.e. it is non-Markovian. We make the Markov approximation, i.e. that it only depends on a i at time t (which is less restrictive than the approximation that a i ≈ ≈ 1 used above, and allows the perturbation to be strong) d a i ( t ) d t = ∫ ∫ C d ε ε | Ω Ω i ε ε | 2 a i ( t ) ∫ ∫ 0 t d T e − − i Δ Δ T , where T = t − − t ′ and Δ Δ = ω ω − − ω ω i . Integrating over T , d a i ( t ) d t = − − 2 π π ℏ ℏ ∫ ∫ C d ε ε | Ω Ω i ε ε | 2 a i ( t ) e − − i Δ Δ t / 2 sin ⁡ ⁡ ( Δ Δ t / 2 ) π π ℏ ℏ Δ Δ , The fraction on the right is a nascent Dirac delta function, meaning it tends to δ δ ( ε ε − − ε ε i ) as t → → ∞ ∞ (ignoring its imaginary part which leads to an unimportant energy shift, while the real part produces decay ). Finally d a i ( t ) d t = − − 2 π π ℏ ℏ | Ω Ω i ε ε i | 2 a i ( t ) , which can have solutions: a i ( t ) = exp ⁡ ⁡ ( − − Γ Γ i → → ε ε i t / 2 ) , i.e., the decay of population in the initial discrete state is P i ( t ) = | a i ( t ) | 2 = exp ⁡ ⁡ ( − − Γ Γ i → → ε ε i t ) where Γ Γ i → → ε ε i = 2 π π ℏ ℏ | Ω Ω i ε ε i | 2 = 2 π π ℏ ℏ | ⟨ ⟨ i | H ′ | ε ε ⟩ ⟩ | 2 .