For states k ≠ ≠ i , a k ( t ) becomes non-zero due to H ′ ≠ ≠ 0 , and these are assumed to be small due to the weak perturbation. The coefficient a i ( t ) which is unity in the unperturbed state, will have a weak contribution from H ′ . Hence, one can plug in the zeroth-order form a n ( t ) = δ δ n , i into the above equation to get the first correction for the amplitudes a k ( t ) : i ℏ ℏ d a k ( t ) d t = ⟨ ⟨ k | H ′ | i ⟩ ⟩ e i t ( E k − − E i ) / ℏ ℏ , whose integral can be expressed as i ℏ ℏ a k ( t ) = ∫ ∫ 0 t ⟨ ⟨ k | H ′ ( t ′ ) | i ⟩ ⟩ e i ω ω k i t ′ d t ′ with ω ω k i ≡ ≡ ( E k − − E i ) / ℏ ℏ , for a state with a i (0) = 1, a k (0) = 0, transitioning to a state with a k (t).