Consider now the case where the perturbation frequency is such that E k − − E n = ℏ ℏ ( ω ω + ε ε ) where ε ε is a small quantity. Unlike the previous case, not all terms in the sum over n in the above exact equation for a k ( t ) matters, but depends only on a n ( t ) and vice versa. Thus, omitting all other terms, we can write i ℏ ℏ d a k d t = ⟨ ⟨ k | F | n ⟩ ⟩ e i ε ε t a n , i ℏ ℏ d a n d t = ⟨ ⟨ n | F † † | k ⟩ ⟩ e − − i ε ε t a k .