Expanding the Hamiltonian and the state, we see that, to first order, ( H 0 + H ′ − − i ℏ ℏ ∂ ∂ ∂ ∂ t ) ∑ ∑ n a n ( t ) | n ⟩ ⟩ e − − i t E n / ℏ ℏ = 0 , where E n and | n ⟩ are the stationary eigenvalues and eigenfunctions of H 0.