It is important to study a periodic perturbation with a given frequency ω ω since arbitrary perturbations can be constructed from periodic perturbations of different frequencies. Since H ′ ( t ) must be Hermitian, we must assume H ′ ( t ) = F e − − i ω ω t + F † † e i ω ω t , where F is a time independent operator. The solution for this case is a f ( t ) = − − ⟨ ⟨ f | F | i ⟩ ⟩ e i ( ω ω f i − − ω ω ) t ℏ ℏ ( ω ω f i − − ω ω ) − − ⟨ ⟨ f | F † † | i ⟩ ⟩ e i ( ω ω f i + ω ω ) t ℏ ℏ ( ω ω f i + ω ω ) . This expression is valid only when the denominators in the above expression is non-zero, i.e., for a given initial state with energy E i , the final state energy must be such that E f − − E i ≠ ≠ ± ± ℏ ℏ ω ω . Not only the denominators must be non-zero, but also must not be small since a f is supposed to be small.