The two independent solutions are a n = A e i α α 1 t , a k = − − A ℏ ℏ α α 1 e i α α 1 t / ⟨ ⟨ n | F † † | k ⟩ ⟩ a n = B e − − i α α 2 t , a k = B ℏ ℏ α α 2 e − − i α α 2 t / ⟨ ⟨ n | F † † | k ⟩ ⟩ where α α 1 = − − 1 2 ε ε + Ω Ω , α α 2 = 1 2 ε ε + Ω Ω , Ω Ω = 1 4 ε ε 2 + | η η | 2 , η η = 1 ℏ ℏ ⟨ ⟨ k | F | n ⟩ ⟩