An important special case of the equation above is obtained if the Hamiltonian H S does not vary with time. Then the time-evolution operator can be written as U ( t ) = e − − i ℏ ℏ t H S , and hence H H ≡ ≡ H S ≡ ≡ H since U ( t ) now commutes with H . Therefore, ⟨ ⟨ A ⟩ ⟩ t = ⟨ ⟨ ψ ψ ( 0 ) | e i ℏ ℏ t H A S ( t ) e − − i ℏ ℏ t H | ψ ψ ( 0 ) ⟩ ⟩ and following the previous analyses, d d t A H ( t ) = i ℏ ℏ [ H , A H ( t ) ] + e i ℏ ℏ t H ( ∂ ∂ A S ∂ ∂ t ) e − − i ℏ ℏ t H .