Taking the time derivative of the expectation value of Q requires use of the product rule, and results in d d t ⟨ Q ⟩ = d d t ⟨ ψ ψ | Q | ψ ψ ⟩ = ( d d t ⟨ ψ ψ | ) Q | ψ ψ ⟩ + ⟨ ψ ψ | d Q d t | ψ ψ ⟩ + ⟨ ψ ψ | Q ( d d t | ψ ψ ⟩ ) = − − 1 i ℏ ℏ ⟨ H ψ ψ | Q | ψ ψ ⟩ + ⟨ ψ ψ | d Q d t | ψ ψ ⟩ + 1 i ℏ ℏ ⟨ ψ ψ | Q | H ψ ψ ⟩ = − − 1 i ℏ ℏ ⟨ ψ ψ | H Q | ψ ψ ⟩ + ⟨ ψ ψ | d Q d t | ψ ψ ⟩ + 1 i ℏ ℏ ⟨ ψ ψ | Q H | ψ ψ ⟩ = − − 1 i ℏ ℏ ⟨ ψ ψ | [ H , Q ] | ψ ψ ⟩ + ⟨ ψ ψ | d Q d t | ψ ψ ⟩