Heisenberg Picture

Operators have time dependence. $| α, t_{0}; t ⟩_{H} = | α, t_{0} ⟩$ . $A_{H} (0) = A_{S} (0)$ . $A_{H} (t) = {\hat{U}}^{†} (t, t_{0}) A_{S} (0) \hat{U} (t, t_{0})$ .

$⟨ b | A^{'} | a ⟩ = ⟨ b | U^{†} A U | a ⟩$ .

Equation of Motion

$i ℏ \frac{\partial}{\partial t} (\hat{U}) | Ψ ⟩ = \hat{H} \hat{U} | Ψ ⟩ \Rightarrow - i ℏ \frac{\partial}{\partial t} ({\hat{U}}^{†}) = {\hat{U}}^{†} \hat{H}$

$\frac{d}{d t} A_{H} = \frac{\partial}{\partial t} ({\hat{U}}^{†}) A_{S} \hat{U} + {\hat{U}}^{†} \frac{\partial}{\partial t} (A_{S}) \hat{U} + {\hat{U}}^{†} A_{S} \frac{\partial}{\partial t} (\hat{U}) = \frac{\partial}{\partial t} ({\hat{U}}^{†}) A_{S} \hat{U} + {\hat{U}}^{†} A_{S} \frac{\partial}{\partial t} (\hat{U}) = \frac{1}{i ℏ} (- \hat{H} {\hat{U}}^{†} A_{S} \hat{U} + {\hat{U}}^{†} A_{S} \hat{U} \hat{H}) = - \frac{i}{ℏ} [A_{H}, H]$ (Note that the time evolution operator uses the Hamiltonian as its generator, i.e. it is a function of the Hamiltonian and because any function of an operator commutes with the operator).

$\frac{d}{d t} A_{H} = - \frac{i}{ℏ} [A_{H}, H]$ .