Heisenberg Picture
Operators have time dependence. \(|\alpha,t_0;t\rangle_H = |\alpha,t_0\rangle\). \(A_H(0) = A_S(0)\). \(A_H(t) = \hat{U}^\dagger(t,t_0)A_S(0)\hat{U}(t,t_0)\).
\(\langle b|A'|a\rangle = \langle b|U^\dagger AU|a\rangle\).
Equation of Motion
\(i\hbar\frac{\partial}{\partial t}(\hat{U})|\Psi\rangle = \hat{H}\hat{U}|\Psi\rangle \Rightarrow -i\hbar\frac{\partial}{\partial t}(\hat{U}^\dagger) = \hat{U}^\dagger\hat{H}\)
\(\frac{d}{dt}A_H = \frac{\partial}{\partial t}(\hat{U}^\dagger)A_S\hat{U} + \hat{U}^\dagger\frac{\partial}{\partial t}(A_S)\hat{U} + \hat{U}^\dagger A_S\frac{\partial}{\partial t}(\hat{U}) = \frac{\partial}{\partial t}(\hat{U}^\dagger)A_S\hat{U} + \hat{U}^\dagger A_S\frac{\partial}{\partial t}(\hat{U}) = \frac{1}{i\hbar}(-\hat{H}\hat{U}^\dagger A_S\hat{U} + \hat{U}^\dagger A_S\hat{U}\hat{H}) = -\frac{i}{\hbar}[A_H,H]\) (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).
\(\frac{d}{dt}A_H = -\frac{i}{\hbar}[A_H,H]\).