\(i\hbar\frac{\partial A_{H\ }\left(t\right)}{\partial t}\ =\ -H_H\left(t\right)A_H\left(t\right)\ +\ A_H\left(t\right)H_H\left(t\right)\ +\ i\hbar\left(\frac{\partial A_s\left(t\right)}{\partial t}\right)_H\)
\(i\hbar\frac{\partial A_H\left(t\right)}{\partial t\ }\ =\ \left[A_H\left(t\right),H_H\left(t\right)\right]\ +\ i\hbar\left(\frac{\partial A_s\left(t\right)}{\partial t}\right)_H\)
c\(A_s:\ \left[A_s,\ H_s\right]\ =0\ \)  \(\frac{\ \partial A_H\left(t\right)}{\partial t}=0\)
Comments 
1. \(i\hbar\frac{\partial A_H\left(t\right)}{\partial t}\ =\ \left[A_H\left(t\right),\ H_H\left(t\right)\right]\)
2. \(i\hbar<\psi,t\left|A_s\right|\psi,t>\ =i\hbar<\psi,0\left|A_H\left(t\right)\right|\psi,0>\)                                                       \(=\ <\psi,0\left|i\hbar\frac{\partial A_H\left(t\right)}{\partial t}\right|\psi,0>\)
= \(=<\psi,0\left|\left[A_H\left(t\right),H_H\left(t\right)\right]\right|\psi,0>\)
                                                            imp => \(i\hbar\frac{\partial\ }{\partial t}<A_H\left(t\right)>\ =\ <\left[\left[A_H\left(t\right),H_H\left(t\right)\right]\right]>\)
\(i\hbar\frac{\partial\ }{\partial t}<A_s>\ =\ <\left[\left[A_s,\ H_s\right]\right]>\) <- most imp\(\ \ \ \left[\ \ \vec{A}\ ,\ \vec{B}\ \ \right]\ =\vec{\ \ \ A.\vec{B}}\ -\ \vec{\ \ \ B.\ \vec{A}}\)