\(A_H\)(t) is defined as a Heisenberg operator associated with \(A_s\) as the object in between the time equal zero states :
\(A_H\left(t,0\right)\ =\ U^+\left(t,0\right)A_sU\left(t,0\right)\)
Now we will take some important consequences of the given definition :
1. At t = 0 Heisenberg operator becomes equal to Schrödinger operator :
\(A_H\left(0\right)=A_s\)
2. The Heisenberg operator associated with the product of Schrödinger operators is equal to the product of the corresponding Heisenberg operators:
\(C_S\ =\ A_SB_S\) -> \(C_H\left(t\right)=A_H\left(t\right)B_H\left(t\right)\)
{ \(C_H\left(t\right)=U^+\left(t,0\right)C_sU\left(t,0\right)\ =\ U^+\left(t,0\right)A_sB_sU\left(t,0\right)\)
\(=U^+\left(t,0\right)A_sU\left(t,0\right)U^+\left(t,0\right)B_sU\left(t,0\right)\ =\ A_H\left(t\right)B_H\left(t\right)\)
}
3.
Heisenberg equation of motion
c\(i\hbar\frac{\partial U\left(t,t_0\right)}{\partial t}\ =\ \ H_s\left(t\right)U\left(t,t_0\right)\)( 1)
c\(i\hbar\frac{\partial U^+\left(t,t_0\right)}{\partial t}\) = \(-U^+\left(t,t_0\right)H_s\left(t\right)\) (2)
c\(i\hbar\frac{\partial A_H\left(t\right)\ }{\ \partial t}\) = \(\left(i\hbar\ \frac{\partial U^+\left(t,0\right)}{\ \partial t}\right)A_s\left(t\right)U\left(t,0\right)\)
+ \(\)\(U^+\left(t,0\right)A_s\left(t\right)\left(i\hbar\frac{\partial U\left(t,0\right)}{\partial t}\right)\)
+ \(U^+\left(t,0\right)i\hbar\frac{\partial A_S\left(t\right)}{\partial t}U\left(t,0\right)\)
\(i\hbar\frac{\partial\ }{\partial t}A_H\left(t\right)\ =\ -U^+\left(t,0\right)H_s\left(t\right)A_s\left(t\right)U\left(t,0\right)\)
\(+\ U^+\left(t,0\right)A_s\left(t\right)H_s\left(t\right)U\left(t,0\right)\)
\(+\ \ U^+\left(t,0\right)i\hbar\frac{\partial A_s\left(t\right)}{\partial t}U\left(t,0\ \right)\)