Time-dependent Schrieffer-Wolff Transformation The time-independent Schrieffer-Wolff Transformation has the form: $H' = e^{S}He^{-S}$. I see that the time-dependent form is: $H' = e^{S}He^{-S} +i\hbar\frac{\delta}{\delta t}(e^{S})e^{-S}$. Why exactly does the time-dependent form have this extra term?
 A: Consider the transformation
$$|\psi'\rangle=e^S|\psi\rangle$$
We are looking for the transformed Hamiltonian $H'$ such that
$$H'|\psi'\rangle=i\hbar{d\over dt}|\psi'\rangle
\ \Leftrightarrow\ H|\psi\rangle=i\hbar{d\over dt}|\psi\rangle$$
Replacing by the expression of $|\psi'\rangle$, one gets
$$\eqalign{
H'e^S|\psi\rangle
&=i\hbar{d\over dt}\left(e^S|\psi\rangle\right)\cr
&=i\hbar \left({de^S\over dt}+e^S{d\over dt}\right)|\psi\rangle\cr
&=\left(i\hbar{de^S\over dt}+e^SH\right)|\psi\rangle\cr
}$$
Multiplying both sides on the left by $e^{-S}$,
$$e^{-S}H'e^S|\psi\rangle
 =\left(i\hbar e^{-S}{dS\over dt}+H\right)|\psi\rangle$$
for any $|\psi\rangle$, so
$$e^{-S}H'e^S=i\hbar e^{-S}{dS\over dt}+H$$
and finally
$$H'=i\hbar {dS\over dt}e^{-S}+e^SHe^{-S}$$
A: The idea is to transfer all time dependence into part of H which contains small parameter. In general  S will then be time dependent as well.
I think the origin of this term can be traced back to the principle of least action and canonical transformations. It can no longer be assumed that Hamilton's equations are unchanged when the generating function of the canonical transformation is time dependent. If you consider a  general canonical transformation for time dependent H, you can see that you cannot just express H in terms of new variables but there is an additive term which is the time derivative of the generating function of this transformation.
