In class we were shown an adiabatic approximation where we started with (or worked our way towards) a representation of the form
where were normalized energy eigenkets for the (slowly) evolving Hamiltonian
In the problem sets we were shown a different adiabatic approximation, where are starting point is
For completeness, here’s a walk through of the general amplitude derivation that’s been used.
We operate with our energy identity once again
Bra’ing with , and split the sum into and parts
In this form we can make an “Adiabatic” approximation, dropping the terms, and integrate
Evaluating at , fixes the integration constant for
Observe that this is very close to the starting point of the adiabatic approximation we performed in class since we end up with
So, to perform the more detailed approximation, that started with 1.1, where we ended up with all the cross terms that had both and Berry phase dependence, we have only to generalize by replacing with .