## Second form of adiabatic approximation.

Posted by peeterjoot on December 11, 2011

# Motivation.

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.

# Guts

We operate with our energy identity once again

where

Bra’ing with , and split the sum into and parts

Again writing

We have

In this form we can make an “Adiabatic” approximation, dropping the terms, and integrate

or

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 .

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