## One more adiabatic pertubation derivation.

Posted by peeterjoot on December 8, 2011

# Motivation.

I liked one of the adiabatic pertubation derivations that I did to review the material, and am recording it for reference.

# Build up.

In time dependent pertubation we started after noting that our ket in the interaction picture, for a Hamiltonian , took the form

Here we have basically assumed that the time evolution can be factored into a portion dependent on only the static portion of the Hamiltonian, with some other operator , providing the remainder of the time evolution. From 2.1 that operator is found to behave according to

but for our purposes we just assumed it existed, and used this for motivation. With the assumption that the interaction picture kets can be written in terms of the basis kets for the system at we write our Schr\”{o}dinger ket as

where are the energy eigenkets for the initial time equation problem

# Adiabatic case.

For the adiabatic problem, we assume the system is changing very slowly, as described by the instantanious energy eigenkets

Can we assume a similar representation to 2.3 above, but allow to vary in time? This doesn’t quite work since are no longer eigenkets of

Operating with does not give the proper time evolution of , and we will in general have a more complex functional dependence in our evolution operator for each . Instead of an dependence in this time evolution operator let’s assume we have some function to be determined, and can write our ket as

Operating on this with our energy operator equation we have

Here I’ve written . In our original time dependent pertubaton the term was , so this killed off the . If we assume this still kills off the , we must have

and are left with

Bra’ing with we have

or

The LHS is a perfect differential if we introduce an integration factor , so we can write

This suggests that we want to form a new function

or

Plugging this into our assumed representation we have a more concrete form

Writing

this becomes

## A final pass.

Now that we have what appears to be a good representation for any given state if we wish to examine the time evolution, let’s start over, reapplying our instantaneous energy operator equality

Bra’ing with we find

Since the first and third terms cancel leaving us just

where and .

# Summary

We assumed that a ket for the system has a representation in the form

where and are given or to be determined. Application of our energy operator identity provides us with an alternate representation that simplifes the results

With

we find that our dynamics of the coefficients are related by

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