## A different derivation of the adiabatic perturbation coefficient equation

Posted by peeterjoot on October 27, 2011

# Motivation.

Professor Sipe’s adiabatic perturbation and that of the text [1] in section 17.5.1 and section 17.5.2 use different notation for and take a slightly different approach. We can find Prof Sipe’s final result with a bit less work, if a hybrid of the two methods is used.

# Guts

Our starting point is the same, we have a time dependent slowly varying Hamiltonian

where our perturbation starts at some specific time from a given initial state

We assume that instantaneous eigenkets can be found, satisfying

Here I’ll use instead of the that we used in class because its easier to write.

Now suppose that we have some arbitrary state, expressed in terms of the instantaneous basis kets

where

and (using the notation in the text, not in class) is to be determined.

For this state, we have at the time just before the perturbation

The question to answer is: How does this particular state evolve?

Another question, for those that don’t like sneaky bastard derivations, is where did that magic factor of come from in our superposition state? We will see after we start taking derivatives that this is what we need to cancel the in Schr\”{o}dinger’s equation.

Proceeding to plug into the evolution identity we have

We are free to pick to kill the second and third terms

or

which after integration is

as in class we can observe that this is a purely real function. We are left with

where

The task is now to find solutions for these coefficients, and we can refer to the class notes for that without change.

# References

[1] BR Desai. *Quantum mechanics with basic field theory*. Cambridge University Press, 2009.

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