## Second order time evolution for the coefficients of an initially pure ket with an adiabatically changing Hamiltonian.

Posted by peeterjoot on November 6, 2011

# Motivation.

In lecture 9, Prof Sipe developed the equations governing the evolution of the coefficients of a given state for an adiabatically changing Hamiltonian. He also indicated that we could do an approximation, finding the evolution of an initially pure state in powers of (like we did for the solutions of a non-time dependent perturbed Hamiltonian ). I tried doing that a couple of times and always ended up going in circles. I’ll show that here and also develop an expansion in time up to second order as an alternative, which appears to work out nicely.

# Review.

We assumed that an adiabatically changing Hamiltonian was known with instantaneous eigenkets governed by

The problem was to determine the time evolutions of the coefficients of some state , and this was found to be

where the coefficient must satisfy the set of LDEs

where

Solving these in general doesn’t look terribly fun, but perhaps we can find an explicit solution for all the ‘s, if we simplify the problem somewhat. Suppose that our initial state is found to be in the th energy level at the time before we start switching on the changing Hamiltonian.

We therefore require (up to a phase factor)

latex s \ne m$}.\end{array}\end{aligned} \hspace{\stretch{1}}(2.8)$

Equivalently we can write

# Going in circles with a expansion.

In class it was hinted that we could try a expansion of the following form to determine a solution for the coefficients at later times

I wasn’t able to figure out how to make that work. Trying this first to first order, and plugging in, we find

equating powers of yields two equations

Observe that the first identity is exactly what we started with in 2.5, but has just replaced the ‘s with ‘s. Worse is that the second equation is only satisfied for , and for we have

So this power series only appears to work if we somehow had always orthonormal to the derivative of . Perhaps this could be done if the Hamiltonian was also expanded in powers of , but such a beastie seems foreign to the problem. Note that we don’t even have any explicit dependence on the Hamiltonian in the final differential equations, as we’d probably need for such an expansion to work out.

# A Taylor series expansion in time.

What we can do is to expand the ‘s in a power series parametrized by time. That is, again, assuming we started with energy equal to , form

The first order term we can grab right from 2.5 and find

latex s = m$} \\ – {\left.{{{\left\langle {\hat{\psi}_s(t)} \right\rvert} \frac{d{{}}}{dt} {\left\lvert {\hat{\psi}_m(t)} \right\rangle}}}\right\vert}_{{t=0}} \\ & \quad \mbox{} \\ \end{array}\right.\end{aligned} $

Let’s write

So we can write

and form, to first order in time our approximation for the coefficient is

Let’s do the second order term too. For that we have

For the derivative we note that

So we have

Again for , all terms are killed. That’s somewhat surprising, but suggests that we will need to normalize the coefficients after the perturbation calculation, since we have unity for one of them.

For we have

So we have, for

It’s not particularly illuminating looking, but possible to compute, and we can use it to form a second order approximate solution for our perturbed state.

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