D.34 The adiabatic theorem

Consider the Schrödinger equation

If the Hamiltonian is independent of time, the solution can be written in terms of the Hamiltonian energy eigenvalues

Here

However, the Hamiltonian varies with time for the systems of interest
here. Still, at any given time its eigenfunctions form a complete
set. So it is still possible to write the wave function as a sum of
them, say like

To get an equation for their variation, plug the expression for

where the primes indicate time derivatives. The middle sum in the left hand side and the right hand side cancel against each other since by definition

In the first sum only the term

This is still exact.

However, the purpose of the current derivation is to address the
adiabatic approximation. The adiabatic approximation assumes that the
entire evolution takes place very slowly over a large time interval

Consider now first the case that there is no degeneracy, in other
words, that there is only one eigenfunction

(Recall that the square magnitudes of the coefficients give the
probability for the corresponding energy. So the magnitude of the
coefficients is bounded by 1. Also, for simplicity it will be assumed
that the number of eigenfunctions in the system is finite. Otherwise
the sums over good enough

to approximate an infinite system by
a large-enough finite one. That makes life a lot easier, not just
here but also in other derivations like {D.18}.)

It is convenient to split up the sum in (D.19):

However, that is not because it is small due to the time derivative in
it, as one reference claims. While the time derivative of

To show that more precisely, note that the formal solution of the full
equation (D.20) is, [40, 19.2]:

All the integrals are negligibly small because of the rapid variation of the first exponential in them. To verify that, rewrite them a bit and then perform an integration by parts:

The first term in the right hand side is small of order

And that means that in the adiabatic approximation

The underbar used to keep

Note that

So the sum of the inner product plus its complex conjugate are zero. That makes it purely imaginary, so

Since both

So far it has been assumed that there is no degeneracy, at least not for the considered state. However it is no problem if at a finite number of times, the energy of the considered state crosses some other energy. For example, consider a three-dimensional harmonic oscillator with three time varying spring stiffnesses. Whenever any two stiffnesses become equal, there is significant degeneracy. Despite that, the given adiabatic solution still applies. (This does assume that you have chosen the eigenfunctions to change smoothly through degeneracy, as perturbation theory says you can, {D.79}.)

To verify that the solution is indeed still valid, cut out a time
interval of size

Things change if some energy levels are permanently degenerate.
Consider an harmonic oscillator for which at least two spring
stiffnesses are permanently equal. In that case, you need to solve
for all coefficients at a given energy level fundamental solution matrix,

a matrix
consisting of independent solution vectors. And

More recent derivations allow the spectrum to be continuous, in which
case the nonzero energy gaps