Condition for adiabatic approximation, derivation? In quantum mechanics it is said that an adiabatic approximation is valid when
$$T\gg \frac{\hbar}{\Delta E},$$
where $T$ is the time scale of variation of the Hamiltonian and $\Delta E$ is the typical difference in energies. Is there a way to mathematically show that in this limit the adiabatic approximation holds without having to go through the whole typical derivation of coefficients of basis functions?
 A: If you're willing to accept a perturbative argument, Shankar's QM book has a nice bit about this:
Take a Hamiltonian of the form:
$H(t)=H_0+e^{t/\tau} H_1$
Time-dependent perturbation theory says that to first order in $H_1$, the amplitude $\langle m | n \rangle$ ($E_m \neq E_n$) is:
$\langle m(t_1) | n(t_0) \rangle= \frac {-i} {\hbar} \int_{t_0}^{t_1} \langle m_0 | n_0 \rangle e^{t/\tau+i \omega_{mn} t } dt$
So if we start at time minus infinity and evolve to time zero, we get:
$\langle m(0) | n(- \infty) \rangle= \frac {-i} {\hbar} \frac{\langle m_0 | n_0 \rangle}{1/\tau+i\omega_{mn}}$
Now observe that if you let $\tau$ go to infinity, you recover the result for the overlaps $\langle m | n \rangle$ in time-independent perturbation, which are used to find the $n$th eigenstate of $H_0+H_1$ in terms of the eigenstates of the original Hamiltonian $H_0$. Putting this another way, a state starting as the $n$th eigenstate of $H_0$ has time-evolved to the $n$th eigenstate of $H_0+H_1$, which is the adiabatic theorem. Furthermore, you get the relevant timescale out as well: you want $1/\tau$ to be much larger than the smallest possible value for $\omega_{mn}$, which is the gap between the initial eigenstate and the nearest state.
Shankar actually uses this while assuming the truth of the adiabatic theorem to show that time-dependent perturbation reduces to time-independent perturbation. But if you instead will accept those two expressions as given then this shows the validity of the adiabatic theorem in this limit.
A: Write $H=H_0 + H'$ where $H_0$ is constant and $H'$ is how it changes.
Let's take a superficially very different situation: Assume $H'$ is sinusoidally oscillating, e.g. $H' \propto \cos(\omega t)$. Then $H'$ will be very good at exciting transitions between states whose energy differs by $\approx \hbar \omega$. For details on this, see any QM textbook description of optical absorption.
OK and now we're almost done: Just take the time-domain Fourier transform of $H'$! ....And then apply the results of the previous paragraph. See how that works?
So we conclude that $H$ will fail to excite transitions to the extent that it has negligible frequency-domain weight near $\Delta E/\hbar$. This happens in particular (though not exclusively) when $H$ varies slowly compared to $\hbar/\Delta E$, i.e. the adiabatic situation. 
I don't promise that this can be turned into a rigorous mathematical derivation, I haven't worked through it myself, but I bet it can. :-)
