Where can a good treatment of the 'sudden' perturbation approximation be found? Where can a good treatment of the 'sudden' perturbation approximation be found?
A lot of quantum mechanics books have very brief discussions of it but I want to see it in some detail and preferably with as many examples as possible. 
Is there a book, paper, or some other place that details this approximation?
 A: Migdal's book recommended in comments is good. If you cannot find it, some Migdal's problems can be found in the standard textbook L.D. Landau and E.M. Lifshitz, Quantum mechanics, Non-relativistic theory, §41. Transitions under a perturbation acting for a finite time. There are five problems considered in the end of the section:

  
*
  
*A uniform electric field is suddenly applied to a charged oscillator in the  ground state. Determine the probabilities of
  transitions of the oscillator to excited states under the action of
  this perturbation.
  
*The nucleus of an atom in the normal state receives an impulse which gives  it a velocity $v$; the duration $\tau$ of the impulse is
  assumed short in comparison both with the electron periods and with
  $a/v$, where $a$ is the dimension of the atom. Determine the
  probability  of excitation of the atom under the influence of such a
  "jolt" (A. B. Migdal 1939).
  
*Determine the total probability of excitation and ionization of an atom of hydrogen which receives a sudden "jolt" (see Problem 2). 
  
*Determine the probability that an electron will leave the $K$-shell of an atom  with large atomic number $Z$ when the nucleus undergoes
  $\beta$-decay. The velocity of the $\beta$-particle is assumed large
  in comparison with that of the $K$-electron (A. B. Migdal and  E. L.
  Feinberg 1941).
  
*Determine the probability of emergence of an electron from the $K$-shell  of an atom with large $Z$ in $\alpha$-decay of the nucleus.
  The velocity of the $\alpha$-particle is small compared with that of
  the $K$-electron, but the time which it takes to leave the nucleus is
  small  in comparison with the time of revolution of the electron (A.
  B. Migdal 1941, J. S. Levinger 1953).

Another application, which is very close connected to the problem 4, is the so called molecular/atomic effects in tritium beta decay. One can measure the electron neutrino mass by studying the $\beta$-electron energy spectrum in the process $\,^{3}H\to \,^{3}He^{+}+e^{-}+\bar{\nu}_{e}$  near the end-point, where the electron energy is very close to 18.6 keV. 
The electron is very fast so it can hardly influence all environment around. Thus the main effect is the sudden change of the charge of nucleus. There were a lot of studies of possible molecular excitations due to such sudden perturbation. Google finds a good thesis about this subject: Natasha Doss, Calculated final state probability distributions for $T_{2}$ $\beta$-decay measurements.
