Simple real-life examples of Fermi's golden rule? I want to teach my students some simple applications of Fermi's Golden Rule. Unfortunately, most examples I can think of are in scattering theory, which they have not learned yet. 
Are there any simple applications of Fermi's Golden Rule to real problems (for example in solid state physics) that undergraduates can completely understand and work out?
 A: The original use of the algorithm known as the Fermi rule is Dirac's calculation of Einstein's absorption coefficient of an ensemble of atoms:
P.A.M. Dirac, On the theory of quantum mechanics, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences (1926), 112, p. 661-677 http://dx.doi.org/10.1098/rspa.1926.0133
http://royalsocietypublishing.org/content/royprsa/112/762/661.full.pdf
This raises difficult questions like in what sense is it consistent to assume atom's state can be any wave function in the beginning when single atom is referred and only one of the Hamiltonian eigenfunctions in the end when set of atoms is referred and why is the averaging done over phases only, why the amplitude is first squared then averaged and not vice versa etc.
Since you are not sure about physical applications of the Fermi rule yourself, I recommend you reconsider teaching this to your students. There are other ways to exercise calculations based on Schroedinger's equation, both exact and approximative, which are easier to understand. For example, Schroedinger's calculation of frequencies of emission and the corresponding intensities based on the oscillating expected average of electric moment calculated from Schroedinger's equation for one atom is physically close but much easier to understand. No ensembles, no averaging and no change of rhetoric towards preferring eigenfunctions to general wavefunctions. Check out this paper
E. Schroedinger, Quantization as a problem of proper values, part IV, Annalen der Physik (4), vol. 81, 1926
which may be found on p. 102 of the book
Collected Papers on Wave Mechanics, Blackie & Son 1928
http://www.ebookee.net/Collected-Papers-on-Wave-Mechanics_143811.html
A: The most obvious application of Fermi's Golden Rule is the transition probablity from 2p to the 1s state in the hydrogen atom. The rate of transition is given by product of the two wave functions with the Hamiltonian, which we can take to be an oscillating potential in the z direction:
$$\langle 2 \, p |\,  z \sin \omega t \, | 1 \, s\rangle $$
This is the same calculation you end up with if you analyze the superposition of the two states as an oscillating charge density, and calculate the dipole moment (in the z direction):
$$\frac{1}{2}\langle 2 \, p + 1 \, s \, | \, z \sin \omega t \, | \, 2 \, p + 1 \, s \, \rangle$$
The extra factor of 1/2 is for normalization. It's easy to see from the symmetries that these two calculations give the same result, since the s^2 and the p^2 terms obviously have no dipole moment. So the Fermi Golden Rule calculation actually gives you the oscillating dipole moment of the mixed atomic state, which intuitively should be proportional to the transition probability.
In fact, not everyone knows this, but if you apply classical antenna theory to the oscillating charge density as calculated above, you get the correct value for the transition rate. I show how this works in this series of blogposts.
EDIT: Actually, my answer is more complicated than necessary. To understand the point of Fermi's Golden Rule, you really just need to look at the integrals for calculating the dipole moment between any to states of the hydrogen atom. You don't actually have to DO the integrals...you just have to see that from the symmetries, the integrals with zero dipole moment are the same ones that have zero transition probability according to Fermi's Golden Rule. It's an extra bonus to show that if you apply Maxwell's Equations to those oscillating dipole moments, you get the correct quantum-mechanical values for radiation rate (as expressed in expected transitions per second).
A: Well, could you start just with perturbation theory, time dependent, and transition amplitudes? Simple interaction and transition between states. If they are familiar with time evolution operator and Dirac bra-ket notation of course. In solid state physics as far as I know, you can use it to calculate scattering of electrons off phonons, so again, its scattering but I dont see why couldn't you just explain transition amplitudes first, probability using scalar products with bras and kets?
