Recurring themes in (condensed matter/nano) physics: Fermi's Golden Rule

Very often in condensed matter (or atomic) physics we are interested in trying to calculate the rate of some quantum process - this could be the absorption of photons by an isolated atom or a solid, for example.  In (advanced) undergraduate quantum mechanics, we can apply time-dependent perturbation theory to do such a calculation.  Typically you assume that the system starts in some initial state \( |i\rangle \), is subjected to some perturbation \(V\) that turns on at time \(t = 0\), and ends up in final state \( |f\rangle \).  If \(V\) has a harmonic time dependence with some (angular) frequency \(\omega\), then you can do a nice bit of math that calculates the rate at which this process happens.  You discover that at long times the only allowed transitions are the ones where the energies of the initial and final states differ by \(\hbar \omega\), and that the rate of that process is \( (2\pi/\hbar) |\langle i |V| f\rangle|^{2} \rho \), where \(\rho\) is the number of states per unit energy per unit volume that satisfy the energy constraint.

This result, associated with Enrico Fermi, shows up over and over, with some common motifs in condensed matter and nanoscale physics, at least in spirit (that is, sometimes people apply heuristically even though the perturbation may not be harmonic, for example).  First, the \( |\langle i |V| f\rangle|^{2} \) term is what gives us selection rules.  If you think about optical transitions in atoms, this is why you get electric dipole transitions from the 2\(p\) state of hydrogen to the 1\(s\) state, rather than from the 2\(s\) state; in the latter, this quantity is zero.  In crystalline solids, if the initial and final states are Bloch waves, it's the periodicity of the lattice that makes this quantity zero unless (crystal) momentum is conserved.  This is the root of the idea that processes ordinarily forbidden in macroscopic crystals can sometimes take place in nanocrystals or at surfaces.

Similarly, meso- and nanoscale systems can greatly constrain \( \rho \).  One reason that you can get very long mean free paths for charge carriers in semiconductor nanowires, carbon nanotubes, graphene, at the edges of quantum Hall systems, etc., is that the density of states available into which carriers can scatter is very restricted.  Similarly, enhancing \(\rho\) can pay dividends - this is the source of the Purcell effect, where radiative transition rates can be greatly enhanced by increasing the photon density of states, and is part of the reason for enhanced rates of optical processes near plasmonic nanostructures.