What are some "real" world applications of the particle in a box (PIB) and the finite square well (FSW) which are discussed in an intro quantum mechanics class? For instance, I know that the PIB can applied to quantum dots and the FSW to the Ramsauer-Townsend effect. How about other applications?
A very important real-world application of quantum mechanics is the laser, in a multitude of ways. For start, the whole operating principle is of quantum mechanic origin, and there is a host of other quantum mechanical phenomena involved.
Specifically, in this case a good example is the quantum-well semiconductor laser, in which the different materials are sandwiched during manufacturing in such a way that they serve both as an optical waveguide and providing confinement for charge carriers. This electron confinement, which is a potential well, results in discrete energy levels. The discretisation permits the quantum well laser to emit a lot narrower spectrum of light than what the energy gap of the host material allows.
Another interesting application of quantum mechanics in real life is the quantum cascade laser (QCL). QCL is composed of a sandwich structure, where there are periodic layers of varying composition. These layers result in a series of quantum wells, through which the carriers can tunnel. This tunnelling effect enables tailoring the output wavelength of the QCL, almost independent of the host material.
For more information, see text books on lasers (my personal favorite is Orazio Svelto: Principles of Lasers, but there are of course others) or Wikipedia (and references therein).