Recurring themes in (condensed matter/nano) physics: boundary conditions

This is the first in a series of posts about tropes that recur in (condensed matter/nano) physics.  I put that qualifier in parentheses because these topics obviously come up in many other places as well, but I run across them from my perspective.

Very often in physics we are effectively solving boundary value problems.  That is, we have some physical system that obeys some kind of differential equations describing the spatial dependence of some variable of interest.  This could be the electron wavefunction \( \psi(\mathbf{r})\), which has to obey the Schroedinger equation in some region of space with a potential energy \( V(\mathbf{r})\).  This could be the electric field \( \mathbf{E}(\mathbf{r})\), which has to satisfy Maxwell's equations in some region of space that has a dielectric function \( \epsilon(\mathbf{r})\).  This could be the deflection of a drumhead \( u(x,y) \), where the drumhead itself must follow the rules of continuum elasticity.  This could be the pressure field \( p(z) \) of the air in a pipe that's part of a pipe organ. 

The thread that unites these diverse systems is that, in the absence of boundaries, these problems allow a continuum of solutions, but the imposition of boundaries drastically limits the solutions to a discrete set.  For example, the pressure in that pipe could (within reasonable limits set by the description of the air as a nice gas) have any spatial periodicity, described by some wavenumber \(k\), and along with that it would have some periodic time dependence with a frequency \(\omega\), so that \( \omega/k = c_{\mathrm{s}}\), where \(c_{\mathrm{s}}\) is the sound speed.  However, once we specify boundary conditions - say one end of the pipe closed, one end open - the rules that have to be satisfied at the boundary force there to be a discrete spectrum of allowed wavelengths, and hence frequencies.   Even trying to have no boundary, by installing periodic boundary conditions, does this. This general property, the emergence of discrete modes from the continuum, is what gives us the spectra of atoms and the sounds of guitars.