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.

## 2 comments:

I think one of my favorite results in this whole field was the fact that the spectrum does not uniquely define the boundary conditions--the famous "can you hear the shape of a drum" problem by Mark Kac.

http://en.wikipedia.org/wiki/Hearing_the_shape_of_a_drum

Anzel - That's cute. I think we already see an example of this when looking at the free electron gas, at least in the statistical limit of density of states for higher levels. It doesn't matter whether you choose hard wall or periodic boundary conditions - the resulting dos is the same.

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