## 1D Infinite well with an ‘extra’ dimension

A friend and I are reading through parts of Zweibach’s introductory String theory book in anticipation for a course in the fall. In chapter two he introduces a couple of things I found interesting. The first are light cone coordinates that I’ll talk about in a later post. The second is the 1D infinite well with a ‘curled up’ extra dimension. Here’s how it works.

Let the potential be infinite everywhere except the rectangle $[0,a]\times[0,2\pi R]$ where the interval [0,a] is on the x-axis and the interval $[0,2\pi R]$ is on the y-axis, but let the lines $[0,a]\times{0}$ and $[0,a]\times {2\pi R}$ be identified. So we’re imposing periodic boundary conditions in the y-direction. This problem isn’t hard to solve via separation of variables and yields the following solution:

$\Psi(x,y)=A\sin(kx)(B\cos(ly)+C\sin(ly))$
where,
$k=\frac{n_x \pi}{a}$ and $l=\frac{n_y}{R}$
so the stationary states are labeled by two quantum numbers $n_x,n_y$ and the energy is given as
$E_{n_x,n_y}=\frac{\hbar^2\pi^2}{2m}(\frac{n_x^2}{a^2}+\frac{n_y^2}{(\pi R)^2})$

So if the extra dimension is small in comparison to the usual 1D dimension $a>>R$ then the $1/R^2$ term in the energy means that the energy associated with some oscillation in the ‘direction’ of the extra dimension must be very large. If $n_y=0$ then we have the usual energy spectrum for the infinite well. In fact the lowest energy states of this tweaked 1D problem are just the same as the usual 1D problem until high enough energies are reached.

An interesting toy example, no doubt a ridiculously trivial one compared with what string theorists actually wrestle with, but a start nevertheless.