# How seriously should one take the energy eigenvalues of an electron's wavefunction in a compactified space?

In chapter 2 (section 2.10) of Zwiebach's string theory book, there is a neat derivation of the energy eigenvalues for the two dimensional square well in which one dimension is compactified to a length $2\pi R$. One finds the energy eigenvalues to be $$E_{nt} = \frac{\hbar^2}{2m}\left(\frac{n^2\pi^2}{a^2} + \frac{t^2}{R^2}\right), \tag{2.122}$$ where $n\in \mathbb{Z}_{>0}$, $t\in \mathbb{Z}_{\geq 0}$, and $a$ is the length of the square well along the non-compact dimension.

My question is how seriously one should take this result, especially in the context of string theory. I agree it's a tidy justification for why compact dimensions are difficult to probe (we may suppose, for instance, that $a \gg R$). But if compact dimensions do exist, will electrons with sufficient energy truly infiltrate them? Or is the notion of a compact dimension more complicated than a mere identification (e.g., $x\sim x + 2\pi R$), so much so that electrons (or other fermions for that matter) are forbidden from accessing them?

• – Qmechanic Mar 15 '18 at 19:30