C.1 The Infinite Potential Well

   The infinite potential well is a simple model for a  quantum dot (see Fig. C.1).

  

Figure C.1: The infinite potential well of width d.

We are interested in the characteristic spacing between energy levels, especially the height of the first level. Starting with the Ansatz

and the boundary conditions
$$\begin{gather}\psi(0) = \psi(d) = 0 \end{gather}$$
non-trivial solutions are only possible for discrete wave vector values
$$\begin{gather}k=\frac{N\pi}{d}. \end{gather}$$
And consequently the energy levels are
 $$\begin{gather} E_N = \frac{\hbar^2k^2}{2m^*} = \frac{\hbar^2N^2\pi^2}{2m^*d^2} \qquad N = 1, 2, \ldots \end{gather}$$