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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
\psi &= 0 &\qquad &x<0, x>d\\
\psi &= Ae^{\text{i}kx} + Be^{-\text{i}kx} &\qquad &0<x<d
and the boundary conditions
\begin{gather}\psi(0) = \psi(d) = 0
non-trivial solutions are only possible for discrete wave vector values
And consequently the energy levels are
E_N = \frac{\hbar^2k^2}{2m^*} = \frac{\hbar^2N^2\pi^2}{2m^*d^2} \qquad
N = 1, 2, \ldots

Christoph Wasshuber