Interference experiments

Davisson and Germer performed experiments where electrons were scattered at the surface of a crystal. They observed interference, similar to the scattering of light, e.g. X-rays. In particular, destructive interference cannot be explained in a corpuscular picture.

After Schrödinger had set up his equation for the wave function $\Psi(x,t)$, it was not clear at all how to interpret this object. We have seen that as a superposition of plane waves, $\Psi(x,t)$ obviously must have wave properties. On the other hand, the photo effect and other effects like the Compton effect showed that electrons act as particles as well.

It became clear that a statistical interpretation of $\Psi(x,t)$ was a consistent way to combine both wave and particle aspects within one picture. In fact, in the experiments where interference was observed, always many particles were involved. This does not mean that the interaction between the particles is required to lead to interference. Even at very low intensities of particle beams, where in the extreme case only one electron at a time scatters from the surface of the crystal, in the end an interference pattern is observed on a screen (or when plotting a histogram of the electrons counted by different detectors). This suggested that the physical content of the wave function is related to a probability. It is clear, on the other hand, that a probability must be positive. One could imagine this probability as a kind of intensity which for waves $\Psi$ is given by $\vert\Psi\vert^2$. This is only a heuristic argument, one could argue that also $\vert\Psi\vert^4$ could do it. We therefore have another look at the Schrödinger equation.