Please Describe Waves and Particles

As we have seen, the evidence that matter is wave-like is very strong. However, the evidence that matter is particle like is equally as strong. How can we mix the wave and particle aspects into a consistent description of matter? We have discussed several techniques for determining the wavelength of a matter wave, but what does its amplitude represent?

We will be a more rigorous mathematical discussion of the amplitude of the matter wave but for now we will assume that it carries information about the location of the particle. The wave has large amplitude where the particle is likely to be found, and it has small amplitude where the particle is unlikely to be found. If the wave has constant amplitude throughout a region of space, the particle is equally likely to be found anywhere in the region. If the wave has zero amplitude throughout a region, the particle is never found there.

One property that we would like for particle (even particle with a wave-like nature) to have is the ability to be localized. That is, we would like to know where the particle is located. We may not know exactly, but we can get a rough idea. For example, we may know that an electron is attached to an atom of diameter 0.1 nm, indicating that we have localized the electron in that region of space. Or a neutron may pass through a narrow slit, allowing us (at least for an instant) to localize its position to a region the size of the slit width.

Let us see what insights we can gain by trying to put together a collection of waves in such a way that we obtain something that resembles that description of a localized particle.