In Newtonian physics, it seemed that there were two very different kinds of entities: particles and waves. Light was a wave; thus it could be focused with a lens. It also exhibited the phenomenon of 'cancellation'; if a crest and a trough were superimposed, the wave would cancel out. This cancellation property leads to effects like diffraction and interference.
But when the electron was discovered, it was thought to be a particle: something that travelled like a billiard ball. The same was assumed about other particles like protons and neutrons.
The first remarkable discovery of quantum mechanics was that everything was a wave. Thus the electron was also a wave, and could be focused by a suitably constructed lens. In typical situations the wavelength would be much smaller than the wavelength of visible light, so care would need to be taken to see the wave nature of the electron. But this smallness could be turned into an advantage: once we were able to focus electrons in an 'electron microscope', we could get much better resolutions than were possible with optical microscopes.
But what determines the wavelength of the electron wave? Electromagnetic waves can have any wavelength from zero to infinity; similarly electron waves can have any wavelength \( \lambda \) from zero to infinity. The value of the wavelength \(\lambda \) depends on the momentum of the electron, and is given by de Broglie's relation
We have said that in quantum theory there is only one kind of object: waves. So if electron waves satisfy the de Broglie relation, one can ask: should electromagnetic waves also satisfy this relation?
The answer is yes; but to understand that we will have to learn how to describe the momentum carried by electromagnetic waves. We will be able to do that after the next chapter.
Secondly, one may wonder about the nature of these waves. The waves on the ocean are water waves; thus they describe deformations of the surface of the water. If electrons are waves, what are these waves a deformation of? For that matter, what are electromagnetic waves a deformation of?
This is a deep question, and has a deep answer. We will address this question near the end of our course, though a detailed answer can be given only in the course on Quantum Field Theory. For the present we proceed as the scientists of the early twentieth century did: assume the wave behavior seen in experiments and try to resolve the puzzles that arise as a consequence.