Newton believed that light consisted of particles. Later Huygens developed a wave theory of light. So for a long time, the two theories existed side by side. The discovery of diffraction of light settled the argument in favour of waves which Maxwell latter proved to consist of electric and magnetic fields travelling at the speed of light. While the invention of radio lent weight to the electromagnetic wave theory, another discovery called the photo electric effect gave proof that light comes in discrete packets of energy which we call photons. Faced with contradictory evidence, physicists developed the concept of wave-particle duality.
As the science of high energy physics was developed, it was discovered that electrons and other particles could exhibit wave like properties. The science of Quantum Mechanics was developed in which the inner workings of the atom were explained through the wave like properties of electrons. So it came to be that one of the cornerstones of modern physics is wave-particle duality.
Wave-particle duality, like relativity, is taught as a mystery, outside common sense, but within the grasp of the most intelligent minds.
I do not believe in mysteries, only nature working through real causal effects which we can understand and quantify.
Lorentz's theory of electrons shows that electrons are not like billiards balls. They have no substance in the sense that we infer from the hardness and weight of a billiards ball that it has substance. Electrons are nothing but an electric field and the electric charge which exists at the inner end of the field. They have this property of mass because moving electric fields generate magnetic fields and magnetic fields contain energy. The energy in the magnetic field is what we call kinetic energy. To accelerate an electron, we have to do work to generate the kinetic energy. This gives the electron the property we call mass.
Moving electrons have an electric field and a magnetic field. We believe light comes in photons. If we combine the ideas of Newton and Maxwell, then photons consist of electric and magnetic fields. Moving electrons and photons are both made from the same two ingredients. Waves in water have crests and troughs. We do not know how many crests and troughs a photon might have, but if it consisted of just one crest, then the pattern of its electric and magnetic fields would be very similar to those of an electron travelling at near light speed. I can understand that kind of wave-particle duality.
Some of the wave like properties which quantum mechanics associates with particles are nothing of the sort. We cannot use light to look at atoms because the wave length of light is thousands of times bigger than an atom. To see greater detail, we need smaller wavelength which means the photons have more energy. The same applies to electrons, but for a different reason. Low speed electrons get deflected by electric fields before they get near an atom. Only a high speed electron will penetrate deep enough into the surrounding fields to be deflected by an atom. These two different effects are incorrectly attributed to the same thing and interpreted as evidence of the electron's particle like behaviour.
The defining proof of Huygens' wave theory over Newton's particle theory was the phenomena of diffraction. We have all seen diffraction in the rainbow colours reflected from a film of oil on a puddle. The most accurate way to measure the wavelength of light is to pass it through a diffraction grating. Light passing through a diffraction grating forms a diffraction pattern. The definitive proof that electrons are waves is said to be the diffraction pattern formed when electrons are bounced of a crystal. To my mind, the patterns produced with electrons are quite different from those of light.
We can explain wave-particle duality by assuming that electric and magnetic fields are both quantised. A photon is formed from quantum strands of magnetic flux, wrapped around its path and moving at the speed of light. They vary in strength in a way which would give a wave shape when drawn on a graph. The magnetic flux is all in the "crest" of the wave, so to speak. Each magnetic field has an associated electric field radiating outwards. At its centre is the charge on a quantum of electric flux. This theory gives us the relationship between wavelength and energy content of a photon in terms of the quanta of electric and magnetic flux.
However, there is one more phenomena which needs explaining! Diffraction patterns can still be accumulated by a camera when only one photon at a time is passing through the diffraction grating. One explanation could be the fact that photons come in showers which are all in phase and that we are actually seeing interference between a number of photons. If single photon diffraction is a fact, then we need an explanation. The explanation would apply equally to photons and high speed electrons.
The possible explanation is that the passing of the moving electric and magnetic fields of photons and high speed electrons causes a wave like variation of the electric fields through which they pass. There is an analogy with water waves, though analogy is not science. Photons and high speed electrons are like ship creating waves. In confined spaces, the bow and stern waves of a boat reflect and can have a powerful effect changing its direction. I think that we might be able to analyse similar effects taking place in diffraction, but I have not yet developed this to a rigorous mathematical proof.