How to picture wave nature of matter? I have started learning about the wave nature of matter. But unlike classical physics, I am having trouble imagining the wave nature. I am unable to imagine that a particle with mass, can also have a frequency. I asked this question to my teacher and he said that I do not have the necessary skills to imagine the situation and said to simply stick to the formulae without putting much imagination into it. But I don't like that. So, is there some way to picture this situation?
 A: The EM field governs every thing we see, feel, remember, it governs all the chemical interactions of matter including all the reactions that cause our brains to function.
Water waves show how energy can move from one place to another and there are some similarities as well as differences to the EM field. In water we have many many particles that form the waves, the waves spread, superimpose, usually caused by wind they eventually transfer their energy by crashing to the shore. In the EM field we have virtual photons and real photons, enmasse they spread like water (like radio waves) but we can also study them as single particles to better know their behaviour.  Every real photon emerges from at atom and is eventually absorbed by an atom. Virtual photons are force carriers, like when you hold 2 magnets apart or feel static electricity, no energy is transferred.
Maxwell gave us an equation for the propagation of light in the EM field, its solution was based on the fact that a magnetic force is generated at 90 degrees whenever there is a electric field generated, the solution was a sinusoidal.  You can think of this as trying to run down a road on a day with a mysterious wind. When you try and run forward the wind blows with an equal force at 90  degrees, you would end up going in circles, but if you try and run in a sine wave pattern you can actually have a net vector down the road!
Now combine the fact that most of our scientific experiments are based on observation using the EM field, and that most of the experiments involve interaction of matter which is also molecules and atoms surrounded by electrons using the EM field to govern all the interactions.  The EM field, i.e. photons, can only act sinusoidally, that puts a lot of wave behahior into the nature of interaction of matter.
Consider an electron in the DSE, before it even leaves the emitter it has already caused virtual photons and is feeling out a path to travel before it even gets started. Certain paths are ideal, they resonate with the field and the eventual absorbing atom, resonance is also ideal when the path length is a multiple of the wavelength (Feynman path integral).
So do not think of matter as some inert chargeless ball of mass, matter is something that exists in the EM field and is even made up of particles the have EM properties of their own.
A: In the article about the electron in wikipedia there is this paragraph:

In 1897, the British physicist J. J. Thomson, with his colleagues John
S. Townsend and H. A. Wilson, performed experiments indicating that
cathode rays really were unique particles, rather than waves, atoms or
molecules as was believed earlier.

Surely they had no idea of how weird this micro-world could be. When they saw that the rays had charge because they were deflected by electric and magnetic fields, and the charge had a minimum value, the conclusion was that the rays were a bunch of particles.
The double slit experience with electrons only happened in 1927, showing its wave behaviour. By that time, the concept of electrons as particles were well consolidated.
The sequence of experiments was different for light. Thomas Young made the double slit experiment in 1801, and only in the end of the 19th century the photoelectric effect was noted. The concept of light as a wave was then well consolidated.
Until today, light waves seems more intuitive than electron waves. And electrons more intuitive than photons (at least for me).
A: The best advice is to stop thinking of it as "a particle" like a tiny grain of sand.  They should never have come up with that word to describe it.  Another word used, but also with some difficulty, is "a quantum".  I prefer to just think of everything as a wave. This is the best I can type a wave:  ~~~~~~   But of course you have to imagine it in 3 dimensions, and with it moving in time.   There are some interesting moving images here: http://www.rhythmodynamics.com/Gabriel_LaFreniere/matter.htm  But again think of them moving outwards in 3 dimensions.  Caltech theoretical physicist Sean Carroll put it this way: “To understand what is going on, you actually need to give up a little bit on the notion of particles..... The universe is full of fields, and what we think of as particles are just excitations of those fields, like waves in an ocean. An electron, for example, is just an excitation of an electron field.
So everything that we can think of as a particle is just a wave. That can be things with mass like electrons and protons, and things without mass like photons.
Now imagine a calm pool of water.  You can drop in big stones that cause big waves, and little stones that cause little waves.  Sometimes these waves interact and interfere with each other, and that would be a good example of an electron and a photon which easily interact.   But other times they simply pass over each other without interacting.
As for mass, this is where it gets down to Einstein.  You have to remember his formula $E=Mc^2$.    So you can turn it around to be $M=E/c^2$, or Mass equals Energy divided by the speed of light squared.   Energy and Mass are connected.  If something has energy, like for instance a moving wave, then it has mass.
Now stop thinking of the universe as being built of grains of sand, and start thinking of it as being built of waves.  Once you wrap your head around that, it all becomes a little clearer.
