In 1924, Louis de Broglie published his doctoral thesis in which he put forward a hypothesis today known as "de Broglie's relation". This was one of the integral pieces of the early development of quantum mechanics and earned him the Nobel prize in physics in 1929. The relation says that any particle with momentum p will propagate as though it were a wave of wavelength
λ = h/p
where λ is known as the de Broglie wavelength. Often amongst physicists, it is written in the more convenient form
p = hbar k
where hbar is Dirac's constant (h/(2π)) and k is the wavenumber. It may seem strange that matter would be expected to behave in a wave-like fashion, but this has been observed in experiment (discussed below). While it was quickly clear that particles exhibited wave-like properties, it wasn't immediately clear what exactly the waves meant or what they were propagating in. The quest to answer those questions became known as the interpretation of quantum mechanics, with the Born statistical interpretation of the wavefunction (a wavefunction is the mathematical representation of a wave) being the first major advance in understanding the meaning of de Broglie waves. The statistical interpretation says that the squared amplitude of the wave at a certain location tells you the probability of finding the particle at that location. The fact that particles have both wave-like and particle-like properties is known as the wave-particle duality. Not long after de Broglie made his hypothesis, Schrödinger tried to develop a wave equation that would have de Broglie's relation as its dispersion relation, and he came up with Schrödinger's Equation.
There are many experiments that demonstrate the wave nature of matter. One early experiment of this type was the 1927 experiment of Davisson and Germer in which electrons hitting a nickel crystal exhibited a diffraction pattern. This sort of effect is predicted by wave theories but is not expected for classical particles. Further experiments by Thompson, Rupp, and others confirmed the existence of these effects. Later in 1961, Claus Jönsson demonstrated the same effect in a very clear way by doing Young's double slit experiment with electrons. Though de Broglie's hypothesis seemed pretty radical when he first made it, experimental evidence quickly forced physicists to take it seriously.
de Broglie's relation gives us new relationships between position and momentum that we did not have in classical mechanics. A particle with one momentum will have a sinusoidal wavefunction of a certain wavelength according to the relation, meaning that it will be spread out all over space. In order to get a particle that's localized (i.e. very likely to be in a fairly small region of space), we want a wave packet, a wave form that is only significantly different from zero in a small area. Fourier analysis tells us that we can make a wave packet by adding together waves of several different wavelengths in a superposition. By making the right combination we can use interference to cancel out most of the original wave leaving only a small "wave packet" in one place. In order to get a smaller wave packet we must add more and more wavelengths to make the cancellation more exact. de Broglie's relation tells us that a particle with such a wave packet would have a combination of several different momenta. What does that mean exactly? Well, the statistical interpretation says that that superposition means that if you measure the momentum you could get any one of the values that make up the wave packet.
So, from the properties of waves we have discussed, we can say that a particle having a single, well-defined, momentum will be spread out over many possible positions, while a particles whose wave packet describes a fairly well localized position will have many possible momenta. One may make this idea more precise mathematically by defining the momentum wavefunction, which is the Fourier transform of the position wavefunction and gives the probability of measuring each momentum for that state. From this mathematical framework one may derive the Heisenberg Uncertainty Principle, which then follows simply from known results in Fourier analysis.
Wavelengths and Length Scales
It is a widely applicable principle in wave mechanics that you will only start to see "wave features", like diffraction and interference, when you start looking phenomena happening on distance scales similar to (or smaller than) the wavelength. This is precisely the reason why we can understand a lot about light thinking about it as "rays", even though it is actually a wave. The same applies for "matter waves" that behave according to de Broglie's relation. If you consider the behavior of these waves going through openings much larger than one wavelength or in general if you look at the patterns with a device that can't resolve details as small as a wavelength, then you can understand the behavior of the wave in terms of rays or particles flying along straight paths. This has two different implications. The first is that a macroscopic object with even a little momentum will have a very small de Broglie wavelength, so we would not be able to notice any wave behavior of the object without very precise instruments*. The other implication is that if we want to bounce particles off of an object to determine its structure, we will have to use higher momentum particles in order to see smaller scale structure, so physicists usually draw a connection between high momentum and short length scale.
* This argument is a bit of a swindle. There's actually more to it than this, because of course you could ask how we know a macroscopic object will never have small enough momentum to become delocalized. To really get a satisfactory answer you must turn to quantum decoherence.
- The Physical Principles of the Quantum Theory, Werner Heisenberg
- Introduction to Quantum Mechanics, David J. Griffiths
- Clinton Davisson's Nobel Lecture, http://nobelprize.org/physics/laureates/1937/davisson-lecture.pdf