As you certainly recall from our discussions on interference phenomenon - the most recent of which was standing waves in instruments - we used phase as our main argument.  The idea was that there was a phase difference between waves traveling in opposite directions, bouncing back and forth.  In order for a wave on a string, for instance, to persist in nature it must be the case that the wave and its doubly reflected counterpart meet over and over at the same location on the string constructively, or with integer multiples of  phase between them, or . In general, we related the phase difference to the wave number (which is often called the wave vector in field contexts since it also defines direction) and the path length difference using .  We will start with the same approach for matter waves like electrons.  But before we can do that we have to put two pieces together - one from classical electromagnetic theory and the other one being the new energy quantization condition that Max Planck came up with, E=hf.

A graduate student by the name of Louis de Broglie was aware of the new quantization condition for energy that Max Planck had come up with to successfully describe blackbody radiation, as well as its application by Einstein in describing the photoelectric effect.  In the writing of his 1924 PhD thesis, Planck suggested without any experimental confirmation or testing, that whereas it had been shown that light waves were also comprised of particle-like quanta (later language), that particle-like electrons must also exhibit wave-like properties.  In this way nature would be symmetric, and all entities would have both a wave-like and a particle-like nature.  This is called wave-particle duality.  This bold suggestion seemed to be correct, and was verified by subsequent experiments, and in 1929 de Broglie won the Nobel Prize for it. It took many years to understand that these "particles" were really field quanta. Recall, as we have discussed before, that the "particle-like" nature that all things exhibit is due to them being field quanta rather than solid and delineated like baseballs. We can reassemble a short form of the mathematics that de Broglie used as follows:  It is known from studies of classical electromagnetic theory that light carries both energy and momentum.  They are related by E=pc, where c is the speed of light in vacuum.  Combining this with Planck's energy quantization condition E=hf, we may write pc=hf.  Rearranging gives p=hf/c.  While this doesn't seem useful for massive particles (ones with non-zero mass) since they can't ever travel at light speed c, this expression may also be written in terms of a wavelength.  Louis de Broglie suggested that this wavelength applies to all entities - with or without mass.  So we can attribute a wavelength to a massive "particle" using:

The implications is that all things that carry momentum have an associated wavelength.  That wavelength, however, is very, very small for anything approaching macroscopic in size. If you recall from optics, effects like diffraction require slits similar in size to the wavelength of the light. Because the larger the object (momentum) the shorter the wavelength, we ourselves (thankfully) don't diffract as we walk through single slit doorways into physics classrooms.  To read the original work of DeBroglie, see this link: DeBroglie’s thesis.