In some circumstances, matter behaves like a wave and in others it behaves like a particle, and that seems to create a paradox. Is matter really a wave, or is it really a particle? Or does it switch between waves and particles depending on the circumstances? It might seem that this dichotomy in properties implies that we can never truly understand the nature of matter. But that is not correct. One can show that the wave function (the ‘waves’) can account for both the wave-like and the particle-like properties of matter. Thus matter can consistently be thought of as made up of wave functions alone, and wave-particle duality becomes a duality in the properties—classical wave-like or classical particle-like—of the wave functions.

The classical wave-like properties of matter. The classical wave-like properties of matter are diffraction (single-slit) and interference (double-slit). These two properties hold for the wave function because the equations of motion are linear.

Properties of a fraction of a wave. This differs for classical waves and quantum wave functions. In a classical wave, a water or sound wave, for example, if the wave is five meters long and we look at a five centimeter segment of the wave, then that segment will have only one hundredth of the total energy and momentum of the wave. But in quantum mechanics, strange as it seems, each part of the wave has the full energy and momentum (see Small Parts of the Wave Function). This difference becomes critical in explaining the photoelectric effect and other phenomena.

Particle-like properties. The particle-like properties of the wave are: (1) mass, (2) energy, (3) momentum, (4) spin, (5) charge, (6) localization and (7) particle-like trajectories. But, using group representation theory, we have seen in Mass, Spin, and Charge that the first five are actually properties of the wave function. In addition, we have seen in Localization that the last two are also properties of the wave function.

Thus all the particle-like properties of matter are actually properties of the wave function. Wave-particle duality is simply a duality in the properties of the wave function rather than being a duality in the basic nature of matter.