Introduction

According to classical physics (i.e., physics prior to the 20th century), particles and waves are two completely distinct classes of physical entity that possess markedly different properties. 1) Particles are discrete: i.e., they cannot be arbitrarily divided. In other words, it makes sense to talk about one electron, or two electrons, but not about a third of an electron. Waves, on the other hand, are continuous: i.e., they can be arbitrarily divided. In other words, given a wave whose amplitude has a certain value, it makes sense to talk about a similar wave whose amplitude is one third, or any other fraction whatsoever, of this value. 2) Particles are highly localized in space. For example, atomic nuclei have very small radii of order $10^{-15}\,m$, whilst electrons act like point particles: i.e., they have no discernible spatial extent. Waves, on the other hand, are non-localized in space. In fact, a wave is defined to be a disturbance that is periodic in space, with some finite periodicity length: i.e., wavelength. Hence, it is fairly meaningless to talk about a disturbance being a wave unless it extends over a region of space that is at least a few wavelengths in dimension.

The classical scenario, described above, in which particles and waves are completely distinct from one another, had to be significantly modified in the early decades of the 20th century. During this time period, physicists discovered, much to their surprise, that, under certain circumstances, waves act as particles, and particles act as waves. This bizarre phenomenon is known as wave-particle duality. For instance, the photoelectric effect (see Section 12.2) indicates that electromagnetic waves sometimes act like swarms of massless particles called photons. Moreover, the phenomenon of electron diffraction by atomic lattices (see Section 12.3) implies that electrons sometimes have wave-like properties. Note, however, that wave-particle duality usually only manifests itself on atomic and sub-atomic lengthscales (i.e., on lengthscales less than, or of order, $10^{-10}\,{\rm m}$--see Section 12.3). The classical picture remains valid on significantly longer lengthscales. In other words, on macroscopic lengthscales, waves only act like waves, particles only act like particles, and there is no wave-particle duality. However, on microscopic lengthscales, classical mechanics, which governs the macroscopic behavior of massive particles, and classical electrodynamics, which governs the macroscopic behavior of electromagnetic fields--neither of which take wave-particle duality into account--must be replaced by new theories. The theories in question are called quantum mechanics and quantum electrodynamics, respectively. In the following, we shall discuss a simplified version of quantum mechanics in which the microscopic dynamics of massive particles (i.e., particles with finite mass) is described entirely in terms of wavefunctions. This particular theory is known as wave mechanics.