# Section 3: Waves, Particles, and a Paradox

A particle is an object so small that its size is negligible; a wave is a periodic disturbance in a medium. These two concepts are so different that one can scarcely believe that they could be confused. In quantum physics, however, they turn out to be deeply intertwined and fundamentally inseparable.

Figure 7: A circular wave created by tossing a pebble in a pond.

The electron provides an ideal example of a particle because no attempt to measure its size has yielded a value different from zero. Clearly, an electron is small compared to an atom, while an atom is small compared to, for instance, a marble. In the night sky, the tiny points of starlight appear to come from luminous particles, and for many purposes we can treat stars as particles that interact gravitationally. It is evident that "small" is a relative term. Nevertheless, the concept of a particle is generally clear.

The essential properties of a particle are its mass, m; and, if it is moving with velocity v, its momentum, mv; and its kinetic energy, 1/2mv 2. The energy of a particle remains localized, like the energy of a bullet, until it hits something. One could say, without exaggeration, that nothing could be simpler than a particle.

Figure 8: Two waves interfere as they cross paths.

A wave is a periodic disturbance in a medium. Water waves are the most familiar example (we talk here about gentle waves, like ripples on a pond, not the breakers loved by surfers); but there are numerous other kinds, including sound waves (periodic oscillations of pressure in the air), light waves (periodic oscillations in the electromagnetic field), and the yet-to-be-detected gravitational waves (periodic oscillations in the gravitational field). The nature of the amplitude, or height of the wave, depends on the medium, for instance the pressure of air in a sound wave, the actual height in a water wave, or the electric field in a light wave. However, every wave is characterized by its wavelength (the Greek letter "lambda"), the distance from one crest to the next; its frequency (the Greek letter "nu"), the number of cycles or oscillations per second; and its velocity v, the distance a given crest moves in a second. This distance is the product of the number of oscillations the wave undergoes in a second and the wavelength.

Figure 9: Standing waves on a string between two fixed endpoints.

The energy in a wave spreads like the ripples traveling outward in Figure 7. A surprising property of waves is that they pass freely through each other: as they cross, their displacements simply add. The wave fronts retain their circular shape as if the other wave were not there. However, at the intersections of the circles marking the wave crests, the amplitudes add, producing a bright image. In between, the positive displacement of one wave is canceled by the negative displacement of the other. This phenomenon, called interference, is a fundamental property of waves. Interference constitutes a characteristic signature of wave phenomena.

If a system is constrained, for instance if the medium is a guitar string that is fixed at either end, the energy cannot simply propagate away. As a result, the pattern is fixed in space and it oscillates in time. Such a wave is called a standing wave.

Figure 10: Ripple tank picture of plane waves incident on a slit that is about two wavelengths wide.

Far from their source, in three dimensions, the wave fronts of a disturbance behave like equally spaced planes, and the waves are called plane waves. If plane waves pass through a slit, the emerging wave does not form a perfect beam but spreads, or diffracts, as in Figure 10. This may seem contrary to experience because light is composed of waves, but light waves do not seem to spread. Rather, light appears to travel in straight lines. This is because in everyday experience, light beams are formed by apertures that are many wavelengths wide. A 1 millimeter aperture, for instance, is about 2,000 wavelengths wide. In such a situation, diffraction is weak and spreading is negligible. However, if the slit is about a wavelength across, the emerging disturbance is not a sharp beam but a rapidly spreading wave, as in Figure 10. To see light diffract, one must use very narrow slits.

Figure 11: Diffraction of laser light through one (top) and two (bottom) small slits.