II.

Waves and Particles

One of the striking differences between quantum theory and classical physics is that quantum theory describes energy and matter both as waves and as particles. The type of energy physicists study most often with quantum theory is light. Classical physics considers light to be only a wave, and it treats matter strictly as particles. Quantum theory acknowledges that both light and matter can behave like waves and like particles.

It is important to understand how scientists describe the properties of waves in order to understand how waves fit into quantum theory. A familiar type of wave occurs when a rope is tied to a solid object and someone moves the free end up and down. Waves travel along the rope. The highest points on the rope are called the crests of the waves. The lowest points are called troughs. One full wave consists of a crest and trough. The distance from crest to crest or from trough to trough—or from any point on one wave to the identical point on the next wave—is called the wavelength. The frequency of the waves is the number of waves per second that pass by a given point along the rope.

If waves traveling down the rope hit the stationary end and bounce back, like water waves bouncing against a wall, two waves on the rope may meet each other, hitting the same place on the rope at the same time. These two waves will interfere, or combine (see Interference). If the two waves exactly line up—that is, if the crests and troughs of the waves line up—the waves interfere constructively. This means that the trough of the combined wave is deeper and the crest is higher than those of the waves before they combined. If the two waves are offset by exactly half of a wavelength, the trough of one wave lines up with the crest of the other. This alignment creates destructive interference—the two waves cancel each other out and a momentary flat spot appears on the rope. See also Wave Motion.

A.

Light as a Wave and as a Particle

Like classical physics, quantum theory sometimes describes light as a wave, because light behaves like a wave in many situations. Light is not a vibration of a solid substance, such as a rope. Instead, a light wave is made up of a vibration in the intensity of the electric and magnetic fields that surround any electrically charged object.

Like the waves moving along a rope, light waves travel and carry energy. The amount of energy depends on the frequency of the light waves: the higher the frequency, the higher the energy. The frequency of a light wave is also related to the color of the light. For example, blue light has a higher frequency than that of red light. Therefore, a beam of blue light has more energy than an equally intense beam of red light has.

Unlike classical physics, quantum theory also describes light as a particle. Scientists revealed this aspect of light behavior in several experiments performed during the early 20th century. In one experiment, physicists discovered an interaction between light and particles in a metal. They called this interaction the photoelectric effect. In the photoelectric effect, a beam of light shining on a piece of metal makes the metal emit electrons. The light adds energy to the metal’s electrons, giving them enough energy to break free from the metal. If light was made strictly of waves, each electron in the metal could absorb many continuous waves of light and gain more and more energy. Increasing the intensity of the light, or adding more light waves, would add more energy to the emitted electrons. Shining a more and more intense beam of light on the metal would make the metal emit electrons with more and more energy.

Scientists found, however, that shining a more intense beam of light on the metal just made the metal emit more electrons. Each of these electrons had the same energy as that of electrons emitted with low intensity light. The electrons could not be interacting with waves, because adding more waves did not add more energy to the electrons. Instead, each electron had to be interacting with just a small piece of the light beam. These pieces were like little packets of light energy, or particles of light. The size, or energy, of each packet depended only on the frequency, or color, of the light—not on the intensity of the light. A more intense beam of light just had more packets of light energy, but each packet contained the same amount of energy. Individual electrons could absorb one packet of light energy and break free from the metal. Increasing the intensity of the light added more packets of energy to the beam and enabled a greater number of electrons to break free—but each of these emitted electrons had the same amount of energy. Scientists could only change the energy of the emitted electrons by changing the frequency, or color, of the beam. Changing from red light to blue light, for example, increased the energy of each packet of light. In this case, each emitted electron absorbed a bigger packet of light energy and had more energy after it broke free of the metal. Using these results, physicists developed a model of light as a particle, and they called these light particles photons.

In 1922 American physicist Arthur Compton discovered another interaction, now called the Compton effect, that reveals the particle nature of light. In the Compton effect, light collides with an electron. The collision knocks the electron off course and changes the frequency, and therefore energy, of the light. The light affects the electron in the same way a particle with momentum would: It bumps the electron and changes the electron’s path. The light is also affected by the collision as though it were a particle, in that its energy and momentum changes.

Momentum is a quantity that can be defined for all particles. For light particles, or photons, momentum depends on the frequency, or color, of the photon, which in turn depends on the photon’s energy. The energy of a photon is equal to a constant number, called Planck’s constant, times the frequency of the photon. Planck’s constant is named for German physicist Max Planck, who first proposed the relationship between energy and frequency. The accepted value of Planck’s constant is 6.626 × 10^-34 joule-second. This number is very small—written out, it is a decimal point followed by 33 zeroes, followed by the digits 6626. The energy of a single photon is therefore very small.

The dual nature of light seems puzzling because we have no everyday experience with wave-particle duality. Waves are everyday phenomena; we are all familiar with waves on a body of water or on a vibrating rope. Particles, too, are everyday objects—baseballs, cars, buildings, and even people can be thought of as particles. But to our senses, there are no everyday objects that are both waves and particles. Scientists increasingly find that the rules that apply to the world we see are only approximations of the rules that govern the unseen world of light and subatomic particles.

B.

Matter as Waves and Particles

In 1923 French physicist Louis de Broglie suggested that all particles—not just photons—have both wave and particle properties. He calculated that every particle has a wavelength (represented by λ, the Greek letter lambda) equal to Planck’s constant (h) divided by the momentum (p) of the particle: λ = h/p. Electrons, atoms, and all other particles have de Broglie wavelengths. The momentum of an object depends on its speed and mass, so the faster and heavier an object is, the larger its momentum (p) will be. Because Planck’s constant (h) is an extremely tiny number, the de Broglie wavelength (h/p) of any visible object is exceedingly small. In fact, the de Broglie wavelength of anything much larger than an atom is smaller than the size of one of its atoms. For example, the de Broglie wavelength of a baseball moving at 150 km/h (90 mph) is 1.1 × 10^-34 m (3.6 × 10^-34 ft). The diameter of a hydrogen atom (the simplest and smallest atom) is about 5 × 10^-11 m (about 2 × 10^-10 ft), more than 100 billion trillion times larger than the de Broglie wavelength of the baseball. The de Broglie wavelengths of everyday objects are so tiny that the wave nature of these objects does not affect their visible behavior, so their wave-particle duality is undetectable to us.

De Broglie wavelengths become important when the mass, and therefore momentum, of particles is very small. Particles the size of atoms and electrons have demonstrable wavelike properties. One of the most dramatic and interesting demonstrations of the wave behavior of electrons comes from the double-slit experiment. This experiment consists of a barrier set between a source of electrons and an electron detector. The barrier contains two slits, each about the width of the de Broglie wavelength of an electron. On this small scale, the wave nature of electrons becomes evident, as described in the following paragraphs.

Scientists can determine whether the electrons are behaving like waves or like particles by comparing the results of double-slit experiments with those of similar experiments performed with visible waves and particles. To establish how visible waves behave in a double-slit apparatus, physicists can replace the electron source with a device that creates waves in a tank of water. The slits in the barrier are about as wide as the wavelength of the water waves. In this experiment, the waves spread out spherically from the source until they hit the barrier. The waves pass through the slits and spread out again, producing two new wave fronts with centers as far apart as the slits are. These two new sets of waves interfere with each other as they travel toward the detector at the far end of the tank.

The waves interfere constructively in some places (adding together) and destructively in others (canceling each other out). The most intense waves—that is, those formed by the most constructive interference—hit the detector at the spot opposite the midpoint between the two slits. These strong waves form a peak of intensity on the detector. On either side of this peak, the waves destructively interfere and cancel each other out, creating a low point in intensity. Further out from these low points, the waves are weaker, but they constructively interfere again and create two more peaks of intensity, smaller than the large peak in the middle. The intensity then drops again as the waves destructively interfere. The intensity of the waves forms a symmetrical pattern on the detector, with a large peak directly across from the midpoint between the slits and alternating low points and smaller and smaller peaks on either side.

To see how particles behave in the double-slit experiment, physicists replace the water with marbles. The barrier slits are about the width of a marble, as the point of this experiment is to allow particles (in this case, marbles) to pass through the barrier. The marbles are put in motion and pass through the barrier, striking the detector at the far end of the apparatus. The results show that the marbles do not interfere with each other or with themselves like waves do. Instead, the marbles strike the detector most frequently in the two points directly opposite each slit.

When physicists perform the double-slit experiment with electrons, the detection pattern matches that produced by the waves, not the marbles. These results show that electrons do have wave properties. However, if scientists run the experiment using a barrier whose slits are much wider than the de Broglie wavelength of the electrons, the pattern resembles the one produced by the marbles. This shows that tiny particles such as electrons behave as waves in some circumstances and as particles in others.

C.

Uncertainty Principle

Before the development of quantum theory, physicists assumed that, with perfect equipment in perfect conditions, measuring any physical quantity as accurately as desired was possible. Quantum mechanical equations show that accurate measurement of both the position and the momentum of a particle at the same time is impossible. This rule is called Heisenberg’s uncertainty principle after German physicist Werner Heisenberg, who derived it from other rules of quantum theory. The uncertainty principle means that as physicists measure a particle’s position with more and more accuracy, the momentum of the particle becomes less and less precise, or more and more uncertain, and vice versa.

Heisenberg formally stated his principle by describing the relationship between the uncertainty in the measurement of a particle’s position and the uncertainty in the measurement of its momentum. Heisenberg said that the uncertainty in position (represented by Δx) times the uncertainty in momentum (represented by Δp;) must be greater than a constant number equal to Planck’s constant (h) divided by 4p (p is a constant approximately equal to 3.14). Mathematically, the uncertainty principle can be written as Δx Δp > h / 4p. This relationship means that as a scientist measures a particle’s position more and more accurately—so the uncertainty in its position becomes very small—the uncertainty in its momentum must become large to compensate and make this expression true. Likewise, if the uncertainty in momentum, Δp, becomes small, Δx must become large to make the expression true.

One way to understand the uncertainty principle is to consider the dual wave-particle nature of light and matter. Physicists can measure the position and momentum of an atom by bouncing light off of the atom. If they treat the light as a wave, they have to consider a property of waves called diffraction when measuring the atom’s position. Diffraction occurs when waves encounter an object—the waves bend around the object instead of traveling in a straight line. If the length of the waves is much shorter than the size of the object, the bending of the waves just at the edges of the object is not a problem. Most of the waves bounce back and give an accurate measurement of the object’s position. If the length of the waves is close to the size of the object, however, most of the waves diffract, making the measurement of the object’s position fuzzy. Physicists must bounce shorter and shorter waves off an atom to measure its position more accurately. Using shorter wavelengths of light, however, increases the uncertainty in the measurement of the atom’s momentum.

Light carries energy and momentum, because of its particle nature (described in the Compton effect). Photons that strike the atom being measured will change the atom’s energy and momentum. The fact that measuring an object also affects the object is an important principle in quantum theory. Normally the affect is so small it does not matter, but on the small scale of atoms, it becomes important. The bump to the atom increases the uncertainty in the measurement of the atom’s momentum. Light with more energy and momentum will knock the atom harder and create more uncertainty. The momentum of light is equal to Planck’s constant divided by the light’s wavelength, or p = h/λ. Physicists can increase the wavelength to decrease the light’s momentum and measure the atom’s momentum more accurately. Because of diffraction, however, increasing the light’s wavelength increases the uncertainty in the measurement of the atom’s position. Physicists most often use the uncertainty principle that describes the relationship between position and momentum, but a similar and important uncertainty relationship also exists between the measurement of energy and the measurement of time.