by CHAD ORZEL
“So, you’re saying I need a new plan?”
“You need a new plan.”
Her tail droops, and we walk in silence for a few seconds. “Hey,” she says, “can you help me with my new plan?”
“I can try.”
“How do I use my Buddha nature to go around both sides of the pond at the same time?”
I really can’t think of anything to say to that, but a flash of gray fur saves me. “Look! A squirrel!” I say.
“Oooooh!” And we’re off in pursuit.
Quantum physics has many strange and fascinating aspects, but the discovery that launched the theory was particle-wave duality, or the fact that both light and matter have particle-like and wavelike properties at the same time. A beam of light, which is generally thought of as a wave, turns out to behave like a stream of particles in some experiments. At the same time, a beam of electrons, which is generally thought of as a stream of particles, turns out to behave like a wave in some experiments. Particle and wave properties seem to be contradictory, and yet everything in the universe somehow manages to be both a particle and a wave.
The discovery in the early 1900s that light behaves like a particle is the launching point for all of quantum mechanics. In this chapter, we’ll describe the history of how physicists discovered this strange duality. In order to appreciate just what a strange development this is, though, we need to talk about the particles and waves that we see in everyday life.
PARTICLES AND WAVES AROUND YOU: CLASSICAL PHYSICS
Everybody is familiar with the behavior of material particles. Pretty much all the objects you see around you—bones, balls, squeaky toys—behave like particles in the classical sense, with their motion determined by classical physics. They have different shapes, but you can predict their essential motion by imagining each as a small, featureless ball with some mass—a particle—and applying Newton’s laws of motion.* A tennis ball and a long bone tumbling end over end look very different in flight, but if they’re thrown in the same direction with the same speed, they’ll land in the same place, and you can predict that place using classical physics.
A particle-like object has a definite position (you know right where it is), a definite velocity (you know how fast it’s moving, and in what direction), and a definite mass (you know how big it is). You can multiply the mass and velocity together, to find the momentum. A great big Labrador retriever has more momentum than a little French poodle when they’re both moving at the same speed, and a fast-moving border collie has more momentum than a waddling basset hound of the same mass. Momentum determines what will happen when two particles collide. When a moving object hits a stationary one, the moving object will slow down, losing momentum, while the stationary object will speed up, gaining momentum.
The other notable feature of particles is something that seems almost too obvious to mention: particles can be counted. When you have some collection of objects, you can look at them and determine exactly how many of them you have—one bone, two squeaky toys, three squirrels under a tree in the backyard.
Waves, on the other hand, are slipperier. A wave is a moving disturbance in something, like the patterns of crests and troughs formed by water splashing in a backyard pond. Waves are spread out over some region of space by their nature, forming a pattern that changes and moves over time. No physical objects move anywhere—the water stays in the pond—but the pattern of the disturbance changes, and we see that as the motion of a wave.
If you want to understand a wave, there are two ways of looking at it that provide useful information. One is to imagine taking a snapshot of the whole wave, and looking at the pattern of the disturbance in space. For a single simple wave, you see a pattern of regular peaks and valleys, like this:
As you move along the pattern, you see the medium moving up and down by an amount called the “amplitude” of the wave. If you measure the distance between two neighboring crests of the wave (or two troughs), you’ve measured the “wavelength,” which is one of the numbers used to describe a wave.
The other thing you can do is to look at one little piece of the wave pattern, and watch it for a long time—imagine watching a duck bobbing up and down on a lake, say. If you watch carefully, you’ll see that the disturbance gets bigger and smaller in a very regular way—sometimes the duck is higher up, sometimes lower down—and makes a pattern in time very much like the pattern in space. You can measure how often the wave repeats itself in a given amount of time—how many times the duck reaches its maximum height in a minute, say—and that gives you the “frequency” of the wave, which is another critical number used to describe the wave. Wavelength and frequency are related to each other—longer wavelengths mean lower frequency, and vice versa.
You can already see how waves are different from particles: they don’t have a position. The wavelength and the frequency describe the pattern as a whole, but there’s no single place you can point to and identify as the position of the wave. The wave itself is a disturbance spread over space, and not a physical thing with a definite position and velocity. You can assign a velocity to the wave pattern, by looking at how long it takes one crest of the wave to move from one position to another, but again, this is a property of the pattern as a whole.
You also can’t count waves the way you can count particles—you can say how many crests and troughs there are in one particular area, but those are all part of a single wave pattern. Waves are continuous where particles are discrete—you can say that you have one, two, or three particles, but you either have waves, or you don’t. Individual waves may have larger or smaller amplitudes, but they don’t come in chunks like particles do. Waves don’t even add together in the same way that particles do—sometimes, when you put two waves together, you end up with a bigger wave, and sometimes you end up with no wave at all.
Imagine that you have two different sources of waves in the same area—two rocks thrown into still water at the same time, for example. What you get when you add the two waves together depends on how they line up. If you add the two waves together such that the crests of one wave fall on top of the crests of the other, and the troughs of one wave fall in the troughs of the other (such waves are called “in phase”), you’ll get a larger wave than either of the two you started with. On the other hand, if you add two waves together such that the crests of one wave fall in the troughs of the other and vice versa (“out of phase”), the two will cancel out, and you’ll end up with no wave at all.
This phenomenon is called interference, and it’s perhaps the most dramatic difference between waves and particles.
“I don’t know . . . that’s pretty weird. Do you have any other examples of interference? Something more . . . doggy?”
“No, I really don’t. That’s the point—waves are dramatically different than particles. Nothing that dogs deal with on a regular basis is all that wavelike.”
“How about, ‘Interference is like when you put a squirrel in the backyard, and then you put a dog in the backyard, and a minute later, there’s no more squirrel in the backyard.’ ”
“That’s not interference, that’s prey pursuit. Interference is more like putting a squirrel in the backyard, then putting a second squirrel in the backyard one second later, and finding that you have no squirrels at all. But if you wait two seconds before putting in the second squirrel, you find four squirrels.”
“Okay, that’s just weird.”
“That’s my point.”
“Oh. Well, good job, then. Anyway, why are we talking about this?”
“Well, you need to know a few things about waves in order to understand quantum physics.”
“Yeah, but this just sounds like math. I don’t like math. When are we going to talk about physics?”
“We are talking about physics. The whole point of physics is to use math to describe the universe.”
“I don’t want to describe the universe, I want to catch squirrels.”
“Well, if you know how to desc
ribe the universe with math, that can help you catch squirrels. If you have a mathematical model of where the squirrels are now, and you know the rules governing squirrel behavior, you can use your model to predict where they’ll be later. And if you can predict where they’ll be later . . .”
“I can catch squirrels!”
“Exactly.”
“All right, math is okay. I still don’t see what this wave stuff is for, though.”
“We need it to explain the properties of light and sound waves, which is the next bit.”
WAVES IN EVERYDAY LIFE: LIGHT AND SOUND
We deal with two kinds of waves in everyday life: light and sound. Though these are both examples of wave phenomena, they appear to behave very differently. The reasons for those differences will help shed some light (pardon the pun) on why it is that we don’t see dogs passing around both sides of a tree at the same time.
Sound waves are pressure waves in the air. When a dog barks, she forces air out through her mouth and sets up a vibration that travels through the air in all directions. When it reaches another dog, that sound wave causes vibrations in the second dog’s eardrums, which are turned into signals in the brain that are processed as sound, causing the second dog to bark, producing more waves, until nearby humans get annoyed.
Light is a different kind of wave, an oscillating electric and magnetic field that travels through space—even the emptiness of outer space, which is why we can see distant stars and galaxies. When light waves strike the back of your eye, they get turned into signals in the brain that are processed to form an image of the world around you.
The most striking difference between light and sound in everyday life has to do with what happens when they encounter an obstacle. Light waves travel only in straight lines, while sound waves seem to bend around obstacles. This is why a dog in the dining room can hear a potato chip hitting the kitchen floor, even though she can’t see it.
The apparent bending of sound waves around corners is an example of diffraction, which is a characteristic behavior of waves encountering an obstacle. When a wave reaches a barrier with an opening in it, like the wall containing an open door from the kitchen into the dining room, the waves passing through the opening don’t just keep going straight, but fan out over a range of different directions. How quickly they spread depends on the wavelength of the wave and the size of the opening through
On the left, a wave with a short wavelength encounters an opening much larger than the wavelength, and the waves continue more or less straight through. On the right, a wave with a long wavelength encounters an opening comparable to the wavelength, and the waves diffract through a large range of directions.
which they travel. If the opening is much larger than the wavelength, there will be very little bending, but if the opening is comparable to the wavelength, the waves will fan out over the full available range.
Similarly, if sound waves encounter an obstacle like a chair or a tree, they will diffract around it, provided the object is not too much larger than the wavelength. This is why it takes a large wall to muffle the sound of a barking dog—sound waves bend around smaller obstacles, and reach people or dogs behind them.
Sound waves in air have a wavelength of a meter or so, close to the size of typical obstacles—doors, windows, pieces of furniture. As a result, the waves diffract by a large amount, which is why we can hear sounds even around tight corners.
Light waves, on the other hand, have a very short wave-length—less than a thousandth of a millimeter. A hundred wavelengths of visible light will fit in the thickness of a hair. When light waves encounter everyday obstacles, they hardly bend at all, so solid objects cast dark shadows. A tiny bit of diffraction occurs right at the edge of the object, which is why the edges of shadows are always fuzzy, but for the most part, light travels in a straight line, with no visible diffraction.
If we don’t readily see light diffracting like a wave, how do we know it’s a wave? We don’t see diffraction around everyday objects because they’re too large compared to the wavelength of light. If we look at a small enough obstacle, though, we can see unmistakable evidence of wave behavior.
In 1799 an English physicist named Thomas Young did the definitive experiment to demonstrate the wave nature of light. Young took a beam of light and inserted a card with two very narrow slits cut in it. When he looked at the light on the far side of the card, he didn’t just see an image of the two slits, but rather a large pattern of alternating bright and dark spots.
Young’s double-slit experiment is a clear demonstration of the diffraction and interference of light waves. The light passing
An illustration of double-slit diffraction. On the left, the waves from two different slits travel exactly the same distance, and arrive in phase to form a bright spot. In the center, the wave from the lower slit travels an extra half-wavelength (darker line), and arrives out of phase with the wave from the upper slit. The two cancel out, forming a dark spot in the pattern. On the right, the wave from the lower slit travels a full extra wavelength, and again adds to the wave from the upper slit to form a bright spot.
through each of the slits diffracts out into a range of different directions, and the waves from the two slits overlap. At any given point, the waves from the two slits have traveled different distances, and have gone through different numbers of oscillations. At the bright spots, the two waves are in phase, and add together to give light that is brighter than light from either slit by itself. At the dark spots, the waves are out of phase, and cancel each other out.
Prior to Young’s experiment, there had been a lively debate about the nature of light, with some physicists claiming that light was a wave, and others (including Newton) arguing that light was a stream of tiny particles. Interference and diffraction are phenomena that only happen with waves, though, so after Young’s experiment (and subsequent experiments by the French physicist Augustin Fresnel), everybody was convinced that light was a wave. Things stayed that way for about a hundred years.
• • •
“How does this relate to going around both sides of a tree? I’m not interested in going through slits, I want to catch bunnies.”
“The same basic process happens when you put small solid obstacles into the path of a light beam. You can think of the light that goes around to the left and the light that goes around to the right of the obstacle as being like the waves from two different slits. They take different paths to their destination, and thus can be either in phase or out of phase when they arrive. You get a pattern of bright and dark spots, just like when you use slits.”
“Oh. I guess that makes sense. So, I just need to get the bunnies to stand at the spots where I’m in phase with me?”
“No, because of the wavelength thing. We’ll get to that in a minute. I need to talk about particles, first.”
“Okay. I can be patient. As long as it doesn’t take too long.”
THE BIRTH OF THE QUANTUM: LIGHT AS A PARTICLE
The first hint of a problem with the wave model of light came from a German physicist named Max Planck in 1900. Planck was studying the thermal radiation emitted by all objects. The emission of light by hot objects is a very common phenomenon (the best-known example is the red glow of a hot piece of metal), and something so common seems like it ought to be easy to explain. By 1900, though, the problem of explaining how much light of different colors was emitted (the “spectrum” of the light) had thus far defeated the best physicists of the nineteenth century.
Planck knew that the spectrum had a very particular shape, with lots of light emitted at low frequencies and very little at high frequencies, and that the peak of the spectrum—the frequency at which the light emitted is brightest—depends only on the object’s temperature. He had even discovered a formula to describe the characteristic shape of the spectrum, but was stymied when he tried to find a theoretical justification for the formula. Every method he tried predicted much more light at high frequencies than was observed
. In desperation, he resorted to a mathematical trick to get the right answer.
Planck’s trick was to imagine that all objects contained fictitious “oscillators” that emit light only at certain frequencies. Then he said that the amount of energy (E) associated with each oscillator was related to the frequency of the oscillation (f) by a simple formula:
E = hf
where h is a constant. When he first made this odd assumption, Planck thought he would use it just to set up the problem, and then use a common mathematical technique to get rid of the imaginary oscillators and this extra constant h. Much to his surprise, though, he found that his results made sense only if he kept the oscillators around—if h had a very small but nonzero value.
Today, h is known as Planck’s constant in his honor, and has the value 6.626 × 10-34 kg m2/s (that’s 0.000000000000000 0000000000000000006626 kg m2/s). It’s a very small number indeed, but definitely not zero.
Planck’s trick amounts to treating light, which physicists thought of as a continuous wave, as coming in discrete chunks, like particles. Planck’s “oscillators” could only emit light in discrete units of brightness. This is a little like imagining a pond where waves can only be one, two, or three centimeters high, never one and a half or two and a quarter. Everyday waves don’t work that way, but that’s what Planck’s mathematical model requires.
These “oscillators” are also what puts the “quantum” in “quantum physics.” Planck referred to the specific levels of energy in his oscillators as “quanta” (the plural of “quantum,” from the Latin word for “how much”), so an oscillator at a given frequency might contain one quantum (one unit of energy, hf), two quanta, three quanta, and so on, but never one and a half or two and a quarter. The name for the steps stuck, and came to be applied to the entire theory that grew out of Planck’s desperate trick.