by Seth Lloyd
Borges asked me to repeat the question in a more comprehensible fashion. When he understood that I was asking whether or not the foundations of quantum mechanics had influenced his writing, he answered, “No.” He went on to say that although he had not been influenced by work on quantum mechanics, he was not surprised that the laws of physics mirrored ideas from literature. After all, physicists were readers, too.
And in fact “The Garden of Forking Paths” was published in 1941, years before John Wheeler’s student Hugh Everett introduced the Many Worlds interpretation of quantum mechanics. So if there was influence, it was from literature to physics, not the other way around.
Wave-Particle Duality
Quantum mechanics is in fact very much like one of Borges’s Ficciones. But its weirdness, as I called it earlier, accurately reflects the fundamental structure of the universe. In the early days of the twentieth century, the Danish physicist Niels Bohr used quantum mechanics to successfully explain the structure of the hydrogen atom, but—like most workers in quantum mechanics, including Einstein—Bohr found the theory counterintuitive. Einstein’s response was to reject quantum mechanics (“God does not play dice,” he said). Bohr’s, on the other hand, was to develop an almost mystical philosophy of the quantum world. Whatever attitude you choose to adopt toward quantum mechanics, though, if you get dizzy contemplating it, that’s a good sign. Of course, dizziness in itself doesn’t guarantee that you have understood quantum mechanics, but it’s a start.
To attain an understanding of quantum mechanics at the intuitive (or, more precisely, the counterintuitive) level, a good start is to contemplate the principle that Bohr called wave-particle duality. “Wave-particle duality” refers to the fact that things we normally think of as waves, like light or sound, are actually made up of particles, or quanta (quantum is the Latin word for “how much”). Particles of light are called photons (“-on” being the usual suffix denoting a particle); particles of sound are called phonons.
A simple experiment demonstrates the quantum nature of light. A photodetector is a device that detects light. It produces an electrical current whose magnitude is proportional to the amount of light it absorbs. A photodetector in a bright room produces a lot of current. When the light is turned down, the current goes down, too. When the lights are turned off and the shades are drawn, the current gets close to zero. Finally, if one takes pains to exclude almost all light, covering windows and doors with plywood and taping all the cracks, the photodetector starts to exhibit a different sort of behavior. The current is zero most of the time, but every now and then it spikes in a short burst. The photodetector is detecting individual photons.
The notion that waves are actually made of particles is very old. That sound is made of waves has been known since Pythagoras, but the ancient Greeks thought light was composed of particles and argued over whether these particles came from the eye or from the object viewed. Newton proposed a “corpuscular theory” of light in terms of particles. But Newton’s own experiments with prisms were more easily explained if light was made of waves, and the wave theory of light dominated from the seventeenth century until the end of the nineteenth, culminating in Maxwell’s equations, which explained all known electromagnetic phenomena in terms of waves of light.
There was a problem with thinking of light only in terms of waves, however, and that problem had to do with heat. At the end of the nineteenth century, Max Planck analyzed the light emitted from a stove that was heated until it glowed red. Such light is called “blackbody radiation.” Black objects absorb and emit light of all frequencies (frequency being the rate at which a light wave wiggles up and down). Planck pointed out that if light was indeed made of waves, then the amount of energy and entropy in the radiation emerging from hot objects should be infinite, a serious problem for both the first and second laws of thermodynamics. He was able to resolve this problem by assuming that light was made out of particles whose energy was proportional to the frequency of the wave. Planck called these particles photons. Each photon carried a small amount of energy—a quantum. Planck found that if the energy of each of these particles (measured in joules) was equal to 6.63 × 10-34 times the wave’s frequency per second, then energy was conserved by the radiant heat. Planck’s constant relates energy to frequency. It is so ubiquitous in physics that it has been given its own special symbol, h.
Things we think of as waves correspond to particles; this is the first aspect of wave-particle duality. The second, complementary aspect of wave-particle duality is that things we think of as particles correspond to waves. Just as every wave is made up of particles, every particle—an electron, an atom, a pebble—has a wave associated with it. The wave is associated with the position of the particle: the particle is more likely to be found in places where the wave is big. The distance between the peaks of the wave is related to the particle’s speed: the smaller the distance from peak to peak, the faster the particle is going. Finally, the wave’s frequency is proportional to the energy of the particle. In fact, the particle’s energy is exactly equal to the frequency times Planck’s constant.
The Double-Slit Experiment
The double-slit experiment demonstrates the wave nature of particles. Waves superpose, or interfere, with each other. If my daughter Emma, sitting at one end of the bathtub, sets a wave going toward her sister Zoe, sitting at the other end, and Zoe sets a wave going toward Emma at the same time, then when the waves meet in the middle, they splash out all over me. Light waves will combine to interfere with each other in the same way. If you shine a beam of light at a screen that has two slits in it and look at the pattern the light makes on the wall beyond the screen, you see alternating bands of light and dark. This is called an “interference pattern.” The light waves, moving like water waves through pilings, go through both slits at once; each wave thus splits in two and then combines on the wall. The bands of light occur in places where the peaks and troughs of the wave from one slit coincide with the peaks and troughs of the wave from the other and reinforce them, a phenomenon called “positive interference.” The bands of dark occur where the peaks of the wave from one slit coincide with the troughs of the other wave and the two waves cancel each other out, a phenomenon called “negative interference.” If you cover one of the slits, the interference pattern goes away, because there is no wave to interfere with the wave that goes through the remaining slit. Interference, crucially, requires the wave to go through both slits at once.
The double-slit experiment can be performed with particles as well. Shoot a beam of particles—electrons, say—at a screen with two slits and put a photographic plate on the wall to record where the particles land. Each particle leaves a spot on the photographic plate. If you close up the left slit and let the electrons go only through the right slit, then you get one pattern of spots. If you close up the right slit and let the electrons go only through the left slit, you get another pattern.
Figure 8. The Double-Slit Experiment
In the double-slit experiment, particles are sent first through a single slit, then through a double slit, before landing on a screen. The pattern that the particles make on a screen exhibits an “interference pattern,” evidence for the underlying wavelike nature of the particles.
Now open both slits. What sort of pattern do you expect to see? Classical reasoning says that each electron will go through either one slit or the other. Thus you expect to see a distribution of spots on the photographic plate that is just the two single-slit distributions of the previous paragraph combined. You don’t expect to see an interference pattern, because each of the particles should go through only one slit. An interference pattern is a wave phenomenon; it arises because waves can and do go through both slits at once. But a particle is a particle: it can’t go through both slits at once.
Perform the double-slit experiment with particles. What do you see? An interference pattern! The spots made by the individual particles fall across the photographic plate in a series of bands. When
you cover one of the slits, the interference pattern disappears. Evidently, the particles behave as if they were waves.
What’s going on? Maybe the electrons going through one slit are knocking into the electrons going through the other slit and making some kind of pattern. OK, decrease the number of electrons in the beam to minimize collisions. The interference pattern remains. Now shoot a single electron at the screen. The interference pattern is still there, but now it simply governs the probability for where the electron arrives on the photographic plate: it preferentially lands where the band of spots lies, thus the interference pattern can’t be caused by multiple particles interfering with one another. There is only one electron, but somehow it still behaves as if it were a wave. The experiment reveals that the particle goes through both slits at once. An electron, a proton, a photon, an atom can be in two places at the same time.
The double-slit experiment illustrates the fact that a particle doesn’t have to be either “here” or “there.” Because of its underlying wavelike nature, a particle can be both “here” and “there” at the same time. This ability of things to be in many places at once is responsible for the power of quantum computation, which we will explore further later on.
Decoherence
If things can be in two places at once, then why don’t we see pebbles, people, and planets showing up in more places than one? The Austrian physicist Anton Zeilinger has performed the double-slit experiment successfully with so-called buckyballs, soccer-ball-like constructions made of sixty carbon atoms. He plans next to do the double-slit experiment with bacteria about 100 times larger. The bigger something is, however, the harder it is to coax it into existing in two places at once. (Big things tend to behave more “classically,” and less quantum-mechanically.) The reason lies not so much with the physical size of the object as with its visibility. The bigger something is, the more interactions it tends to have with its surroundings, thus the easier it is to detect. In order to go through both slits at once and produce an interference pattern, a particle must pass through the slits undetected.
Suppose you place a detector on the right-hand slit. The detector registers the presence or absence of a particle at the slit, letting the particle pass through otherwise unchanged. When the detector detects a particle, it clicks. Now perform the double-slit experiment with the detector operating. Look at the screen. The interference pattern has disappeared!
What happened? Recall that an interference pattern stems from the wave associated with a particle. That wave naturally goes through both slits at once. When the detector is operating, a particle going through the right slit will cause it to click. A particle going through the left slit will not. (Whether or not the detector clicks is random: the particle goes through one slit or the other with equal probability.)
When the detector clicks and detects the particle, the wave corresponding to the particle has been localized to the right slit. When the detector fails to click, the particle has been localized to the left slit. This process of localization of the wave is sometimes called “collapse of the wave function.” That is, when the detector is “observing” the right slit, the particle has to go through either one slit or the other; it no longer goes through both slits at once. And since the wave corresponding to the particle no longer goes through both slits at once, it cannot interfere with itself to produce the interference pattern’s alternating bands of light and dark.
Observation (or measurement, as it is conventionally called) destroys interference. Without measurement, the particle merrily goes through both slits at once; with measurement, it goes through one or the other. In other words, measurement intrinsically disturbs the particle. When you ask the particle where it is, it is forced to confess that it is in one place or another and no longer in both places at once.
It is interesting to note, in the above experiment, that the measurement disturbs the particle’s wave whether or not the detector clicks. The detector clicks only if the particle goes through the right-hand slit, where the detector is located. But when the detector fails to click, meaning that the particle has gone through the left-hand slit, the interference pattern is still destroyed—that is, the measurement still disturbs the particle’s wave. The particle need not ever come close to the detector. (Are you dizzy yet?) Nor does the detector have to be a macroscopic device: All that is required to destroy the interference pattern is for some system, no matter how small, to get information about the position of the particle. If the particle knocks a passing electron or molecule of air, for example, that, too, will destroy the interference pattern.
It is now clear why big things tend to show up in one place or another, but not both. Pebbles, people, and planets are constantly interacting with their surroundings. Each interaction with an electron, a molecule of air, a particle of light tends to localize a system. Big things interact with lots of little things, each of which gets information about the location of the big thing. As a result, big things tend to appear here or there instead of here and there at the same time.
The process by which the environment destroys the wavelike nature of things by getting information about a quantum system is called “decoherence.” Decoherence is a common process. Remember the argument given earlier for entropy increase: almost any interaction between one thing and another causes the first thing to get information about the second, and vice versa. As the spread of ignorance shows, these interactions cause the entropies of the things taken on their own to increase. The same mechanism operates to make quantum objects behave in a more classical way.
Quantum Bits
In the previous chapter, each mechanism by which information was conserved, spread about, erased, or increased was illustrated by a simple example presented in terms of bits. To understand how quantum mechanics work, then, wouldn’t it be nice to have a similar, quantum-mechanical device? A good example of a quantum-mechanical bit, or qubit, is a nuclear spin, like that of protons and neutrons in the spin-echo effect. “Spin up” is conventionally given a bit value of 0 and “spin down” a value of 1. The bit value of a nuclear spin can be determined by putting the spin through a device called a Stern-Gerlach apparatus, which discriminates between 0 and 1 by moving spin-up nuclei in one direction and spin-down nuclei in the opposite direction (their positions are recorded on a photographic plate). Both possible values for spin correspond to waves: a wave moving counterclockwise for spin-up (or 0) and a wave moving clockwise for spin-down (or 1). The wave corresponding to 0 is customarily represented by the symbol |0> and the wave corresponding to 1 by the symbol |1>. The “| >” or “bracket” notation has a mathematical significance, but for our purposes here it simply serves to indicate that whatever appears within the brackets is a quantum-mechanical object—a wave.
It is possible to combine waves. The resulting combination is referred to as a “superposition.” What is the state of the system corresponding to the sum—or superposition—of the wave for spin-up and the wave for spin-down? That is, what wave corresponds to the state |0> + |1>? In the case of spins, this state turns out to be easy to visualize: it is a state of spin along an axis perpendicular to the axis defining spin-up and spin-down. Spin-up plus spin-down is spin sideways!
It’s also possible to subtract waves from each other. The wave designated -|1> is a wave whose troughs correspond to the peaks of the wave |1> and whose peaks correspond to the troughs of the wave |1>. That is, -|1> wiggles where |1> waggles, and vice versa. Now look at the superposition |0> - |1>. This state is also easily visualized. It is a state of spin along the same axis as the state |0> + |1>, but in the opposite direction. Thus the direction of the spin depends crucially on the sign (or phase) of each wave in the superposition. We could distinguish between these states along the sideways axis by taking the Stern-Gerlach apparatus and turning it on its side.
Figure 9abc. Quantum Bits
a
b
c
A nuclear spin is a quantum bit. Spin counterclockwise, or “up,
” registers the logical state 0 (figure 9a). Spin clockwise, or “down,” registers the logical state 1 (figure 9b). Spin “sideways” is a quantum state that registers 0 and 1 at the same time.
The state |0> + |1> has a definite value of spin along the sideways axis. If you measure which direction it is spinning about that axis, you always find that it is spinning clockwise. But when you take this same spin and try to determine its value of spin about the vertical axis, the result will be completely random; half the time you will find that it is clockwise (that is, you find the state spin-up, or |0>) and half the time you will find that it is counterclockwise (spin-down, or |1>). When the value of spin about the sideways axis is completely certain, the value of spin about the vertical axis is completely uncertain.
Similarly, the state |0> has a definite value of spin along the vertical axis. If you measure the spin, you find that it is clockwise (spin-up). But now the value of spin about the sideways axis is completely uncertain; if you measure the spin about the sideways axis, half the time you’ll find it spinning clockwise about this axis and half the time you’ll find it spinning counterclockwise. Now that the value of spin about the vertical axis is certain, the value of spin about the sideways axis is uncertain.
The Heisenberg Uncertainty Principle
Apparently it is not possible to have a definite value of spin about two different axes at the same time. This intrinsically chancy nature of quantum mechanics was immortalized by Werner Heisenberg, one of the founders of quantum mechanics, as the “uncertainty principle.” The uncertainty principle states that if the value of some physical quantity is certain, then the value of a complementary quantity is uncertain. Spin about the vertical axis and spin about the sideways axis are just such complementary quantities: if you know one, you can’t know the other. Another pair of complementary quantities are position and speed: if you know the position of a particle exactly, then you know nothing of how fast it’s going. (Traffic cop pulls over Heisenberg’s car: “Professor Heisenberg, do you have any idea how fast you were going?” Heisenberg: “No, but I know exactly where I am.”)
