The Fabric of the Cosmos: Space, Time, and the Texture of Reality (12 page)

If you shine a laser pointer on a little piece of black, overexposed 35mm film from which you have scratched away the emulsion in two extremely close and narrow lines, you will see direct evidence that light is a wave. If you've never done this, it's worth a try (you can use many things in place of the film, such as the wire mesh in a fancy coffee plunger). The image you will see when the laser light passes through the slits on the film and hits a screen consists of light and dark bands, as in Figure 4.1, and the explanation for this pattern relies on a basic feature of waves. Water waves are easiest to visualize, so let's first explain the essential point with waves on a large, placid lake, and then apply our understanding to light.

A water wave disturbs the flat surface of a lake by creating regions where the water level is higher than usual and regions where it is lower than usual. The highest part of a wave is called its
peak
and the lowest part is called its
trough.
A typical wave involves a periodic succession: peak followed by trough followed by peak, and so forth. If two waves head toward each other—if, for example, you and I each drop a pebble into the lake at nearby locations, producing outward-moving waves that run into each other—when they cross there results an important effect known as
interference,
illustrated in Figure 4.2a. When a peak of one wave and a peak of the other cross, the height of the water is even greater, being the sum of the two peak heights. Similarly, when a trough of one wave and a trough of the other cross, the depression in the water is even deeper, being the sum of the two depressions. And here is the most important combination: when a peak of one wave crosses the trough of another, they tend to cancel each other out, as the peak tries to make the water go up while the trough tries to drag it down. If the height of one wave's peak equals the depth of the other's trough, there will be perfect cancellation when they cross, so the water at that location will not move at all.

Figure 4.1 Laser light passing through two slits etched on a piece of black film yields an interference pattern on a detector screen, showing that light is a wave.

The same principle explains the pattern that light forms when it passes through the two slits in Figure 4.1. Light is an electromagnetic wave; when it passes through the two slits, it splits into two waves that head toward the screen. Like the two water waves just discussed, the two light waves interfere with each other. When they hit various points on the screen, sometimes both waves are at their peaks, making the screen bright; sometimes both waves are at their troughs, also making it bright; but sometimes one wave is at its peak and the other is at its trough and they cancel, making that point on the screen dark. We illustrate this in Figure 4.2b.

When the wave motion is analyzed in mathematical detail, including the cases of partial cancellations between waves at various stages between peaks and troughs, one can show that the bright and dark spots fill out the bands seen in Figure 4.1. The bright and dark bands are therefore a telltale sign that light is a wave, an issue that had been hotly debated ever since Newton claimed that light is not a wave but instead is made up of a stream of particles (more on this in a moment). Moreover, this analysis applies equally well to
any
kind of wave (light wave, water wave, sound wave, you name it) and thus, interference patterns provide the metaphorical smoking gun: you know you are dealing with a wave if, when it is forced to pass through two slits of the right size (determined by the distance between the wave's peaks and troughs), the resulting intensity pattern looks like that in Figure 4.1 (with bright regions representing high intensity and dark regions being low intensity).

Figure 4.2
(
a
)
Overlapping water waves produce an interference pattern.
(
b
)
Overlapping light waves produce an interference pattern.

In 1927, Clinton Davisson and Lester Germer fired a beam of electrons—particulate entities without any apparent connection to waves—at a piece of nickel crystal; the details need not concern us, but what does matter is that this experiment is equivalent to firing a beam of electrons at a barrier with two slits. When the experimenters allowed the electrons that passed through the slits to travel onward to a phosphor screen where their impact location was recorded by a tiny flash (the same kind of flashes responsible for the picture on your television screen), the results were astonishing. Thinking of the electrons as little pellets or bullets, you'd naturally expect their impact positions to line up with the two slits, as in Figure 4.3a. But that's not what Davisson and Germer found. Their experiment produced data schematically illustrated in Figure 4.3b: the electron impact positions filled out an interference pattern characteristic of waves. Davisson and Germer had found the smoking gun.
They had shown that the beam of particulate electrons must, unexpectedly, be some
kind of wave.

Figure 4.3
(
a
)
Classical physics predicts that electrons fired at a barrier with two slits will produce two bright stripes on a detector.
(
b
)
Quantum physics predicts, and experiments confirm, that electrons will produce an interference pattern, showing that they embody wavelike features.

Now, you might not think this is particularly surprising. Water is made of H
₂
O molecules, and a water wave arises when many molecules move in a coordinated pattern. One group of H
₂
O molecules goes up in one location, while another group goes down in a nearby location. Perhaps the data illustrated in Figure 4.3 show that electrons, like H
₂
O molecules, sometimes move in concert, creating a wavelike pattern in their overall, macroscopic motion. While at first blush this might seem to be a reasonable suggestion, the actual story is far more unexpected.

We initially imagined that a flood of electrons was fired continuously from the electron gun in Figure 4.3. But we can tune the gun so that it fires fewer and fewer electrons every second; in fact, we can tune it all the way down so that it fires, say, only one electron every ten seconds. With enough patience, we can run this experiment over a long period of time and record the impact position of each individual electron that passes through the slits. Figures 4.4a-4.4c show the resulting cumulative data after an hour, half a day, and a full day. In the 1920s, images like these rocked the foundations of physics.
We see that even individual, particulate
electrons, moving to the screen independently, separately, one by one, build
up the interference pattern characteristic of waves.

This is as if an individual H
₂
O molecule could still embody something akin to a water wave. But how in the world could that be? Wave motion seems to be a collective property that has no meaning when applied to separate, particulate ingredients. If every few minutes individual spectators in the bleachers get up and sit down separately, independently, they are
not
doing the wave. More than that, wave interference seems to require a wave from
here
to cross a wave from
there.
So how can interference be at all relevant to single, individual, particulate ingredients? But somehow, as attested by the interference data in Figure 4.4, even though individual electrons are tiny particles of matter, each and every one also embodies a wavelike character.

Figure 4.4 Electrons fired one by one toward slits build up an interference pattern dot by dot. In
(
a
)-(
c
)
we illustrate the pattern forming over time.

If an individual electron is also a wave, what is it that is waving? Erwin Schrödinger weighed in with the first guess: maybe the stuff of which electrons are made can be smeared out in space and it's this smeared electron essence that does the waving. An electron particle, from this point of view, would be a sharp spike in an electron mist. It was quickly realized, though, that this suggestion couldn't be correct because even a sharply spiked wave shape—such as a giant tidal wave—ultimately spreads out. And if the spiked electron wave were to spread we would expect to find part of a single electron's electric charge over here or part of its mass over there. But we never do. When we locate an electron, we always find all of its mass and all of its charge concentrated in one tiny, pointlike region. In 1927, Max Born put forward a different suggestion, one that turned out to be the decisive step that forced physics to enter a radically new realm. The wave, he claimed, is not a smeared-out electron, nor is it anything ever previously encountered in science. The wave, Born proposed, is a
proba
bility wave.

To understand what this means, picture a snapshot of a water wave that shows regions of high intensity (near the peaks and troughs) and regions of low intensity (near the flatter transition regions between peaks and troughs). The higher the intensity, the greater the potential the water wave has for exerting force on nearby ships or on coastline structures. The probability waves envisioned by Born also have regions of high and low intensity, but the meaning he ascribed to these wave shapes was unexpected:
the size of a wave at a given point in space is proportional to the
probability that the electron is located at that point in space.
Places where the probability wave is large are locations where the electron is most likely to be found. Places where the probability wave is small are locations where the electron is unlikely to be found. And places where the probability wave is zero are locations where the electron will not be found.

Figure 4.5 gives a "snapshot" of a probability wave with the labels emphasizing Born's probabilistic interpretation. Unlike a photograph of water waves, though, this image could not actually have been made with a camera. No one has ever directly seen a probability wave, and conventional quantum mechanical reasoning says that no one ever will. Instead, we use mathematical equations (developed by Schrödinger, Niels Bohr, Werner Heisenberg, Paul Dirac, and others) to figure out what the probability wave should look like in a given situation. We then test such theoretical calculations by comparing them with experimental results in the following way. After calculating the purported probability wave for the electron in a given experimental setup, we carry out identical versions of the experiment over and over again from scratch, each time recording the measured position of the electron.
In contrast to what Newton would have
expected, identical experiments and starting conditions do not necessarily
lead to identical measurements.
Instead, our measurements yield a variety of measured locations. Sometimes we find the electron here, sometimes there, and every so often we find it
way
over there. If quantum mechanics is right, the number of times we find the electron at a given point should be proportional to the size (actually, the square of the size), at that point, of the probability wave that we calculated. Eight decades of experiments have shown that the predictions of quantum mechanics are confirmed to spectacular precision.

Figure 4.5 The probability wave of a particle, such as an electron, tells us the likelihood of finding the particle at one location or another.

Only a portion of an electron's probability wave is shown in Figure 4.5: according to quantum mechanics, every probability wave extends throughout all of space, throughout the entire universe.
⁶
In many circumstances, though, a particle's probability wave quickly drops very close to zero outside some small region, indicating the overwhelming likelihood that the particle is in that region. In such cases, the part of the probability wave left out of Figure 4.5 (the part extending throughout the rest of the universe) looks very much like the part near the edges of the figure: quite flat and near the value zero. Nevertheless, so long as the probability wave somewhere in the Andromeda galaxy has a nonzero value, no matter how small, there is a tiny but genuine—nonzero—chance that the electron could be found there.

Thus, the success of quantum mechanics forces us to accept that the electron, a constituent of matter that we normally envision as occupying a tiny, pointlike region of space, also has a description involving a wave that, to the contrary, is spread through the entire universe. Moreover, according to quantum mechanics this particle-wave fusion holds for all of nature's constituents, not just electrons: protons are both particlelike and wavelike; neutrons are both particlelike and wavelike, and experiments in the early 1900s even established that light—which demonstrably behaves like a wave, as in Figure 4.1—can also be described in terms of particulate ingredients, the little "bundles of light" called photons mentioned earlier.
⁷
The familiar electromagnetic waves emitted by a hundred-watt bulb, for example, can equally well be described in terms of the bulb's emitting about a hundred billion billion photons each second. In the quantum world, we've learned that everything has both particlelike and wavelike attributes.

Over the last eight decades, the ubiquity and utility of quantum mechanical probability waves to predict and explain experimental results has been established beyond any doubt. Yet there is still no universally agreed-upon way to envision what quantum mechanical probability waves actually are. Whether we should say that an electron's probability wave
is
the electron, or that it's
associated
with the electron, or that it's a
mathematical device
for describing the electron's motion, or that it's the
embodiment of what we can know
about the electron is still debated. What is clear, though, is that through these waves, quantum mechanics injects probability into the laws of physics in a manner that no one had anticipated. Meteorologists use probability to predict the likelihood of rain. Casinos use probability to predict the likelihood you'll throw snake eyes. But probability plays a role in these examples because we haven't all of the information necessary to make definitive predictions. According to Newton, if we knew in complete detail the state of the environment (the positions and velocities of every one of its particulate ingredients), we would be able to predict (given sufficient calculational prowess) with certainty whether it will rain at 4:07 p.m. tomorrow; if we knew all the physical details of relevance to a craps game (the precise shape and composition of the dice, their speed and orientation as they left your hand, the composition of the table and its surface, and so on), we would be able to predict with certainty how the dice will land. Since, in practice, we can't gather all this information (and, even if we could, we do not yet have sufficiently powerful computers to perform the calculations required to make such predictions), we set our sights lower and predict only the probability of a given outcome in the weather or at the casino, making reasonable guesses about the data we don't have.

The probability introduced by quantum mechanics is of a different, more fundamental character. Regardless of improvements in data collection or in computer power, the best we can ever do, according to quantum mechanics, is predict the probability of this or that outcome. The best we can ever do is predict the probability that an electron, or a proton, or a neutron, or any other of nature's constituents, will be found here or there. Probability reigns supreme in the microcosmos.

As an example, the explanation quantum mechanics gives for individual electrons, one by one, over time, building up the pattern of light and dark bands in Figure 4.4, is now clear. Each individual electron is described by its probability wave. When an electron is fired, its probability wave flows through both slits. And just as with light waves and water waves, the probability waves emanating from the two slits interfere with each other. At some points on the detector screen the two probability waves reinforce and the resulting intensity is large. At other points the waves partially cancel and the intensity is small. At still other points the peaks and troughs of the probability waves completely cancel and the resulting wave intensity is exactly zero. That is, there are points on the screen where it is very likely an electron will land, points where it is far less likely that it will land, and places where there is no chance at all that an electron will land. Over time, the electrons' landing positions are distributed according to this probability profile, and hence we get some bright, some dimmer, and some completely dark regions on the screen. Detailed analysis shows that these light and dark regions will look exactly as they do in Figure 4.4.