One of the most stubborn enigmas of modern physics is neatly summarized by playwright Tom Stoppard in his spy drama Hapgood: There is a straight ladder from the atom to the grain of sand, and the only real mystery is the missing rung. Below it, particle physics. Above it, classical physics. But in between, metaphysics.
Classical physics, in the tradition of Isaac Newton, refers to the description of the world in terms of objects--atoms, marbles, planets, and galaxies--that move along precise trajectories through space and time. It is the world of the baseball diamond and of the solar system, predictable, familiar, readily visualized. Particle physics, on the other hand, describes the quantum world, populated with such subatomic objects as quarks and electrons, whose existence can be detected only through the most indirect means. Though every bit as successful in its own sphere as classical physics is in the ordinary world, quantum physics describes a world that is unimaginable. It deals as much with potentiality and possibility as with actuality; it speaks not of certainty but of chance and randomness. Its abstract mathematical equations defy translation into visual terms.
When physicists step across the gap that separates these two realms, they guide themselves more by instinct than by reason--picking some concepts from quantum mechanics and combining them with others from classical mechanics, as convenience and intuition suggest. Metaphysics, Stoppard contemptuously calls this haphazard approach, and with the fine insight of the artist touches a sensitive nerve. Physicists detest the accusation of engaging in speculative philosophy, although they deserve it in this instance. From the birth of quantum theory in 1925 to this day, much theoretical effort has been devoted to finding the missing rung, but a universally satisfactory reconciliation of quantum theory with classical physics has yet to be discovered.
Recently experimentalists have joined the quest by opening a new window on this forbidding territory. The focus of their attention has been a class of objects known as Rydberg atoms, named after nineteenth-century Swedish physicist Johannes Robert Rydberg. These are ordinary atoms in which the outermost electron has been promoted to an immensely large orbit. (To gain some idea of just how large that orbit is, you may imagine that by analogy, a Rydberg solar system would look like the real one, except that Pluto would somehow have been pushed out a thousand times farther from the sun than it is now.) Rydberg atoms occur in nature, but they are extremely delicate--even a small disturbance can tear the distant electron from its orbit and leave behind the positively charged rump of the atom (the ion). The precision of modern lasers, however, is allowing physicists to manipulate these exotic atoms; and, as it turns out, they function as a natural magnifying glass focused on the quantum-classical boundary.
Imagine that such an atom is placed in an external electric or magnetic field that guides the outermost electron along a cometlike orbit, one that periodically dips into the core of the atom between long stretches at a great distance. In its travel, the electron crosses and recrosses the gap in the ladder: far from the nucleus, it obeys classical mechanics, but inside the atom--as it comes barreling around the nucleus the way Halley’s comet races around the sun once every 76 years--the electron mingles with the other electrons in a quantum mechanical cloud. The electron behaves like an atomic amphibian, as it were, sprinting along the firm ground of classical mechanics before it plunges back into the swirling waves of quantum mechanics. A full understanding of this hybrid system requires the seamless splicing of the classical and quantum descriptions.
The beguiling image of an electron as a comet has its roots in the first viable model of the interior of the atom, proposed by Niels Bohr in 1913. In a bold leap of the imagination, Bohr compared the orbit of the electron in an invisible hydrogen atom to the billion-trillion-times-larger orbit of Earth, with the tiny just-discovered nucleus playing the role of the sun and gravity replaced by an electrical attraction. By extension, the lithium atom with its three electrons was represented by three oval orbits around a central dot. As the universal icon of the atom, this image adorns stamps, coins, cartoons, and the letterheads of countless electronics companies. In the form of a five-foot aluminum sculpture it graces the physics building at the College of William and Mary, in which I work. It is a powerful, instantly recognizable symbol--its only drawback being that it is utterly wrong.
As a serious scientific model, the planetary atom lasted only about six years. By 1919 its creator had already abandoned it. I am quite prepared, or rather more than prepared, to give up all ideas of electronic arrangements in ‘rings,’ Bohr wrote to a colleague in reaction to the mounting evidence of conflicts between his model and the experimental evidence. In the case of hydrogen, for example, the planetary analogy implied that the electron always remains at a distance from the nucleus, the way Earth always maintains its separation from the sun, whereas in fact the most likely place to find the electron is in the immediate vicinity of the nucleus. Furthermore, the model is as flat as a pancake, while experiment shows that the real hydrogen atom is a perfectly spherical, partially translucent cloud. These and countless other pieces of hard evidence began to discredit the planetary model soon after its birth. What little scientific currency it managed to hang on to was destroyed completely with the advent of quantum mechanics in 1925.
And yet the icon lives on, 76 years after its own author repudiated it and 70 years after it was officially declared dead. Physicists are partly to blame for this, because they have not been nearly vociferous enough in their objections to its continued use. But the more fundamental problem in disposing of this outdated image is that there just doesn’t seem to be a reasonable substitute. The human imagination craves pictures; quantum theory steadfastly refuses to furnish them.
No one has been more insistent on this austere point of view than the inventor of the quantum theory, Werner Heisenberg. The atom of modern physics can be symbolized only through a partial differential equation in an abstract space of many dimensions, he wrote in 1945. All its qualities are inferential; no material properties can be directly attributed to it. That is to say, any picture of the atom that our imagination is able to invent is for that very reason defective. An understanding of the atomic world in that primary sensuous fashion . . . is impossible.
This opinion contrasted sharply with that of Erwin Schrödinger, who developed quantum mechanics independently of Heisenberg. Schrödinger insisted that the goal of atomic physics is to create images of the interior of the atom that appeal to our intuition and make sense in terms of ordinary experience. Pictures, he thought, were indispensable. But in spite of his best efforts to interpret his theory visually, Schrödinger never succeeded in improving upon Bohr’s nifty, though obsolete, icon.
Modern images of atoms tend, if anything, to exacerbate the problem. Scanning-tunneling microscopes, which in the early 1980s for the first time rendered atoms on the surfaces of solid materials visible to the eye, show them as beautiful, multicolored lumps. But the lumps are invariably covered by what appear to be impenetrable veils. Although these shrouds are mere artifacts of the computers that re-create the three- dimensional structure of the surfaces, they underscore the failure of the microscope to penetrate to the interior of the atom. The veiled lumps we see belong indisputably to the classical world. The underlying quantum mechanical atoms remain resolutely invisible.
The first hint of the way in which Rydberg atoms can mediate between the classical and quantum worlds appeared unexpectedly in 1969. Frank Tomkins and William Garton at the Argonne National Laboratory near Chicago were investigating how light is absorbed as it shines through a bottle of barium gas. As they looked at the light exiting the bottle, they saw that the barium atoms had, as expected, absorbed light only of particular frequencies, or energies. As they slowly increased the energy of the light particles, called photons, they observed, at regular intervals, telltale peaks, or spikes, in the amount of light the barium atoms absorbed.
The origin of these spikes was well known. Every atom incorporates its own peculiar staircase of discrete quantum mechanical energy levels--corresponding to orbits, in Bohr’s obsolete analogy--upon which electrons may rest. If a passing photon carries just the right amount of energy, the atom will swallow it up and use its energy to raise an electron to a higher step on the staircase. If it carries anything other than the right amount of energy, however, the photon will continue on its way unhindered. As Tomkins and Garton had expected, their barium atoms absorbed only those photons whose energies corresponded to the known steps on barium’s energy staircase. Eventually, as the energy of the photons increased, the outermost electron of the barium atom was torn away, leaving behind a positively charged ion. Beyond this ionization energy, the barium atom could absorb nothing, and Tomkins and Garton observed no more spikes in the gas’s absorption of light.
The two physicists then placed their flask of barium gas between the jaws of a powerful magnet to study the effects of a magnetic field on the atoms’ behavior. This time when they turned their laser up beyond the ionization energy, they encountered a surprise: even though the outermost electrons should have slipped their moorings and left the barium atoms behind, the atoms were somehow still absorbing energy. To be sure, the scientists didn’t see sharp spikes indicating definite quantum energy levels, but they saw something even more confusing: minute, slightly rounded ripples.
What could account for this puzzling behavior? The inner electrons were too tightly bound to the nucleus to cause absorption spikes. The results suggested that a new mechanism of absorption was at work: apparently the rogue outer electron was still influencing the atom in some way. Theorists later suspected that the magnetic field had converted ordinary barium atoms into exotic Rydberg atoms, in which the outermost electron left the atom and moved in huge, cometlike orbits. When it returned periodically to the nucleus, the electron somehow modified the atom’s normal pattern of allowed energy levels, giving it the ability to absorb photons when it would normally have been saturated. In 1969, however, the theoretical machinery was not yet in place to confirm this hypothesis.
Even as theorists began to propose tentative explanations for the phenomenon, the mystery deepened. More precise experiments--made possible by the development of lasers that could offer dramatically improved fine- tuning of photon energy--were carried out at MIT and at the University of Bielefeld in Germany. Under this closer scrutiny, scientists once again observed atoms absorb photons beyond their ionization energy--but now they saw the smooth, regular ripple of the Argonne experiment dissolved into a messy, seemingly irregular oscillation that appeared to bear no relation whatsoever to the original observations. The experimentalists were justifiably alarmed. They might not know precisely what was allowing the atom to absorb extra energy, but they certainly expected to see that absorption portrayed as a neat, regular staircase. Instead they saw the craggy side of a cliff.
A satisfactory explanation did not appear until 1988 when John Delos, a colleague of mine at William and Mary, proposed a theory that removed the apparent disagreement between the old and the new observations. When I asked John what attracted him to the problem of the amphibious Rydberg atoms, he reminded me that he is something of an amphibian himself. Although he is a professor of physics, his Ph.D. is in chemistry. I guess with that background I was destined to stumble into the gap between the atom and the grain of sand, he joked.
In climbing out of that gap, however, Delos came up with an intuitively appealing way of visualizing the behavior of these odd atoms, and in doing so he went a fair way toward realizing Schrödinger’s goal of creating images of the atomic world that make sense to us. Delos’s outermost electron behaves like a classical particle that travels along a real orbit through space and time, away from the central ion and back again, like a comet. But as it repeatedly crosses the line into the nonclassical world of the atomic nucleus, it manifests all the bizarre attributes of the quantum particle it remains at heart. Amazingly, Delos found a way to derive one character from the other.
In quantum mechanics, an electron is described according to the principle of wave-particle duality, which holds that every moving particle can be regarded as a wave whose characteristics depend on the particle’s energy. Delos’s insight was to realize that interpreting the departing and arriving electron as a wave meant that its outgoing and incoming portions will inevitably display the symptoms of interference. In other words, when conditions are right for crest to meet crest, and trough to line up with trough, the wave will reinforce its own tail in mutually enhancing cooperation. On the other hand, if each crest happens to meet a trough, and vice versa, the wave will cancel itself out. Ultimately, Delos saw, the survival of some of these quantum mechanical waves and the canceling out of others result in only certain trajectories’ being allowed for the electron in its classical cometlike ramblings far from the nucleus.
When the laser light impinges on the atom, it causes the wave- electron to leave the nucleus and travel outward, much as ripples travel away from a stone dropped into a pond. But the ripples on the water’s surface are essentially a two-dimensional phenomenon, while the electron takes the form of a three-dimensional spherical wave. That wave’s movement outward marks the infinite possible trajectories that the electron, as a classical particle, might take as it moves away from the atom. At some distance away, the electron shakes off the effects of quantum mechanics and follows winding trajectories that eventually lead back to the nucleus. As it arrives, it once again takes on the character of a wave, this time traveling in toward the nucleus from all directions (you can think of this, again in two dimensions, as the inward-traveling ripples produced on the surface of your morning coffee by tapping the side of the cup). However, the elaborate orbits that the electron follows while it is far from the atom mean that the returning wave-electron is not a near-perfect sphere but a highly irregular, complex wave.
Once Delos established that only some trajectories are produced, he had effectively explained the new mechanism that caused the mysterious ripples. The Rydberg electron is allowed to continue to absorb energy, so long as that energy is precisely of an amount that will propel the electron to the next trajectory allowed by the interference pattern.
When Delos calculated the precise shape of the simple, classical orbit of the outermost electron that gave rise to the ripples seen in the Argonne experiments, he found himself back at the point where Bohr had abandoned his model in 1919: he was assigning definite trajectories to atomic electrons. But unlike Bohr’s electrons, these were not electrons in the core of the atom. They were Rydberg electrons, far from the nucleus and uninfluenced by interactions with other particles. Furthermore, as they approached their parent atom, they shifted between their classical incarnation to their quantum mechanical one.
Encouraged by his success in explaining the ripples found by Tomkins and Garton, Delos pressed on, but he quickly ran into a problem. He knew that in addition to the short, simple orbits he had started with, an electron departing from the vicinity of an ion and deflected by a magnetic field can also describe fancier orbits with increasingly longer size and duration. So he and his colleagues began to compute the shapes of hundreds of them in increasing order of duration, using nothing more complicated than classical mechanics. Isaac Newton could have done the same thing given infinite patience or access to a computer. The plots of these special orbits display wonderfully intricate swoops and folds like the tracks of a figure skater’s pirouettes. Delos had ignored these long-duration orbits when he explained the Argonne results. If he refined his calculation to include them, however, wouldn’t they create new energy levels and cause new absorption peaks? And wouldn’t his fine model cease to explain the Argonne results? If Delos’s classical orbits, which in complexity far surpassed Bohr’s simple ellipses, really exist, these difficult questions would have to be answered.
The solution turned out to lie in another bit of quantum theory, Heisenberg’s uncertainty principle. This fundamental law asserts that certain pairs of variables cannot be simultaneously determined with great accuracy. Thus, for example, if the speed of an electron is measured with great precision, its position will necessarily be uncertain, and vice versa. The variables relevant to Rydberg orbits are not so much position and speed as energy and time. A rough measurement of energy will pick out only those features of the atom that occur over a very short time. Conversely, a narrower pinning down of the energy automatically includes longer-lasting effects.
The original experiment of Garton and Tomkins, performed with relatively unsophisticated equipment, measured absorption with a crude energy resolution. This meant in turn that the ripples due only to the shortest orbits were discernible. Later, when the scientists began to use lasers to make more precise measurements of the photon energy, orbits with much longer durations contributed to the interference pattern. In fact, when Delos added the effects of 64 orbits with periods ranging from 2 to 20 times that of the shortest one, he began to recover the complex, messy signal of the MIT and Bielefeld experiments. For each trajectory the attribute that entered the analysis in a crucial way was the period--a strictly classical quantity that has no place in conventional quantum mechanics. The striking agreement between theory and observation therefore demonstrated impressively that the best way to analyze Rydberg atoms is in terms of classical orbits. That the uncertainty principle rendered the early, crude experiments more comprehensible than the later, more precise ones represents a remarkable triumph of serendipity.
Although Delos hasn’t completely solved the mysteries of the missing rung, his theory gives a clue as to how to reconcile quantum mechanics with the everyday world of classical mechanics. At first glance the universe governed by Newton’s laws, with its particle trajectories that sweep elegantly through space and time, appears to be a realm of order and predictability. The unruly swarm of random events described by quantum mechanics, on the other hand, might seem to signify chaos. In fact, as Delos’s theory dramatically illustrates, the exact opposite is true.
Chaos is what happens when two marbles, or two atoms, or two electrons, whose motions differ by imperceptible amounts at the outset, and which are exposed to identical influences, nevertheless diverge and wander far apart. It can be thought of as an overly sensitive response to initial conditions. Such unruly behavior effectively frustrates most attempts at prediction. Since there are no trajectories in quantum mechanics, however, chaos never arises. Indeed, it has been proved quite generally that chaos cannot exist in a quantum mechanical system. On the other hand, in the last two decades physicists have finally realized what they should have understood long ago--that classical mechanics is almost always chaotic. To be sure, at low energies the orbits of Rydberg electrons are as simple and predictable as the ellipses of the Bohr model, but as the energy increases, and the motion becomes more agitated, classical chaos suddenly sets in. The question is, how can the chaotic classical theory tell us anything about the orderly, nonchaotic architecture of a Rydberg atom?
The virtue of Delos’s theory is precisely that it does refer to orbits and is therefore capable of capturing some of the effects of chaos. In particular, it dramatically illuminates the crucial role of timing in understanding chaos. Chaos is a long-term phenomenon, Delos explains. In the short term, there is always order. As an example he cites the weather. It is easy to predict precisely what will happen in the next five minutes, and meteorologists are even fairly successful in forecasting for a day or two. But after that, chaos sets in, and specific predictions become unreliable. Conversely, statistical predictions gain in accuracy as the time interval increases. Thus the average temperature in New York City for the month of May is predictable with moderate precision, for the entire year with great confidence.
The same relationship holds for Rydberg atoms. Classical orbits are more predictable and regular when they are short. When an atom is observed at a crude energy resolution, these short orbits predominate and the atom appears to absorb energy in a regular pattern. On the other hand, improving the energy resolution means that longer, more chaotic orbits have an effect on the way the atom absorbs energy. The result is a hopelessly confused and chaotic absorption pattern. During the last seven years Delos, together with his students and colleagues, has refined his theory to the point where it can follow with precision a Rydberg atom’s approach to chaos.
Although the quantum-classical boundary is still a place of mystery and ambiguity, the Rydberg atom has helped physicists come a step closer to understanding the true nature of the missing rung. For the rest of us it is reassuring to learn that in the face of the forbidding abstractness of quantum mechanics, it is sometimes still useful to picture the atom as a miniature solar system.