by James Gleick
Forces in Molecules
Thirteen physics majors completed senior theses in 1939. The world of accumulated knowledge was still small enough that MIT could expect a thesis to represent original and possibly publishable work. The thesis should begin the scientist’s normal career and meanwhile supply missing blocks in the wall of organized knowledge, by analyzing such minutiae as the spectra of singly ionized gadolinium or hydrated manganese chloride crystals. (Identifying the telltale combinations of wavelengths emitted by such substances still required patience and good experimental technique, and science seemed to be engendering new substances as fast as spectroscopists could analyze them.) Seniors could devise new laboratory instruments or investigate crystals that produced electrical currents when squeezed. Feynman’s thesis began as a circumscribed problem like these. It ended as a fundamental discovery about the forces acting within the molecules of any substance. If it bore little connection to his greater work that followed—and Feynman himself dismissed it as an obvious result that he should have written in “half a line”—it nevertheless found its way into the permanent tool kit of the physics of solids.
Although he did not know it, his quantum-mechanics professor, Morse, had recommended in his junior year that the department graduate him a year early. The suggestion was turned down, and Slater himself became Feynman’s thesis adviser. Slater proposed a problem that at first seemed not much deeper than most senior theses. The question could almost have come from a physics and chemistry handbook: Why does quartz expand so little when heated? Compared to metals, for example, why is its coefficient of expansion so small? Any substance expands because heat agitates its molecules—heat is the agitation of its molecules—but in a solid the details of the expansion depend on the actual molecular layout. A crystal, with its molecules in a regular geometrical array, can expand more along one axis than another. Typically scientists would represent a crystalline structure with a Tinkertoy model, balls stuck on rods, but real matter is not so rigid. Atoms may be more or less locked in an array, or they may swing or float more or less freely from one place to another. Electrons in a metal will swarm freely about. The color, the texture, the rigidity, the frangibility, the conductivity, the softness, the taste of a substance all depend on the local habits of atoms. Those habits in turn depend on the forces at work within a substance—forces both classical and quantum mechanical—and when Feynman began his thesis work those forces were not well understood, even in quartz, the most common mineral on earth.
An old-fashioned steam engine was regulated by a mechanical governor: a pair of iron balls swinging outward from a spinning shaft. The faster it spun, the farther outward they would swing. But the farther they would swing, the harder they would make it to spin the shaft. Feynman started by imagining some analogous effect in the atoms of quartz, silicon dioxide, a pair of oxygen atoms clinging to each atom of silicon. Instead of spinning, the silicon atoms were vibrating; as the quartz grew warmer, he thought that the oxygen atoms might provide a mechanical force that would pull inward against the increasing agitation of the molecules, thus compensating somehow for the ordinary expansion. But how could the forces within each molecule—forces that varied in different directions—be calculated? No straightforward method seemed to exist.
He had never thought about molecular structure in such detail before. He taught himself everything he could about crystals, their standard arrangements, the geometries and the symmetries, the angles between atoms. It all came down to one unknown, he realized: the nature of the forces pressing the molecules into particular alignments. In its search for fundamental laws ever farther down the hierarchy of sizes, physics had now reached a level where molecular forces should be coming into focus. Scientists could measure how much pressure it took to squeeze quartz a given distance in a given direction. With the still-new technique of X-ray diffraction, they could look at the shadow patterns of a regular crystal and deduce its structure. As some theorists continued to look even deeper toward the atom’s core, others now tried applying the quantum techniques to questions of structure and chemistry. “A science of materials as distinct from matter became possible,” a scholar of structure, Cyril Stanley Smith, who worked with Feynman a few years later as the chief metallurgist on the secret project at Los Alamos, said of this time. From atomic forces to the stuff that feeds our senses—that was the connection waiting to be made. From abstract energy levels to three-dimensional forms. As Smith added epigrammatically, “Matter is a holograph of itself in its own internal radiation.”
Forces or energy—that was the choice for those seeking to apply the quantum understanding of the atom to the workings of real materials. At stake was not mere terminology but a root decision about how to conceive of a problem and how to proceed in calculating.
The conception of nature in terms of forces went back to Newton. It was a direct way of dealing with the world, envisioning firsthand interactions between objects. One exerts a force on another. A distinction between force and energy did not emerge clearly until the nineteenth century, and then, gradually, energy began to take over as the fulcrum of scientists’ thinking. Force is, in modern terms, a vector quantity, with both a magnitude and a direction. Energy is directionless, scalar—meaning that it has a magnitude only. With the rise of thermodynamics energy came to the fore. It began to seem more fundamental. Chemical reactions could be neatly computed as operations designed to minimize energy. Even a ball rolling down a hill—moving from a state of higher to lower potential energy—was seeking to minimize its energy. The Lagrangian approach that Feynman resisted in his sophomore-year physics class also used a minimum of energy to circumvent the laborious calculation of direct interactions. And the law of conservation of energy provided a tidy bookkeeping approach to a variety of calculations. No comparable law existed for forces.
Yet Feynman continued to seek ways of using the language of forces, and his senior thesis evolved beyond the problem Slater had posed. As Feynman conceived the structure of molecules, forces were the natural ingredients. He saw springlike bonds with varying stiffness, atoms attracting and repelling one another. The usual energy-accounting methods seemed secondhand and euphemistic. He titled his thesis—grandly—“Forces and Stresses in Molecules” and began by arguing that it would be more illuminating to attack molecular structure directly by means of forces, intractable though that approach had been considered in the past.
Quantum mechanics had begun with energy for two reasons, he contended. One was that the original quantum theorists had habitually tested their formulas against a single type of application, the calculation of the observed spectra of light emitted by atoms, where forces played no obvious part. The other was that the wave equation of Schrödinger simply did not lend itself to the calculation of vector quantities; its natural context was the directionless measurement of energy.
In Feynman’s senior year, just over a decade after the three-year revolution of Heisenberg, Schrödinger, and Dirac, the applied branches of physics and chemistry had been drawn into an explosion of activity. To outsiders quantum mechanics might have seemed a nuisance, with its philosophical entanglements and computational nightmares. In the hands of those analyzing the structures of metals or chemical reactions, however, the new physics was slicing through puzzles that classical physics found impenetrable. Quantum mechanics was triumphing not because a few leading theorists found it mathematically convincing, but because hundreds of materials scientists found that it worked. It gave them insights into problems that had languished, and it gave them a renewed livelihood. One had only to understand the manipulation of a few equations and one could finally compute the size of an atom or the precise gray sheen of a pewter surface.
Chief in the new handbook was Schrödinger’s wave equation. Quantum mechanics taught that a particle was not a particle but a smudge, a traveling cloud of probabilities, like a wave in that the essence was spread out. The wave equation made it possible to compute with smudges and accommodate the pr
obability that a feature of interest might appear anywhere within a certain range. This was essential. No classical calculation could show how electrons would arrange themselves in a particular atom: classically the negatively charged electrons should seek their state of lowest energy and spiral in toward the positively charged nuclei. Substance itself would vanish. Matter would crumple in on itself. Only in terms of quantum mechanics was that impossible, because it would give the electron a definite pointlike position. Quantum-mechanical uncertainty was the air that saved the bubble from collapse. Schrödinger’s equation showed where the electron clouds would find their minimum energy, and on those clouds depended all that was solid in the world.
Often enough, it became possible to gain an accurate picture of where the electrons’ charge would be distributed in the three-dimensional space of a solid crystal lattice of molecules. That charge distribution in turn held the massive nuclei of the atoms in place—again, in places that kept the overall energy at a minimum. If a researcher wanted to calculate the forces working on a given nucleus, there was a way to do it—a laborious way. He had to calculate the energy, and then calculate it again, this time with the nucleus slightly shifted out of position. Eventually he could draw a curve representing the change in energy. The slope of that curve represented the sharpness of the change—the force. Each varied configuration had to be computed afresh. To Feynman this seemed wasteful and ugly.
It took him a few pages to demonstrate a better method. He showed that one could calculate the force directly for a given configuration, without having to look at nearby configurations at all. His computational technique led directly to the slope of the energy curve—the force—instead of producing the full curve and deriving the slope secondarily. The result caused a small sensation among MIT’s physics faculty, many of whom had spent enough time working on applied molecular problems to appreciate Feynman’s remark, “It is to be emphasized that this permits a considerable saving of labor of calculations.”
Slater made him rewrite the first version. He complained that Feynman wrote the way he talked, hardly an acceptable style for a scientific paper. Then he advised him to submit a shortened version for publication. The Physical Review accepted it, with the title shortened as well, to “Forces in Molecules.”
Not all computational devices have analogues in the word pictures that scientists use to describe reality, but Feynman’s discovery did. It corresponded to a theorem that was easy to state and almost as easy to visualize: The force on an atom’s nucleus is no more or less than the electrical force from the surrounding field of charged electrons—the electrostatic force. Once the distribution of charge has been calculated quantum mechanically, then from that point forward quantum mechanics disappears from the picture. The problem becomes classical; the nuclei can be treated as static points of mass and charge. Feynman’s approach applies to all chemical bonds. If two nuclei act as though strongly attracted to each other, as the hydrogen nuclei do when they bond to form a water molecule, it is because the nuclei are each drawn toward the electrical charge concentrated quantum mechanically between them.
That was all. His thesis had strayed from the main line of his thinking about quantum mechanics, and he rarely thought about it again. When he did, he felt embarrassed to have spent so much time on a calculation that now seemed trivial and self-evident. As far as he knew, it was useless. He had never seen a reference to it by another scientist. So he was surprised to hear in 1948 that a controversy had erupted among physical chemists about the discovery, now known as Feynman’s theorem or the Feynman-Hellmann theorem. Some chemists felt it was too simple to be true.
Is He Good Enough?
A few months before graduation, most of the thirty-two brothers of Phi Beta Delta posed for their portrait photograph. Feynman, seated at the left end of the front row, still looked smaller and younger than his classmates. He clenched his jaw, obeyed the photographer’s instruction to rest his hands on his knees, and leaned gravely in toward the center. He went home at the end of the term and returned for the ceremony in June 1939. He had just learned to drive an automobile, and he drove his parents and Arline to Cambridge. On the way he became sick to his stomach—from the tension of driving, he thought. He was hospitalized for a few days, but he recovered in time to graduate. Decades later he remembered the drive. He remembered his friends teasing him when he donned his academic robe—Princeton did not know what a rough guy it was getting. He remembered Arline.
“That’s all I remember of it,” he told a historian. “I remember my sweet girl.”
Slater left MIT not many years after Feynman. By then the urgency of war research had brought I. I. Rabi from Columbia to become the vigorous scientific personality driving a new laboratory, the Radiation Laboratory, set up to develop the use of shorter and shorter radio wavelengths for the detection of aircraft and ships through night and clouds: radar. It seemed to some that Slater, unaccustomed to the shadow of a greater colleague, found Rabi’s presence unbearable. Morse, too, left MIT to take a role in the growing administrative structure of physics. Like so many scientists of the middle rank, both men saw their reputations fade in their lifetimes. Both published small autobiographies. Morse, in his, wrote about the challenges in guiding students toward a career as esoteric as physics. He recalled a visit from the father of a graduating senior named Richard. The father struck Morse as uneducated, nervous merely to be visiting a university. He did not speak well. Morse recalled his having said (“omitting his hesitations and apologies”):
My son Richard is finishing his schooling here next spring. Now he tells me he wants to go on to do more studying, to get still another degree. I guess I can afford to pay his way for another three or four years. But what I want to know is, is it worth it for him? He tells me you’ve been working with him. Is he good enough to deserve the extra schooling?
Morse tried not to laugh. Jobs in physics were hard to get in 1939, but he told the father that Richard would surely do all right.
PRINCETON
The apostle of Niels Bohr at Princeton was a compact, gray-eyed, twenty-eight-year-old assistant professor named John Archibald Wheeler who had arrived the year before Feynman, in 1938. Wheeler had Bohr’s rounded brow and soft features, as well as his way of speaking about physics in oracular undertones. In the years that followed, no physicist surpassed Wheeler in his appreciation for the mysterious or in his command of the Delphic catchphrase:
A black hole has no hair was his. In fact he coined the term “black hole.”
There is no law except the law that there is no law.
I always keep two legs going, with one trying to reach ahead.
In any field find the strangest thing and then explore it.
Individual events. Events beyond law. Events so numerous and so uncoordinated that, flaunting their freedom from formula, they yet fabricate firm form.
He dressed like a businessman, his tie tightly knotted and his white cuffs starched, and he fastidiously pulled out a pocket watch when he began a session with a student (conveying a message: the professor will spare just so much time …). It seemed to one of his Princeton colleagues, Robert R. Wilson, that behind the gentlemanly façade lay a perfect gentleman—and behind that façade another perfect gentleman, and on and on. “However,” Wilson said, “somewhere among those polite façades there was a tiger loose; a reckless buccaneer … who had the courage to look at any crazy problem.” As a lecturer he performed with a magnificent self-assurance, impressing his audience with elegant prose and provocative diagrams. When he was a boy, he spent many hours poring over the drawings in a book called Ingenious Mechanisms and Mechanical Devices. He made adding machines and automatic pistols with gears and levers whittled from wood, and his blackboard illustrations of the most foggy quantum paradoxes retained that ingenious flavor, as though the world were a wonderful silvery machine. Wheeler grew up in Ohio, the son of librarians and the nephew of three mining engineers. He went to college in Baltimore, got his grad
uate degree at Johns Hopkins University, and then won a National Research Council Fellowship that brought him to Copenhagen in 1934 via freighter (fifty-five dollars one way) to study with Bohr.
He and Bohr worked together again, as colleagues this time, in the first months of 1939. Princeton had hired Wheeler and promoted the distinguished Hungarian physicist Eugene Wigner in a deliberate effort to turn toward nuclear physics. MIT had remained deliberately conservative about rushing to board the wagon train; Slater and Compton preferred to emphasize well-roundedness and links to more applied fields. Not so Princeton. Wheeler still remembered the magic of his first vision of radioactivity: how he had sat in a lightless room, staring toward the black of a zinc sulfide screen, counting the intermittent flashes of individual alpha particles sent forth by a radon source. Bohr, meanwhile, had left the growing tumult of Europe to visit Einstein’s institute in Princeton. When Wheeler met his ship at the pier in New York, Bohr was carrying news about what would now rapidly become the most propitious object in physics: the uranium atom.