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like a wave in that the essence was spread out. The wave equation made it possible to compute with smudges and accommodate the probability 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.

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