Unknown to Dirac, in the summer of 1925 Werner Heisenberg had figured out the skeleton of a new and abstract “quantum mechanics” that promised to be fundamental, logically consistent, and not plagued by the difficulties of the existing quantum theory of atomic structure. Only eight months older than Dirac, Heisenberg was a graduate student of Max Born in Göttingen. Like Dirac, he had not yet obtained his Ph.D. degree.
The young German reasoned that a truly fundamental theory had to contain observable quantities only. By this criterion, electron orbits were no longer legitimate, whereas the frequencies of light emitted by atoms were. After all, who had ever observed an electron orbiting around a nucleus? Transforming this general idea into an abstract mathematical scheme, Heisenberg expressed physical quantities by arrays of symbols soon recognized to be matrices. One of the arrays might represent the position of an electron (Q) and another array its momentum (P, which equals mass times velocity). Heisenberg’s theory led to a mysterious law of multiplication according to whichQP differed fromPQ, that is,QP ≠PQ. It was as if 3 × 2 was not always equal to 2 × 3, or that the two numbers do not “commute.” Noting the puzzling fact that in general physical quantities did not commute, Heisenberg, at first, thought it was a flaw in his theory that might disappear when it was further developed.
Heisenberg’s seminal paper was published in a German journal on September 18, 1925, but Dirac knew of it ahead of publication. In August, Heisenberg had sent proofs of his forthcoming paper to Ralph Fowler, who sent them on to Dirac with a note saying, “What do you think of this? I shall be glad to hear.” Dirac first thought that it was of no interest, but a closer study told him a different story. He now realized that far from being a flaw, the strange appearance of non-commuting physical variables was the key element in the new mechanics and, consequently, had to be understood. Some versions of classical mechanics, he remembered, operated with non-commuting variables, which indicated that Heisenberg’s idea might be expressed in formal analogy with classical theory. The crucial insight came to Dirac “in a flash” (as he recalled) one afternoon at the beginning of October. A couple of weeks later he had a paper ready with the ambitious title “The Fundamental Equations of Quantum Mechanics.” This was the first paper that mentioned the term “quantum mechanics” in its title and with a meaning recognizable by modern physicists.
Among the fundamental equations was a significant sharpening of Heisenberg’s non-commuting variables.PQ andQP did not only differ, but, according to Dirac, they differed by a precise amount given by the tiny constant of nature Max Planck had introduced in 1900:PQ –QP ~h. Planck’s constant is tiny indeed(h = 6.6 × 10-34 in units of joule × second), which explains why position and momentum commute for a canon-ball but not for an electron. Dirac’s paper also contained a quantum analogue of the classical equation of motion, that is, an expression of how a physical quantity varies in time. Without an equation of motion, quantum mechanics would not be of much value, just as classical mechanics would be of little use without Newton’s second law of motion.
Even back in the 1920s, physics was a very competitive field (as it remains to this day). Dirac was aware that he was in competition with the German physicists, but the question of priority of the laws of quantum mechanics was not of great concern to him. Yet, he must have been disappointed when Heisenberg informed him in a letter of November 20 that most of the results in Dirac’s “extraordinarily beautiful paper” had already been derived by Born and another talented young Göttingen physicist, Pascual Jordan. The two Germans had extended and clarified Heisenberg’s theory, recogniz
