Der Comptoneffekt nach der Schrödingerschen Theorie.

Berlin: Springer, 1926.

First edition of Gordon’s account of the ‘Klein-Gordon equation’ (‘KG equation’ in what follows), the first attempt at unifying wave mechanics with special relativity. The KG equation was discovered by several authors independently, the first being the Swedish physicist Oskar Klein, although the equation appeared only incidentally in his paper. Gordon gave a detailed account of the KG equation in the present paper, and he was the first to apply it to a concrete physical problem – the Compton effect (the decrease in energy of photons when scattered from electrons). “Soon after Erwin Schrödinger’s publication of his first papers on wave mechanics in 1926, Gordon made several important contributions to the relativistic generalization of non-relativistic quantum mechanics: the current-density vector of the scalar wave equation, and the quantitative formula for the Compton effect. That these results were not applicable to the electron—as was generally believed at that time—but to particles obeying Bose statistics, which, however, were discovered later, does not detract from the quality of this work” (DSB). “Since the mid-thirties the KG equation had been recognised to belong to the fundamental equations of quantum field theory” (Kragh 1984, p. 1031). “The work of Walter Gordon provided the most condensed mathematical presentation of the various versions of Schrödinger’s equation, especially of the fully relativistic form. Although he was nearly the last to publish the results, this fact justifies the association of his name with [the KG equation]. Finally, Gordon did not stop at the relativistic wave equation, but went on to use it in a crucial problem: to calculate the Compton effect in wave mechanics. We conclude, therefore, that Oskar Klein – who, besides Schrödinger, first worked on the relativistic wave equation – and Walter Gordon are indeed the appropriate ‘patrons’ of the equation that bears their names” (Mehra & Rechenberg, pp. 819-820). “The Klein-Gordon equation and its applications led to many further important developments … in particular, the relativistic equation is connected with the physical interpretation of quantum mechanics and with the origin of quantum field theory” (Mehra & Rechenberg, p. 808, n. 271). In addition, the KG equation was later shown to correctly describe spin-zero particles, such as mesons. “On July 4, 2012 CERN announced the discovery of the Higgs boson. Since the Higgs boson is a spin-zero particle, it is the first observed ostensibly elementary particle to be described by the Klein–Gordon equation” (Wikipedia). Not on OCLC. No copies in auction records.

The KG equation is a relativistic wave equation, a quantized version of the relativistic energy-momentum relation. Like the Schrödinger equation, it is second order in space and time, but unlike the Schrödinger equation it is manifestly relativistically covariant. Another difference from the Schrödinger equation is that the wave function in the KG equation cannot be interpreted as a probability amplitude – the norm squared of the wave function must be interpreted as a charge density rather than a probability. The equation describes all spinless particles with positive, negative as well as zero charge.

“On Schrödinger’s original road to wave mechanics, relativistic considerations were of crucial importance. In fact, he first derived a relativistic eigenvalue equation, which he did not publish, mainly because he realized that it did not reproduce the hydrogen spectrum with acceptable accuracy … Consequently, Schrödinger reported only the non-relativistic approximation of [the KG equation] in his first publication on wave mechanics.

“Independently of Schrödinger, the relativistic second-order equation was found in the spring of 1926 by Oskar Klein, who was the first to publish it. During the next half year, it was investigated by several other physicists, including Fock, Gordon, de Broglie, Schrödinger, and Kudar, and eventually became known as the Klein-Gordon equation” (Kragh 1990, pp. 51-2).

Klein (1894–1977) was the first to publish the KG equation, in April 1926, in his ‘Quantentheorie und fünfdimensionale Relativitätstheorie’ (Zeitschrift für Physik 37, pp. 895-906). This work was a development of Theodor Kaluza’s five-dimensional ‘unified’ theory of gravity and electromagnetism presented in ‘Zum Unitätsproblem der Physik’ (Sitzungsberichte der Preussischen Akademie der Wissenschaften (Berlin) (1921), pp. 966-972). The KG equation appeared only incidentally in Klein’s paper and no applications of it were given. In December Klein published a second paper, ‘Elektrodynamik und Wellenmechanik vom Standpunkt der Korrespondenzprincips’ (Zeitschrift für Physik 41 (1927), pp. 407-442), which was principally designed to support Bohr’s physical interpretation of quantum mechanics. But in Section 1 of the paper Klein stated the KG equation explicitly, and in Section 4 gave applications of it to the Zeeman effect and dispersion phenomena. (See Mehra & Rechenberg, pp. 810-813.)

“The fact that Schrödinger had abandoned his relativistic wave equation to avoid disagreement with experiment was commented on extensively by Dirac … The disagreement between [the KG equation] and the hydrogen spectrum was, Dirac said,

‘most disappointing to Schrödinger. It was an example of a research worker who is hot on the trail and finding all his worst fears realized. A theory which was so beautiful, so promising, just did not work out in practice. What did Schrödinger do? He was most unhappy. He abandoned the thing for some months, as he told me … Schrödinger had really been too timid in giving up his first relativistic wave equation … Klein and Gordon published the relativistic equation which was really the same as the equation which Schrödinger had discovered previously. The only contribution of Klein and Gordon in this respect was that they were sufficiently bold not to be perturbed by the lack of agreement of the equation with observations’” (Kragh 1990, pp. 50-1).

“It was Dirac’s preoccupation with the general principles of quantum mechanics, and the transformation theory in particular, that led him to realize that the formal structure of the Schrödinger equation had to be retained even in a future unification of quantum theory and relativity. Since the KG equation is of second order in d/dt, it seemed to Dirac to be in conflict with the general formalism of quantum mechanics … During the Solvay Congress in October 1927, Dirac mentioned to Bohr his concern about a relativistic wave equation:

‘Then Bohr answered that the problem had already been solved by Klein. I tried to explain to Bohr that I was not satisfied with the solution of Klein, and I wanted to give him reasons, but I was not able to do so because the lecture started just then and our discussion was cut short. But it rather opened my eyes to the fact that so many physicists were quite complacent with a theory that involved a radical departure from the basic laws of quantum mechanics, and they did not feel the necessity of keeping to these basic laws in the way that I felt.’

“After his return from Brussels, Dirac concentrated on the problem of formulating a first-order relativistic theory of the electron. Within two months he had solved the whole matter” (Kragh 1990, pp. 54-7). This was the famous ‘Dirac equation’.

“Although the KG theory virtually disappeared from the scene of physics after Dirac’s alternative theory made its entry, it reappeared some years later. In 1934 Pauli and Weisskopf reconsidered the KG equation in the light of current theories of quantization of wave fields. They emphasized that their new interpretation of the KG equation allowed them to account for the existence of anti-particles in a much more natural way than Dirac’s questionable hole theory which was, they said, based on a priori arguments. Indeed, Pauli referred to the work as the ‘anti-Dirac paper’. However, the Pauli-Weisskopf version of the KG equation applies to bosons only and is thus not truly an alternative to Dirac’s theory [of the electron]. In 1934 no elementary particles of spine zero were known. It was only later, with the discovery of mesons, that the KG equation proved to apply to reality. In Yukawa’s classic theory of 1935 in which the meson was predicted, he made use of a version of the KG equation. Since the mid-thirties the KG equation had been recognised to belong to the fundamental equations of quantum field theory” (Kragh 1984, p. 1031).

Walter Gordon (1893–1940) obtained his PhD from the University of Berlin in 1921. He remained there until 1929, when he became Privatdozent—and later associate professor—at the University of Hamburg. He lost his position there, like other professors of Jewish origin, in the spring of 1933. He became a member of the Institute of Mathematical Physics at the University of Stockholm in the fall of the same year, but poor conditions for science at the University of Stockholm, together with a general lack of understanding of existing German political conditions, prevented his obtaining a regular position. During almost all of his stay in Sweden, Gordon participated eagerly in the seminars at the Institute of Mathematical Physics, to which his erudition, not merely in physics and mathematics, and his caustic but friendly humour gave a characteristic touch. He also gave lectures, among them a valuable course in group theory. But Gordon’s forced exile, taking him from the congenial and inspiring circle at the Hamburg Institute of Physics, and the uncertainty of his future brought an end to his creative powers. Early in 1937 his health declined, and inoperable stomach cancer was diagnosed. Good medical treatment and the care of his wife enabled him to live a reasonably normal life until the last months of 1940.

Kragh, ‘Equation with many fathers: The Klein-Gordon equation in 1926,’ American Journal of Physics 52 (1984), pp. 1024-1033. Kragh, Dirac: A Scientific Biography, 1990. Mehra & Rechenberg, The Historical Development of Quantum theory, Vol. 5, 1987.

Offprint from: Zeitschrift für Physik, Band 40, Heft 1/2, 29 November 1926. 8vo (230 x 155 mm), pp. 117-133. Original printed wrappers. A very fine copy.

Item #4497

Price: $2,800.00

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