## ‘Anwendung der Quantenmechanik auf das Problem der anomalen Zeemaneffekte,’ pp. 263-277 in Zeitschrift fur Physik 37 Band, 4/5 Heft, 5 May 1926.

Berlin: Springer, 1926.

First edition of the explanation of the anomalous Zeeman effect on the basis of matrix mechanics. “By including the spin property of the electron, Heisenberg and Jordan obtained perhaps the greatest triumph of matrix mechanics: they were able to derive all observed phenomena connected with the anomalous Zeeman effect” (Rechenberg, p. 211). When an atom is placed in a magnetic field, its spectral lines split into a series of equidistant lines – always an odd number - whose separation is proportional to the field strength. This, the normal Zeeman effect, was explained in 1916 by Debye and Sommerfeld in terms of the ‘old’ quantum theory: the splitting was due to the interaction between the magnetic field and the orbital magnetic moment of the electrons in the atom. However, there is also an anomalous Zeeman effect, observed particularly in atoms with odd atomic number, in which the lines split in a more complex fashion. “During 1920-24, many physicists attacked the problem [of the anomalous Zeeman effect], including Landé, who was able to give a phenomenological explanation of the observed splitting of spectral lines. However, neither Landé, Sommerfeld, Pauli, Heisenberg nor other physicists occupied with the problem could justify their results in terms of quantum theory. “It’s a great misery with the theory of anomalous Zeeman effect,” Pauli wrote to Sommerfeld on July 19, 1923” (Kragh, p. 158). This paper was of crucial importance in the early history of quantum mechanics because its success in explaining the hitherto mysterious anomalous Zeeman effect validated not only the new quantum mechanics itself but also the highly controversial concept of electron spin, discovered by Uhlenbeck and Goudsmit in the previous year.

After Heisenberg’s introduction of matrix (quantum) mechanics in 1925, one of the first problems he wanted to address using his new theory was the anomalous Zeeman effect. The crucial ingredient was electron spin, which Uhlenbeck and Goudsmit had discovered by studying the regularities in the anomalous Zeeman effect documented by Landé. “Although based originally upon the classical concept of a rotating electron, electron spin is a purely quantum mechanical property intrinsic to the electron. Opinions were strongly divided about the validity of the concept, Pauli taking a strongly negative position, while Bohr, Heisenberg and Jordan took a more positive view. The challenge taken up by Heisenberg was to find a quantum mechanical solution for the anomalous Zeeman effect using the concept of a spin-½ particle within the context of their recently completed matrix formalism.

“Despite the less than encouraging views of Pauli, in November 1925 Heisenberg set about [finding] the stationary states and line splittings associated with the anomalous Zeeman effect. Disappointingly, he almost reproduced Landé’s formula for the anomalous Zeeman effect, but the crucial spin-orbit coupling term resulted in a factor of 2 discrepancy from Landé’s expression, a result which cast doubt on the whole scheme.

“The solution was, however, at hand thanks to the insight of Llewellyn Thomas who had arrived recently at Bohr’s Institute in Copenhagen as a visiting graduate student … Thomas was aware of the fact that there is an additional kinematic effect associated with the orbital motion of a vector, such as the spin vector of the electron, according to the special theory of relativity… This purely kinematic effect results in an additional contribution to the precession, and hence interaction energy of the electron… and can account completely for the discrepant factor of 2. After considerable debate, even Pauli was convinced and the paper on the quantum mechanical explanation for the anomalous Zeeman effect was published by Heisenberg and Jordan in June 1926. Rechenberg has written in his summary of the history of quanta and quantum mechanics that the explanation of the anomalous Zeeman effect was one of the greatest triumphs of matrix mechanics” (Longair, pp. 312-5).

Heisenberg and Jordan described their results in the abstract of the paper as follows: “For explaining the anomalous Zeeman effect, Uhlenbeck and Goudsmit have applied Compton’s hypothesis of the rotating electron. In the present paper [we] investigate the quantum mechanical behaviour of the atomic model characterized by this hypothesis. The result is that the Zeeman effect and the fine structure of the doublet spectra can be explained completely by the said hypothesis” (Mehra & Rechenberg, p. 273).

Helge Kragh, *Quantum Generations*, 1999; Malcolm Longair, *Quantum Concepts in Physics*, 2013; Helmut Rechenberg, Ch. 3, ‘Quanta and Quantum Mechanics,’ in *Twentieth Century Physics*, Vol. 1, L. Brown, B. Pippard & A. Pais (eds.), 1995. For a detailed analysis of the paper, see Jagdish Mehra & Helmut Rechenberg, *The Historical Development of Quantum Theory*, Vol. 3, 1982, pp. 272-281.

8vo (229 x 156 mm), pp. 235-394. Original printed wrappers.

Item #3528

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