Anwendung der Quantenmechanik auf das Problem der anomalen Zeemaneffekte. Offprint from: Zeitschrift fur Physik 37 Band, 4/5 Heft, 5 May 1926.
Berlin: Springer, 1926. First edition, very rare offprint, 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). 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). 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. OCLC lists Oregon State only. Not on RBH. 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. “In early January 1926, Heisenberg became aware of the new operator formulation of quantum mechanics of Born and Wiener (1926) which opened up the route for extending matrix mechanics to the more general operator formalism. He was then able to re-evaluate the problem using action–angle variables, but nonetheless, the stubborn factor of 2 remained, causing general disappointment among the proponents of electron spin. “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). “The main theoretical task which Heisenberg wanted to achieve in the paper with Jordan, was to calculate the eigenvalues of the quantum-theoretical formulation of the Hamiltonian … and to obtain from them the anomalous Zeeman effects and fine structure of spectral lines. To achieve this goal, the authors proceeded in several steps. First, they studied the question of whether the well-known selection and intensity rules, which had been obtained from the analysis of the anomalous Zeeman effect data, could be derived from the matrix mechanical treatment of the problem. The calculation of the amount of magnetic splitting turned out to be strenuous work, involving as it did the exact evaluation of certain inverse powers of the matrix of the radial variable for realistic three-dimensional atoms. With the extension of the matrix scheme to include angle variables, such as the one developed by Heisenberg and Pauli in January 1926, the authors succeeded in obtaining the desired results… “The actual calculations of Heisenberg and Jordan were straightforward, for they followed exactly the pattern of matrix methods which had been developed in the three-man paper and in Pauli’s treatment of the hydrogen atom … [The formulae they obtained] were identical with the ones obtained by Sommerfeld when he rinterpreted Woldmar Voigt’s equations for the anomalous Zeeman effects in accordance with the Bohr-Sommerfeld theory of atomic structure. The theoretical understanding of the anomalous Zeeman effects in the old approach was, however, fundamentally different from the present calculation based on quantum mechanics; the coincidence of the results obtained by so different means was, therefore, deemed fortunate. Heisenberg was really ‘excited and enthusiastic about the fact that the old formulae of Voigt actually did come out of quantum mechanics’. “The difference between the methods used by Sommerfeld, on the one hand, and by Heisenberg and Jordan, on the other, showed up in particular in the calculation of the intensities of the Zeeman lines. Voigt had discussed the question of line intensities on the basis of the classical electromagnetic theory of emission of light by bound charges, and Sommerfeld had not changed this aspect in his treatment of 1922. In Heisenberg and Jordan’s calculation the intensities had to be obtained from the matrix elements of the position vector matrix of the orbiting electron … “Heisenberg and Jordan’s matrix calculation, in the case of spin = 1⁄2, accounted perfectly for the observations of the anomalous Zeeman effect in alkali doublets, and all the expressions obtained were equivalent to Sommerfeld's successful formulae of 1922. The new theory described equally well the situation for the spectra of alkaline earths. The latter have series of singlet and triplet spectral lines when no external magnetic field is present, which may be readily explained by assuming that the spin angular momenta of the two outer electrons (responsible for the radiation) either subtract from each other to give [total spine] = 0 or add up to give [total spin] = 1 … Heisenberg was also happy to derive another result, a ‘summation principle’ (‘Summationsprinzip’), from the equations of perturbation theory … Heisenberg and Jordan drew further conclusions applicable to all multiplets, that is, independent of the specific system under investigation … “The methods of matrix mechanics thus turned out to be very successful in dealing with the multiplet structure and the anomalous Zeeman effects. Especially, Landé’s empirical rules followed automatically. The question remained, however, whether also certain violations of Landé’s rules, which had been observed in connection with the so-called partial Paschen-Back effect, could be explained in the new theory … Lucie Mensing then took up the problem … Mensing’s calculation showed that the transition elements were always nonzero, and the experimental data did not indeed exhibit any anomalies in this case. Moreover, with respect to the nonzero intensities of the lines, experiments and the quantum-mechanical results agreed completely. Lucie Mensing, the young lady who proved herself capable of performing the most delicate theoretical calculations of anomalous Zeeman effects, was offered a post-doctoral research position at Tübingen where Back and Landé were the leading lights of atomic spectroscopy” (Mehra & Rechenberg, pp. 273-282). 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.
8vo (229 x 156 mm), pp. pp. 263-277. Original printed wrappers.
Item #6418
Price: $9,500.00
