BIBLIOGRAPHY

A. Sommerfeld, Atombau und Spektrallinien, II Band (2., umgearbeitete und erweiterte Auflage des “Wellenmechanischen Ergänzungsbandes”), F. Vieweg, Braunschweig, 1939.

A. Sommerfeld, The Structure of the Atom and Spectra, Vol. 2 (2nd supplemented and revised edition of the “Supplementary Volume on Wave Mechanics”), with 62 figures. Published by Vieweg, Braunschweig, 1939.

In the autumn of 1938, scholars of all countries marked the seventieth birthday of one of the great masters of theoretical physics of our day—Professor Arnold Sommerfeld of the University of Munich. As though in answer to the wishes for further fruitful activity, exactly one year later the indefatigable author of the famous book from which a number of generations studied Bohr’s theory is bringing out a book devoted to wave mechanics. Nominally this is the 2nd edition of the “Supplementary Volume,” but in substance it is a new book, containing much entirely new material, while the old material is set forth in revised form. The increase in size from 300 to 800 pages speaks for itself. Acquaintance with the new work justifies expectations: we have a new, excellent book on quantum mechanics, which will become one of the basic aids for research workers and students. The material is distributed as follows: the fundamental problems and general theory occupy about 200 pages, Dirac’s theory—130 pages, the theory of perturbations—100 pages. These chapters naturally constitute the first part of the book. There then follow chapters on the photoelectric effect, the Compton effect, and bremsstrahlung, helium—the hydrogen molecule (approximately 60 pages each), approximate methods (40 pages), and mathematical supplements (100 pages).

As in the first edition, Sommerfeld sets himself the task above all of communicating to the reader the methods for solving Schrödinger’s equation in various concrete cases, referring the reader for a detailed exposition of questions of principle (axiomatics, the theory of transformations, etc.) to other books (Heisenberg, Cramer, Dirac, etc.). Sommerfeld likes to carry calculations through to the end; he tastefully analyzes individual integrals and instills this love in the reader. But the sense of “understanding” does not abandon the reader all the time. This impression of a certain sic veniat verba—an extremely good-natured craftsmanship and old-fashionedness—is strengthened by the absence from the book of such newer methods as second quantization, the neglect of Dirac’s $\delta$-function (in the general theory of the continuous spectrum and in individual calculations where the $Zackenfunktion$, p. 75, is considered), etc. Thus there is lost the peculiar abstractness inherent—whether for ill or for good—in modern quantum theory. On the other hand, of course, one cannot reproach the author for the absence of one or another item in an enormous book which one involuntarily wishes to divide into two volumes.

Without giving the titles of the sections, let us pass to a survey of the individual chapters, noting the new material (cf. the Russian translation, Wave Mechanics, L.–M., GTTI, 1934).

In Chapter I, a treatment is introduced of the simplest barriers, in particular those described by Eckart’s potential, which is later applied to Dirac’s equation.

In Chapter II a detailed exposition of the continuous spectrum of hydrogen has been added. Unfortunately, the number of graphs of wave functions, though now somewhat increased, is insufficient and is inferior to the illustrative material of the American course by Pauling and Wilson.

Chapter III, written almost anew, gives a very good introduction to general questions: the recognition of packets, the derivation of Heisenberg’s uncertainty formula, and the beginnings of operator theory. Here, too, the solution of the oscillator by matrix mechanics is given, together with the earlier brief objection to the thesis that unobservable quantities should be banished. Sommerfeld gives an example, at least of one unobservable element of the present theory, namely the arbitrary phase of the wave function. Unfortunately, from the first edition there has remained a certain opposition between the matrix and wave methods, which of course are essentially completely equivalent (historical parallel: the kinetic theory of gases and the justly condemned energetics).

Chapter IV, devoted to the Dirac equation, is one of the most interesting new features of the book. Not only the general derivation of the Dirac equation, spin, the magnetic moment, and invariant properties, but also concrete problems of the free electron and Kepler’s problem (following Sauter–Franz) are given in a hypercomplex exposition. Indeed, only acquaintance with the Cayley–Klein parameters, quaternions, and hypercomplex numbers enables the student to penetrate into the essence of the new apparatus of Dirac’s theory. We tried, in this sense, to give a number of indications in the notes to the first Russian edition of Dirac’s book several years ago. For the first time in the literature Sommerfeld gives an expanded exposition of this whole circle of questions; this is its enormous merit. Sommerfeld’s polemic against the matrix exposition of Dirac’s theory on the grounds that matrices are “impossible to remember without error,” that they give a “somewhat cumbersome representation,” and so on, should be recognized as a certain hypercomplex “excess,” often obscuring the essence of the calculations. Here the author gives a detailed exposition of the continuous spectrum, the introduction to the theory of the positron, in particular Klein’s paradox, and concludes the chapter with a splendid analysis of the question of the polarization of matter waves (analysis of the analogues of Barkla’s experiment with double scattering of X-rays and of Malus’s experiment with the polarization of light upon reflection—the latter has no place for \(\psi\)-waves!)\(^1\).

The important Chapter V on perturbation theory has been considerably supplemented with details in the form of the Rezsöford diagrams, diagrams on the Born approximation in collision theory, the Dirac theory of nonstationary processes, and various applications of the perturbation method to the Dirac equation. The exposition of this chapter seems to us especially successful.

The next three chapters treat problems of radiation (photoeffect, bremsstrahlung, Compton effect). Each of them begins with a historical exposition of the question in the résumé.

All phenomena are given both in stationary and in nonstationary description. It is significant that detailed calculations are also given for the relativistic case of the description of the electron by the Dirac equation.

In Chapter VI on the photoeffect, a derivation is given of the nonrelativistic formula of Fischer and Sauter for the \(K\)-shell; the angular distribution is discussed in detail, as are the photoeffect from the \(L\)-shell and, finally, the relativistic formula. Bremsstrahlung is also worked out in detail (Chapter VII), down to the derivation of the now famous Sauter–Bethe–Heitler formula, which has played such an important—

\(^1\) In this chapter, as in others, references to the works of other authors are often lacking, although, for example, at least in the small paragraph on pp. 226–227, where, without any mention of the literature, the “interesting connection between Dirac’s equation and the relativistic metric” is set forth. The matter concerns the linearization of \(ds^2\), first pointed out by us in DAN SSSR in 1929 (cf., for example, I. B. P e k u n o and F o k, Physik. Z., 30, 648, 1929, and Z. Physik, 54, 798, 1929; numerous continuations in the papers of mathematicians of Hiroshima University, etc.).

role in cosmic rays, in particular in the discovery of the meson. The chapter ends with remarks on astrophysical applications.

Chapter VIII gives a complete derivation of the Klein–Nishina formula for the scattering of light by a free electron according to Dirac’s theory; moreover, essentially the original, semiclassical method of these authors is used (contrary to the modern presentation by the method of secondary quantization, applied to this case for the first time by Tamm; see, for example, Heitler’s book on radiation). The Compton effect on bound electrons and the shape of the line are then considered in detail.

The extensive material of these three chapters allowed Sommerfeld once again to demonstrate the best aspects of his style of physical analysis: an abundance of direct calculations, and immediate contact with experiment. A large number of points in these chapters are based on works of the Sommerfeld school.

Chapter IX considers characteristic phenomena of exchange degeneracy using the examples of helium and the hydrogen molecule. This is followed by paragraphs on para- and orthohydrogen, nuclear statistics, and a detailed derivation of Mott’s correction (taking statistics into account) to Rutherford’s formula. The exchange integral in the Heitler–London problem of the hydrogen molecule is calculated in detail.

Chapter X, on approximate methods, is devoted to solving the Hilleraas problem of helium by the Ritz variational method, and the problems of the hydrogen ion and the hydrogen molecule. The Thomas–Fermi method with Sommerfeld’s additions is set out in detail, and an introduction is given to the Wentzel–Brillouin–Kramers method.

Finally, there follows a chapter entitled “Mathematical Supplements.” In seventeen separate sections it discusses a variety of questions, both purely mathematical (recurrence relations for spherical functions, the polynomial method, transformations to curvilinear coordinates, etc.) and physical (the Morse formula in the theory of molecules, a detailed exposition of the theory of multipole radiation, and a number of other interesting and important problems for mastering the apparatus of wave mechanics).

Such, in general outline, is the rich material of Sommerfeld’s new book. It would be highly desirable, without waiting for a Russian translation, to make this course available as soon as possible to all who study quantum mechanics. Speaking of a university course in quantum theory, we have in mind a complete, unabridged program, to which, without Bohr’s theory, 80–100 hours are allotted.

Our somewhat unusually long review was prompted by the desire to provide, as far as possible, detailed information to the many readers of the first edition.

D. Ivanenko, Sverdlovsk