A. MARX. Foundations of Quantum Mechanics, translated from the German by T. A. Kontorova, E. B. Kofman, and N. V. Kashtanov, edited by Prof. Ya. I. Frenkel, Leningrad—Moscow, GTTI, 1933, price 4 rub. 50 kop., binding 1 rub. 50 kop.

The book under review is, as the author says in his preface, an attempt to compile a textbook of quantum mechanics understandable to a beginning student, with the principal attention, in the author’s view, to be given to questions of fundamental importance—namely the so-called theory of transformations and the question of “secondary quantization.” At the same time the author considers it possible and correct to present these “basic things” while ignoring the “applications” of the theory, referring the reader to A. Sommerfeld’s well-known book. In the reviewer’s opinion, such an attitude on the author’s part constitutes the fundamental defect of the whole book, for such a division of quantum mechanics is scholastic, and, of course, one can teach the reader and thereby make quantum mechanics understandable to him only by securing on his part a practical mastery of the subject. But this cannot be done by presenting the mathematical skeleton.

In the first chapter the wave mechanics of a single particle is considered. The author begins with an exposition of Heisenberg’s uncertainty relation for the coordinate and the quantity of motion. Passing then to an analysis of the uncertainty relation for time and energy, the author limits himself to its generally accepted, erroneous interpretation. The whole exposition of these questions, as well as of a number of others, bears the stamp of Machist philosophy. Fortunately, in the Russian translation many of the author’s “profound” arguments have been omitted by the editor of the translation, Prof. Ya. I. Frenkel, without detriment to the book.

Further, in the first chapter the transition from quantum mechanics to classical mechanics is also considered in considerable detail. The static element of quantum processes is appropriately emphasized from the very beginning of the book; however, the moment of its inclusion is not marked sufficiently clearly.

The second chapter considers the wave mechanics of an individual light quantum, the author transferring to the photon all the properties of a massy small particle and formulating Schrödinger’s equation as an ordinary wave equation. However, as the development of quantum mechanics has shown, such a formulation of the question is unsuitable for the photon, and therefore this chapter should in the main be regarded as incorrect. Nevertheless, it may be read with profit as an example of the development of the ideas of the first chapter.

It should be especially noted that the probability of the location of the photon, in the form in which it is treated by the author, leads to logical contradictions.

In the third chapter the wave mechanics of stationary states is studied. Particularly valuable is the emphasis placed on the importance of the statistical element of the theory in such, for example, questions as states with “indefinite energy” and the difference between “mixtures” and “pure cases.” In this same chapter the question of classical mechanics as a limiting case of quantum mechanics is again considered, and in a rather elegant manner (Ehrenfest’s theorem and analogous ones). It is very valuable that the author constantly draws a parallel with the study of phenomena in the so-called configuration space of momenta. As abstract examples, the harmonic oscillator and the hydrogen atom are investigated. Naturally, with this purely quantitative method, the richness of the physical world in quantum systems disappears without a trace for the author. The chapter ends with a consideration of the radiation of a quantum system in the sense of the correspondence principle, i.e., the connection is determined between the matrix elements of the coordinate and the coefficients of the Fourier series. However, unfortunately, this is done very sketchily.

Chapter four considers essentially the same questions, but in the form of matrix mechanics. Here, in a non-rigorous mathematical form, which formerly had heuristic and now has pedagogical significance, the theory of the representation of relations between physical quantities by means of relations between matrices is set forth—the so-called theory of transformations. The author sees in this the foundation of all quantum mechanics. One cannot agree with this point of view. Indeed, the method of noncommuting quantities developed by Dirac had, and still has, enormous heuristic significance, but it is impossible to see in this apparatus the essence of the matter, if only because the basic thesis of the method—the bringing of a matrix to diagonal form—physically signifies the finding of the values of a quantity, and this thesis is in need of serious corrections.

Chapter five studies perturbation theory both in matrix and in “wave” form, and only from the general formal side, ignoring many essential physical details, such as, for example, the question of the degree of applicability of one or another approximation, the question of how properties not inherent in the system but inevitably introduced with its approximate description are reproduced, the question of the conservation of momenta and energy, etc. In §§ 48 and 49 the application of perturbation theory to the phenomena of the scattering of light by atoms is briefly considered, and a connection is established between the constants characterizing the medium and the quantum description of atoms (the Kramers–Heisenberg formula).

Chapter six—the many-body problem. Pauli’s principle and its formul—

... in the zero approximation by means of wave functions—determinants. Quantum resonance is discussed. However, a defect of this analysis should be considered to be that the author does not distinguish what is inherent only in the approximation being used from what is inherent in the real system described by the quantum scheme. The reader may therefore mistakenly see analogies where there are none, and overlook a truly important fact—the degeneracy of quantum systems where classically there was only one state.

Section 58 deals with Bose statistics.

The last chapter, the seventh, describes the wave field as a quantum system. Here the system subject to Bose–Einstein statistics is first studied (§§ 59–65), i.e., photons, and then in § 66 a system of electrons subject to Fermi–Dirac statistics. The author sees in the method of “second” quantization the most essential thing in quantum mechanics: “Mit der Diracschen Theorie der Emission und Absorption ist in das Lehrgebäude der Quantenmechanik der Schlussstein gefügt, und wir es aufgebaut haben, jesst abtragen. Der Bau bedarf keiner Stütze mehr, und es scheint uns trotz seiner Kühnheit fest genug zu stehen, um jedem Ansturm der Zweifel standzuhalten,” p. 279.

The following should be noted. The author has in mind the method of second quantization and the quantum electrodynamics constructed by means of it, and, alas!—in the quotation given—he proves to be the victim of a fine illusion. One must distinguish quantum mechanics in the proper sense of the word, which generalizes the conception of mechanical motion into a peculiar quantum kinematics, and then quantum electrodynamics, which analyzes the continuous transmission of electromagnetic interactions. In application to the former, Marx’s words are indeed quite just, whereas in the latter we clearly see that attempts to treat this sphere of phenomena only by the methods of quantum mechanics are successful only insofar as the atomism of electric charge can be ignored. In this domain something new is required. Let us note, in particular, that the method of second quantization is only an auxiliary tool, and the essence of quantum properties does not lie in it.

Finally, it should be noted that, in expounding Dirac’s theory, the author does not indicate that Dirac’s method (as was indicated by Dirac himself from the very beginning) neglects the electrostatic field in analyzing the actions of light. Taking it into account, we encounter fundamental difficulties—the point model of the electron or the absence of relativistic invariance.

At present, after a number of profound investigations (Landau, Peierls, Bohr, Rosenfeld, and others), we see the boundary of the domain within which one may rely on the methodology of quantum mechanics, and outside which a very deep investigation is still needed before one can say what the author says in the quotation cited.

K. Nikolsky

¹ This paragraph has been omitted by the editor in the Russian translation.