Crafting the Quantum: Sommerfeld, Bohr, and the Older Quantum Theory

Kamal Barley, Andreas L. Ruffing, Sergeĭ K. Suslov

Richard H. Cushman, Jędrzej Śniatycki

The seminal paper by Niels Bohr followed by a paper by Arnold Sommerfeld led to a revolutionary Bohr–Sommerfeld theory of atomic spectra. We are interested in the information about the structure of quantum mechanics encoded in this theory. In particular, we want to extend Bohr–Sommerfeld theory to a full quantum theory of completely integrable Hamiltonian systems, which is compatible with geometric quantization. In the general case, we use geometric quantization to prove analogues of the Bohr–Sommerfeld quantization conditions for the prequantum operators Pf. If a prequantum operator Pf satisfies the Bohr–Sommerfeld conditions and if it restricts to a directly quantized operator Qf in the representation corresponding to the polarization F, then Qf also satisfies the Bohr–Sommerfeld conditions. The proof that the quantum spherical pendulum is a quantum system of the type we are looking for requires a new treatment of the classical action functions and their properties. For the sake of completeness we have provided an extensive presentation of the classical spherical pendulum. In our approach to Bohr–Sommerfeld theory, which we call Bohr–Sommerfeld–Heisenberg quantization, we define shifting operators that provide transitions between different quantum states. Moreover, we relate these shifting operators to quantization of functions on the phase space of the theory. We use Bohr–Sommerfeld–Heisenberg theory to study the properties of the quantum spherical pendulum, in particular, the boundary conditions for the shifting operators and quantum monodromy.

Makoto Katsumori

Anthony Duncan A, Michel Janssen B

Abstract This is the first of two volumes on the genesis of quantum mechanics. It covers the key developments in the period 1900–1923 that provided the scaffold on which the arch of modern quantum mechanics was built in the period 1923–1927 (covered in the second volume). After tracing the early contributions by Planck, Einstein, and Bohr to the theories of black‐body radiation, specific heats, and spectroscopy, all showing the need for drastic changes to the physics of their day, the book tackles the efforts by Sommerfeld and others to provide a new theory, now known as the old quantum theory. After some striking initial successes (explaining the fine structure of hydrogen, X‐ray spectra, and the Stark effect), the old quantum theory ran into serious difficulties (failing to provide consistent models for helium and the Zeeman effect) and eventually gave way to matrix and wave mechanics. Constructing Quantum Mechanics is based on the best and latest scholarship in the field, to which the authors have made significant contributions themselves. It breaks new ground, especially in its treatment of the work of Sommerfeld and his associates, but also offers new perspectives on classic papers by Planck, Einstein, and Bohr. Throughout the book, the authors provide detailed reconstructions (at the level of an upper‐level undergraduate physics course) of the cental arguments and derivations of the physicists involved. All in all, Constructing Quantum Mechanics promises to take the place of older books as the standard source on the genesis of quantum mechanics.

Paul K. Feyerabend

“Bohr was primarily a philosopher, not a physicist, but he understood that natural philosophy ... carries weight only if its every detail can be subjected to the ... test of experiment” (Heisenberg in [59], p. 95). As a result his approach differed from that of theschool-philosopherswhom he regarded with a somewhat “sceptical attitude, to say the least” ([59], p. 129) and whose lack of interest in “the important viewpoint which had emerged during the development of atomic physics” he noticed with regret ([59], p. 183).But it also differed, and to a considerable degree, from the spirit of what Professor T. S. Kuhn has called a “normal science.” Looking at Bohr's method of research we see that technical problems, however remote, are always related to a philosophical point of view; they are never treated as “tiny puzzles” whose solution is valuable in itself, even if one has not the faintest idea what it means, and where it leads: “For me” Bohr writes to Sommerfeld in 1922 ([59], p. 71) “[the quantum theory] is not a matter for petty didactic details, but a serious attempt to reach ... an inner coherence.” Emphasis is put on matters of principle ([59], p. 36) and minor discrepancies, or “puzzles” in the sense of Kuhn, instead of being deemphasized, and assimilated to the older paradigm,are turned into fundamental difficultiesby looking at them from a new direction, and by testing their background “in its furthest consequences by exaggeration” ([59], p. 329). A noteworthy example of Bohr's “non-normal” and rather metaphysical approach to physics is his scepticism in the face of the success of his own atomic model.

Carl S. Helrich