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**Foundations of quantum mechanics II. Transl. from the German by Carl A. Hein.**
*(English)*
Zbl 0574.46057

Texts and Monographs in Physics. New York etc.: Springer-Verlag. XVI, 416 p., 54 figs. DM 228.00 (1985).

This book is the second volume of a two-volume work on the foundations of quantum mechanics which is written as an enlarged version of the author’s former book ”Die Grundlagen der Quantenmechanik”, Berlin (1954; Zbl 0058.228) as to contain the development of the latter thirty years. Whereas the fundamental concepts of quantum mechanics are presented in the first volume (1983; Zbl 0509.46057), this second volume mainly deals with the applications of the theory.

The essential notions of the whole representation are the ensembles and effects. In Volume I it was shown that these concepts are related to macroscopical preparation and registration devices. Microsystems were introduced as ”interaction carriers” between these devices. Moreover, the derivation of Hilbert space structure was sketched.

Volume II starts with representation theory of Hilbert spaces by function spaces and with the quantum mechanical equations of motion. In the following chapters atoms with one and more electrons and the corresponding spectra are treated. These discussions and calculations involve the representation of important topics like e.g. perturbation theory, symmetry, the addition of angular momenta, selection rules and the intensity of spectral lines, and the periodic systems of elements. As a further topic, molecules and the chemical bond are discussed. The final application of quantum mechanics presented in the text is scattering theory. The S-operator formalism, stationary scattering theory, and some examples are studied.

The last two chapters lead back to fundamental considerations. Measuring and preparing processes on which quantum mechanics is based are reformulated by quantum mechanics itself, and it is investigated whether such a description is consistent with previously introduced assumptions. In this context the author gives a contradiction-free explanation of the so-called Einstein-Podolksy-Rosen paradox. The book closes with a discussion about the relation between quantum mechanics, macrophysics, and other physical theories. An appendix about group theory is added.

This second volume on the foundations of quantum mechanics treats the most important applications of the theory by consequent use of the methodology developed in Volume I. Both volumes present a self- consistent, physically well founded, and mathematically rigorous formulation of quantum mechanics unifying axiomatics and applications. The entire book should be of great interest to all readers concerned with quantum theory as well as to graduate students.

The essential notions of the whole representation are the ensembles and effects. In Volume I it was shown that these concepts are related to macroscopical preparation and registration devices. Microsystems were introduced as ”interaction carriers” between these devices. Moreover, the derivation of Hilbert space structure was sketched.

Volume II starts with representation theory of Hilbert spaces by function spaces and with the quantum mechanical equations of motion. In the following chapters atoms with one and more electrons and the corresponding spectra are treated. These discussions and calculations involve the representation of important topics like e.g. perturbation theory, symmetry, the addition of angular momenta, selection rules and the intensity of spectral lines, and the periodic systems of elements. As a further topic, molecules and the chemical bond are discussed. The final application of quantum mechanics presented in the text is scattering theory. The S-operator formalism, stationary scattering theory, and some examples are studied.

The last two chapters lead back to fundamental considerations. Measuring and preparing processes on which quantum mechanics is based are reformulated by quantum mechanics itself, and it is investigated whether such a description is consistent with previously introduced assumptions. In this context the author gives a contradiction-free explanation of the so-called Einstein-Podolksy-Rosen paradox. The book closes with a discussion about the relation between quantum mechanics, macrophysics, and other physical theories. An appendix about group theory is added.

This second volume on the foundations of quantum mechanics treats the most important applications of the theory by consequent use of the methodology developed in Volume I. Both volumes present a self- consistent, physically well founded, and mathematically rigorous formulation of quantum mechanics unifying axiomatics and applications. The entire book should be of great interest to all readers concerned with quantum theory as well as to graduate students.

Reviewer: W.Stulpe

### MSC:

46N99 | Miscellaneous applications of functional analysis |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

81U20 | \(S\)-matrix theory, etc. in quantum theory |

81P10 | Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) |

46E20 | Hilbert spaces of continuous, differentiable or analytic functions |

46C99 | Inner product spaces and their generalizations, Hilbert spaces |