Mathematical Foundations of Quantum Mechanics explained

The book Mathematical Foundations of Quantum Mechanics (1932) by John von Neumann is an important early work in the development of quantum theory.^[1]

Publication history

The book was originally published in German in 1932 by Julius Springer, under the title German: Mathematische Grundlagen der Quantenmechanik.^[2] An English translation by Robert T. Beyer was published in 1955 by Princeton University Press.A Russian translation, edited by N. Bogolyubov, was published by Nauka in 1964.A new English edition, edited by Nicholas A. Wheeler, was published in 2018 by Princeton University Press.^[3]

Significance and Criticism

The book mainly summarizes results that von Neumann had published in earlier papers.^{[4] [5] [6] [7] [8]} Its main significance may be its argument against the idea of hidden variables, on thermodynamic grounds. However, this idea was refuted as early as 1935 by Grete Hermann, but the critique remained unknown until J.S. Bell's rediscovery in 1966.^[9]

von Neumann's claim rested on the assumption that any linear combination of Hermitian operators represents an observable and the expectation value of such combined operator follows the combination of the expectation values of the operators themselves.Bell shows that the consequences of that assumption are at odds with results of incompatible measurements, which are not explicitly taken into von Neumann's considerations.

See also

• Mathematical formulation of quantum mechanics

External links

• Full online text of the 1932 German edition (facsimile) at the University of Göttingen.

Notes and References

1. Van Hove. Léon. Von Neumann's contributions to quantum theory. Bull. Amer. Math. Soc.. 1958. 64. 3. 10.1090/s0002-9904-1958-10206-2. 95–100. free. Léon Van Hove.
2. 10.1090/S0002-9904-1933-05665-3. Book Review: Mathematische Grundlagen der Quantenmechanik. 1933. Margenau. Henry. Henry Margenau. Bulletin of the American Mathematical Society. 39. 7. 493–495. 1562667. free.
3. Book: Mathematical Foundations of Quantum Mechanics. New Edition . John von Neumann . Robert T. Beyer . Nicholas A. Wheeler . Princeton University Press . 2018 . 9781400889921 . John von Neumann.
4. von Neumann. J.. Mathematische Begründung der Quantenmechanik [Mathematical Foundation of Quantum Mechanics]. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse. 1927. 1–57.
5. von Neumann. J.. Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik [Probabilistic Theory of Quantum Mechanics]. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse. 1927. 245–272.
6. von Neumann. J.. Thermodynamik quantenmechanischer Gesamtheiten [Thermodynamics of Quantum Mechanical Quantities]. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse. 1927. 102. 273–291.
7. von Neumann. J.. Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren [General Eigenvalue Theory of Hermitian Functional Operators]. Mathematische Annalen. 1929. 49–131.
8. von Neumann. J.. Die Eindeutigkeit der Schrödingerschen Operatoren [The uniqueness of Schrödinger operators]. Mathematische Annalen. 1931. 104. 570–578. 10.1007/bf01457956. 120528257.
9. Bell . John S. . 1966-07-01 . On the Problem of Hidden Variables in Quantum Mechanics . Reviews of Modern Physics . en . 38 . 3 . 447–452 . 10.1103/RevModPhys.38.447 . 0034-6861.