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Mathematical Foundations of Quantum Mechanics

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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

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The book was originally published in German in 1932 by Julius Springer, under the title 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

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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.[9] 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

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References

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  1. ^ Van Hove, Léon (1958). "Von Neumann's contributions to quantum theory". Bull. Amer. Math. Soc. 64 (3): 95–100. doi:10.1090/s0002-9904-1958-10206-2.
  2. ^ Margenau, Henry (1933). "Book Review: Mathematische Grundlagen der Quantenmechanik". Bulletin of the American Mathematical Society. 39 (7): 493–495. doi:10.1090/S0002-9904-1933-05665-3. MR 1562667.
  3. ^ John von Neumann (2018). Nicholas A. Wheeler (ed.). Mathematical Foundations of Quantum Mechanics. New Edition. Translated by Robert T. Beyer. Princeton University Press. ISBN 9781400889921.
  4. ^ von Neumann, J. (1927). "Mathematische Begründung der Quantenmechanik [Mathematical Foundation of Quantum Mechanics]". Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 1–57.
  5. ^ von Neumann, J. (1927). "Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik [Probabilistic Theory of Quantum Mechanics]". Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 245–272.
  6. ^ von Neumann, J. (1927). "Thermodynamik quantenmechanischer Gesamtheiten [Thermodynamics of Quantum Mechanical Quantities]". Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse. 102: 273–291.
  7. ^ von Neumann, J. (1929). "Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren [General Eigenvalue Theory of Hermitian Functional Operators]". Mathematische Annalen: 49–131.
  8. ^ von Neumann, J. (1931). "Die Eindeutigkeit der Schrödingerschen Operatoren [The uniqueness of Schrödinger operators]". Mathematische Annalen. 104: 570–578. doi:10.1007/bf01457956. S2CID 120528257.
  9. ^ a b Bell, John S. (1966-07-01). "On the Problem of Hidden Variables in Quantum Mechanics". Reviews of Modern Physics. 38 (3): 447–452. doi:10.1103/RevModPhys.38.447. ISSN 0034-6861.
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