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The Quantum CookbookMathematical Recipes for the Foundations for Quantum Mechanics$
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Jim Baggott

Print publication date: 2020

Print ISBN-13: 9780198827856

Published to Oxford Scholarship Online: January 2020

DOI: 10.1093/oso/9780198827856.001.0001

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Dirac, Von Neumann, and the Derivation of the Quantum Formalism

Dirac, Von Neumann, and the Derivation of the Quantum Formalism

State Vectors in Hilbert Space

(p.203) 10 Dirac, Von Neumann, and the Derivation of the Quantum Formalism
The Quantum Cookbook

Jim Baggott

Oxford University Press

The evolution of quantum mechanics through the 1920s was profoundly messy. Some physicists believed that it was necessary to throw out much of the conceptual baggage that early quantum mechanics tended to carry around with it and re-establish the theory on much firmer ground. It was at this critical stage that the search for deeper insights into the underlying reality was set aside in favour of mathematical expediency. All the conceptual problems appeared to be coming from the wavefunctions. But whatever was to replace them needed to retain all the properties and relationships that had so far been discovered. Dirac and von Neumann chose to derive a new quantum formalism by replacing the wavefunctions with state vectors operating in an abstract Hilbert space, and formally embedding all the most important definitions and relations within a system of axioms.

Keywords:   Paul Dirac, John von Neumann, state vectors, Hilbert space, quantum formalism, axioms, classical unit vectors, Hamiltonian operator, time evolution operator

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