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Von Neumann’s axiomatic treatment of non-relativistic quantum mechanics is the archetypal example of the dual interaction between physical theories and the development of mathematical ideas. We examine this interaction by first building up the necessary parts of the theory of unbounded self-adjoint operators on a Hilbert space, emphasizing the physical intuition that motivates the mathematical concepts. We then present a version of the Dirac-von Neumann axioms on a quantum system and deduce some of their elementary consequences, illustrating the converse effect of the mathematical formalism on the physical theory.

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