Abstract
In this paper, we discuss the time evolution of the quantum mechanics formalism. Starting from the heroic beginnings of Heisenberg and Schrödinger, we cover successively the rigorous Hilbert space formulation of von Neumann, the practical bra-ket formalism of Dirac, and the more recent rigged Hilbert space approach.
Highlights
One morning about the 10 July 1925 I suddenly saw light: Heisenberg’s symbolic manipulation was nothing but the matrix calculus well-known to me since my student days [2]. This approach opened the way to an algebraic formulation of quantum mechanics, that flourished much later in statistical mechanics
Von Neumann derives the whole structure of quantum mechanics (QM): expectation values, statistical operator, measurability and simultaneous measurability, uncertainty relations, and so on
Each physical system has a complete system of commuting observables (CSCO), and the eigenvectors of the elements of this CSCO constitute a basis of E
Summary
In 1900, Planck explained it by introducing artificial entities, called quanta, that were supposed to be elementary constituents of energy, each of them proportional to a given frequency. As. Born recalled, One morning about the 10 July 1925 I suddenly saw light: Heisenberg’s symbolic manipulation was nothing but the matrix calculus well-known to me since my student days [2]. One morning about the 10 July 1925 I suddenly saw light: Heisenberg’s symbolic manipulation was nothing but the matrix calculus well-known to me since my student days [2] This approach opened the way to an algebraic formulation of quantum mechanics, that flourished much later in statistical mechanics. The time was ripe for a precise mathematical overhaul of the whole theory, and this was the achievement of John von Neumann [7]
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