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Quantum Mechanics and Its Evolving Formulations.

Jean-Pierre Antoine1

  • 1Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium.

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Summary

This paper traces the evolution of quantum mechanics formalism, from early concepts to modern approaches. It covers key developments like Hilbert spaces and Dirac

Keywords:
bra-ket formalismquantum mechanicsrigged Hilbert spacevon Neumann

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Area of Science:

  • Quantum Mechanics
  • Mathematical Physics

Background:

  • The foundational principles of quantum mechanics were established by Heisenberg and Schrödinger.
  • The need for a rigorous mathematical framework to describe quantum phenomena became apparent early on.

Purpose of the Study:

  • To provide a historical overview of the development of quantum mechanics formalism.
  • To detail the progression from early formulations to more advanced mathematical structures.

Main Methods:

  • Review of historical literature and foundational papers in quantum mechanics.
  • Analysis of the mathematical formalisms developed by key figures: von Neumann, Dirac.
  • Discussion of the rigged Hilbert space approach as a modern development.

Main Results:

  • The evolution of quantum mechanics formalism involved increasing mathematical rigor and practical utility.
  • Key milestones include the Hilbert space formulation and Dirac's bra-ket notation.
  • The rigged Hilbert space approach offers a more advanced framework for contemporary quantum mechanics.

Conclusions:

  • The formalism of quantum mechanics has undergone significant development, enhancing its descriptive power.
  • Understanding this evolution is crucial for appreciating the mathematical underpinnings of quantum theory.