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arXiv:2603.15638

The Birth of Quantum Mechanics and the Dirac Equation

Volodimir Simulik, Denys I. Bondar

physics.hist-phmath-phquant-phSubmitted Feb 27, 2026• Announced Mar 3, 2026• 0 citations
v1 · Feb 27, 2026v2 · Apr 18, 2026
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Abstract

This paper commemorates the 100th anniversary of quantum mechanics as a theoretical model of atomic phenomena. We discuss briefly the foundational contributions of Werner Heisenberg, Wolfgang Pauli, Erwin Shrodinger, and Paul Dirac to the development of quantum theory. Special attention is given to the often-overlooked contributions of Charles Galton Darwin and Hendrik Anthony Kramers. We examine the three periods of quantum theory development: foundation, development, and the modern era connected with information science. The paper also highlights current challenges facing quantum theory, including the quantum-classical boundary, quantum gravity, and the mysteries of dark matter and dark energy. One of the goals of this article is to extend the recent review in J. Phys. A: Math. Theor. 58 (2025) 053001 primarily by a a more detailed consideration of the period of formation of quantum mechanics.

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This paper commemorates the 100th anniversary of quantum mechanics as a theoretical model of atomic phenomena. We discuss briefly the foundational contributions of Werner Heisenberg, Wolfgang Pauli, Erwin Shrodinger, and Paul Dirac to the development of quantum theory. Special attention is given to the often-overlooked contributions of Charles Galton Darwin and Hendrik Anthony Kramers. We examine the three periods of quantum theory development: foundation, development, and the modern era connected with information science. The paper also highlights current challenges facing quantum theory, including the quantum-classical boundary, quantum gravity, and the mysteries of dark matter and dark energy. One of the goals of this article is to extend the recent review in J. Phys. A: Math. Theor. 58 (2025) 053001 primarily by a a more detailed consideration of the period of formation of quantum mechanics.
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In this chapter, we investigate the Dirac equation, which is named after P. A. M. Dirac, who is one of the fathers of quantum field theory. The Dirac equation is a relativistic quantum mechanical wave equation for spin-1/2 particles (e.g. electrons), which was derived by Dirac in 1928. The difficulties in finding a consistent single-particle theory from the Klein–Gordon equation led Dirac to search for an equation that had a positive-definite conserved probability density and was first order both in time and space. One can show that these two conditions imply that a matrix equation is required. The reason why the Klein–Gordon equation did not yield a positive-definite probability density is connected with the second-order time derivative in this equation, which arises because the Klein–Gordon equation is related to the relativistic energy–momentum relation E 2 = m 2 + p 2 via the correspondence principle that includes a term E 2 . Thus, a ‘better’ Lorentz covariant wave equation with a positive-definite probability density should have a first-order time derivative only. However, the equivalence of time and space coordinates in Minkowski space requires that such an equation also have only first-order space derivatives.
Purpose. The Dirac equation is one of the fundamental equations of modern theoretical physics. It is in service more than 90 years (1928–2018). The application today is much wider than the areas of quantum mechanics, quantum field theory, atomic and nuclear physics, solid state physics. The successful derivation of some equation of mathematical physics is the first step to successful application. In such process the essence of the corresponding model of nature, the mathematical principles and the physical foundations are visualized. Here we deal with the different approaches to the problem of the Dirac equation derivation. Methods. The quantum-mechanical, quantum field-theoretical, group-theoretical, algebraic, symmetrical, statistical, stochastical and numerical approaches are used. Results. The 26 different ways of the Dirac equation derivation are presented. The various physical principles and mathematical formalisms are used. Three original approaches of the authors to the problem are given. They are (i) the generalization of H. Sallhofer derivation, (ii) the obtaining of the massless Dirac equation from the Maxwell equations in maximally symmetrical form, (iii) the derivation of the Dirac equation with nonzero mass from the relativistic quantum mechanics of the fermion-antifermion spin s = 1/2 doublet. In some sense the role of the Dirac equation today is demonstrated. Conclusions. The original investigation of the problem of the Dirac equation derivation is presented. The different approaches, which are based on the various mathematical and physical principles, are considered (26 methods). Therole of the scientists from Uzhhorod is shown. The importance of place of the Dirac equation in modern theoretical physics is discussed.
Before examining the Dirac equation that gives a relativistic quantum mechanical description of the electron, we will examine the Klein-Gordon equation. Compared to the Schrödinger equation, the Dirac equation should have an appreciable modification in the quantum description of the properties of the electron. In the non-relativistic quantum mechanical formulation, we begin with the classical relation for the energy in the absence of interaction, i.e., the kinetic energy: 15.1 https://www.w3.org/1998/Math/MathML"> Η = p 2 2 m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math15_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/>
Chapter 19 introduces relativistic quantum mechanics, with the aim of providing the student with a basic knowledge of the subject and a link to more advanced courses in relativistic quantum theory. It begins by showing that when the Schrödinger equation is modified to make it consistent with special relativity, the result is the Klein–Gordon equation, which gives better qualitative agreement with experiment. Introducing the electron spin into the Schrödinger equation leads to the Pauli equation, which gives rise to important spin effects. By considering both special relativity and spin, one obtains the Dirac and the van der Waerden equations. The main properties and implications of Dirac’s equation are discussed, including the electron’s zitterbewegung. The solution of Dirac’s equation for the free particle is presented in detail, the hydrogen-like atom is shown to be exactly solvable, and the particle in an external electromagnetic field is included as a complementary material.