Quantum Mechanics: Dirac 40 Years On

Tim Coker

This is the first of a series of short papers exploring various aspects of quantum mechanics and quantum field theory. The intent of the full series of articles is to take the reader/student from a basic starting point (somewhere around high school / first year undergraduate maths/ physics/ engineering) to an understanding of relativistic quantum mechanics that would be appropriate for a third/fourth year undergraduate or early stage postgraduate. This particular article is a derivation of Dirac's Equation.

Vilas Y. Rajopadhye

Malcolm S. Longair

Göttingen and Copenhagen were the undoubted capitals of the new discipline of quantum mechanics. The expertise in experimental and mathematical physics and in pure mathematics made Göttingen the epicentre of the revolution which was taking place in the mathematical physics of quanta. Whilst this was to remain the case for the next few years, other actors soon appeared on the scene who were to contribute to Born's ‘tangle of interconnected alleys’. What was truly remarkable was how quickly the different approaches to the problems of quantum theory were developed and the rapid assimilation of all of them into a coherent and self-consistent theory of quantum mechanics. Whilst the theory itself was completed relatively quickly, the understanding of its physical content was to take many more years. The new players on the scene included Paul Dirac at Cambridge, Erwin Schrödinger in Vienna and Norbert Wiener at the Massachusetts Institute of Technology. Each of them brought quite new approaches to the development of quantum theory – their innovations were to supersede the matrix mechanics of Born, Heisenberg and Jordan, but there can be no doubt that the success of that theory indicated clearly the route ahead. They were however to involve the introduction of new mathematical techniques into the description of quantum phenomena. Dirac's approach to quantum mechanics Paul Dirac was trained as an electrical engineer at Bristol University, but he had a very strong mathematical bent. He was a solitary character who was notoriously quiet and self-effacing. He simply worked things out on his own.

Helge Kragh

Volodimir Simulik, Denys I. Bondar

The year 2025 marked the centennial of quantum mechanics, inaugurated by Heisenberg's matrix formulation and the foundational contributions of Pauli, Schrodinger, and Dirac. Concurrently, 2026 marks the centennial of the Klein - Gordon equation, the second-order relativistic wave equation from which both the Schrodinger and Dirac equations were derived. This article supplements the recent review published in J.Phys. A: Math.Theor.,58 (2025) 053001 by providing a more detailed examination of the formative period 1925 - 1928, with particular attention to contributions that have received insufficient recognition in the standard narrative. We reconstruct Kramers' independent derivation of the Dirac equation - obtained essentially simultaneously with Dirac's own result yet unpublished for seven years - and discuss its relation to Van der Waerden's group-theoretical approach. The role of Charles Galton Darwin in elucidating the physical content of the Dirac equation is also highlighted. In addition, we present two modern derivations not catalogued in the earlier review: one based on Operational Dynamical Modeling, which deduces the Dirac equation from relativistic Ehrenfest relations and the canonical commutation algebra, and one rooted in the Madelung hydrodynamic formulation. Three broad periods of quantum theory development -- foundational, consolidation, and the modern era of quantum information -- are briefly surveyed.