The Birth of Quantum Mechanics and the Dirac Equation

TL;DR

This paper reviews the early development of quantum mechanics and the Dirac equation, highlighting overlooked contributions and presenting new derivations and historical insights.

Contribution

It provides a detailed historical analysis of 1925-1928, reconstructs Kramers' independent derivation of the Dirac equation, and introduces two modern derivations not previously catalogued.

Findings

Reconstructed Kramers' independent derivation of the Dirac equation.

Highlighted Charles Galton Darwin's role in understanding the Dirac equation.

Presented two modern derivations based on operational modeling and hydrodynamic formulation.

Abstract

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Kramers' independent derivation of the Dirac equation - obtained essentially simultaneously with Dirac's own result yet unpublished for seven years - and discuss its relation to Van der Waerden's group-theoretical approach.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.