Abstract We discuss the discovery of the relativistic wave equation for a spin-zero charged particle in the Coulomb field by Erwin Schrödinger (presumably during the Christmas holidays of 1925–1926). However, in this new approach, an essential discrepancy was found with the fine structure formula for energy levels already obtained by Sommerfeld in the framework of the ‘old’ quantum mechanics. As a result, Schrödinger had to withdraw the original ‘relativistically framed’ article, a draft of which has never been found, from a journal and start all over with his centennial article on the nonrelativistic stationary Schrödinger equation. Our goal here is to follow the original ‘relativistic idea’ from a modern mathematical viewpoint and elaborate on why Schrödinger did not publish it. We hope that this consideration will encourage readers to study quantum physics starting from one of the crucial moments of its creation.
Abstract We discuss the discovery of the relativistic wave equation for a spin-zero charged particle in the Coulomb field by Erwin Schrödinger (presumably during the Christmas holidays of 1925–1926). However, in this new approach, an essential discrepancy was found with the fine structure formula for energy levels already obtained by Sommerfeld in the framework of the ‘old’ quantum mechanics. As a result, Schrödinger had to withdraw the original ‘relativistically framed’ article, a draft of which has never been found, from a journal and start all over with his centennial article on the nonrelativistic stationary Schrödinger equation. Our goal here is to follow the original ‘relativistic idea’ from a modern mathematical viewpoint and elaborate on why Schrödinger did not publish it. We hope that this consideration will encourage readers to study quantum physics starting from one of the crucial moments of its creation.
Schrödinger’s wave mechanics, developed in a series of papers published one century ago, offered an exceptionally intuitive route into the new atomic physics. The later axiomatic formulation of quantum mechanics brought indispensable mathematical precision, but it can also conceal the physical images that guided the original discovery. We revisit the optical–mechanical analogy behind Schrödinger’s construction and use it as a visual guide to stationary states, superposition, field modes, and gauge interactions. The analogies are deliberately heuristic and their limitations are stated: a quantum wavefunction is not, in general, a classical wave in ordinary space, and discrete classical modes do not by themselves imply quantized energy. Nevertheless, the wave viewpoint makes it possible to see the operator language of first and second quantization as the compact algebraic expression of an underlying mode structure. The purpose of this article is therefore not to replace the modern formalism, but to recover the physical intuition that makes that formalism intelligible.
This article is an introduction to and advertisement of Erwin Schrödinger’s little-known real-valued wave equation, the first published time dependent Schrödinger equation. I argue that this equation is not merely a historical curiosity. Not only does it show that quantum mechanics need not be viewed as essentially complex-valued, but the real formalism also provides a deep insight into the puzzling nature of time reversal in a quantum world. It is hoped that this observation will stimulate the discovery of other areas where ‘keeping it real’ will prove pedagogically or foundationally useful.
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.
On 13 March 1926, the first of six papers on wave mechanics by Erwin Schrödinger was published in the Annalen der Physik with the title Quantisation as an eigenvalue problem (Part 1) (Schrödinger, 1926b). The startling first paragraph reads: ‘In this paper, I wish to consider, first, the simple case of the hydrogen atom (nonrelativistic and unperturbed), and show that the customary quantum conditions can be replaced by another postulate, in which the concept of “whole numbers”, merely as such, is not introduced. Rather, when integralness does appear, it arises in the same natural way as it does in the case of node-numbers of a vibrating string. The new conception is capable of generalisation, and strikes, I believe, very deeply at the true nature of quantum rules.’ These papers were the fruits of an extraordinary burst of creativity on Schrödinger's part which resulted from his interactions with Einstein in the latter part of 1925. Central to these exchanges were de Broglie's remarkable researches which culminated in his famous PhD dissertation and published papers of 1924. These events have already be recounted in Chap. 9. The subsequent developments which led to Schrödinger's discovery of the equation which bears his name will be taken up in Sect. 14.2, but let us first understand more about Schrödinger's background. Schrödinger's background in physics and mathematics Education and career up to 1925 Unlike Heisenberg, Jordan, Pauli and Dirac, Erwin Schrödinger was not one of the young Turks who developed Knabenphysik , young man's physics.
Abstract Wave mechanics triumphed when Schrödinger published his now famous equation and showed how to describe hydrogen-like atoms using it. However, while looking for the right equation, Schrödinger first explored, but did not publish, the equation that we today call the Klein–Gordon equation. An alternative possible choice is explored in this work. It is shown a quasi-relativistic wave equation which solutions match the Schrödinger’s results at electron energies much smaller than the energy associated to the electron’s mass, but include, at higher energies, the relativity energy’s correction calculated in traditional first order perturbation theory. A discussion is presented about several consequences that would follow from using this quasi-relativistic wave equation as a quantum mechanics foundational equation. It is also suggested the academic use of this equation for introducing the students to the implications of the special theory of relativity in introductory quantum mechanics courses.
The major problem in quantum physics is to find the wave function of a given physical system and to understand (predict) how the wave function of the system evolves in time, or how it changes under external influences. In 1926, Erwin Schrodinger predicted that the wave function of a given physical system might be completely determined if the total energy of the system was known. Since the equation represents a real physical system, it must satisfy the following conditions: the equation must be linear and coefficients appearing in this equation should only depend on the parameters characteristic of the particle. This chapter focuses on to the non-relativistic case only. It considers the case of a free particle moving in one dimension. The chapter shows that the Schrodinger equation can be obtained from the energy of the free particle. It shows that the Schrodinger equation guarantees the conservation of normalization of the wave function.