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.
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.
There has been much research on Schrodinger's route to what we now call Schrodinger's equation. Various authors disagree as to the exact nature of the influence of each of the physicists he cites—and of some that he does not. This paper, intended for graduate students of and researchers in quantum theory, clarifies Schrodinger's original aims in formulating a wave equation for matter, discusses how far he fulfilled his original aspirations and in what respects he fell short of his goal. An analysis of Schrodinger's foundational paper enables us to distinguish between a formal and an epistemological part, and consider the input of the physicists cited on the basis of the part in which each reference occurs. It turns out, for instance, that Hamilton's optical-mechanical analogy belongs entirely to the epistemological part. Indeed, no element of this analogy plays any role in the formal part of Schrodinger's argument. Instead of basing his theory on this analogy, as is often done nowadays in the physics literature and even in the history of science, we maintain that the aim of Schrodinger's project was to represent wave phenomena by a ‘wave’ in configuration- or q-space, paralleling Hertz's treatment in his Mechanics (1894). The influence of this book on Schrodinger's foundational paper is demonstrated by an analysis of his unpublished paper: Hertz's Mechanics and Einstein's Theory of Gravitation. This approach enables us to dispense with the optical-mechanical analogy in tracing the route to Schrodinger's equation. We also discuss the curious role of the Van Vleck determinant as the ‘missing link’ in taking the classical limit of Schrodinger's wavefunction. A concluding section discusses the relation of some of Schrodinger's earlier and later work to the development of quantum field theory.
Purpose of the study: This study aims to provide a comprehensive pedagogical review of the Schrödinger equation by integrating its physical derivations, mathematical structure, and applications to support advanced undergraduate and beginning graduate students in understanding quantum mechanics coherently. Methodology: Literature review, pedagogical synthesis, canonical model analysis, Hilbert-space formalism, self-adjoint operator framework, spectral theory approach, quantum mechanics textbooks and journal sources, mathematical physics methods, conceptual analysis, and visualization of wave packets and quantum phenomena were used as tools and methods in this study. Main Findings: The study shows that multiple derivations of the Schrödinger equation converge to a unified structure based on linear, unitary evolution with a self-adjoint Hamiltonian. Key quantum phenomena such as superposition, tunnelling, and quantization emerge consistently from canonical models, while mathematical conditions ensure physical consistency and probability conservation. Novelty/Originality of this study: This study uniquely integrates physical derivations, rigorous mathematical structure, and pedagogical organization into a single coherent framework. It bridges conceptual gaps between theory and application, offering a unified reference that enhances understanding of quantum mechanics and supports both self-study and instructional practices.
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.
Courses about Quantum Mechanics are generally developed at the level of the mathematical formalism, its syntactic level, seldom treating the interpretation of the theory, its semantic level. In the majority of cases, this first formal approach is not followed by any other course addressing the numerous interpretations of the theory. It has become apparent that this strategy leads to a deficit in the professional formation of the students, exceedingly driving them to technical areas of application of Quantum Mechanics. However, applications such as those on quantum computing and entanglement, as they become more and more encrusted in the deepest foundations of the theory, are asking more and more for the development of this kind of ability. In this paper we develop an approach to deal with second quantization in the realm of Quantum Mechanics. We show that the second quantization Schrodinger equation can be mathematically derived from a classical Hamiltonian written in the phase space (Q,P), obtained by a canonical transformation upon the original phase space (q,p), using an axiomatic quantization procedure developed elsewhere. We apply this quantization process to a bosonic system, modelled by the harmonic oscillator problem. We then make reverse engineering to show that Schwinger's second quantization approach to fermionic systems furnishes the path to derive a Schrodinger equation explicitly written in terms of the usual momenta and positions operators (q^,p^) for such systems. Finally, we use these mathematical developments to address the commonly accepted statement that 'the spin has no classical analog'.
The Schrodinger equation has been considered to be a postulate of quantum physics, but it is also perceived and derived heuristically as the quantum equivalent of the classical energy relation. We indicate that the Schrodinger equation cannot be a physical postulate, and we show that the heuristic derivations wrongly consider that the particle kinetic energy is proportional to the square of the momentum operator acting on the wavefunction. The correct term is proportional to the square of the momentum. Analyzing particle interactions, we realize that particles have multiple virtual motions and that each motion is accompanied by a wave that has constant amplitude. Accordingly, we define the wavefunction as the superposition of the virtual waves of the particle. As a result, quantum mechanics becomes a local and global theory, while in the traditional formulation it is a local theory. In several interaction settings we can tell what particle motions arise and explain the outcomes in simple and tangible terms. Most importantly, the mathematical foundation of quantum mechanics becomes clear and justified, and we derive the Schrodinger, Dirac, etc. equations as the conditions the wavefunction must satisfy at each space-time point in order to fulfill the respective total energy equation.
An axiomatic approach to quantum mechanics that has, as a theorem, the Schrödinger equation may be of enormous value to cope with interpretation issues, since all the interpretation constructs must be present in the axioms, or directly derived by them from their mathematical unfolding. Thus, it is critical to show that this axiomatic derivation is reliable beyond any possible doubt. To show this, it is possible to make generalizations and extensions of the axioms to derive the Schrödinger equation in the underlying generalized or extended formats. In previous papers, we have shown that the axiomatic approach we propose can be used to derive the Schrödinger equation as a direct axiom. Since then, we have also shown that it was possible to generalize that derivation to coordinate systems other than the Cartesian, as well as its relativistic extensions that lead to the relativistic wave equations. An extension to dissipative systems was also performed, allowing us to mathematically derive the Caldirola–Kanai equation from first principles. All these derivations were performed using pure states and in the absence of the electromagnetic field. This means that we can further generalize the approach to embrace these two possibilities. Being an axiomatic approach, we show that we need only to slightly modify the axioms to derive the Schrödinger equation for these two contexts. Despite being quite direct, the algebraic complexity of these derivations should give the reader the desired confidence in the proposed axioms.