Abstract This chapter explores Schrödinger’s introduction of wave mechanics. After briefly covering Schrödinger’s earlier work, it turns to the investigations that led him to write a four-part paper in 1926 introducing and applying wave mechanics. The chapter provides a detailed analysis of these four papers, in which the analogy of the new theory to the Hamilton-Jacobi formulation of classical mechanics plays a central role, as well as the paper written in between the second and the third installment on the relation between wave and matrix mechanics. The chapter also discusses Schr "odinger’s original interpretation of his wave functions and the reception of his theory by the physics community.
Abstract This chapter explores Schrödinger’s introduction of wave mechanics. After briefly covering Schrödinger’s earlier work, it turns to the investigations that led him to write a four-part paper in 1926 introducing and applying wave mechanics. The chapter provides a detailed analysis of these four papers, in which the analogy of the new theory to the Hamilton-Jacobi formulation of classical mechanics plays a central role, as well as the paper written in between the second and the third installment on the relation between wave and matrix mechanics. The chapter also discusses Schr "odinger’s original interpretation of his wave functions and the reception of his theory by the physics community.
In the first part of this chapter, the Hamiltonian function and Hamilton-Jacobi equation are presented. The latter is the starting point of Schrödinger’s work to obtain his wave equation. Next, the derivation of the wave equation is given along with the foundations of Schrödinger’s wave mechanics, i.e., the Hamilton’s analogy between optics and mechanics. It is also introduced some applications of the Schrödinger’s new quantum theory to several problems (radial function of the hydrogen atom, Compton effect and quantum harmonic oscillator). Other applications are also indicated in this chapter. The Dirac notation is introduced followed by the derivation of the expectation value in quantum mechanics and its derivation with respect to time. Next, Born’s probability theory and the superposition principle are introduced followed by the derivation of the quantum linear momentum operator. Afterwards, the normalization of a wave function and the orthogonality of a wave function are given followed by the properties of the Hermitian operator. In the end of the chapter, several examples and exercises are presented (168 words).
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.
Chapter 4 shows that, just as Heisenberg’s matrix mechanics represented a break with classical physics and the beginning of a new physics, Schrödinger’s wave mechanics represented another, no less radical, break with classical physics and at the same time opened up another path to the quantum world. Schrödinger’s theory is shown to make substantial use of de Broglie’s waves and related phenomena. The Schrödinger equation, being wave-like, constitutes mathematically an eigenvalue problem for bounded systems, and thus naturally leads to quantization, as is illustrated with a couple of simple examples. The chapter ends with Dirac’s abstract formulation, which has the advantage of being suitable for any description, be it Heisenberg’s, Schrödinger’s or any other.
Schrödinger's work on wave mechanics in 1926 appears to have been driven by the idea that one could give a purely wave-theoretical description of matter. Key elements in this picture were the idea of particles as wave packets (Section 4.3) and the possible implications for the problem of radiation (Section 4.4). This pure wave theory, in contrast to de Broglie's work, did away with the idea of particle trajectories altogether (Section 4.5). The main conflict, however, was between Schrödinger and the proponents of quantum mechanics (in particular Heisenberg, Section 4.6), both in its form at the time of Schrödinger's papers and in its further developments as sketched in the previous chapter. For reference, we provide a brief chronology of Schrödinger's writings relating to wave mechanics up to the Solvay conference: Paper on Einstein's gas theory, submitted 15 December 1925, published 1 March 1926 (Schrödinger 1926a). First paper on quantisation, submitted 27 January 1926, addendum in proof 28 February 1926, published 13 March 1926 (Schrödinger 1926b). Second paper on quantisation, submitted 23 February 1926, published 6 April 1926 (Schrödinger 1926c). Paper on the relation between wave and matrix mechanics (‘equivalence paper’), submitted 18 March 1926, published 4 May 1926 (Schrödinger 1926d). Paper on micro- and macromechanics (coherent states for the harmonic oscillator), published 9 July 1926 (Schrödinger 1926e). […]
Abstract Following the path by which humanity learned quantum mechanics can lead to an improved understanding of the theory and the origins of its perceived limitations. Our goal is to retrace the development by investigating primary sources, including original published papers and letters, with attention to their timing and influence. Quantum mechanics began in 1900 with the introduction of Planck’s constant, which lead to the “old” quantum theory and Bohr’s model of the atom. With the introduction of matter waves in 1924, a second more intense period began. By 1928 the complete fundamental structure of quantum mechanics was established with the introduction of the Schrödinger equation, the proof of its equivalence to Heisenberg’s matrix mechanics, and the introduction of the relativistic Dirac equation. The theory retains the same essential form today, fueling the technological revolution that began in the last century. We culminate with an introduction to quantum information and computing. This completely new physical scientific paradigm was developed largely in Germany during a period of enormous social, economic, and political upheaval—in a hostile intellectual environment—that parallels the overturning of classical physics. It is striking that all the problems debated today were immediately recognized by the founders of the theory and instructive to see how they dealt with the various issues. We emphasize the centrality of the often neglected second quantization form of the theory for questions of interpretation and give a detailed examination of von Neumann’s widely misunderstood hidden variable theorem.
During 1925 three different but equivalent versions of quantum theory were proposed, Schrodinger proposed wave mechanics; Heisenberg developed matrix mechanics and Dirac-introduced operator theory. Considering the de Broglie's matter waves Erwin Schrödinger, an Austrian physicist, argued that if a particle like an electron behaves as a wave then the equation of wave motion could be successfully applied to it. He postulated a function varying in both space and time in a wave-like manner. Schrodinger set up a linear and time-dependent wave-like equation, called Schrodinger wave equation, to describe the wave aspect of a particle taking account of de Broglie's relation for wavelength. This chapter discusses the Schrodinger equation and explores the general properties of the solution of the Schrodinger equation. It verifies the Ehrenfest's theorem in certain interesting systems and solves equation for linear harmonic oscillator and shows that their solutions are identical to the solutions of classical equation of motion.