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Schrödinger and the development of wave mechanics

Robert Golub, S. K. Lamoreaux

Published Feb 28, 2023• 0 citations
Publisher

Abstract

Abstract While, as Feynman said, nobody can really understand the origin of the Schrödinger equation, we investigate several lines of thought leading to such an equation. We show some simple applications of the equation. An important application was Schrödinger’s solution for the hydrogen atom. This provided a theoretical basis for Bohr’s ad hoc model. We conclude with Schrödinger’s demonstration of the equivalence of his wave mechanics with Heisenberg’s matrix mechanics.

DOI
10.1093/oso/9780198822189.003.0007

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Abstract While, as Feynman said, nobody can really understand the origin of the Schrödinger equation, we investigate several lines of thought leading to such an equation. We show some simple applications of the equation. An important application was Schrödinger’s solution for the hydrogen atom. This provided a theoretical basis for Bohr’s ad hoc model. We conclude with Schrödinger’s demonstration of the equivalence of his wave mechanics with Heisenberg’s matrix mechanics.
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.
The aim of this paper is to give a brief account of the development of the mathematical equivalence of quantum mechanics. In order to deal with atomic systems, Heisenberg developed matrix mechanics in 1925. Some time later, in the winter of 1926, Schrodinger established his wave mechanics. In the spring of 1926, quantum physicists had two theoretical models that allowed them to predict the same behaviour of the quantum systems, but both of them were very different. Schrodinger thought that the empirical equivalence could be explained by means of a proof of mathematical equivalence.
In 1926, Schrödinger [365] recognized that the variational theory of elliptical differential equations with fixed boundary conditions could produce a discrete eigenvalue spectrum in agreement with the energy levels of Bohr's model of the hydrogen atom. This conceptually startling amalgam of classical ideas of particle and field turned out to be correct. Within a few years, the new wave mechanics almost completely replaced the ad hoc quantization of classical mechanics that characterized the “old” quantum theory initiated by Bohr. Although the matrix mechanics of Heisenberg was soon shown to be logically equivalent, the variational wave theory became the standard basis of methodology in the physics of electrons. The nonrelativistic Schrödinger theory is readily extended to systems of N interacting electrons. The variational theory of finite N -electron systems (atoms and molecules) is presented here. In this context, several important theorems that follow from the variational formalism are also derived.
Schrödinger's wave equation The new theory was given in Schrödinger's first paper in much the same casual way as was that of Planck's radiation formula in his earliest papers. Both were arrived at by a process for which no particular justification was given; but the wave equation in the one case and the radiation formula in the other were so striking in their immediate consequences that a real theoretical basis had to be sought for them. In this first paper the wave equation was found by a variation principle (it was rather like Hamilton's variation principle, but not much more could be said for it); but in the second paper Schrödinger shows it to be a real generalisation of the classical mechanics suggested by the waves of Louis de Broglie (§ 114). In the former paper Schrödinger remarks that in his theory the ‘quantum numbers’ appear as naturally as do ‘integers’ in the theory of a vibrating string, where they are determined by certain boundary conditions to be satisfied by the solution of a differential equation; in quantum mechanics the corresponding differential equation is Schrödinger's wave equation.
Abstract In June 1925 Heisenberg made the breakthrough that would lead to matrix mechanics. The theory was abstract, offered no means for visualizing the mechanics of the atom, and its application required a lot of heavy lifting. Erwin Schrödinger subsequently made the breakthrough that would lead to wave mechanics, building on Louis de Broglie’s association of a wave with an electron. In Schrödinger’s wave equation the quantum numbers of the hydrogen atom arise naturally. He argued that his ‘wavefunction’ offered a realistic description of the electron in an atom, and sought to eliminate Bohr’s discontinuous quantum jumps. But in the spring of 1926, Max Born offered an enduring alternative. The modulus-square of the wavefunction is a measure of probability in an unvisualizable microphysics. In an intense debate with Bohr, Schrödinger demanded a theory based on logical concepts free from contradictions. Bohr was left to ponder what this might mean.