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Cambridge University Press eBooks

Schrödinger and wave mechanics

Malcolm S. Longair

Published Jan 31, 2013• 0 citations

Abstract

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.

DOI: 10.1017/cbo9781139062060.015

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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.

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). […]

Bohr’s theory of the atom was in trouble as soon as it was formulated. Further detailed spectroscopy studies encouraged a proliferation of quantum numbers and ‘selection rules’ in what became known as the Bohr–Sommerfeld model. It couldn’t last, and by 1925 the theory was in crisis. The immediate concern was with the quantum numbers themselves. Where did they come from? Could de Broglie’s hypothesis shed any light? In October 1925 the attentions of Erwin Schrödinger were drawn to a footnote in one of Einstein’s recent papers. Intrigued, Schrödinger acquired a copy of de Broglie’s PhD thesis. Although he eventually published a more obscure derivation, Schrödinger essentially applied the de Broglie relation to the classical equation of wave motion. He applied the result to the hydrogen atom, and showed that the quantum numbers emerge ‘in the same natural way as the integers specifying the number of nodes in a vibrating string’.

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.

Quantum theory originated from the constant introduced by Max Planck in his radiation law in 1900. Later Niels Bohr used the idea of matter waves for the explanation of atomic spectra. Heisenberg gave an algebraic treatment, called matrix mechanics, while Erwin Schrödinger formulated his wave mechanics. He published the first of four papers containing the celebrated equation in the Annalen der Physik in 1927, and showed that the atomic spectra resulted from solving his wave equation for the vibrational states. Schrödinger moved from Zürich to Berlin as Planck’s successor. In the year 1933 he shared the Nobel Prize for Physics with Dirac. With Hitler’s coming to power, however, Schrödinger was outraged and decided to leave Germany to take up a fellowship in Oxford. In 1934 he was offered a permanent position at Princeton University, but he did not accept it. He took a position at the Institute for Advanced Studies in Dublin in 1938, where he stayed till his retirement in 1955.

Abstract In addition to recounting some contemporary scientific history, Chapter 6 describes the hypothesis that matter, like light, can display wavelike properties, and the creation of the various formulations of quantum mechanics. That matter could have a wavelength was proposed in 1924 by Louis de Broglie, who presented a specific formula for calculating it, one that was verified experimentally in 1927. However, de Broglie’s hypothesis was overshadowed by the creation of three versions of quantum mechanics in 1925/26. The first, denoted matrix mechanics, was proposed by Werner Heisenberg. It was quickly and successfully applied by Wolfgang Pauli to the hydrogen atom. Paul Dirac introduced the next version, which was followed by that of Erwin Schrödinger via a wave equation whose solutions, denoted wave functions, were soon interpreted byMax Born to be related to the probability that certain outcomes or events will occur: classical-physics determinism was thereby removed from quantum mechanics.

Bohr’s model of the hydrogen atom was successful in that it gave us a radically new way to look at atoms. However, it has serious shortcomings. It could not be used to explain the spectra of He or any multielectron atom. It could not predict the intensities of the H absorption and emission lines. With de Broglie’s hypothesis that matter was also wavelike,1 there arose a question at the 1925 Solvey conference: What is the wave equation? De Broglie could not answer this; however, over the next year Erwin Schro¨dinger, working in Vienna, published a series of papers in which he deduced the general form of the equation that bears his name and applied it successfully to the hydrogen atom.2,3 What emerged was a new set of postulates, much like Newton’s, that laid the foundations of quantum theory.