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’.
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’.
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 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.
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.
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.
Where did we get that [equation] from? Nowhere. It is not possible to derive it from anything you know. It came out of the mind of Schrödinger. Richard Feynman De Broglie's matter waves The early struggles of physicists towards a quantum theory were mostly concerned with attempts to understand the nature of light. The traditional picture of light as a wave motion had been challenged by Planck and Einstein. They had shown that certain experimental results that were impossible to understand in terms of a wave picture could be easily explained if light was thought of as a stream of particles, now called photons. William Bragg, who, with his son, won the 1915 Nobel Prize for studies of crystal structure using X-rays, summarized this dilemma for physics by exclaiming in despair that he was teaching the corpuscular theory of light on Mondays, Wednesdays and Fridays, and the undulatory theory on Tuesdays, Thursdays and Saturdays! Physicists were still wrestling with these apparently contradictory properties of light when, in 1924, Prince Louis de Broglie (pronounced ‘de Broy’) suggested that all matter, even objects that we usually think of as particles – such as electrons – should also display wavelike behaviour! This revolutionary idea was completely unexpected and, what is more, was included by de Broglie in his Ph. D. thesis. Like most people, physicists are generally reluctant to accept any wild new idea, especially if there is not a shred of evidence to support it.
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.