by Manjit Kumar
As the years passed, Schrödinger began to wonder if he would ever make a major contribution that would establish him among the first rank of contemporary physicists. At the beginning of 1925 he was 37, long having celebrated the 30th birthday that was said to be the watershed in the creative life of a theorist. Doubts over his worth as a physicist were compounded by a marriage in trouble because of affairs on both sides. By the end of the year Schrödinger’s marriage was shakier than ever, but he made the breakthrough that would ensure his place in the pantheon of physics.
Schrödinger was taking an ever more active interest in the latest developments in atomic and quantum physics. In October 1925, he read a paper that Einstein had written earlier in the year. A footnote that flagged up Louis de Broglie’s thesis on wave-particle duality caught his eye. As with most footnotes, virtually everyone ignored it. Intrigued by Einstein’s stamp of approval, Schrödinger set about acquiring a copy of the thesis, unaware that papers by the French prince had been in print for nearly two years. A couple of weeks later, on 3 November, he wrote to Einstein: ‘A few days ago I read with the greatest interest the ingenious thesis of de Broglie, which I finally got hold of.’9
Others were also beginning to take note, but in the absence of any experimental support, few were as receptive to de Broglie’s ideas as Einstein and Schrödinger. In Zurich, every fortnight, physicists from the university got together with those from the Eidgenossische Technische Hochschule (ETH), for a joint colloquium. Pieter Debye, the ETH professor of physics, ran the meetings and asked Schrödinger to give a talk on de Broglie’s work. In the eyes of his colleagues, Schrödinger was an accomplished and versatile theoretician who had made solid but unremarkable contributions in his 40-odd papers that spanned areas as diverse as radioactivity, statistical physics, general relativity and colour theory. Among these were a number of well-received review articles that demonstrated his ability to absorb, analyse and organise the work of others.
On 23 November Felix Bloch, a 21-year-old student, was present when ‘Schrödinger gave a beautifully clear account of how de Broglie associated a wave with a particle and how he could obtain the quantization rules of Niels Bohr and Sommerfeld by demanding that an integer number of waves should be fitted along a stationary orbit’.10 With no experimental confirmation of wave-particle duality, which would come in 1927, Debye found it all far-fetched and ‘rather childish’.11 The physics of a wave – any wave, from sound to electromagnetic, even a wave travelling along a violin string – has an equation that describes it. In what Schrödinger had outlined there was no ‘wave equation’ de Broglie had never tried to derive one for his matter waves. Nor had Einstein after he read the French prince’s thesis. Debye’s point ‘sounded quite trivial and did not seem to make a great impression’, Bloch still remembered 50 years later.12
Schrödinger knew that Debye was right: ‘You cannot have waves without a wave equation.’13 Almost at once he decided to find the missing equation for de Broglie’s matter waves. After returning from his Christmas holiday, Schrödinger was able to announce at the next colloquium held early in the New Year: ‘My colleague Debye suggested that one should have a wave equation; well, I have found one!’14 Between one meeting and the next, Schrödinger had taken de Broglie’s nascent ideas and developed them into a fully-blown theory of quantum mechanics.
Schrödinger knew exactly where to start and what he had to do. De Broglie had tested his idea of wave-particle duality by reproducing the allowed electron orbits in the Bohr atom as those in which only a whole number of standing electron wavelengths could fit. Schrödinger knew that the elusive wave equation he sought would have to reproduce the three-dimensional model of the hydrogen atom with three-dimensional standing waves. The hydrogen atom would be the litmus test for the wave equation he needed to find.
Not long after starting the hunt, Schrödinger thought he had bagged just such an equation. However, when he applied it to the hydrogen atom, the equation churned out the wrong answers. The root of the failure lay in the fact that de Broglie had developed and presented wave-particle duality in a manner consistent with Einstein’s theory of special relativity. Following de Broglie’s lead, Schrödinger started out by looking for a wave equation that was ‘relativistic’ in form, and found one. In the meantime, Uhlenbeck and Goudsmit had discovered the concept of electron spin, but their paper did not appear in print until the end of November 1925. Schrödinger had found a relativistic wave equation, but unsurprisingly it did not include spin and therefore failed to agree with experiments.15
With the Christmas vacation fast approaching, Schrödinger began to concentrate his efforts on finding a wave equation without worrying about relativity. He knew that such an equation would fail for electrons travelling at speeds close to that of light where relativity could not be ignored. But for his purposes such a wave equation would do. Soon, however, there was more than just physics on his mind. He and his wife Anny were having another of their sustained bouts of marital turbulence, one that was lasting longer than most. Despite the affairs and talk of divorce, each seemed incapable and unwilling to permanently part from the other. Schrödinger wanted to escape for a couple of weeks. Whatever excuse he gave his wife, he left Zurich for the winter wonderland of his favourite Alpine resort, Arosa, and a rendezvous with an ex-lover.
Schrödinger was delighted to be back in the familiar and comfortable surroundings of the Villa Herwig. It was here that he and Anny had spent the previous two Christmas holidays, but there was hardly time enough over the next two weeks to feel guilty as Schrödinger spent his passion with his mysterious lady. However distracted he may have been, Schrödinger made time to continue the search for his wave equation. ‘At the moment I am struggling with a new atomic theory’, he wrote on 27 December.16 ‘If only I knew more mathematics! I am very optimistic about this thing and expect that if I can only…solve it, it will be very beautiful.’ Six months of sustained creativity were to follow during this ‘late erotic outburst’ in his life.17 Inspired by his unnamed Muse, Schrödinger had discovered a wave equation, but was it the wave equation he was seeking?
Schrödinger did not ‘derive’ his wave equation; there was just no way to do it from classical physics that was logically rigorous. Instead he constructed it out of de Broglie’s wave-particle formula that linked the wavelength associated with a particle to its momentum, and from well-established equations of classical physics. As simple as it sounds, it required all of Schrödinger’s skill and experience to be the first to write it down. It was the foundation on which he built the edifice of wave mechanics in the months ahead. But first he had to prove that it was the wave equation. When applied to the hydrogen atom, would it generate the correct values for the energy levels?
After returning to Zurich in January, Schrödinger found that his wave equation did reproduce the series of energy levels of the Bohr-Sommerfeld hydrogen atom. More complicated than de Broglie’s one-dimensional standing electron waves fitted into circular orbits, Schrödinger’s theory obtained their three-dimensional analogues – electron orbitals. Their associated energies were generated as part and parcel of the acceptable solutions of Schrödinger’s wave equation. Banished once and for all were the ad hoc additions required by the Bohr-Sommerfeld quantum atom – all the previous tinkering and tweaking that sat uneasily now emerged naturally from within the framework of Schrödinger’s wave mechanics. Even the mysterious quantum jumping between orbits by an electron appeared to be eliminated by the smooth and continuous transitions from one permitted three-dimensional electron standing wave to another. ‘Quantization as an Eigenvalue Problem’ was received by the Annalen der Physik on 27 January 1926.18 Published on 13 March, it presented Schrödinger’s version of quantum mechanics and its application to the hydrogen atom.
In a career that spanned some 50 years, Schrödinger’s average annual output of research papers amounted to 40 printed pages. In 1926 he published 256 pages in which he demonstrated how wave mechanics could
successfully solve a range of problems in atomic physics. He also came up with a time-dependent version of his wave equation that could tackle ‘systems’ that changed with time. Among them were processes involving the absorption and emission of radiation and the scattering of radiation by atoms.
i. The fifth Solvay conference, 24 to 29 October 1927, devoted to the new quantum mechanics and to questions connected with it. Auguste Piccard; E. Henriot; Paul Ehrenfest; E. Herzen; T. de Donder; Erwin Schrödinger; J.E. Verschaffelt; Wolfgang Pauli; Werner Heisenberg; Ralph Fowler; Léon Brillouin. Pieter Debye; Martin Knudsen; William L. Bragg; Hendrik Kramers; Paul Dirac; Arthur H. Compton; Louis de Broglie; Max Born; Niels Bohr. Irving Langmuir; Max Planck; Marie Curie; Hendrik Lorentz; Albert Einstein; Paul Langevin; Charles-Eugène Guye; C.T.R. Wilson; Owen Richardson. (Photograph by Benjamin Couprie, Institut International de Physique Solvay, courtesy AIP Emilio Segrè Visual Archives)
ii. Max Planck, the conservative theorist who unwittingly started the quantum revolution in December 1900 when he unveiled his derivation for the distribution of electromagnetic radiation emitted by a blackbody. (AIP Emilio Segrè Visual Archives, W. F. Meggers Collection)
iii. Ludwig Boltzmann, the Austrian physicist and foremost advocate of the atom until his suicide in 1906. (University of Vienna, courtesy AIP Emilio Segrè Visual Archives)
iv. ‘The Olympia Academy’. Conrad Habicht, Maurice Solovine and Albert Einstein. (© Underwood & Underwood/CORBIS)
v. Albert Einstein in 1912, seven years after the annus mirabilis in which he published five papers, including his quantum solution to the photoelectric effect and his special theory of relativity. (© Bettmann/CORBIS)
vi. The first Solvay conference, Brussels, 30 October to 3 November 1911 – a summit meeting on the quantum. Walther Nernst; Marcel-Louis Brillouin; Ernest Solvay; Hendrik Lorentz; Emil Warburg; Jean-Baptiste Perrin; Wilhelm Wien; Marie Curie; Henri Poincaré. Robert B. Goldschmidt; Max Planck; Heinrich Rubens; Arnold Sommerfeld; Frederick Lindemann; Maurice de Broglie; Martin Knudsen; Friedrich Hasenohrl; G. Hostelet; E. Herzen; Sir James Jeans; Ernest Rutherford; Heike Kamerlingh-Onnes; Albert Einstein; Paul Langevin. (Photograph by Benjamin Couprie, Institut International de Physique Solvay, courtesy AIP Emilio Segrè Visual Archives)
vii. Niels Bohr, the ‘golden Dane’ who introduced the quantum into the atom. This photo was taken in 1922, the year he won the Nobel Prize. (Emilio Segrè Visual Archives, W. F. Meggers Collection)
viii. Ernest Rutherford, the charismatic New Zealander whose inspirational style motivated Bohr to run his own institute in Copenhagen along similar lines. Eleven of Rutherford’s students would win the Nobel Prize. (AIP Emilio Segrè Visual Archives)
ix. Always known as the Bohr Institute, the Universitetets Institut for Teoretisk Fysik was formally opened on 3 March 1921. (Niels Bohr Archive, Copenhagen)
x. Einstein and Bohr walking together in Brussels during the 1930 Solvay conference. They are almost certainly discussing Einstein’s light box thought experiment, which temporarily got the better of Bohr, leading him to fear the ‘end of physics’ if Einstein’s ideas proved correct. (Photograph by Paul Ehrenfest, courtesy AIP Emilio Segrè Visual Archives, Ehrenfest Collection)
xi. Einstein and Bohr at Paul Ehrenfest’s home in Leiden sometime after the 1930 Solvay conference. (Photograph by Paul Ehrenfest, courtesy AIP Emilio Segrè Visual Archives)
xii. Prince Louis Victor Pierre Raymond de Broglie, a member of one of France’s leading aristocratic families, who dared to ask the simple question: If light waves can behave like particles, can particles such as electrons behave like waves? (AIP Emilio Segrè Visual Archives, Brittle Books Collection)
xiii. Wolfgang Pauli, the discoverer of the exclusion principle, was noted for his acerbic wit, but was also regarded as ‘a genius comparable only with Einstein’. (©CERN, Geneva)
xiv. A moment to relax at the ‘Bohr Festspiele’, Göttingen University, June 1922. Left to right standing: Carl Wilhelm Oseen, Niels Bohr, James Franck and Oskar Klein. Max Born is seated. (AIP Emilio Segrè Visual Archives, Archive for the History of quantum Physics)
xv. Oskar Klein and the two ‘spin doctors’, George Uhlenbeck and Samuel Goudsmit, at Leiden University, summer 1926. (AIP Emilio Segrè Visual Archives)
xvi. Werner Heisenberg, aged 23. Two years later, he was responsible for one of the greatest and most profound achievements in the history of the quantum – the uncertainty principle. (AIP Emilio Segrè Visual Archives/Gift of Jost Lemmerich)
xvii. Bohr, Heisenberg and Pauli deep in discussion over lunch at the Bohr Institute in the mid-1930s. (Niels Bohr Institute, courtesy AIP Emilio Segrè Visual Archives)
xviii. The quiet Englishman, Paul Dirac, who helped to reconcile Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics. (AIP Emilio Segrè Visual Archives)
xix. Erwin Schrödinger, whose discovery of wave mechanics was described as the product of ‘a late erotic outburst’. (AIP Emilio Segrè Visual Archives)
xx. Heisenberg’s mother, Schrödinger’s wife, Dirac’s mother, Dirac, Heisenberg, Schrödinger at Stockholm train station in 1933. It was the year that Schrödinger and Dirac shared the Nobel Prize, and Heisenberg was awarded the deferred prize for 1932. (AIP Emilio Segrè Visual Archives)
xxi. Albert Einstein seated in his book-filled study at home in Princeton in 1954. (© Bettmann/CORBIS)
xxii. The last drawing by Niels Bohr on the blackboard in his study, made the night before he died in November 1962, was of Einstein’s 1930 light box. To the very end, Bohr continued to analyse the debate with Einstein about quantum mechanics and the nature of reality. (AIP Emilio Segrè Visual Archives)
xxiii. David Bohm, who produced an alternative to the Copenhagen interpretation, seen here after refusing to testify whether or not he was a member of the Communist party before the House Un-American Activities Committee. (Library of Congress, New York World-Telegram and Sun Collection, courtesy AIP Emilio Segrè Visual Archives)
xxiv. John Stewart Bell, the Irish physicist who discovered what Einstein and Bohr could not: a mathematical theorem that could decide between their two opposing philosophical worldviews. (©CERN, Geneva)
On 20 February, as the first paper was being readied for the printers, Schrödinger used the name Wellenmechanik, wave mechanics, for the first time to describe his new theory. In stark contrast to the cold and austere matrix mechanics that proscribed even the hint of visualisability, Schrödinger offered physicists a familiar and reassuring alternative that offered to explain the quantum world in terms closer to those of nineteenth-century physics than Heisenberg’s highly abstract formulation. In place of the mysterious matrices, Schrödinger came bearing differential equations, an essential part of every physicist’s mathematical toolbox. Heisenberg’s matrix mechanics gave them quantum jumps and discontinuity, and nothing to picture in their mind’s eye as they sought to glimpse the inner workings of the atom. Schrödinger told physicists they no longer needed to ‘suppress intuition and to operate only with abstract concepts such as transition probabilities, energy levels, and the like’.19 It was hardly surprising that they greeted wave mechanics with enthusiasm and quickly rushed to embrace it.
As soon as he received complimentary copies of his paper, Schrödinger sent them out to colleagues whose opinions mattered most to him. Planck wrote back on 2 April that he had read the paper ‘like an eager child hearing the solution to a riddle that had plagued him for a long time’.20 Two weeks later, Schrödinger received a letter from Einstein, who told him ‘the idea of your work springs from true genius’.21 ‘Your approval and Planck’s mean more to me than that of half the world’, Schrödinger wrote back.22 Einstein was convinced that Schrödinger had made a decisive advance, ‘just as I am convinced that the Heisenberg-Born method is misleading’.23
Others took longer to fully appreciate the product of Schrödinger’s ‘late erotic outburst’. Sommerfeld initially believed that wave mechanics was ‘totally
crazy’, before changing his mind and declaring: ‘although the truth of matrix mechanics is indubitable, its handling is extremely intricate and frighteningly abstract. Schrödinger has now come to our rescue.’24 Many others also breathed easier as they learnt and began using the more familiar ideas embodied in wave mechanics rather than having to struggle with the abstract and alien formulation of Heisenberg and his Göttingen colleagues. ‘The Schrödinger equation came as a great relief,’ wrote the young spin doctor George Uhlenbeck, ‘now we did not any longer have to learn the strange mathematics of matrices.’25 Instead Ehrenfest, Uhlenbeck and the others in Leiden spent weeks ‘standing for hours at a time in front of the blackboard’ in order to learn all the splendid ramifications of wave mechanics.26
