by Manjit Kumar
With Born as his guide, Jordan soon began concentrating on problems of atomic structure. Somewhat insecure and with a stutter, he appreciated Born’s patience whenever they discussed the latest papers touching on atomic theory. Fortuitously, he had moved to Göttingen in time to attend the Bohr Festspiele and, like Heisenberg, was inspired by the lectures and the discussions that followed. After his doctoral dissertation in 1924, Jordan worked briefly with others before being asked by Born to collaborate with him on an attempt to explain the width of spectral lines. Jordan is ‘exceptionally intelligent and astute and can think far more swiftly and confidently than I’, Born wrote to Einstein in July 1925.52
By then Jordan had already heard of Heisenberg’s latest ideas. Before he left Göttingen at the end of July, Heisenberg gave a talk to a small circle of students and friends about his attempt to construct a quantum mechanics based solely on the relations between observable properties. When Born asked him to collaborate, Jordan jumped at the chance to recast and extend Heisenberg’s original ideas into a systematic theory of quantum mechanics. Unknown to Born, as he sent Heisenberg’s paper to the journal Zeitschrift für Physik, Jordan was well versed in matrix theory through his background in mathematics. Applying these methods to quantum physics, in two months Born and Jordan laid the foundations for a new quantum mechanics that others would call matrix mechanics.53
Once Born identified Heisenberg’s multiplication rule as a rediscovery of matrix multiplication, he quickly found a matrix formula that connected position q and momentum p using an expression that included Planck’s constant: pq–qp=(ih/2)I, where I is what mathematicians call a unit matrix. It allowed the right-hand side of the equation to be written as a matrix. It was from this fundamental equation using the methods of matrix mathematics that all of quantum mechanics was constructed in the months that followed. Born was proud to be ‘the first person to write a physical law in terms of non-commuting symbols’.54 But it ‘was only a guess, and my attempts to prove it failed’, he recalled later.55 Within days of being shown the formula, Jordan came up with the rigorous mathematical derivation. No wonder Born was soon telling Bohr that, aside from Heisenberg and Pauli, he considered Jordan ‘to be the most gifted of the younger colleagues’.56
In August, Born went on his summer holiday to Switzerland with his family while Jordan stayed in Göttingen to write up a paper by the end of September for publication. Before it appeared in print they sent a copy to Heisenberg, who was in Copenhagen at the time. ‘Here, I got a paper from Born, which I cannot understand at all’, Heisenberg said to Bohr as he handed him the paper.57 ‘It is full of matrices, and I hardly know what they are.’
Heisenberg was hardly alone in not being familiar with matrices, but he set about learning the new mathematics with gusto and mastered enough to begin collaborating with Born and Jordan while still in Copenhagen. Heisenberg returned to Göttingen in the middle of October in time to help write the final version of what became known as the Drei-Männer-Arbeit, the ‘three-man paper’ in which he, Born and Jordan presented the first logically consistent formulation of quantum mechanics – the long-sought-after new physics of the atom.
However, there were already reservations being expressed about Heisenberg’s initial work. Einstein wrote to Paul Ehrenfest: ‘In Göttingen they believe it (I don’t).’58 Bohr believed it was ‘a step probably of fundamental importance’ but ‘it has not yet been possible to apply [the] theory to questions of atomic structure’.59 While Heisenberg, Born and Jordan had been concentrating on developing the theory, Pauli had been busy using the new mechanics to do just that. By early November, while the ‘three-man paper’ was still being written, he had successfully applied matrix mechanics in a stunning tour de force. Pauli had done for the new physics what Bohr had done for the old quantum theory – reproduced the line spectrum of the hydrogen atom. For Heisenberg, to add insult to injury, Pauli had also calculated the Stark effect – the influence of an external electric field on the spectrum. ‘I myself had been a bit unhappy that I could not succeed in deriving the hydrogen spectrum from the new theory’, Heisenberg recalled.60 Pauli had provided the first concrete vindication of the new quantum mechanics.
‘The Fundamental Equations of quantum Mechanics’ read the title. Born had been in Boston for a nearly a month, as part of a five-month lecture tour of the United States, when one December morning he opened his post and received ‘one of the greatest surprises’ of his scientific life.61 As he read the paper by one P.A.M. Dirac, a senior research student at Cambridge University, Born realised that ‘everything was perfect in its way’.62 Even more remarkably, Born soon discovered that Dirac had sent his paper to the Proceedings of the Royal Society containing the nuts and bolts of quantum mechanics a whole nine days before the ‘three-man paper’ was finished. Who was Dirac and how had he done it, wondered Born?
Paul Adrien Maurice Dirac was 23 years old in 1925. The son of a Swiss, French-speaking father, Charles, and an English mother, Florence, he was the second of three children. His father was such an overbearing and dominant figure that when he died in 1935, Dirac wrote: ‘I feel much freer now.’63 It was the trauma of having to remain silent in the presence of his father, a teacher of French, as he grew up that made Dirac a man of few words. ‘My father made the rule that I should only talk to him in French. He thought it would be good for me to learn French in that way. Since I found that I couldn’t express myself in French, it was better for me to stay silent than to talk in English.’64 Dirac’s preference for silence, the legacy of a deeply unhappy childhood and adolescence, would become legendary.
Although interested in science, in 1918, Dirac acted on his father’s advice and enrolled to study electrical engineering at the University of Bristol. Three years later, despite graduating with a first-class honours degree, he could not find a job as an engineer. With his employment prospects looking bleak as Britain’s post-war depression continued, Dirac accepted the offer of free tuition for two years to study mathematics back at his old university. He would rather have gone to Cambridge, but the scholarship he had won did not cover all the expenses of studying at the university. However, in 1923, after gaining his mathematics degree and receiving a government grant, he finally arrived in Cambridge as a PhD student. His supervisor was Ralph Fowler, Rutherford’s son-in-law.
Dirac had a thorough grasp of Einstein’s theory of relativity, which had generated a firestorm of publicity around the world in 1919 while he was still an engineering student, but he knew very little about Bohr’s decade-old quantum atom. Until his arrival in Cambridge, Dirac always considered atoms ‘as very hypothetical things’, hardly worth bothering about.65 He soon changed his mind and set about making up for lost time.
The quiet, secluded life of a budding Cambridge theoretical physicist was tailor-made for the shy and introverted Dirac. Research students were largely left to work alone in either their college rooms or in the library. While others might have struggled with a lack of human contact day after day, Dirac was perfectly happy to be left alone in his room to think. Even on a Sunday as he relaxed by walking in the Cambridgeshire countryside, Dirac preferred to do it alone.
Like Bohr, whom he met for the first time in June 1925, Dirac chose his words, written or spoken, very carefully. If he gave a lecture and was asked to explain a point that had not been understood, Dirac would often repeat word for word what he had said before. Bohr had gone to Cambridge to lecture on the problems of quantum theory and Dirac had been impressed by the man, but not by his arguments. ‘What I wanted was statements which could be expressed in terms of equations,’ he said later, ‘and Bohr’s work very seldom provided such statements.’66 Heisenberg, on the other hand, arrived from Göttingen to give a lecture having spent months doing just the sort of physics that Dirac would have found stimulating. But he did not hear about it from Heisenberg, who chose not to mention it as he spoke about atomic spectroscopy.
It was Ralph Fowler who alerted Dirac to Heisenberg
’s work by giving him a proof copy of the German’s soon-to-be-published paper. Heisenberg had been Fowler’s house-guest during his brief visit and had discussed his latest ideas with his host, who asked for a copy of the paper. When it arrived, Fowler had little time to study it thoroughly and so passed it on to Dirac, asking him for his opinion. When he first read it in early September, he found it difficult to follow and failed to appreciate what a breakthrough it represented. Then, as one week turned into two, Dirac suddenly realised that the fact that A×B did not equal B×A lay at the very heart of Heisenberg’s new approach and ‘provided the key to the whole mystery’.67
Dirac developed a mathematical theory that also led him to the formula pq–qp=(ih/2)I by distinguishing between what he called q-numbers and c-numbers, between those quantities that do not commute (AB does not equal BA) and those that do (AB=BA). Dirac showed that quantum mechanics differs from classical mechanics in that the variables, q and p, representing the position and momentum of a particle, do not commute with one another but obey the formula that he had found independently of Born, Jordan and Heisenberg. In May 1926, he received his PhD with the first-ever thesis on the subject of ‘quantum mechanics’. By then physicists were beginning to breathe a little easier after being confronted by matrix mechanics, which was difficult to use and impossible to visualise, even though it generated the right answers.
‘The Heisenberg-Born concepts leave us all breathless, and have made a deep impression on all theoretically orientated people’, Einstein wrote in March 1926. ‘Instead of dull resignation, there is now a singular tension in us sluggish people.’68 They were roused out of their stupor by an Austrian physicist who found time while conducting an affair to produce an entirely different version of quantum mechanics that avoided what Einstein called Heisenberg’s ‘veritable calculation by magic’.69
Chapter 9
‘A LATE EROTIC OUTBURST’
‘I do not even know what a matrix is’, Heisenberg had lamented when told of the origins of the strange multiplication rule that lay at the heart of his new physics. It was a reaction widely shared among physicists when they were presented with his matrix mechanics. Within a matter of months, however, Erwin Schrödinger offered them an alternative that they eagerly embraced. His friend, the great German mathematician Hermann Weyl, later described Schrödinger’s astonishing achievement as the product of ‘a late erotic outburst’.1 A serial womaniser, the 38-year-old Austrian discovered wave mechanics while enjoying a secret tryst during Christmas 1925 at the Swiss ski resort of Arosa. Later, after fleeing Nazi Germany, he first scandalised Oxford and then Dublin when he set up home with his wife and yet another mistress under the same roof.
‘His private life seemed strange to bourgeois people like ourselves’, Born wrote some years after Schrödinger’s death in 1961. ‘But all this does not matter. He was a most lovable person, independent, amusing, temperamental, kind and generous, and he had a most perfect and efficient brain.’2
Erwin Rudolf Josef Alexander Schrödinger was born in Vienna on 12 August 1887. His mother wanted to name him Wolfgang, after Goethe, but allowed her husband to honour an older brother of his who had died in childhood. This brother was the reason why Schrödinger’s father inherited the thriving family business manufacturing linoleum and oilcloth, ending his hopes of being a scientist after studying chemistry at Vienna University. Schrödinger knew that the comfortable and carefree life he enjoyed before the First World War was possible only because his father had sacrificed his personal desires on the altar of duty.
Even before he could read or write, Schrödinger kept a record of the day’s activities by dictating it to a willing adult. Precocious, he was educated at home by private tutors until the age of eleven when he began attending the Akademisches Gymnasium. Almost from the very first day until he left eight years later, Schrödinger excelled at the school. He was always first in his class without appearing to make much of an effort. A classmate recalled that ‘especially in physics and mathematics, Schrödinger had a gift for understanding that allowed him, without any homework, immediately and directly to comprehend all the material during the class hours and to apply it’.3 In truth, he was a dedicated student who worked hard in the privacy of his own study at home.
Schrödinger, like Einstein, had an intense dislike of rote learning and being forced to memorise useless facts. Nevertheless, he enjoyed the strict logic that underpinned the grammar of Greek and Latin. With a maternal grandmother who was English, he began learning the language early and spoke it almost as fluently as German. Later he learnt French and Spanish and was able to lecture in these languages whenever the occasion demanded. Well versed in literature and philosophy, he also loved the theatre, poetry and art. Schrödinger was just the sort of person to leave Werner Heisenberg feeling inadequate. Paul Dirac, when asked once if he played an instrument, replied that he did not know. He had never tried. Nor had Schrödinger, who shared his father’s dislike of music.
After graduating from the Gymnasium in 1906, Schrödinger looked forward to studying physics at Vienna University under Ludwig Boltzmann. Tragically, the legendary theoretician committed suicide weeks before Schrödinger started his course. With his grey-blue eyes and shock of swept-back hair, Schrödinger made quite an impression despite being only 5ft 6in. Having shown himself to be an exceptional student at the Gymnasium, much was now expected from him. He did not disappoint, coming top of the class in one exam after another. Surprisingly, given his interest in theoretical physics, Schrödinger gained his doctorate in May 1910 with a dissertation entitled ‘On the conduction of electricity on the surface of insulators in moist air’. It was an experimental investigation, showing that Schrödinger was, unlike Pauli and Heisenberg, perfectly at ease in the laboratory. Twenty-three-year-old Dr Schrödinger had a summer of freedom before reporting for military service on 1 October 1910.
All able-bodied young men in Austria-Hungary were required to do three years of military service. But as a university graduate he was able to choose a year’s officer training, leading to a commission in the reserve ranks. When he returned to civilian life in 1911, Schrödinger secured a position as an assistant to the professor of experimental physics at his old university. He knew he was not cut out to be an experimenter, but never regretted the experience. ‘I belong to those theoreticians who know by direct observation what it means to make a measurement’, he later wrote.4 ‘Methinks it were better if there were more of them.’
In January 1914, Schrödinger, aged 26, became a privatdozent. Like everywhere else, opportunities in theoretical physics in Austria were few. The road to the professorship he desired seemed a long and difficult one. So he toyed with the idea of abandoning physics. Then in August that year the First World War began and he was called up to fight. He had luck on his side from the very beginning. As an artillery officer, he served in fortified positions high on the Italian front. The only real danger he faced during his various postings was boredom. Then he began receiving books and scientific journals that helped to relieve the tedium. ‘Is this a life: to sleep, to eat, and to play cards?’ he wrote in his diary before the first consignment arrived.5 Philosophy and physics were the only things that kept Schrödinger from total despair: ‘I no longer ask when will the war be over? But: will it be over?’6
Relief came when he was transferred back to Vienna in the spring of 1917 to teach physics at the university and meteorology at an anti-aircraft school. Schrödinger ended the war, as he wrote later, ‘without getting wounded and without illness and with little distinction’.7 As for most others, the early post-war years were difficult for Schrödinger and his parents, with the family business ruined. As the Habsburg Empire fell apart, the situation was made worse as the victorious allies maintained a blockade that cut off food supplies. As thousands starved and froze during the winter of 1918–19 in Vienna, with little money to buy food on the black market, the Schrödingers were often forced to eat at a local soup kitchen. Things began to improv
e slowly after March 1919 when the blockade was lifted and the emperor went into exile. Salvation for Schrödinger arrived early the following year with the offer of a job at the University of Jena. The salary was just enough for him to marry 23-year-old Annemarie Bertel.
Arriving in Jena in April, the couple stayed just six months before Schrödinger was appointed to an extraordinary professorship in October at the Technische Hochschule in Stuttgart. The money was better, and after the experiences of the past few years that mattered to him. By spring 1921 the universities of Kiel, Hamburg, Breslau and Vienna were all looking to appoint theoretical physicists. Schrödinger, who had by then earned a solid reputation, was being seriously considered by all of them. He accepted the offer of a professorship at Breslau.
At the age of 34, Schrödinger might have achieved the ambition of every academic; however, in Breslau he had the title but not the salary to go with it, and he left when the University of Zurich came calling. Not long after arriving in Switzerland in October 1921, Schrödinger was diagnosed with bronchitis and possibly tuberculosis. Negotiations surrounding his future, and the deaths of his parents during the previous two years, had taken their toll. ‘I was actually so kaput that I could no longer get any sensible ideas’, he later told Wolfgang Pauli.8 On doctor’s orders, Schrödinger went to a sanatorium in Arosa. It was in this high-altitude Alpine resort not far from Davos that he spent the next nine months recuperating. He was not idle during this time, but found the energy and enthusiasm to publish several papers.
