Quantum: Einstein, Bohr and the Great Debate About the Nature of Reality

by Manjit Kumar

  In 1892, improved equipment appeared to show that the red alpha and blue gamma Balmer lines of the hydrogen spectrum were not single lines at all, but were each split in two. For more than twenty years, it remained an open question whether these lines were ‘true doublets’ or not. Bohr thought not. It was at the beginning of 1915 that he changed his mind as new experiments revealed that the red, blue and violet Balmer lines were all doublets. Using his atomic model, Bohr could not explain this ‘fine structure’, as the splitting of the lines was called. As he settled into his new role as a professor in Copenhagen, Bohr found a batch of papers waiting for him from a German who had solved the problem by modifying his atom.

  Arnold Sommerfeld was a 48-year-old distinguished professor of theoretical physics at Munich University. Over the years, some of the most brilliant young physicists and students would work under his watchful eye as he turned Munich into a thriving centre of theoretical physics. Like Bohr, he loved skiing and would invite students and colleagues to his house in the Bavarian Alps to ski and talk physics. ‘But let me assure you that if I were in Munich and had the time, I would sit in on your lectures in order to perfect my knowledge of mathematical physics’, Einstein had written to Sommerfeld in 1908 while still at the Patent Office.48 It was some compliment coming from a man described as a ‘lazy dog’ by his maths professor in Zurich.

  To simplify his model, Bohr had confined electrons to move only in circular orbits around the nucleus. Sommerfeld decided to lift this restriction, allowing electrons to move in elliptical orbits, like the planets in their journey around the sun. He knew that, mathematically speaking, circles were just a special class of ellipse, therefore circular electron orbits were only a subset of all possible quantised elliptical orbits. The quantum number n in the Bohr model specified a stationary state, a permitted circular electron orbit, and the corresponding energy level. The value of n also determined the radius of a given circular orbit. However, two numbers are required to encode the shape of an ellipse. Sommerfeld therefore introduced k, the ‘orbital’ quantum number, to quantise the shape of an elliptical orbit. Of all the possible shapes of an elliptical orbit, k determined those that were allowed for a given value of n.

  In Sommerfeld’s modified model, the principal quantum number n determined the values that k could have.49 If n=1, then k=1; when n=2, k=1 and 2; when n=3, k=1, 2 and 3. For a given n, k is equal to every whole number from 1 up to and including the value of n. When n=k, the orbit is always circular. However, if k is less than n, then the orbit is elliptical. For example, when n=1 and k=1, the orbit is circular with a radius r, called the Bohr radius. When n=2 and k=1, the orbit is elliptical; but n=2 and k=2 is a circular orbit with a radius 4r. Thus, when the hydrogen atom is in the n=2 quantum state, its single electron can be in either the k=1 or k=2 orbits. In the n=3 state, the electron can occupy any one of three orbits: n=3 and k=1, elliptical; n=3, k=2, elliptical; n=3 and k=3, circular. Whereas in Bohr’s model n=3 was just one circular orbit, in Sommerfeld’s modified quantum atom there were three permitted orbits. These extra stationary states could explain the splitting of the spectral lines of the Balmer series.

  Figure 8: Electron orbits for n=3 and k=1, 2, 3 in the Bohr-Sommerfeld model of the hydrogen atom

  To account for the splitting of the spectral lines, Sommerfeld turned to Einstein’s theory of relativity. Like a comet in orbit about the sun, as an electron in an elliptical orbit heads towards the nucleus its speed increases. Unlike a comet, the speed of the electron is great enough for its mass to increase as predicted by relativity. This relativistic mass increase gives rise to a very small energy change. The n=2 states, the two orbits, k=1 and k=2, have different energies because k=1 is elliptical and k=2 circular. This minor energy difference leads to two energy levels that yield two spectral lines where only one was predicted by Bohr’s model. However, the Bohr-Sommerfeld quantum atom was still unable to explain two other phenomena.

  In 1897 the Dutch physicist Pieter Zeeman discovered that in a magnetic field, a single spectral line split into a number of separate lines or components. This was called the Zeeman effect, and once the magnetic field was switched off, the splitting disappeared. Then in 1913 the German physicist Johannes Stark found that a single spectral line splits up into several lines when atoms are placed in an electric field.50 Rutherford contacted Bohr as Stark published his findings: ‘I think it is rather up to you at the present time to write something on the Zeeman and electric effects, if it is possible to reconcile them with your theory.’51

  Rutherford was not the first to ask. Soon after the publication of Part I of his trilogy, Bohr had received a letter of congratulation from Sommerfeld. ‘Will you also apply your atomic model to the Zeeman effect?’ he asked. ‘I want to tackle this.’52 Bohr was unable to explain it, but Sommerfeld did. His solution was ingenious. Earlier he had opted for elliptical orbits and thereby increased the number of possible quantised orbits that an electron could occupy when an atom was in a given energy state, such as n=2. Bohr and Sommerfeld had both pictured orbits, whether circular or elliptical, as lying in a plane. As he tried to account for the Zeeman effect, Sommerfeld realised that the orientation of an orbit was the vital missing component. In a magnetic field, an electron can select from more permitted orbits pointing in various directions with respect to the field. Sommerfeld introduced what he called the ‘magnetic’ quantum number m to quantise the orientation of those orbits. For a given principal quantum number n, m can only have values that range from –n to n.53 If n=2, then m has the values: –2, –1, 0, 1, 2.

  ‘I do not believe ever to have read anything with more joy than your beautiful work’, Bohr wrote to Sommerfeld in March 1916. The orientation of electron orbits, or ‘space quantisation’ as it became known, was experimentally confirmed five years later in 1921. It made available extra energy states, now labelled by the three quantum numbers n, k and m, which an electron could occupy in the presence of an external magnetic field, leading to the Zeeman effect.

  Necessity being the mother of invention, Sommerfeld had been forced to introduce his two new quantum numbers k and m to explain facts revealed by experiments. Leaning heavily on the work of Sommerfeld, others explained the Stark effect as resulting from the changes in the spacing between energy levels due to the presence of an electric field. Although there were still weaknesses, such as the inability to reproduce the relative intensity of the spectral lines, the successes of the Bohr-Sommerfeld atom further enhanced Bohr’s reputation and earned him an institute of his own in Copenhagen. He was on his way to becoming, as Sommerfeld called him later, ‘the director of atomic physics’ through his work and the inspiration he gave others.54

  It was a compliment that would have pleased Bohr, who had always wanted to replicate the way in which Rutherford had run his laboratory, and the spirit he had succeeded in creating among all those who worked there. Bohr had learnt more than just physics from his mentor. He saw how Rutherford was able to galvanise a group of young physicists into producing their best. In 1917 Bohr set out to replicate what he had been fortunate enough to experience in Manchester. He approached the authorities in Copenhagen about establishing an institute for theoretical physics at the university. The institute was approved, as friends raised the money necessary for buildings and land. Construction began the following year, soon after the end of the war, at a site on the edge of a beautiful park not far from the city centre.

  Work had only just begun when a letter arrived that unsettled Bohr. It was from Rutherford, who was offering him a permanent professorship in theoretical physics back in Manchester. ‘I think the two of us could try and make physics boom’, wrote Rutherford.55 It was tempting, but Bohr could not leave Denmark just as he was about to be given everything that he wanted. Maybe if he had gone, Rutherford would not have left Manchester in 1919 to take over from J.J. Thomson as the director of the Cavendish Laboratory at Cambridge.

  Always known as the Bohr Institute, the Universitetets Inst
itut for Teoretisk Fysik was formally opened on 3 March 1921.56 The Bohrs had already moved into the seven-room flat on the first floor with their growing family. Following the upheavals of war and the hardship of the years that followed in its wake, the institute was soon the creative haven Bohr hoped it would be. It quickly became a magnet for many of the world’s brightest physicists, but the most talented of them all would always remain an outsider.

  Chapter 5

  WHEN EINSTEIN MET BOHR

  ‘Those are the madmen who do not occupy themselves with quantum theory’, Einstein told a colleague as they looked out of the window of his office in the Institute of Theoretical Physics at the German University in Prague.1 After his arrival from Zurich in April 1911, he had been puzzled as to why only women used the grounds in the mornings and only men in the afternoons. As he struggled with his own demon he discovered that the beautiful garden next door belonged to a lunatic asylum. Einstein was finding it difficult to live with the quantum and the dual nature of light. ‘I wish to assure you in advance that I am not the orthodox light-quantizer for whom you take me’, he told Hendrik Lorentz.2 It was a faulty impression that arose, he claimed, ‘from my imprecise way of expressing myself in my papers’.3 Soon he gave up even asking if ‘quanta really exist’.4 By the time he returned from the first Solvay conference in November 1911 on ‘The Theory of Radiation and the Quanta’, Einstein had decided that enough was enough and pushed the lunacy of the quantum to one side. Over the next four years, as Bohr and his atom took centre stage, Einstein effectively abandoned the quantum to concentrate on extending his theory of relativity to encompass gravity.

  Founded in the mid-fourteenth century, Prague University was divided in 1882 along lines of nationality and language into two separate universities, one Czech and the other German. It was a division that reflected a society where Czechs and Germans harboured a deep-seated suspicion and mistrust of each other. After the easy-going, tolerant atmosphere of Switzerland and the cosmopolitan mix of Zurich, Einstein was ill at ease in spite of the full professorship and the salary that enabled him to live in some comfort. It provided just a quantum of solace against the creeping sense of isolation.

  By the end of 1911, as Bohr contemplated his move from Cambridge to Manchester, Einstein desperately wanted to return to Switzerland, and it was then that an old friend came to his rescue. Recently appointed as the dean of the mathematics and physics section of the Swiss Federal Technical University (ETH), Marcel Grossmann offered Einstein a professorship in Zurich at the renamed former Polytechnic. Although the job was his, there were formalities that Grossmann had to observe. High on the list was seeking the advice of eminent physicists about Einstein’s possible appointment. One of those asked was France’s premier theorist, Henri Poincaré, who described Einstein as ‘one of the most original minds’ he knew.5 The Frenchman admired the ease with which he adapted to new concepts, his ability to see beyond classical principles, and when ‘faced with a physics problem, [he] promptly envisages all possibilities’.6 Where Einstein had once failed to get a job as an assistant, in July 1912 he returned as a master physicist.

  It was inevitable that sooner rather than later Einstein would become a prime target for the men in Berlin. In July 1913 Max Planck and Walther Nernst boarded the train to Zurich. They knew that it would not be easy to persuade Einstein to return to a country he had left almost twenty years ago, but they were prepared to make him an offer he simply could not refuse.

  As Einstein met them off the train, he knew why Planck and Nernst had come, but not the details of the proposal they were about to make. Having just been elected a member of the prestigious Prussian Academy of Sciences, he was being offered one of its two salaried positions. This alone was a great honour, but the two emissaries of German science also offered a unique research professorship without any teaching duties and the directorship of the Kaiser Wilhelm Institute of Theoretical Physics once it was established.

  He needed time to mull over the unprecedented package of three jobs. Planck and Nernst went on a short sightseeing train ride as he considered whether or not to accept. Einstein told them they would have his answer when they returned by the colour of the rose he carried. If red, he would go to Berlin; if white, he would stay in Zurich. As they got off the train, Planck and Nernst knew they had got their man when they saw Einstein clutching a red rose.

  Part of the lure of Berlin for Einstein was the freedom to ‘give myself over completely to rumination’ with no obligations to teach.7 But with it came the pressure of having to deliver the sort of physics that made him the hottest property in science. ‘The Berliners are speculating with me as with a prize-winning laying hen,’ he told a colleague after his farewell dinner, ‘but I don’t know if I can still lay eggs.’8 After celebrating his 35th birthday in Zurich, Einstein moved to Berlin at the end of March 1914. Whatever reservations he might have had about returning to Germany, he was soon enthusing: ‘Intellectual stimulation abounds here, there is just too much of it.’9 The likes of Planck, Nernst and Rubens were all within easy reach, but there was another reason why he found ‘odious’ Berlin exciting – his cousin Elsa Löwenthal.10

  Two years earlier, in March 1912, Einstein had begun an affair with the 36-year-old divorcee with two young daughters – Ilse, aged thirteen, and Margot, eleven. ‘I treat my wife like an employee whom I cannot fire’, he told Elsa.11 Once in Berlin, Einstein would often disappear for days without a word of explanation. Soon he moved out of the family home altogether and drew up a remarkable list of conditions under which he was willing to return. If Mileva accepted his terms she would indeed become an employee, and one her husband was determined to fire.

  Einstein demanded: ‘1. that my clothes and laundry are kept in good order and repair; 2. that I receive my three meals regularly in my room; 3. that my bedroom and my office are always kept neat, in particular, that the desk is available to me alone.’ Further, she was to ‘renounce all personal relations’ and refrain from criticising him ‘either in word or deed in front of my children’. Finally he insisted that Mileva adhere to ‘the following points: 1. You are neither to expect intimacy from me nor reproach me in any way. 2. You must desist immediately from addressing me if I request it. 3. You must leave my bedroom or office immediately without protest if I so request.’12

  Mileva agreed to his demands and Einstein returned. But it could not last. At the end of July, after just three months in Berlin, Mileva and the boys went back to Zurich. As he stood on the platform waving goodbye, Einstein wept, if not for Mileva and the memories of what had been, then for his two departing sons. But within a matter of weeks he was happily enjoying living alone ‘in my large apartment in undiminished tranquillity’.13 It was a tranquillity that few would enjoy as Europe descended into war.

  ‘One day the great European war will come out of some damned foolish thing in the Balkans’, Bismarck was once reported as saying.14 That day was Sunday, 28 June 1914, and it was the assassination in Sarajevo of Archduke Franz Ferdinand, the heir to the crowns of Austria and Hungary. Austria, supported by Germany, declared war on Serbia. Germany declared war on Serbia’s ally Russia on 1 August and on France two days later. Britain, who guaranteed Belgian independence, declared war on Germany on 4 August after it had violated Belgium’s neutrality.15 ‘Europe in its madness has now embarked on something incredibly preposterous’, Einstein wrote on 14 August to his friend Paul Ehrenfest.16

  While Einstein felt ‘only a mixture of pity and disgust’, Nernst at 50 volunteered as an ambulance driver.17 Planck, unable to contain his patriotism, declared: ‘It is a great feeling to be able to call oneself a German.’18 Believing that it was a glorious time to be alive, as rector of Berlin University, Planck sent his students to the trenches in the name of a ‘just war’. Einstein could hardly believe it when he discovered that Planck, Nernst, Röntgen and Wien were among the 93 luminaries who signed the Appeal to the Cultured World.

  This manifesto was published on 4 Octob
er 1914 in leading German newspapers and in others abroad, its signatories protesting against ‘the lies and defamations with which our enemies are trying to besmirch Germany’s pure cause in the hard life-and-death struggle forced upon it’.19 They asserted that Germany bore no responsibility for the war, had not violated Belgian neutrality, and had committed no atrocities. Germany was ‘a cultured nation to whom the legacy of Goethe, Beethoven and Kant is fully as sacred as its hearths and plots of land’.20

  Planck quickly regretted having signed, and in private began apologising to his friends among foreign scientists. Of all those that lent their names to the falsehoods and half-truths of the Manifesto of the Ninety-Three, as it became known, Einstein had expected better from Planck. Even the German chancellor had publicly admitted that Belgium’s neutral status had been violated: ‘The wrong that we are committing, we will endeavour to make good as soon as our military goal is reached.’21