Heisenberg on Helgoland

The sun was setting on a cloudless sky, the gulls screeching in the distance. The air was bracing and clear. Land rose from the blue ocean, a vague apparition on the horizon.

He breathed the elixir of pure evening air in and heaved a sigh of relief. This would help the godforsaken hay fever which had plagued him like a demon for the last four days. It had necessitated a trip away from the mainland to this tiny outcrop of flaming red rock out in the North Sea. Here he could be free not just of the hay fever but of his mentor, Niels Bohr.

For the last several months, Bohr had followed him like a shadow, an affliction that seemed almost as bad as the hay fever. It had all started about a year earlier, but really, it started when he was a child. His father, an erudite scholar but unsparing disciplinarian, made his brother and him compete mercilessly with each other. Even now he was not on the best terms with his brother, but the cutthroat competition produced at least one happy outcome: a passion for mathematics and physics that continued to provide him with intense pleasure.

He remembered those war torn years when Germany seemed to be on the brink of collapse, when one revolution after another threatened to tear apart the fabric of society. Physics was the one refuge. It sustained him then, and it promised to sustain him now.

If only he could understand what Bohr wanted. Bohr was not his first mentor. That place of pride belonged to Arnold Sommerfeld in Munich. Sommerfeld, the man with the impeccably waxed mustache who his friend Pauli called a Hussar officer. Sommerfeld, who would immerse his students not only in the latest physics but in his own home, where discussions went on late into the night. Discussions in which physics, politics and philosophy co-existed. His own father was often distant; Sommerfeld was the father figure in his life. It was also in Sommerfeld’s classes that he met his first real friend – Wolfgang Pauli. Pauli was still having trouble attending classes in the morning when there were all those clubs and parties to frequent at night. He always enjoyed long discussions with Pauli, the ones during which his friend often complimented him by telling him he was not completely stupid. It was Pauli who had steered him away from relativity and toward the most exciting new field in physics – quantum theory.

Quantum theory was the brainchild of several people, but Bohr was its godfather, the man who everyone looked up to. It was Bohr who had first applied the notion of discontinuity to the interior of the atom. It was Bohr who had explained the behavior of the simplest of atoms, hydrogen. But much more than that, it was Bohr who had an almost demonic obsession both with the truths of quantum theory and the dissemination of its central tenets to young physicists like him.

Darkness was approaching as he descended the rock and started walking back to his inn. He smiled as he remembered his first meeting with Bohr. After the war, Germany was the world’s most hated nation. Nobody wanted to deal with her. The Versailles treaty had imposed draconian measures on her already devastated economy. How could they do this? Bohr was one of those very few who had extended a statesmanlike hand toward his country. War is war, Bohr had said, but science is science. Its purity cannot be violated by the failings of humanity. The University of Göttingen had invited Bohr to inaugurate a new scientific relationship between Germany and the rest of the world. The day was as clear in his memory as the air around him. The smell of roses wafting through the windows, the audience standing or sitting on the windowsills, the medieval churches chiming in the distance.

Bohr was explaining one of the finer points related to spectroscopy which his quantum theory explained. But there was clearly a mathematical error. Had anyone else seen it? The error was an elementary one, and it did not seem worthy of Bohr. Later as he found out, Bohr was a competent but not particularly noteworthy mathematician. Physical and philosophical intuition was his forte. The mathematics he left to lesser souls, to young men who he called scientific assistants. At Göttingen he pointed out the mistake from the back and offered some other comments. He was all of twenty. Bohr graciously admitted the mistake. After the talk, when he was leaving, Bohr caught up with him. Walk with me, said Bohr. Walk, and talk. It was what Bohr did best.

They climbed up the hill near the university, then discussed the problems of atomic physics in a nearby cafe. He felt he could pledge his soul to Bohr. After Munich he had been tempted to go to Copenhagen right away, but Sommerfeld had cautioned him otherwise. Bohr was an excellent physicist, Sommerfeld had said, but at this stage in his career he would be much better served by a more rigorous and mathematical immersion in atomic physics. The best man to mentor him in this regard was Max Born in Göttingen. Born was hesitant, sometimes too sensitive to perceived slights, often in undue awe of his own students, but there was no one else who combined physical insights with mathematical rigor the way he did. Born could acquaint him much better with the formal techniques; he could always spend time with the philosophical Niels. His friend Pauli had already served as Born’s assistant and had vouched for Born’s first-rate mentorship. However he had cautioned him about Born’s insistence on early morning meetings, an expectation that had been so hard for him to meet that Born had had to send a maid to wake him up.

The moonlight illuminated the path in front of him, but there were few other lights on the tiny island. This was what he liked best about it though. Very few people, very few lights, almost nobody to talk with, but plenty of opportunities for walking and swimming in the cool water. And the air, the air. Crystal clear and seemingly designed for clearing both his nasal passages and the cobwebs in his mind. His hay fever seemed almost gone already. He could read Goethe and think about physics as much as he wanted. When he arrived at the inn he greeted the innkeeper, who when he arrived four days ago, had seemed horrified at his swollen face. She had asked him if he had been in a brawl. Sadly, political brawls and beatings were not uncommon in Germany. After a light meal of sausages and potato dumplings, he retired to his room.

In Munich, for his doctoral dissertation, he had chosen an uncontroversial topic in fluid dynamics. The final exam had been a fiasco though, and he wrinkled his brow as he thought about it. One of the examiners, Wilhelm Wien, had asked him a question from elementary physics about the resolving power of a microscope. He had forgotten the formula and had gotten hopelessly entangled in trying to work it out. He was trying to solve problems at the forefront of quantum theory; why was he being asked to answer questions that were better suited to a second-rate undergraduate? Wien would not let up, however, and Sommerfeld finally had to step in, assuring the examiners that his student was certainly promising enough to be awarded his doctorate. He had still barely escaped with a passing grade. It still rankled.

He had packed his bags and gone straight to Göttingen from Munich. It was partly to start off on quantum theory right away, but also to escape the depressing pessimism that gripped German society. The past year had seen unprecedented inflation cripple his beloved country. At its height an American dollar had been worth a trillion marks. People were carrying entire carts full of money to trade for a load of bread or for some potatoes. They were using it as insulating wallpaper in their homes. Is this what his country really deserved? As he pondered the situation he felt a spring of resentment welling up inside him. If nothing else, he would show them that Germany was still not lacking in scientific talent.

After spending some time with Born and becoming familiar with the fundamental mathematical tools of atomic physics, he had finally made it to Copenhagen. The past few months there had been among the happiest of his life. Bohr had created an atmosphere whose spirit of camaraderie exceeded even Sommerfeld’s seminars. The days would be filled with deep scientific and philosophical discussions, long walks in the Faelledparken behind the institute and games of ping-pong. Evenings were spent in entertaining Bohr and his kind wife Margarethe with Beethoven and Schubert on the piano, which after physics had been his main passion. Even more than Sommerfeld Bohr had become a father figure to him. His avuncular nature, his obsession with quantum theory and his physical agility; all of these were impressive. He would take stairs two at a time, and it seemed nobody could beat him at ping-pong.

But he had also encountered aspects of Bohr’s personality that had not been apparent before. Bohr was very gentle in personal relations, but when it came to divining scientific truth he could be ferocious, unremittingly persistent, a fanatic without scruples. He had been arguing the validity of some rather well known facts of atomic physics, but Bohr’s relentless questioning of even the basic existence of the properties of electrons and photons - questioning that continued well into the night even after he had expressed his fatigue - had almost reduced him to tears. As if Bohr’s inquisition-style interrogation had not been enough, another hitherto unobserved particle had entered Bohr’s orbit since he last met him. His name was Hendrik Kramers. Kramers was Dutch, voluble, mathematically sophisticated, could speak four languages and could play both the piano and the cello. He had been struggling with Danish and English for some time and it was difficult not to be jealous of Kramers. A kind of sibling rivalry had developed between them, both vying for the attention of the father figure.

While he had been putting the finishing touches on his mundane dissertation on fluid dynamics, Bohr, Kramers, and a young American postdoctoral fellow named John Slater had created a compelling picture of electrons in the atom as a set of pendulum-like objects. The technical term for this was harmonic oscillators. The oscillators would vibrate with certain frequencies that would correspond to transitions of electrons between different states in the atoms. Bohr and Kramers were using these oscillators as convenient representations to picture what goes on inside an atom, but they were still concerned with the well-known basic properties of atoms like their positions and velocities. He had been asked to see what he could do with Bohr and Kramers’s model.

This was where the problems had started. He liked the idea of using oscillators to represent electrons. The oscillators expressed themselves in the form of a well-known mathematical device called a Fourier series. His time with Born had made him quite familiar with Fourier series. But when he had inserted formulas for the series into the basic equations of motion, single numbers had grotesquely multiplied into entire lists of numbers. Every time he got rid of certain numbers others would mushroom, like the heads of a Hydra. He had played algebraic games, filled tables upon tables with numerical legerdemain, had gotten not an inch closer to expressing any physical quantity. And then, suddenly, like a gale from the North Sea, he had been swept off his feet by the worst bout of hay fever he remembered. It kept him awake at night. It made him feel groggy during the day. It made the morass of numbers appear even bigger than what it was.

He had finally had enough. Time for resetting the mental gears, he had told himself. The little rocky outcrop with its very low pollen count had been a favored destination for sufferers. That’s where he would go, away from the stifling hay fever and the intellectual hothouse, to the ocean, mountains and clear air which he loved best. He had known this part of the country during expeditions with his youthful Pfadfinder classmates. There they had sung songs about the fatherland and had had fervent patriotic discussions about the spiritual and political revival of Germany. He felt at home there.

The light on the ceiling was flickering as he started thinking about oscillators, about frequencies, about electrons. How does one ever know what goes inside an atom? And that’s when it struck him. It seemed like a bolt out of the blue then, but later on he realized that it was part of a continuum of mental states, a flash of insight that only seemed discontinuous like the transitions of electrons. Once again, how does one ever know what goes on inside an atom? Nobody has seen an atom or electron; they are unobservable. And yet we know they are real because we observe their tangible effects. Unobservable entities have been part of science for a very long time. Nobody knew what went on inside the sun. But scientists – German scientists among them – had figured it out based on the frequencies of spectral lines that indicated the presence of certain elements. Spectroscopy had also been paramount in the development of atomic theory. Bohr himself had demonstrated the success of the theory by using it to explain spectral lines of hydrogen.

He took a step back, looked at the whole picture from a fresh viewpoint, saw the forest for the trees. What we see are spectral lines and nothing else but spectral lines. We do not see the electron’s position; we do not see its momentum. Position and momentum may have been the primary variables in classical physics, but that was because we could measure them. In case of atoms and electrons, all we see are the frequencies of the spectral lines. What we do not see we do not know. Then why pretend to use it? Why pretend to calculate it? The frequencies are the observables. Why not use them as the primary variables, with the positions and momenta as secondary quantities? He had always been a first-rate mathematician, but now he thought about the physics. It was a fundamental shift of a frame of reference, so memorably introduced by Einstein before. The problem was that representing the position and momenta as Fourier series and frequencies still led to a list of numbers rather than a single number obtained by multiplication. But here is where his physical intuition proved pivotal. One could know which numbers from the list to keep and which ones to discard based on whether they represented transitions between real energy states in atoms. That information was available and implicit in the frequency of the spectral lines. Nature could steady that tentative march of numbers.

It was finally time to use his strange calculus to calculate the energy of a real physical system. As his excitement mounted he kept on making mistakes and correcting them, but finally he had it. When he looked at it he was struck with joy and astonishment. Out of the dance of calculations emerged an answer for the energy of the system, but crucially, this energy could only exist in a restricted set of values. In one fell swoop he had rediscovered Max Planck’s original formulation of quantum theory without explicitly using Planck’s energy formula. An answer this correct must be true. An answer this elegant must be true.

It was almost three o’clock in the morning. The night outside seemed to deepen into a deep chasm. He had hardly talked to anyone during his four days on the island, and now it seemed that all that silence was culminating in a full-throated expression of revolutionary insight. The hand of nature and his own dexterous mind had cracked the puzzle in front of him, just as invisible writing is suddenly revealed by the application of the right chemical solution. But the sheer multiplicity of applications that he now foresaw was startling. At first he was deeply alarmed. He had the feeling that, through the surface of atomic phenomena he was looking at a strangely beautiful interior, and now had to probe this wealth of mathematical structures that nature had so generously spread before him.

But that could wait. He now knew that he had a general scheme of quantum theory that could be used to solve any number of old and new problems. Bohr would be pleased, although he would still insist on several modifications to his formulation when it was time to publish. And of course he would show it to his friend Pauli who would provide the most stringent test of the correctness of his theory.


His hay fever seemed to have disappeared. He felt strong again. There did not seem much point in trying to fall asleep at this very late hour. He put on his boots and set out. There was a distant rocky outcrop, the northernmost tip of the island that he had not explored yet. He walked in the predawn light. Not a gull cried around him, not a leaf seemed to tremble. An hour later he was at the base of the rock and scaled it without much effort. There he sat for a long time until he saw the first rays of the sun penetrate the darkness. Photons of light falling on his eyes, stimulating electron transitions in atoms of carbon, nitrogen and oxygen. And at that moment he was the sole human being on earth who knew how this was happening.

Note: This is my latest column for 3 Quarks Daily. It's a piece of historical fiction in which I imagine 24-year-old Werner Heisenberg inventing quantum mechanics on the small island of Helgoland in the North Sea. Heisenberg's formulation was not the easiest to use and was supplanted by Schrödinger's more familiar wave mechanics, but it inaugurated modern quantum theory and was by any reckoning one of the most important discoveries in the history of physics.