Through Two Doors at Once: The Elegant Experiment That Captures the Enigma of Our Quantum Reality

by Ananthaswamy, Anil

  While nature’s discontinuity and discreteness at the smallest scales was becoming ever more obvious, there was the puzzling issue of its concomitant wave nature, emblematic of classical continuity. And, probably most disturbingly, there was the question of indeterminism. It was clear that there are natural phenomena that do not follow the clockwork determinacy of Newton’s classical world. Take, for example, radioactivity: nothing about the current state of a radioactive atom lets you predict exactly when it’ll emit a ray of radioactivity. The process is unpredictable, stochastic. This went against the tenets of the science of the time, according to which full knowledge of a system should let you predict with precision some future event involving that system. The microscopic world seemed to be operating with a different set of rules.

  But it wasn’t obvious what these rules were. What physics lacked was an overarching framework that brought these disparate elements together. All that changed during the mid to late 1920s, when in a few feverish years, brilliant minds forged not one but two frameworks for theorizing about the world at small scales. This effort would culminate in one of the most celebrated scientific conferences in history—the Fifth Solvay International Conference on Electrons and Photons held in October 1927 in Brussels, Belgium. The moment, captured in a now-iconic photograph taken by Belgian photographer Benjamin Couprie, shows all twenty-nine attendees, some standing in the back row still in their twenties and yet to become famous, some already so and seated in the front row, including Einstein, Planck, and Marie Curie, and almost everyone else in between who mattered to the emerging field of quantum physics. If they weren’t already Nobel Prize winners, many would go on to win—turning seventeen of the twenty-nine into Nobel laureates.

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  “The lakes” of Copenhagen are five reservoirs that stretch crescent-shaped not far from the city center. Walk along the northern end of these lakes, go past a stretch of shore lined with horse chestnut trees, down a couple of blocks along an alley named Irmingersgade, and you come up, quite suddenly, on an unassuming building: the Niels Bohr Institute. When it was founded in 1921 by Bohr, it was called the Institute of Theoretical Physics. Bohr had moved from Manchester to the University of Copenhagen, where he became a professor in 1916 at just thirty-one years of age. He then lobbied hard and got the funds to build an institute for theoretical physics. And for a few decades, the institute became a cauldron where great minds stewed over the evolving field of quantum physics, under Bohr’s deeply engaged gaze.

  One of these great minds was a young German physicist named Werner Heisenberg. Bohr first met Heisenberg at Göttingen, Germany, in June 1922. Bohr was there to talk about the current understanding of the model of the atom and the various outstanding problems yet to be solved. During the talk, Heisenberg, still a twenty-year-old student in his fourth semester, questioned Bohr with such clarity that a suitably impressed Bohr took Heisenberg for a walk afterward to discuss atomic theory. He also invited Heisenberg to Copenhagen, and it was there in 1924 that Heisenberg realized, after discussions with Bohr and others, that “ perhaps it would be possible one day, simply by clever guessing, to achieve the passage to a complete mathematical scheme of quantum mechanics.” The word mechanics refers to physics that can explain how something changes with time under the influence of forces.

  Heisenberg’s insight was prophetic. In the spring of 1925, suffering from severe hay fever, he decamped to Helgoland in the North Sea, a rocky island devoid of pollen. There, between long walks and contemplating Goethe’s West-östlicher Divan , he developed the early mathematics that would become the basis for modern quantum theory. Heisenberg recalled later, “ It was almost three o’clock in the morning before the final result of my computations lay before me . . . I could no longer doubt the mathematical consistency and coherence of the kind of quantum mechanics to which my calculations pointed. At first, I was deeply alarmed. I had the feeling that, through the surface of atomic phenomena, I was looking at a strangely beautiful interior, and felt almost giddy at the thought that I now had to probe this wealth of mathematical structures nature had so generously spread out before me. I was far too excited to sleep, and so, as a new day dawned, I made for the southern tip of the island, where I had been longing to climb a rock jutting out into the sea. I now did so without too much trouble, and waited for the sun to rise.”

  Heisenberg wrote up his work, showed it first to Wolfgang Pauli (another of the brilliant young minds) and then to Max Born (an equally brilliant but a more fatherly figure in his forties, with whom Heisenberg was doing his postdoctoral work). Born immediately realized the import of Heisenberg’s paper. “ I thought the whole day and could hardly sleep at night . . . In the morning I suddenly saw the light,” he would say.

  What Born realized was that the symbols Heisenberg was manipulating in his equations were mathematical objects called matrices, and there was an entire field of mathematics devoted to them, called matrix algebra. For example, Heisenberg had found that there was something strange about his symbols: when entity A was multiplied by entity B, it was not the same as B multiplied by A; the order of multiplication mattered. Real numbers don’t behave this way. But matrices do. A matrix is an array of elements. The array can be a single row, a single column, or a combination of rows and columns. Heisenberg had brilliantly intuited a way of representing the quantum world and asking questions about it using such symbols, while being unaware of matrix algebra.

  In a few frenetic months, Born, along with Heisenberg and Pascual Jordan, developed what’s now known as the matrix mechanics formulation of quantum physics. In England, Paul Dirac saw the light too when he encountered Heisenberg’s work, and he too, in a series of papers, independently added tremendous insight and mathematics to the formulation and developed the “Dirac notation” that’s still in use today.

  Most important, it was clear that the formalism worked. For example, the position of, say, an electron, is represented by a matrix. The position in this case is called an observable. The matrix then dictates all the possible positions in which the electron can be found, or observed. The formalism implicitly allows for the electron to be only in certain positions and not in others. And there is no sense of a continuous change from one position to another. Discreteness, or jumps from one state to another, is baked into matrix mechanics.

  In due course, physicists were able to use the formalism to calculate, for example, the energy levels of electrons in atoms, explain the radiation emitted by glowing bits of sodium or other metals, understand how such spectral emissions could be split into slightly different frequencies under the influence of a magnetic field, and better understand the hydrogen atom itself.

  But it wasn’t obvious why the formalism worked. What did these matrices map to, physically speaking? The elements of these matrices could be complex numbers (a complex number has a real part and an imaginary part; the imaginary part is a real number multiplied by the square root of -1 and is imaginary because √-1 doesn’t exist yet turns out to be incredibly useful in certain kinds of mathematics). How could the physical world be represented by things that could only be imagined? Were we at the very limit of human understanding? Was a clear understanding possible?

  Matrix mechanics does not allow physicists to think of electrons as having clear, fixed orbits, even if they are quantized. One can describe an electron’s quantum state using a set of numbers, carry out a whole lot of matrix manipulations to predict things like spectral emissions, but what you lose is the ability to visualize the electron’s orbit in the way that one can visualize, say, Earth’s orbit around the sun.

  Plus, the formalism deals in probabilities. If a particle is in state A and you measure to see if it’s state A, then, of course, the math says you’ll find the particle in state A with 100 percent certainty. The same goes for, say, state B. But matrix mechanics says that a particle can be in some intermediate state, where the state is x parts A and y parts B. Now, if you try and predict whether you’ll find the particle in
state A or state B, 100 percent certainty about reality is no longer possible.

  Matrix mechanics lets you calculate only the probabilities of outcomes of measurements. So, for an electron whose state is x parts A and y parts B, say you want to see if the particle is in state A. The math says that the probability of finding the particle in state A is x 2 . Similarly, if you check to see if the particle is in state B, the probability you’ll find it in state B is y 2 . (The terminology gets tweaked a little bit when you allow x and y to be complex numbers, but for now, it’s easy to see what rules x and y have to follow: the probabilities have to add up to one, so x 2 + y 2 should equal 1.)

  The fact that we are now dealing in probabilities is not, presumably, because we do not know enough about the particle. Matrix mechanics says you have all the information you can possibly have. Yet, if you take a million identically prepared particles in the same state (the same combination of states A and B) and perform a million identical measurements, then, on average, x 2 number of times you will find the particle in state A, y 2 of the time you’ll find it in state B. But you can never predict the answer you’ll get for any single particle. You can only talk statistically. Nature, it seems, is not deterministic in the quantum realm.

  Recall that something similar happens with the double slit. We cannot predict where exactly a single photon will land on the screen—we can only assign probabilities for where it might go.

  Soon after these phenomenal developments, an Austrian physicist named Erwin Schrödinger, whose status as a founding member of quantum physics was yet to be established, expressed his dismay at, even distaste for, Heisenberg’s matrix mechanics. He said he was “ discouraged, if not repelled” by what he saw as “very difficult methods of transcendental algebra, defying any visualization.”

  The battle lines were being drawn. Wave versus particle, continuous versus discrete, old versus new. Schrödinger’s distaste led him to develop a formidable old-school alternative to the upstart, matrix mechanics—one that seemed to restore faith in the classical way of thinking about nature.

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  When Louis de Broglie wrote his 1924 thesis on the wave-particle duality of matter, Schrödinger was already a professor of theoretical physics at the University of Zurich, and compared to the young geniuses elsewhere in Europe, he was practically an old man, approaching forty. But for years Schrödinger had been delving into the same questions that had been tormenting everyone. Schrödinger learned of de Broglie’s work when he read a reference to it in a paper by Einstein. Thinking of matter as waves made sense to Schrödinger’s classically intuitive mind, and he acknowledged as much in a letter to Einstein, dated November 3, 1925: “ A few days ago I read with the greatest interest the ingenious thesis of Louis de Broglie, which I finally got hold of.” Schrödinger wanted to describe the motion of electrons around the nucleus by thinking of them as waves. Instead of Heisenberg’s matrix mechanics, Schrödinger wanted wave mechanics for electrons.

  If Heisenberg’s solo sojourn at Helgoland has become quantum physics lore, so has Schrödinger’s own burst of creativity in isolation—well, almost in isolation. A New York Times book review captures this period in Schrödinger’s life: “ A few days before Christmas, 1925, Schrödinger . . . took off for a two-and-a-half-week vacation at a villa in the Swiss Alpine town of Arosa. Leaving his wife in Zurich, he took along de Broglie’s thesis, an old Viennese girlfriend (whose identity remains a mystery) and two pearls. Placing a pearl in each ear to screen out any distracting noise, and the woman in bed for inspiration, Schrödinger set to work on wave mechanics. When he and the mystery lady emerged from the rigors of their holiday on Jan 9, 1926, the great discovery was firmly in hand.”

  Within weeks, Schrödinger published his first paper in the Annalen der Physik . Three more papers followed in quick succession, and Schrödinger turned the world of Heisenberg and Born upside down. Suddenly, physicists had an intuitive way of understanding what was ostensibly happening to an electron in a hydrogen atom. Schrödinger had come up with his now eponymous wave equation, which treated the electron as a wave, and showed how this wave would change over time. It was wave mechanics. It was almost classical physics, except there were curious and consequential differences.

  In classical physics, solving a wave equation for, say, a sound wave can give you the pressure of the sound wave at a certain point in space and time. Solving Schrödinger’s wave equation gives you what’s called a wavefunction. This wavefunction, denoted by the Greek letter ψ (psi, pronounced “sigh”), is something quite strange. It represents the quantum state of the particle, but the quantum state is not a single number or quantity that reveals, for example, that the electron is at this position at this time and at that position at another time. Rather, ψ is itself an undulating wave that has, at any given moment in time, different values at different positions. Even more weirdly, these values are not real numbers; rather, they can be complex numbers with imaginary parts. So the wavefunction at any instant in time is not localized in a region of space; rather, it is spread out, it’s everywhere, and it has imaginary components. The Schrödinger equation, then, allows you to calculate how the state of the quantum system, ψ, changes with time.

  Schrödinger thought the wavefunction provided a way to visualize what was actually happening to electrons or other inhabitants of the quantum world. But this view was challenged within months of Schrödinger’s papers being published, when Max Born realized that Schrödinger was wrong about the meaning of the wavefunction.

  In a couple of seminal papers published in the summer of 1926, Born showed that when electrons collide and scatter, the resulting wavefunction that represents the state of the electrons only encodes the probability of finding the electrons in one state or another. It took Born a couple of tries to get it right, but he showed that if ψ is the wavefunction of an electron, and if it can be written, for example, in terms of two different possible states of the electron, ψA and ψB, such that ψ = x.ψA + y .ψB, then all you can do is calculate the probability that you’ll find the electron in state A or state B when you do a measurement. (The probability of finding the electron in state A is given by the square of the amplitude of x , also called the square of the modulus of x, denoted as |x |2 , and the probability of finding it in state B is given by |y |2 . If x is, say, a real number, the modulus |x | is simply its absolute value: if it’s positive to start with, it remains positive; if it’s negative, then we multiply it by -1; squaring it gives us a positive number. Of course, x and y can be complex numbers, and calculating the modulus of a complex number is a bit more complicated, but in essence, when you take the modulus-squared of a complex number, you again get a number that is positive and real, without any imaginary parts.)

  Born had, it seemed at first blush, cast doubt on causality, the underpinning of deterministic classical physics, which says any given effect has a cause. Given an initial state of an electron, standard quantum mechanics cannot definitively say what the electron’s next state will be. One can only calculate the probability of an electron transitioning to some new state, using what came to be called the Born rule. An element of randomness, or stochasticity, became an integral part of the laws of nature. As Born put it, “ The motion of particles follows probability laws but the probability itself propagates according to the law of causality.”

  And there it was—one interpretation of the wavefunction. It’s a probability wave. Schrödinger’s equation lets you calculate how this wave changes with time deterministically, but as it evolves and takes on different shapes, what’s changing are the probabilities of finding the quantum system in various states.

  If this sounds like the probabilities of matrix mechanics, you are not mistaken. Schrödinger himself, in another stroke of insight, showed that wave mechanics and matrix mechanics are mathematically equivalent (in hindsight, it was a mathematician called John von Neumann who would really prove the equivalence a few years later). Rather than see this as a validation of matrix
mechanics, Schrödinger claimed victory for wave mechanics, considering his approach to be correct and arguing that anything that was calculated using matrix mechanics could be calculated using wave mechanics. The advantage of wave mechanics, in Schrödinger’s opinion, was the idea that nature even at the smallest scales was continuous, not discrete. There were no quantum jumps.

  Heisenberg, meanwhile, wasn’t enamored of Schrödinger’s ideas. He wrote to Pauli, complaining that he found them “abominable,” calling it “Mist ” (which is German for rubbish, manure, dung, or droppings). Pauli himself alluded to Züricher Lokalaberglauben (local Zurich superstitions, an allusion to the city where Schrödinger worked). Schrödinger, unsurprisingly, wasn’t pleased by Pauli’s assertions. Pauli, in turn, tried to appease Schrödinger by saying, “ Don’t take it as a personal unfriendliness to you but look on the expression as my objective conviction that quantum phenomena naturally display aspects that cannot be expressed by the concepts of continuum physics. But don’t think that this conviction makes life easy for me. I have already tormented myself because of it and will have to do so even more.”

  The torment these titans felt over the nature of reality continued when Schrödinger visited Copenhagen and met Bohr for the very first time.

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  Decades after Schrödinger’s visit to Copenhagen in September 1926, Heisenberg would recount the intensity of their meetings: “ The discussion between Bohr and Schrödinger began at the railway station in Copenhagen and was carried on every day from early morning till late at night. Schrödinger lived at Bohr’s house so that even external circumstances allowed scarcely any interruptions of the talks. And although Bohr as a rule was especially kind and considerate in relations with people, he appeared to me now like a relentless fanatic, who was not prepared to concede a single point to his interlocutor or to allow him the slightest lack of precision. It will scarcely be possible to reproduce how passionately the discussion was carried on from both sides.”