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§ Phase ii · Matter Waves · Chapter 04

Heisenberg's uncertainty

In June 1925, a 23-year-old physicist with hay fever fled to a treeless island in the North Sea and came back with a new mechanics in which quantities refused to commute. Two years later he wrote down a single short inequality that closed the door on the classical world for good. It said: you cannot know where a particle is and how fast it is going, both, beyond a certain limit set by Planck's constant. The limit is not in your meter. It is in the wave.

By the spring of 1925, Niels Bohr’s atom was in trouble. The simple picture of electrons running on circular orbits had given Balmer’s ladder, that one beautiful early victory, but it had stalled on everything else. Helium was a mess. Magnetic field splittings did not add up. The fluctuating intensities of spectral lines could not be derived from anything resembling an orbit. Bohr himself was speaking in cryptic phrases about a forthcoming “rational” theory. He had already started saying things like “perhaps electrons do not have orbits in the ordinary sense at all.” Nobody knew quite what to do with that. In Munich, in Göttingen, in Copenhagen, an entire generation of young theorists was hammering on the same wall, and most of them were under thirty.

One of them was a slight, blond, athletically inclined doctoral student of Arnold Sommerfeld’s named Werner Heisenberg. He had finished his thesis at twenty-one (on turbulence, of all things) and had moved to Göttingen to work as Max Born’s assistant. He was fast, quarrelsome, brilliant, and at that moment intensely miserable. It was June 1925 and his hay fever was so bad that he could not see across the lecture hall. Born sent him on leave. He took a train and then a steamer to Helgoland, a rocky island in the North Sea, more or less treeless and therefore mercifully pollen-free, and rented a room above the cliffs.

What Heisenberg had carried to Helgoland was a riddle. Bohr had been preaching for two years that the right way forward was to give up asking about quantities you cannot measure (the unobservable orbit of an electron around a nucleus) and to build the theory only out of quantities you can measure (the frequencies and intensities of the light it emits). Heisenberg, who took the slogan literally, had been trying to do this for an oscillating electron. He had a table of transition amplitudes, one number for every pair of stationary states, and he wanted to find an algebra for them that would reproduce the classical equations of motion in the right limit. On a sleepless night around June 7, perched on a rock above the surf, he found it. The trick was to multiply his tables row-by-column, the same operation that nineteenth-century mathematicians had called the multiplication of matrices, though Heisenberg, who had never taken a course in matrix algebra, did not yet know the name. The result had one strange feature: A times B did not in general equal B times A. He wrote later that he was so excited he could not sleep, and at dawn climbed a high rock to watch the sunrise.

Back in Göttingen, Born read Heisenberg’s draft, recognised matrices on sight (he had taken Hilbert’s course years earlier), and within weeks had recruited Jordan to formalise the new mechanics. The trio wrote what physicists still call the Dreimännerarbeit, the “three-man paper”, which laid out matrix mechanics as a complete theory of atomic motion. The central formal feature was the canonical commutation relation: position and momentum, written as matrices, obey x · p − p · x = iℏ, where i is the imaginary unit and ℏ = h / 2π is Planck’s constant divided by 2π. In classical physics every pair of variables commutes; the order in which you multiply them does not matter. In the new mechanics, the order mattered, and the difference was exactly ℏ. That little imaginary remainder was the fingerprint of quantisation. It was the rule of the game.

Within six months a second, apparently unrelated theory had appeared from a different direction. Erwin Schrödinger, working in Zurich and inspired by Louis de Broglie, had written down a wave equation that gave the hydrogen spectrum on the first try. His ψ-function smoothly threaded the atom, and his mathematics, partial differential equations with sines and cosines and exponentials, was the daily bread of every physicist of the previous century. Where Heisenberg’s algebra felt alien, Schrödinger’s felt like home. The two camps spent the spring of 1926 trading insults. Schrödinger called matrix mechanics “repugnant”; Heisenberg called wave mechanics “disgusting” and, to a colleague, “Schrödinger sentimentality.” Then in late spring Schrödinger, and independently the young John von Neumann, proved that the two formalisms are exactly the same theory in different languages. Both describe operators on the same Hilbert space. Heisenberg’s matrices are Schrödinger’s wave equation written in a particular basis. The drama collapsed into a footnote: there was only one quantum mechanics, and it could be written either way.

For a year after Helgoland, Heisenberg was still unhappy. The mathematics worked. The hydrogen levels came out. But what did x and p actually mean now, if they were matrices and not numbers? In the old picture, a particle had a position and a momentum at every instant of time. You could talk about its trajectory. In the new picture, position and momentum were operators on an abstract space of states; the very phrase “the particle’s position” no longer pointed at a definite number. Bohr, in Copenhagen, was pushing this confusion further. He had begun teaching that the wave and particle pictures of light, and now of electrons, were complementary aspects of one reality: you could choose which to display in a given experiment, but never both at once. Heisenberg, who had moved to Copenhagen as Bohr’s assistant, fought with Bohr through the winter of 1926-27. The arguments were so intense that, on one occasion, Heisenberg fled the institute to walk alone in Fælled Park, trying to think his way clear.

On one of those walks, late in February 1927, he had the idea that became the famous principle. He returned to a thought experiment he had used in his Habilitation lecture (the so-called gamma-ray microscope) and pressed it harder. Suppose you wanted to see exactly where an electron was. You would have to scatter light off it, the way you locate any small object. The light you used must have a wavelength λ at most equal to the precision Δx you needed; you cannot resolve detail finer than the wavelength you probe with. But by Einstein’s 1905 photon hypothesis, each photon carries momentum p_γ = h / λ, and when it bounces off the electron it transfers an unknown fraction of that momentum, an uncontrollable kick of order h / λ. Thus, after the measurement, the electron’s momentum is uncertain by at least Δp ≈ h / λ ≈ h / Δx, so the product Δx · Δp comes out of order Planck’s constant. He wrote the result in March 1927 in a paper called Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, “On the perceptual content of quantum-theoretical kinematics and mechanics,” and the inequality has been quoted ever since:

Δx · Δp  ≥  ℏ / 2

He sent the paper to Pauli on February 23 with a covering letter asking him to “be severe.” Pauli, who could be merciless, replied with rare warmth: “It begins to dawn in the quantum theory.” Bohr returned from a skiing trip in Norway and was less pleased. He had been working out his own complementarity principle in parallel, and he thought Heisenberg’s emphasis on the gamma-ray microscope, on the disturbance caused by a measurement, was misleading. The thought experiment was a useful illustration, Bohr agreed, but the real reason for the inequality was not that the meter was clumsy. The real reason was that the electron was a wave, and a wave with a sharply defined wavelength must spread over many wavelengths in space. The mathematics was simply that of the Fourier transform: the narrower a wavepacket in position, the wider it must be in wavelength, and therefore (by de Broglie’s p = h / λ) in momentum. Heisenberg’s microscope was a story to illustrate a fact about waves. The fact about waves was the principle.

The mathematical statement is sharp. For any quantum state ψ, define Δx as the standard deviation of position when many copies of the state are measured, and Δp as the standard deviation of momentum when many other copies are measured (you cannot measure both on the same copy; you have to prepare many identical systems). Then, for any ψ whatsoever,

Δx · Δp  ≥  ℏ / 2

The proof, which Earle Kennard worked out a few months after Heisenberg’s paper, takes about ten lines and uses only the Cauchy-Schwarz inequality applied to the commutator [x, p] = iℏ. The minimum, Δx · Δp = ℏ/2, is achieved by exactly one shape of wavefunction, a Gaussian. Every Gaussian wavepacket sits on the boundary; every other wavepacket sits above. There is no wavefunction in the universe that sits below the curve Δx · Δp = ℏ/2. The space below that curve is empty.

⟨φ | φ⟩  =  (ΔA)² + λ² (ΔB)² + i λ ⟨ψ | [Â, B̂] | ψ⟩  ≥  0

(ΔA)² (ΔB)²  ≥  (1/4) | ⟨ [A, B] ⟩ |²

Δx · Δp  ≥  ℏ / 2

By the end of 1927, the European physics establishment had absorbed the new mechanics. At the Fifth Solvay Conference that October (the famous photograph with Einstein in the front row), Bohr and Einstein began their long debate over whether quantum mechanics was a complete description of nature. Einstein kept inventing thought experiments to break the uncertainty relation; Bohr kept finding the flaw. The most celebrated round came two years later at Solvay 1930, when Einstein proposed a clock-in-a-box scheme to violate ΔE · Δt ≥ ℏ/2, and Bohr (the story goes that he spent a sleepless night working it out) showed Einstein the next morning that his own general relativity, the redshift of clocks in a gravitational field, exactly restored the uncertainty he had tried to escape. Bohr had used Einstein’s own theory against him. From then on Einstein conceded the consistency of quantum mechanics but never accepted its finality. “God does not play dice,” he told Bohr more than once, in conversations and in letters. Bohr’s reply was patient: stop telling God what to do.

The uncertainty principle is sometimes presented as a kind of metaphysical limit on knowledge, the universe drawing a veil across its own affairs. It is more useful, and more accurate, to see it the way Bohr taught Heisenberg to see it: as a statement about waves. A wave with a sharply defined wavelength is, by definition, one that repeats forever and has no place; a wave that lives in one place is, by Fourier’s theorem, a superposition of many wavelengths. The world is made of waves. The waves have widths. The product of those widths is bounded below by ℏ/2 because that is what waves are. The same principle controls the diffraction of a laser through a slit, the linewidth of an atomic clock, the recoil of a phonon in a crystal, and the irreducible jitter at the heart of any photodetector. It is the rule that organises every experiment we have ever built. Once you have absorbed it, the strange behaviour of the next several chapters, tunnelling, zero-point energy, the spreading of a free wavepacket, the natural width of every spectral line, will all read as variations on one theme: the wave nature of matter has a price, and the price is ℏ/2.