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§ Phase i · The Quantum Crisis · Chapter 04

Bohr's leap

Niels Bohr was twenty-seven when he proposed something so strange that even he called it 'audacious'. Electrons in atoms didn't spiral into the nucleus because they couldn't. They were stuck on a quantum ladder, and the rungs were sharp.

In the spring of 1911, a New Zealander named Ernest Rutherford was working in a basement laboratory in Manchester with two assistants, Hans Geiger and Ernest Marsden. They were shooting alpha particles (naked helium nuclei, blasted out of a fleck of radium) at a sheet of gold foil so thin you could read newsprint through it. Almost everything went straight through. Almost. About one alpha in eight thousand bounced back, sometimes nearly straight at the source. Rutherford later compared the result to firing a fifteen-inch naval shell at a sheet of tissue paper and having it ricochet into your face. The atom, it turned out, was almost entirely empty space. But at its center sat something tiny and outrageously dense.

The prevailing picture of the atom in 1911 was J. J. Thomson’s plum pudding: a diffuse cloud of positive charge with electrons embedded inside it like raisins in a steamed dessert. Thomson’s model could not produce the kind of violent backscatter Geiger and Marsden were seeing. To kick a five-million-electron-volt alpha particle back the way it came, you needed a target that was concentrated, massive, and electrically charged. Rutherford did the geometry on the back of an envelope: the scatterer had to be no bigger than one ten-thousandth of the atom and had to contain essentially all of its mass. He published the result in 1911 in a paper that mostly nobody read. Within two years it would be the foundation of every atomic model.

Rutherford had handed physics a beautiful picture and a fatal problem in the same gesture. A picture: the atom was a miniature solar system, a positive sun with electrons orbiting at distance. A problem: that picture could not possibly be true. Maxwell’s equations, the proudest achievement of nineteenth-century physics, were unambiguous. Any charged particle that accelerates radiates electromagnetic waves. An electron in a circular orbit is accelerating constantly, sideways, every moment of its dance around the nucleus. So it must radiate. So it must lose energy. So it must spiral inward. The arithmetic was merciless: a classical hydrogen atom should collapse in about ten picoseconds, a hundredth of a billionth of a second. Yet the hydrogen on the table in front of Rutherford had been sitting there for billions of years, perfectly stable, perfectly unbothered.

Into this contradiction walked a quiet, slow-spoken young Dane who mumbled when he lectured and revised every sentence he wrote until it stopped meaning anything to anyone but him.

Bohr spent the summer of 1912 trying to make the planetary atom work. He played with the numbers, dimensional analysis at first, the way a child plays with blocks. He noticed something strange. If you combined Planck’s constant h, the electron mass m_e, and the electron charge e, you could build a quantity with the dimensions of length, and the number you got was roughly the size of an atom. This was not the kind of coincidence a serious physicist could ignore. Whatever the atom was, Planck’s constant (Planck’s act of desperation, dragged into the world to explain the glow of a furnace) had to be inside it.

He returned to Copenhagen in the autumn of 1912 to take up a lectureship and married his fiancée, Margrethe Nørlund, who would type and edit his manuscripts for the rest of his life. He kept thinking about the atom. He drafted a memorandum to Rutherford that summer outlining a model with quantized electron orbits, but the memo was vague and Rutherford had little to say about it. The breakthrough came in February 1913, when Bohr ran into his old friend Hans Marius Hansen at the University of Copenhagen and asked, almost as small talk, whether spectroscopy had ever produced any useful patterns. Hansen, a spectroscopist by trade, pulled out Balmer’s old paper and showed him the simple algebraic formula that fit the visible hydrogen lines. Bohr looked at it and the world reorganized itself. The integers n in Balmer’s expression (the mysterious whole numbers that had no physical interpretation in 1885) were obviously the labels of quantized orbits. The structure of the atom and the structure of its light were the same structure, seen from two sides.

The problem, Bohr realized, was that physicists had been asking the wrong question. Everyone wanted to know why the electron didn’t spiral in. Bohr decided to stop asking. He simply postulated, with the calm audacity of someone who had nothing left to lose, that it didn’t. The electron occupies stationary states: special orbits in which, contrary to everything Maxwell taught, it does not radiate. It can hop from one such orbit to another, and when it hops it gives up a photon whose energy is exactly the difference between the two states. That was the leap. Not a derivation. A postulate. He wrote it down, and he turned the crank, and the universe answered back.

Here is what Bohr actually wrote down. First postulate: of all the orbits an electron could take, only those with quantized angular momentum are allowed, L = n ℏ, where n is a whole number (1, 2, 3, …) and ℏ = h / 2π is Planck’s reduced constant. Second postulate: when the electron is in such an orbit, it does not radiate. Maxwell’s equations are wrong here, or rather, they do not apply. Third postulate: the electron can change orbit, but only by absorbing or emitting a single photon whose energy matches the gap, hν = E_high − E_low. That is the entire model, and from those three lines a high-school algebra exercise drops out the radius of the hydrogen atom, the binding energy, and Balmer’s formula, Rydberg constant and all.

Balance the centripetal force with the Coulomb attraction; substitute the quantization rule for the angular momentum; solve. The bound-state energies come out clean: E_n = −13.6 eV / n². The lowest rung sits at minus thirteen-point-six electron-volts below the free electron. The second rung at minus three-point-four. The third at minus one-and-a-half. The spacing shrinks like 1/n², just as Balmer had noticed thirty years earlier when he was hunting a numerical pattern in a list of red, blue, violet, and ultraviolet hydrogen lines without the faintest idea what they meant.

The four lines Balmer had fitted by intuition (H_α, H_β, H_γ, H_δ) were now transitions from n = 3, 4, 5, 6 down to n = 2. The Lyman lines in the ultraviolet were transitions to n = 1. The Paschen lines in the infrared were transitions to n = 3. The Rydberg constant R, that mysterious empirical number Balmer had been forced to fit, fell out of Bohr’s algebra as a precise combination of fundamental constants:

R = m_e e⁴ / (8 ε₀² h³ c)

Plug in the numbers (the electron mass measured by Thomson, the charge measured by Millikan, the Planck constant measured from black-body curves, the speed of light measured by Michelson) and the predicted value of R agreed with the spectroscopic value to four significant figures. Nobody had fit anything. Nobody had tuned anything. The constants came from completely different experiments, and they conspired to produce the precise color of the red Balmer line. It was, Einstein later said, “an enormous achievement.”

k e² / r² = m_e v² / r

m_e v r = n ℏ

r_n = n² ℏ² / (m_e k e²)

E = (1/2) m_e v² − k e² / r

E_n = − m_e k² e⁴ / (2 ℏ²) · (1 / n²)

hν = E_n − E_n′ = 13.6 eV · ( 1/n′² − 1/n² )

1/λ = R · ( 1/n′² − 1/n² ),    R = m_e e⁴ / (8 ε₀² h³ c)

The model worked perfectly for hydrogen. It worked, with a slight tweak, for singly ionized helium (one electron, two protons), a heavier version of the same problem. The factor of Z² in the binding energies (fourfold deeper for helium-plus than for hydrogen) predicted a spectral series at exactly the wavelengths a Lick Observatory astronomer named Edward Pickering had measured in the light of the star ζ Puppis but failed to identify. Bohr’s model named them. The lines were not hydrogen; they were singly ionized helium. The model had reached into the sky and labeled a star.

Then Bohr tried neutral helium, with its two electrons, and the model fell silent. There was no clean way to add a second electron. The two electrons interacted with each other as well as with the nucleus, and the simple quantization rule L = nℏ did not generalize. Lithium was worse. The heavier atoms were a mess. The Bohr model could draw you the rungs of the hydrogen ladder with impossible precision and then fall apart entirely when you tried to climb the second ladder over.

What Bohr was missing, what nobody had any way of knowing in 1913, was that electrons are not little planets. They are waves, and they obey exclusion rules and exchange statistics that nobody would invent for another decade. The reason helium does not fit Bohr’s formula is the same reason the periodic table has the shape it does: the second electron cannot simply occupy the same state as the first. The Pauli exclusion principle and the antisymmetry of fermion wavefunctions (both still in the future of our story) are what really make multi-electron atoms behave. We will meet them in Phase 09. For now, accept that Bohr had cracked open the door, walked through it confidently for two paces, and run straight into a wall.

The wall did not diminish the achievement. Bohr had shown that the atom is fundamentally quantum, not classical. He had shown that the spectroscopist’s mysterious whole numbers n were not numerology but physics: they were the labels on the rungs of an energy ladder. He had derived, from three audacious postulates and a high-school force balance, a constant that earlier generations had been forced to measure. And he had given the next generation of physicists a target to aim at: explain why these particular postulates, in this particular form, are forced on us by something deeper.

The Bohr model is half-classical and half-quantum, and the seams show. Electrons travel in classical orbits with classical momentum and classical kinetic energy, until they don’t, and then they jump discontinuously, by rules no classical mechanic can write down. The orbit is real until it isn’t. The transition is instantaneous in a theory that has no clock fast enough to describe it. By the mid-1920s, when Schrödinger and Heisenberg had built genuine quantum mechanics, the orbits would dissolve into probability clouds and the jumps would become amplitudes. The energies would remain. The model would be retired. The picture would survive, wrong but beautiful, in every chemistry classroom in the world.

But all of that was a decade away. In 1913, Bohr’s three postulates worked. They explained Balmer. They explained the size of the atom. They explained, for the first time in history, why matter was stable: why electrons did not crash into nuclei in ten picoseconds and snuff out the universe. They did all of this by quantizing angular momentum, by demanding that L = nℏ. And yet nobody (including Bohr) could say why angular momentum should be quantized. Why ℏ? Why this particular constant? What kind of universe rounds the spin of a planet to the nearest integer?

The answer was hiding in plain sight, in a doctoral thesis being written that very year by a French aristocrat named Louis de Broglie. He would propose, with a kind of Gallic insouciance, that the reason L = nℏ is because the electron is a wave. The orbit is a standing wave on a ring. The integer n counts the wavelengths that fit around the circumference. Bohr’s postulate, the audacious leap, is just the resonance condition of a vibrating string, bent into a loop. And when matter itself turned out to be made of waves, the entire universe of position and momentum and orbit and path would dissolve into something far stranger than Bohr ever imagined.