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§ Phase i · The Quantum Crisis · Chapter 02

Einstein's photons

Five years after Planck's reluctant quanta, a 26-year-old patent clerk in Bern took the idea seriously and explained why a beam of light, no matter how dim, could knock electrons free from metal, but only if its color was bluish enough.

In 1887, the German physicist Heinrich Hertz noticed something he had not been looking for. He had been chasing Maxwell’s prediction that light and electricity were two faces of the same coin, and he was about to confirm it spectacularly by generating radio waves in his Karlsruhe laboratory. But while watching the tiny sparks jump across the gap of his spark-gap detector, he noticed that the sparks were brighter, and easier to provoke, when ultraviolet light from a nearby arc lamp fell on the receiving plates. He wrote it up dutifully in Annalen der Physik and moved on. The phenomenon was a curiosity. Hertz died in 1894 at the age of 36, never suspecting that his side observation contained the seed of a revolution.

Fifteen years later, Hertz’s former assistant Philipp Lenard, by then a professor at Heidelberg, set out to study the curiosity properly. Using clean metal surfaces in evacuated glass tubes, Lenard confirmed that the ultraviolet light was ripping electrons clean off the metal. Then he measured their energies, and the data refused to behave. Classical wave theory of light, the proud edifice that James Clerk Maxwell had built and Hertz himself had vindicated, made a clean prediction: brighter light carries more energy, so brighter light should produce more energetic electrons. Lenard found the opposite. Cranking up the intensity of his lamp produced more electrons, but each one came out with exactly the same energy as before. To get faster electrons you had to change the color of the light, not its brightness. Dim ultraviolet beat bright red, every time. Below a certain threshold frequency, no light at all came out, no matter how blindingly bright the source.

This was, by the standards of 1902 physics, deeply weird. A wave is a continuous thing; you can pour as much energy into it as you like by turning up the amplitude. The electron, sitting in the metal, ought to soak that energy up gradually, like a swimmer being pushed harder by a stronger current. Yet here was nature insisting that the swimmer felt nothing at all unless the frequency of the waves crossed a magic line. It was as if a thousand gentle pushes could never add up to one good shove. Lenard published, the journals took note, and the puzzle sat there for three years, undigested. Then, in the spring of 1905, a 26-year-old technical expert third class at the Swiss federal patent office in Bern picked it up.

That spring and summer Einstein wrote four papers, any one of which would have made his career; together they earned the year the nickname annus mirabilis, the miracle year. The first, dated March, was titled “On a Heuristic Viewpoint Concerning the Production and Transformation of Light.” It opens with a careful, almost defensive paragraph noting that the wave theory of light, while superbly successful for optics, may need to be supplemented (at least for processes involving emission and absorption) by the idea that the energy of light is “not continuously distributed over an ever-increasing volume, but consists of a finite number of energy quanta localized at points in space, which move without dividing, and can only be produced and absorbed as complete units.” Einstein himself, writing to a friend, called it “very revolutionary.” He was not exaggerating.

Einstein’s move was simple and audacious. He took Planck’s quantum, the mathematical fudge introduced five years earlier to fix the black-body curve, and he asked: what if it is not a fudge? What if light really is, when it interacts with matter, lumpy? What if a beam of light is not a continuous wave depositing energy smoothly, but a hail of discrete bullets, each carrying exactly the amount Planck had written down (E = hν), one bullet for one quantum of frequency ν?

If that picture is right, the puzzling features of the photoelectric effect fall into place at once. To knock an electron free, you need a single bullet with enough energy to break the electron’s binding to the metal. A red bullet (low frequency) simply does not carry enough punch, no matter how many of them you fire. Doubling the intensity of red light doubles the number of bullets, but each bullet is still too weak. Cross the threshold frequency, though, and every bullet now has enough energy to liberate an electron, with whatever’s left over showing up as the electron’s kinetic energy. Brightness controls the count of electrons coming out. Color controls their speed. The data made sense.

Einstein wrote down the relation in one line. The maximum kinetic energy of the ejected electron equals the photon’s energy minus the energy required to free the electron from the metal. He called that escape cost the Austrittsarbeit, the “work of escape,” now universally written W and called the work function. So K_max = hν − W. Plot K_max against ν for any metal and you should get a straight line with slope h, intercepting the energy axis at −W. The same Planck constant should appear regardless of which metal you choose. It is one of the cleanest, most testable predictions in the history of physics.

The reception was frosty. Wave theory had earned its credibility honestly, over a century of triumphs from Young’s double slit to Maxwell’s equations to Hertz’s own radio waves. To call light a stream of particles in 1905 was to invite ridicule. Planck himself, the man who had first written down E = hν, refused to accept Einstein’s reading. When Planck nominated Einstein for membership in the Prussian Academy of Sciences in 1913, he wrote a warm letter of recommendation that nevertheless contained a famous caveat: “That he may sometimes have missed the target in his speculations, as, for example, in his hypothesis of light quanta, cannot really be held too much against him.” Eight years after the paper, the founder of quantum theory still considered the photon a mistake.

The empirical test took eleven years. The American experimentalist Robert Millikan, who had already nailed the charge of the electron with his oil-drop apparatus at the University of Chicago, set out around 1914 to demolish Einstein’s wild equation. Millikan was a careful, traditional physicist; he disliked the photon hypothesis and expected his measurements to expose it. He built a small machine in vacuum, scraped the surfaces of alkali metals fresh with a tiny knife inside the chamber to keep them oxide-free, and measured stopping voltages across a range of ultraviolet frequencies. The data came in slowly through the war years.

By 1916 Millikan had his answer, and it horrified him. The plots were ruler-straight. The slopes, measured for sodium and potassium and lithium, were identical within experimental error. Translating slope to Planck constant gave h = 6.57 × 10⁻³⁴ joule-seconds, within about 1% of the value Planck had extracted from the black-body curve sixteen years earlier, by a wholly independent method. The photoelectric equation worked. Millikan published his results in 1916 in the Physical Review, prefacing them with an extraordinary confession: “Einstein’s photoelectric equation… cannot in my judgment be looked upon at present as resting upon any sort of a satisfactory theoretical foundation,” he wrote, “yet it actually represents very accurately the behavior of the photoelectric effect.” He had set out to bury the photon and had instead become its most decisive witness.

K_max = hν − W

eV_s = hν − W
V_s  = (h/e) ν − W/e

The triumph was real but the conceptual price was steep. The wave theory of light had not gone away. It still explained diffraction patterns through narrow slits, the colors in soap films, the polarization of skylight. Maxwell’s equations were not wrong. Yet here was the photoelectric effect demanding that the same light, when it crossed a metal boundary, behaved like a stream of particles each carrying a precisely-defined energy. By 1923, when Arthur Compton showed that X-ray photons bouncing off electrons obeyed the same conservation laws as billiard balls, complete with measurable transfer of momentum p = h/λ, the case for the photon as a real particle was effectively closed.

And yet the wave evidence had not vanished either. Light, when nobody was watching it interact, propagated as a wave; when forced to deposit or extract energy from matter, it acted as a particle. The two pictures could not be reconciled within classical physics. They were complementary, in a sense that would not be properly named for another quarter century, when Niels Bohr coined the word. For the moment, in the late 1910s and early 1920s, physicists had to learn to hold both descriptions in their heads at once, using whichever was appropriate for the experiment at hand and refusing to ask, too loudly, what light “really” was when no one was looking.

Einstein himself never made peace with this. To his last years he wrote letters protesting that “fifty years of conscious brooding have brought me no nearer to the answer to the question, ‘What are light quanta?’” His 1905 paper had cracked open a door that, when fully opened, would reveal a quantum mechanics he found philosophically unbearable. But the photon, the bullet he had postulated to explain Lenard’s stubborn data, was here to stay. The next domino was waiting in the hydrogen spectrum: a set of bright sharp lines, known since the 1860s, that no one had been able to explain. A Swiss school-teacher in his sixties had already written down a curious formula for them in 1885, twenty years before Einstein’s miracle year. He had not known what it meant.