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§ Phase i · The Quantum Crisis · Chapter 01

Planck's catastrophe

At the close of the 19th century, the most distinguished physicists in Europe could explain the orbits of comets and the bending of light, but they could not explain why a poker glows red. Their best equation predicted that any hot object should pour out infinite energy in violet, ultraviolet, and beyond. A 42-year-old professor in Berlin fixed the problem on the afternoon of December 14, 1900 with what he called an act of desperation. The fix was a tiny number. The tiny number cracked the universe open.

On the evening of October 19, 1900, a small, dignified man with a neat moustache rose to read a paper at the German Physical Society in Berlin. His name was Max Planck. He was a thoroughgoing classical physicist, raised on Helmholtz and Kirchhoff, the kind of man who arrived at lectures ten minutes early and corrected colleagues’ mathematics with a soft cough. He was, in his own description, “by nature peaceful and disinclined to questionable adventures.” And yet within ten weeks he would commit the most consequential act of theoretical defiance in three hundred years.

The problem on Planck’s desk had a homely name and a homely origin. It was called the black-body problem, and it had begun forty years earlier with another Berliner, Gustav Kirchhoff, who in 1860 had proved a remarkable theorem. Take any sealed cavity (a kiln, an iron box with a pinhole, the inside of a star), heat it to a uniform temperature, and the light leaking out depends on one thing and one thing only: the temperature. Not the material of the walls. Not the shape of the cavity. Not the chemistry of what burns inside. Only T. The spectrum of that light, Kirchhoff said, was a universal function of nature. Find that function and you would have written down a law as basic as gravity.

For forty years, no one could find it. By the 1890s, the German engineers building the new electric streetlamps in Berlin had a practical motive. They needed to know how much of a glowing filament’s energy escaped as visible light and how much was wasted as heat. The Reichsmark and the kilowatt-hour both wanted an answer. So the Physikalisch-Technische Reichsanstalt, the imperial standards lab where Planck consulted, built the world’s most exquisite ovens (porcelain cavities encased in platinum, with pinhole apertures the diameter of a sewing needle) and measured. They measured at red heat. They measured at orange. They pushed the ovens to white. They cross-checked their bolometers against thermopiles and rebuilt their prism spectrometers to chase systematic errors below one part in a thousand. By 1900 they had drawn a beautiful, smooth curve: the rate of energy radiated at each wavelength, by a perfect black body, as a function of temperature. The curve was real. The curve was reproducible. The curve was, on the basis of every classical theory then available, impossible.

The trouble began when theorists tried to derive Kirchhoff’s curve from first principles. The first serious attempt came from Wilhelm Wien in 1896. By analogy with the Maxwell-Boltzmann distribution of molecular speeds, Wien guessed a spectral formula that fit the high-frequency tail of the data (the violet end, where the curve falls off) remarkably well. It even fit the visible. For a few years, Wien’s law looked like the answer.

Then in 1900 two of the Reichsanstalt experimentalists, Heinrich Rubens and Ferdinand Kurlbaum, pushed their measurements out to longer wavelengths than anyone before: deep into the infrared, with rock-salt prisms and bolometers cooled in ice baths. There Wien’s formula failed. The intensity at long wavelengths was systematically higher than Wien predicted, and it grew in a particular way, proportional to the temperature itself, not to anything more exotic. The data at low frequency wanted to be a straight line in T. Wien’s exponential could not bend that way.

Meanwhile, in England, Lord Rayleigh had attacked the same problem from the other end. He treated the cavity as a box of standing electromagnetic waves and counted them, exactly the way one counts the vibrational modes of a stretched string. Sir James Jeans cleaned up the arithmetic in 1905. The argument is so simple it can be reproduced in a paragraph: each standing wave mode is an oscillator, classical thermodynamics says every oscillator at temperature T carries on average kT of energy (the equipartition theorem), and the number of modes per unit volume in a tiny frequency band grows as the square of the frequency. Multiply. You get an energy density that climbs as ν² and never stops. Add up all frequencies and the total energy in any warm box is infinite.

This was, plainly, ridiculous. A teapot does not contain infinite energy. You can warm your hands by a fire without being vaporized by its high-frequency radiation. The X-rays that should pour out of a kitchen kettle by Rayleigh’s argument are not in fact pouring out of any kitchen kettle in human history; if they were, every cook in the world would be dead by lunchtime. And yet the derivation used nothing more controversial than Maxwell’s equations, Newton’s mechanics, and the equipartition theorem, three of the most thoroughly tested pillars of 19th-century physics. The Rayleigh-Jeans law was not a sloppy approximation. It was a rigorous consequence of accepted theory. To deny its conclusion was to deny one of its premises. The only question was which one.

So by October 1900, classical physics had two formulas (Wien’s, which fit the violet and visible but failed in the infrared; Rayleigh’s, which fit the infrared and exploded into nonsense in the ultraviolet) and one beautifully measured curve that lay smoothly between them. Planck’s first move, on the night of October 7, was a piece of pure mathematical opportunism. Rubens had stopped by his house that afternoon and shown him the new long-wavelength data. Planck went into his study, played with the algebra of Wien’s law, and looked for a single interpolation formula that would reduce to Wien’s at high frequency and to Rayleigh’s at low. By midnight he had one. It was a small adjustment to an entropy expression, more bookkeeping than theory. He sent it to Rubens by postcard the next morning. Rubens compared it to every measurement in the lab. It fit. It fit at every temperature, at every wavelength they could measure. It was, as far as the data could tell, exact.

Planck did not consider this a victory. He had a formula but no derivation. As he wrote later, he had pulled “a happy guess” out of the air, and “the search for its theoretical interpretation began the most strenuous work of my life.” For the next two months he worked obsessively. He had to find a physical reason, a model of matter and radiation, from which his interpolation would fall out as a natural consequence.

The standard tool for that job was Boltzmann’s statistical mechanics: count the number of microstates compatible with a given macrostate, take the logarithm, and you have the entropy. But to count, you must have discrete things to count. Boltzmann had used this trick himself when treating gases, slicing energy into tiny pieces ε, doing the combinatorics, and then taking ε → 0 at the end to recover continuous physics. Planck adopted the same trick (slice the oscillator energies in the cavity wall into chunks of size ε) and ground through the combinatorial argument.

And here, on or about December 14, 1900, the strangest moment in modern science took place. When Planck reached the end of the derivation, he found that he could only get his interpolation formula to come out right if he refused to take the final limit. He could not let ε go to zero. Instead, the size of each energy chunk had to be proportional to the oscillator’s frequency:

ε = h ν

with h a small, universal constant whose value he extracted by fitting to Kurlbaum’s data. He got, to within a fraction of a percent, the number we still use today.

What did this mean, physically? Planck himself was honest about his confusion. He had been forced to assume that an oscillator at frequency ν could only hold energy in integer multiples of hν (0, hν, 2hν, 3hν, and so on) but nothing in between. The idea was monstrous. Maxwell’s equations are linear; a classical oscillator can be excited at any amplitude. Pluck a guitar string a little harder and it carries a little more energy; there is no rung-by-rung ladder in classical mechanics. To say otherwise was to throw away two and a half centuries of mechanics, from Galileo’s pendulum to Maxwell’s field equations. Planck spent the next decade trying, and failing, to reconcile his quantization with classical theory. He treated it for years as a calculational trick rather than a fact of nature, a piece of bookkeeping inside the cavity wall that produced the right entropy without committing him to anything radical about light or matter. Only after Einstein’s 1905 paper on the photoelectric effect (which we will read in the next chapter) did Planck slowly accept that he had stumbled onto something real. As late as 1913, when he nominated Einstein for membership in the Prussian Academy, Planck wrote a generous tribute and then added a hedge: “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.” The man who lit the fuse was, in 1913, still hoping that the explosion would somehow be contained.

He never quite stopped apologizing for it. In a 1931 letter to his American colleague Robert Williams Wood, Planck described his December 1900 leap as “an act of desperation. I was ready to sacrifice every one of my previous convictions about physics.” It is a phrase that has earned its place in the textbooks because it is exactly true. Planck was no revolutionary. He was a conservative man who had been chased into a corner by the data and had thrown the only thing left at hand, a discontinuous energy spectrum, at a problem that refused to yield to anything else. The throw worked. The throw started everything that follows in this book.

P(n) = exp(-n h ν / k T) / Z

Z = Σₙ exp(-n h ν / k T) = 1 / (1 - exp(-h ν / k T))

⟨E⟩ = h ν / ( exp(h ν / k T) - 1 )

u(ν, T) = (8 π h ν³ / c³) · 1 / ( exp(h ν / k T) - 1 )

It is worth pausing, before we leave Berlin, to take in the smallness of h. The number 6.626 × 10⁻³⁴ J·s is about as far below human scale as the size of a proton is below the size of a continent. For any object you can hold in your hand, the quantum levels of its energy are so dense and so finely spaced that no laboratory in the world can resolve them. A swinging pendulum at one cycle per second carries energy in steps of about 6.6 × 10⁻³⁴ joules, utterly invisible against the joule-scale energy of the swing. The classical world looks continuous because, at human scale, the quantum grain is finer than any meter we have ever built or ever will build. This is the entire reason that physics worked for three hundred years without anyone noticing.

But push to small things and short times, to molecules and atoms and the electrons inside them, and the energies in play shrink by ten or twenty orders of magnitude. Now hν is no longer negligible. Now the grain begins to show. A hydrogen atom is the size of an angstrom, its binding energy a few electron-volts, its allowed states a discrete ladder. The structure that classical physics could not see because its measurements were too coarse turns out to be the structure that makes the world. Every fact in the rest of this book (the colors of stars, the periodic table, the rigidity of solids, the existence of chemistry, the existence of life) is downstream of the fact that h is small, but not zero. We will see h again in the next chapter, when Einstein takes Planck’s hν and assigns it to a single particle of light. We will see it again in chapter three, when Balmer’s empirical fit to the colors of hydrogen turns out to be a thinly disguised ratio of two energies hν. We will see it in chapter four when Bohr quantizes angular momentum in units of h-bar = h / 2π, and once more in every subsequent chapter where matter, light, or both refuse to behave continuously. The same tiny constant runs through all of it. Get used to its face. You will be seeing a great deal of it in the chapters that follow.

Planck himself, of course, did not see any of that yet. On December 14, 1900, when he presented his derivation to the German Physical Society, he had a formula that fit the data and a hypothesis (quantized oscillator energies) that he did not believe. His audience was politely puzzled. Wien’s law had been tidied; that much they could see. Whether the underlying picture meant anything physical was unclear. The paper was filed alongside a hundred other technical contributions of that month. No newspaper covered it. No memorial plaque marks the lecture hall. The first sentence of the first chapter of the modern world was read in front of perhaps fifty people, several of whom went home and forgot about it. The constant h would have to wait five more years for someone less cautious than Planck, a Swiss patent clerk with no academic post, to take its discontinuity seriously, and turn it into the photon.