Planck's desperate act

He guessed the formula in an evening, spent eight weeks finding out what it meant, and did not like the answer. Then he spent a decade trying to get rid of it.

1900 – 191118 min

By the end you should be able to

Distinguish Planck’s guessed formula from the derivation that forced the quantum
Explain how energy quantisation stops the high-frequency modes from diverging
Say what Planck himself thought he had done, and why he was uncomfortable

On Sunday 7 October 1900, Heinrich Rubens and his wife came to the Plancks' house in Grunewald for tea. Rubens mentioned his newest far-infrared measurements: at long wavelengths the intensity came out simply proportional to temperature, and Wien's law — which everybody had believed exact for four years — plainly failed there. Planck already knew that Wien's law worked beautifully at short wavelengths. So he now had two limits and nothing joining them: - short λ: Wien's exponential form, well tested - long λ: intensity ∝ T, brand new That evening, after his guests left, he wrote down a formula that reduces to the first at one end and the second at the other. He posted it to Rubens on a postcard the same night.

Max Planck in 1933, elderly, in a dark suit and round spectacles. — Planck in 1933, thirty-three years after the quantum. He had spent much of the intervening time trying to derive his own formula without it, and described the assumption as **an act of desperation** — a phrase he meant literally rather than modestly.

A formula that works and cannot be explained is not physics

Twelve days later Planck presented the formula to the German Physical Society. Rubens went home, spent the night checking it against his data point by point, and returned next morning to report perfect agreement. That sounds like the end of the story, and Planck was entirely clear that it was not. He had no derivation. He later called it "a happily guessed interpolation formula" — a curve constructed to have the right behaviour at both ends, with nothing behind it. This distinction matters and it recurs throughout physics. A formula that fits is a fit. Newcomb's modified exponent fitted Mercury too. What makes something an explanation is that it follows from premises you had independent reason to hold — and Planck had none. So he went looking for them.

He afterwards described the following weeks as the most strenuous work of my life. And to get anywhere he had to do something he had spent two decades publicly resisting. Planck disliked Ludwig Boltzmann's statistical approach to thermodynamics. He had argued against it in print, because it makes the second law of thermodynamics probabilistic — overwhelmingly likely rather than absolute — and Planck had built his career on the second law being absolute. Now, to derive his own formula, he had to pick up Boltzmann's method and use it.

Boltzmann's technique works by counting: you find the entropy of a system by counting the number of ways its energy can be distributed among its parts, and the most probable distribution is the one realisable in the most ways. But counting requires discrete things to count. You cannot enumerate the ways of dividing something perfectly continuous — there are infinitely many, and the counting is meaningless. So Planck chopped the total energy into finite elements of size ε, purely so the combinatorics would work. This is a standard manoeuvre. You make the thing discrete, do the counting, and let ε shrink to zero at the end, recovering the continuous answer.

Switch the classical curve on and off, and watch how the two agree where the quantum is small compared with kT.

Loading the spectrum…

The same sim as the previous lesson, now with the mechanism in view. The green curve is what you get when each mode’s energy comes in steps of hν; the red is what you get when every mode receives kT regardless. They coincide wherever hν ≪ kT and part company where the entry fee becomes unaffordable.

An entry fee that rises with frequency

The physical picture is worth holding onto, because it recurs throughout quantum mechanics. Classically, energy is infinitely divisible, so every mode can always take up a little more — and with infinitely many high-frequency modes each taking kT, the total is infinite. Quantised, a mode faces an entry fee of hν, and it must pay in full or not at all. Thermal energy is handed out in typical amounts of kT. So: In the Sun's spectrum at 5772 K, where kT = 0.50 eV: | mode | quantum hν | fraction of kT it holds | |---|---|---| | radio | 0.0001 eV | 100% | | infrared | 0.12 eV | 88% | | green light | 2.5 eV | 3.4% | | ultraviolet | 6.2 eV | 0.005% | | X-ray | 124 eV | 10⁻¹⁰⁴ % | The high-frequency modes are not suppressed a bit. They are switched off.

The experiment

Weighing the electron by looking at a hot box

Max Planck, using the data of Lummer, Pringsheim, Rubens and Kurlbaum · Presented 14 December 1900 · Berlin

The question: If the radiation formula is correct, its two constants can be read off from the measured curve. What values does the blackbody spectrum imply for the constants of nature?
The apparatus: No new apparatus — a fit of the newly derived formula to the Berlin cavity measurements. The curve contains exactly two adjustable constants, h and k, and the shape of the spectrum fixes both: the position of the peak and the steepness of the fall-off constrain them independently.

Theory predicted

Two numbers, h and k. But k unlocks more: the gas constant R was well measured and R = N_A·k, so Avogadro’s number follows. And the Faraday constant F = N_A·e was well measured, so the electron’s charge follows from that.

They measured

h = 6.55 × 10⁻²⁷ erg·s (modern 6.626, off by 1.2%); k = 1.346 × 10⁻¹⁶ erg/K (2.5%); N_A = 6.175 × 10²³ (2.5%); e = 4.69 × 10⁻¹⁰ esu (2.4%).

How sure could they be? One to three per cent on all four, limited by the radiation measurements rather than by the fitting. The value for e is the remarkable one: no direct measurement came close to it in 1900, and none beat it until Millikan’s oil-drop experiment of 1909 reached 0.6%.

Why it mattered

A striking demonstration that the formula was more than a curve fit. It delivered four fundamental constants at once — including the charge on the electron, obtained years before anyone could measure it directly, from nothing but the colour of light leaving a hot cavity. Planck regarded this as the strongest evidence that his derivation had touched something real, even while he disbelieved the quantisation it rested on.

Briefly summarized, what I did can be described as simply an act of desperation. By nature I am peacefully inclined and reject all doubtful adventures. But by then I had been wrestling unsuccessfully for six years with the problem of equilibrium between radiation and matter, and I knew that this problem was of fundamental importance. A theoretical interpretation therefore had to be found at any price, however high that might be.
Max Planckletter to Robert Williams Wood, 7 October 1931

You might think

Planck discovered that energy is quantised and thereby founded quantum theory.

Actually

He did the first, but he did not believe it and he did not intend the second. Planck regarded ε = hν as a formal mathematical device — a piece of scaffolding demanded by the combinatorics, which he expected somebody would eventually explain away. He spent roughly the next decade trying to remove it: deriving the formula without quantisation, or confining the discreteness to emission so that the electromagnetic field itself could remain classical. As late as 1911 he was still proposing a scheme with continuous absorption and quantised emission. He was among the last physicists of his generation to accept the implications of his own result, and he said so himself.

1894Planck takes up the blackbody problem, funded partly by the electric lighting industry.
7 Oct 1900Rubens visits for tea. Planck writes down the formula that evening.
19 Oct 1900He presents it as an interpolation, with no derivation.
14 Dec 1900He presents the derivation, and with it ε = hν. Usually taken as the birthday of quantum theory.
1905Einstein proposes that light itself is quantised — a far stronger claim, which Planck rejects.
1911Planck is still trying to make absorption continuous and only emission quantised.
1918He receives the Nobel Prize for the discovery of energy quanta.

Where this leaves us

Planck quantised the oscillators in the wall of the cavity — the little charged things that emit and absorb. He left the electromagnetic field alone, and he expected the whole business to be temporary. In 1905, Einstein went much further. He proposed that light itself comes in quanta — that the electromagnetic field is grainy, that a beam of light is a hail of localised packets each carrying hν. That is a far more radical claim, and it was rejected by essentially everyone for the better part of twenty years, Planck included. When Planck nominated Einstein for the Prussian Academy in 1913, he added a note asking that it not be held against him that he had "missed the target in his speculations, as for example in his hypothesis of light quanta". But it explained something. There was an experiment involving ultraviolet light and metal plates, discovered by accident in 1887 by Heinrich Hertz — in the middle of the experiment that proved Maxwell right, back in Act 1 — and it had been sitting unexplained ever since.