Continuous or Discrete? The Line Planck Crossed in 1900

Quantum mechanics, the story goes, began with a short talk in Berlin on 14 December 1900. Yet the man who gave it did not believe he had opened a door onto a new physics. For Max Planck, the assumption that energy comes only in discrete amounts was not a conclusion his theory had led him to — it was a last resort for making the numbers agree. This is the first of five pieces on the old quantum theory: the story of a line a physicist crossed without wanting to.

A rare problem with only one possible answer

It starts in 1859. Gustav Kirchhoff showed that the spectrum radiated by an ideal body that absorbs light of every wavelength — a black body — depends only on its temperature and the wavelength, not on what the body is made of. Iron, charcoal, ceramic: it makes no difference. That made the problem unusually attractive to physicists. Without knowing anything about the material, first principles alone ought to fix the answer.

In Germany it also mattered practically. Readings of furnace temperature from colour, estimates of the efficiency of lamp filaments — such demands drove the Physikalisch-Technische Reichsanstalt in Berlin to accumulate the most precise radiation measurements in the world. Theory was expected to produce a formula that could keep up with them.

Two formulas that failed in opposite directions

Wilhelm Wien's distribution law of 1896 matched the measurements well at short wavelengths. The formula Lord Rayleigh derived in June 1900 from the classical equipartition theorem matched at long wavelengths instead — but as the wavelength shrank, the predicted energy ran off to infinity. That difficulty later acquired the name "ultraviolet catastrophe," and it is worth being clear that it was not what moved Planck in 1900. The name itself was coined by Paul Ehrenfest in 1911; the historian Clayton Gearhart calls the standard textbook account pedagogically useful but, as history, a myth.

What moved Planck was something far more concrete: data.

A formula written on a Sunday evening

On the evening of 7 October 1900, Heinrich Rubens of the Reichsanstalt visited Planck at home. Rubens and Ferdinand Kurlbaum had measured out to 51 micrometres — an extreme long-wavelength region for the time — and found the energy density to be very nearly proportional to temperature. Wien's law could not reproduce that behaviour.

That same evening Planck wrote down an interpolation that joined Wien's law to the new long-wavelength data. Twelve days later, on 19 October, he presented it to the German Physical Society as "On an improvement of Wien's spectral equation" — a report of barely three pages. The formula fit the measurements beautifully. It said nothing whatsoever about why that formula, and not another.

"At any cost" — 14 December

Supplying a reason took two months. On 14 December 1900, Planck brought in Boltzmann's statistical counting methods and treated the energy of the oscillators exchanging light not as a continuous quantity but as a whole-number multiple of an element, ε = hν. He chopped energy into pieces. The new constant this introduced came out at h = 6.55×10⁻²⁷ erg·seconds — the same order and very nearly the same digits as today's value (6.62607015×10⁻³⁴ J·s, fixed by definition since the 2019 revision of the SI).

The by-products were substantial. The same framework fixed the Boltzmann constant, and from it Avogadro's number and the charge on a single electron — values more accurate than the direct measurements available at the time.

Planck still did not like what he had done. Writing to Robert W. Wood thirty-one years later, in 1931, he recalled that a theoretical interpretation had to be found at any cost, however high, and that he was ready to sacrifice every one of his previous convictions about physical laws so long as the two laws of thermodynamics survived. That letter is the source of the famous phrase: an act of desperation.

What had been achieved, and what had not

By the end of 1900 there was a formula that agreed with measurement across the whole spectrum, and a new constant of nature, h. What there was not was any explanation of why energy should have to come in pieces at all. Planck himself treated the quantization as a computational device rather than a physical fact, and spent the better part of the next decade trying to push it back inside classical physics.

In 1978 the historian Thomas Kuhn went further, arguing that Planck was not claiming energy quantization in 1900 at all. Few today read him quite that strongly — Gearhart's assessment is that Planck's thinking was considerably more complex and ambiguous than either reading allows. What is certain either way is that "the quantum revolution began in 1900" is a narrative tidied up in hindsight. Almost nobody at the time thought a revolution had occurred.

It would take another five years before someone read hν not as a bookkeeping trick but as a property of light itself. Next: Einstein, 1905, and the light-quantum hypothesis.