Physics 1900

On the Theory of the Energy Distribution Law of the Normal Spectrum

Max Planck

Energy is not poured but counted out — in tiny indivisible grains. The quantum is born.

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In depth · the introduction

Planck discovered that energy doesn't flow in a smooth stream but comes in tiny indivisible grains — the first clue that the world is built from quanta.

The big idea

Heat up a piece of metal and it glows — first red, then orange, then white. Physicists could measure exactly how this glow's colour shifts with temperature, but their best theory gave a ridiculous answer: it predicted that hot objects should blast out unlimited energy in violet and beyond. Reality politely refused.

Planck found the formula that matched the real measurements, but it only worked with one strange assumption: that energy can't be given out in any amount you like. Instead it comes in fixed little chunks, and the chunk size depends on the colour (frequency) of the light. It was as if energy could be paid only in whole coins, never in fractions.

How it came about

Planck was a careful, conservative 42-year-old professor in Berlin, not a revolutionary. In October 1900 he reverse-engineered a formula to fit the latest, very precise measurements of cavity radiation by his colleagues Lummer, Pringsheim and Kurlbaum. It fit beautifully — but he had no explanation for it.

Over the following weeks he found one, at the price of the energy-element idea. He presented the derivation to the German Physical Society on 14 December 1900 — a date now remembered as the birthday of quantum theory. He disliked the assumption intensely and later called it “an act of desperation,” something he resorted to because nothing else worked.

Why it mattered

This single assumption was the seed of quantum physics — the science of the very small. Everything from lasers and LEDs to computer chips and MRI machines rests on the discovery that, deep down, nature comes in grains rather than a smooth, continuous flow.

A way to picture it

Think of a vending machine that takes only whole coins, never fractions — and the coin it demands gets bigger for bluer light. Red light is cheap, paid in small coins; violet light costs a big coin. A warm object simply can't afford to pour out the huge, expensive high-frequency coins, so the runaway “ultraviolet” energy never happens. Slide the temperature in the tool below and watch the glow change colour.

What came next

Planck quantised only the energy traded by matter; in 1905 Einstein made the bolder claim that light itself comes in quanta (later called photons). In 1913 Bohr used quantised energy to explain the atom, and by the mid-1920s these clues had grown into quantum mechanics — the most precisely tested theory in all of science. The strangest theory we have grew from Planck's one reluctant step.

The original document

Original source text

Max Planck · Verhandlungen der Deutschen Physikalischen Gesellschaft 2 (1900): 237–245 · presented 14 December 1900

Gentlemen: some weeks ago I had the honour to draw your attention to a new formula which seemed to me suited to express the law of the distribution of radiation energy over the whole range of the normal spectrum. I remarked then that, in my opinion, the usefulness of this equation was not based solely on its close agreement with the observations available, but rested chiefly on the simple structure of the formula. Today I should like to lay before you the theoretical deduction of that formula.

To obtain it I follow the lead of Boltzmann: the entropy S of a system in a given state is proportional to the logarithm of the probability W of that state. Of the N resonators of frequency ν in the cavity, let the total energy be E. The whole question reduces to determining the probability W — the number of ways in which this energy can be distributed over the N resonators.

The energy element

We consider, however — this is the most essential point of the whole calculation — E to be composed of a very definite number of equal parts, and use thereto the constant of nature h = 6.55 × 10⁻²⁷ erg·sec. This constant, multiplied by the common frequency ν of the resonators, gives us the energy element ε in ergs, and dividing E by ε we get the number P of energy elements which must be divided over the N resonators.

The probability W is then the number of ways in which the P indistinguishable energy elements can be distributed over the N resonators. The energy of a resonator is thus to be regarded as made up of a whole number of finite equal parts, and not as a continuously divisible quantity.

ε = hν.

I therefore call h the elementary quantum of action. It is the presence of this finite element — energy exchanged not continuously, but in whole multiples of hν — that makes the calculation succeed where the classical, continuous treatment had failed.

The radiation law

Carrying the calculation through, the energy distribution of the normal spectrum follows, per unit frequency, as u = (8πhν³/c³) / (e^{hν/kT} − 1). With its help it is possible to derive a radiation formula in complete agreement with the observations made up to the present.

Comparison with the measurements of Kurlbaum and of Lummer and Pringsheim fixes the two constants as h = 6.55 × 10⁻²⁷ erg·sec and k = 1.346 × 10⁻¹⁶ erg·degree⁻¹. From k there follow, in turn, the number of molecules in a gramme-molecule and the elementary electric charge, e = 4.69 × 10⁻¹⁰ electrostatic units — values in satisfactory agreement with those obtained by other means.

Berlin · 14 December 1900