Why Energy Comes in Lumps — Counting It Whole Made It Work

An infinite answer means the calculation is wrong. This story is about how long it took to find the wrong step, and what physics had to give up to fix it.

Five articles follow quantum mechanics in order. The first begins at a stove.

Why hot things glow

Anything with a temperature emits light. Your body does. It happens to be infrared, so you cannot see it — a thermal camera can.

As temperature rises, the centre of the emitted light shifts from the red end towards the blue. That is why colour reports temperature.

So far these are facts you can measure. The difficulty is the next step: why that colour at that temperature. Only a calculation counts as understanding.

The calculation comes out infinite

The method of the day ran like this. Light is a wave, so count every kind of wave that fits inside a hot box, then share the heat energy evenly among them. Sharing energy evenly was a principle that already worked beautifully for gases.

But the shorter the wavelength — the higher the frequency — the more ways there are to fit. Towards very short wavelengths the count grows without limit. Share energy evenly among infinitely many kinds and the total is infinite.

The awkward part was that no step in the calculation looked suspect. Counting the modes was right; sharing energy evenly had served for over a century. Put the two together and the world burns.

Planck's prescription

In 1900 Max Planck had fresh, precise measurements in hand. He first fitted a formula to them. He did not know why it worked; he found the working formula first.

Then he spent two months asking where such a formula could come from. The only route he found was this: energy is not handed over in arbitrary amounts, but only in whole multiples of a smallest lump whose size is proportional to frequency.

Put that assumption in, redo the calculation, and it matched the measurements — not roughly, but to the decimal places.

Energy restricted to lumps like this is said to be quantised, and one such lump is a quantum.

Then why can we not see it

The size of a lump is the frequency multiplied by Planck's constant — and that constant is punishingly small.

A little further in

14 December 1900, the German Physical Society in Berlin. Planck, aged forty-two, presented the assumption. The date is now called the birthday of quantum mechanics. Nobody in the room that evening thought anything of the sort.

Planck himself did not believe it. He had spent his life on thermodynamics and was temperamentally conservative; he took the lumps for a device to make the arithmetic come out. For years afterwards he tried to fold the assumption back into classical physics. He failed.

It was an act of desperation. By nature I am peaceable and disinclined to questionable adventures.Planck, looking back in 1931

The one who went further

In 1905 a twenty-six-year-old patent clerk read Planck's assumption far more boldly. Planck had said energy is granular in the act of exchange. Einstein said light itself is granular.

His evidence was the photoelectric effect. Shine light on metal and electrons come off — except that red light, however intense, produces none, while blue light, however faint, produces them at once.

The claim was resisted for years. That light is a wave had been settled a century earlier by the double-slit experiment, and this sounded like an attempt to unsettle it. Even Planck, recommending Einstein for the Prussian Academy in 1913, added in effect that his overshooting on light quanta should not be held against him.

An experiment meant to refute it

Robert Millikan in America was certain Einstein was wrong. So he measured the photoelectric effect precisely for a decade — in order to show it.

By 1916 his measurements matched Einstein's equation exactly. Publishing them, he noted that he still did not accept the interpretation. The same measurements fixed the value of Planck's constant independently.

1. 1900Planck — assume energy comes in lumps, and the stove's colour follows
2. 1905Einstein — light itself is granular
3. 1913Bohr — the seats inside an atom are granular too
4. 1916Millikan — sets out to refute it and confirms it
5. 1921Einstein's Nobel — for the photoelectric effect, not relativity

That the Nobel citation was not for relativity is often remarked upon. Relativity was still contested at the time, while the photoelectric effect had been nailed down by Millikan's measurements.

The more interesting point lies elsewhere. The man who first insisted that quanta were real became quantum mechanics' most persistent opponent. That story arrives in the last article of this series.

The next question

If energy only moves in lumps, something must follow for whatever is doing the exchanging. If an electron in an atom cannot hold just any energy, then the places it can occupy are fixed too.

The question that remainsIf energy is a staircase, an electron may only stand on the rungs. So when it moves from one rung to another, how does it cross the gap between them? Or does it not cross at all?