Field · Emerged 1900 – 1924

Old Quantum Theory

Why does energy come in lumps, and how can light be both a wave and a stream of particles?

5 chapters4 min read6 turning points 1 open problem

Branched from: Statistical Mechanics + Electromagnetism
Branched into: Astronomical Spectroscopy + Quantum Mechanics
Figures: Max Planck, Albert Einstein, Hans Geiger, Ernest Marsden, Ernest Rutherford, Niels Bohr, Robert Millikan, Arthur Compton

In brief

The old quantum theory is the set of rules, part inspired and part improvised, that physicists used between 1900 and 1925 to explain phenomena classical physics could not. Its central idea is that energy is exchanged in discrete packets, quanta, whose size is set by a new constant of nature, Planck's constant $h$ .

It began when statistical mechanics and electromagnetism together predicted, absurdly, that a hot oven should radiate infinite energy. Planck's fix, Einstein's light quanta and Bohr's atom each worked brilliantly for particular problems. None of them made sense together, and the contradictions forced the creation of quantum mechanics in 1925.

Key ideas

Quantum of energyEnters 1900: Energy of light of frequency $f$ is exchanged only in multiples of $E = hf$ , where $h \approx 6.63 \times 10^{-34}$ J·s is Planck's constant.
PhotonEnters 1905: A particle of light, carrying energy $hf$ . Einstein proposed it in 1905. It was widely accepted only after Compton's experiment in 1923.
Wave–particle dualityEnters 1923: Light spreads and interferes like a wave, yet delivers its energy in particle-like lumps. The old quantum theory had no way to reconcile the two.
Stationary statesEnters 1913: In Bohr's model, electrons in atoms occupy only certain allowed orbits with fixed energies. Light is emitted when an electron jumps between them.
Atomic nucleusEnters 1909 – 1911: Almost all of an atom's mass is concentrated in a positive nucleus about 100,000 times smaller than the atom itself.

Chapter I

An Act of Desperation

Around 1900, statistical mechanics and electromagnetism combined to give a prediction that was obviously wrong. A hot oven's light is a set of electromagnetic waves, and statistical mechanics shares energy equally among all of them. But there are infinitely many possible short waves, so the oven should radiate infinite energy in the ultraviolet. Real ovens glow red, then white, and radiate a finite amount.

In October 1900 Max Planck, a conservative physicist in Berlin, found a formula that matched the measurements exactly. To derive it, which he did in December, he had to assume that the oven's walls exchange energy with light only in lumps of size $hf$ , proportional to the frequency $f$ . High-frequency lumps are too expensive to be produced often, so the ultraviolet catastrophe disappears. Planck was not in fact aiming at that problem, which Rayleigh and Jeans spelled out only in 1900–1905, but his formula removes it. In 1931 Planck called it an act of desperation. He expected the lumps to be a device that would go away.

Chapter II

Light Quanta

In 1905 Albert Einstein took the lumps literally. Light itself, he proposed, is made of quanta. He predicted how electrons should be knocked out of metal by light: the electrons' energy should depend on the light's colour, not its brightness. Hardly anyone believed it. Robert Millikan spent a decade trying to refute it and in 1916 confirmed it precisely. In 1923 Arthur Compton bounced X-rays off electrons and found they recoiled like colliding particles. Light was a wave, as a century of interference experiments showed, and also a stream of particles. Nobody knew how to make sense of that.

Chapter III

The Quantum Atom

Meanwhile the atom had acquired a nucleus. In 1909 Hans Geiger and Ernest Marsden, working for Ernest Rutherford in Manchester, found alpha particles bouncing back from gold foil, and Rutherford concluded in 1911 that atoms have a tiny, heavy nucleus. But by classical physics, an electron orbiting a nucleus should radiate and spiral inwards in a fraction of a second. Atoms should not exist.

In 1913 Niels Bohr simply declared that electrons can occupy only certain orbits, where they do not radiate, and emit light only when they jump between them. The energies of hydrogen's orbits come out as $-13.6/n^2$ electronvolts. A jump from orbit 3 to orbit 2 releases $13.6 \times (\tfrac14 - \tfrac19) \approx 1.89$ eV, a photon of wavelength 656 nanometres, exactly hydrogen's red spectral line. Bohr's rules reproduced the whole spectrum, but they were rules without a reason, and for atoms with more than one electron they failed.

Chapter IV

A Closer Look: Colour, Not Brightness

A photon's energy is $E = hf = hc/\lambda$ . With $hc \approx 1240$ electronvolt-nanometres, a photon's energy in electronvolts is 1240 divided by its wavelength in nanometres:

Light	Wavelength	Photon energy
Red	700 nm	1.77 eV
Green	530 nm	2.34 eV
Violet	400 nm	3.10 eV

To free an electron from sodium takes about 2.3 eV, its work function. Einstein's equation says the fastest electrons leave with energy

E_{\max} = hf - W .

Shine red light on sodium and no electrons come out at all, however bright the light, because no single red photon carries 2.3 eV, and electrons absorb photons one at a time. Violet light releases electrons with up to $3.10 - 2.3 \approx 0.8$ eV, and making it brighter releases more electrons, but none faster. Green light, at 2.34 eV, just barely frees them.

In the wave picture this makes no sense. A brighter wave carries more energy and should shake electrons loose eventually, whatever its colour. The experimental facts, a sharp colour threshold and electron energies set by colour alone, were exactly what Einstein predicted. Millikan's measurements of $E_{\max}$ against frequency formed a straight line whose slope gave Planck's constant to within about one per cent of today's value.

Chapter V

Contradictions

By 1924 the old quantum theory was a patchwork: Planck's lumps, Einstein's photons, Bohr's orbits, and rules for when to use each. It could not explain helium, the intensities of spectral lines or how an electron chooses when to jump. In 1924 Louis de Broglie proposed that if waves can act as particles, particles such as electrons should also act as waves. Within two years the patchwork was replaced by quantum mechanics.

Turning points6

1900DISCOVERY
Planck's quantum
Classical physics predicts that a hot body should radiate unlimited energy at high frequencies. Max Planck finds a formula that fits the measured spectrum exactly, and to justify it assumes that energy is absorbed and emitted in lumps proportional to frequency. He called it "an act of desperation" and thought it a mathematical device. It introduced Planck's constant, $h$ .
Max Planck
› Sources (2)
- Planck, M. (1901). Über das Gesetz der Energieverteilung im Normalspectrum. Annalen der Physik 309(3): 553–563.
- Kuhn, T. S. (1978). Black-Body Theory and the Quantum Discontinuity, 1894–1912. Oxford University Press.
1905DISCOVERY
Einstein's light quanta
Albert Einstein proposes that light itself consists of quanta, and uses the idea to predict the photoelectric effect: the energy of electrons knocked out of a metal should depend on the light's colour, not its brightness. Almost no one believed it for years. His law of the photoelectric effect was the work named when he received the 1921 Nobel prize.
Albert Einstein
› Sources (1)
- Einstein, A. (1905). Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt. Annalen der Physik 17: 132–148.
1909 – 1911EXPERIMENT
The atomic nucleus
Hans Geiger and Ernest Marsden fire alpha particles at thin gold foil, and a few bounce almost straight back. Ernest Rutherford said it was as if a 15-inch shell had bounced off tissue paper. In 1911 he concludes that an atom's positive charge and mass are packed into a tiny nucleus. Classical physics then predicted that orbiting electrons would spiral into it within a fraction of a second.
Hans Geiger, Ernest Marsden, Ernest Rutherford
› Sources (1)
- Rutherford, E. (1911). The scattering of α and β particles by matter and the structure of the atom. Philosophical Magazine 21: 669–688.
1913DISCOVERY
Bohr's atom
Niels Bohr postulates that the electron in hydrogen can occupy only certain orbits and radiates only when jumping between them. The rule reproduces the exact colours of hydrogen's spectral lines, a pattern known since Balmer's formula of 1885 but never explained. The model worked, though it contradicted the physics it was built on.
Niels Bohr
› Sources (1)
- Bohr, N. (1913). On the constitution of atoms and molecules. Philosophical Magazine 26: 1–25.
1916EXPERIMENT
Millikan confirms the photoelectric equation
Robert Millikan spends ten years trying to show that Einstein's photoelectric equation is wrong. His precise measurements confirm it in every detail and give an accurate value of Planck's constant. He still doubted the idea of light particles behind it.
Robert Millikan
› Sources (1)
- Millikan, R. A. (1916). A direct photoelectric determination of Planck's h. Physical Review 7(3): 355–388.
1923EXPERIMENT
Compton scattering
Arthur Compton scatters X-rays off electrons and finds that the X-rays lose energy exactly as if they were particles colliding like billiard balls, with momentum $h/\lambda$ . After this, most physicists accepted that light has a particle character.
Arthur Compton
› Sources (1)
- Compton, A. H. (1923). A quantum theory of the scattering of X-rays by light elements. Physical Review 21(5): 483–502.

Applications

Where it is used

Vision↗ Biology · Systems Neuroscience
The eye can count photons
In 1942 Hecht, Shlaer and Pirenne showed that a dark-adapted human can see a flash of only a few photons, and that a single rod cell responds to one photon. Vision is a quantum measurement, and its sensitivity is limited by the statistics of photon arrival.
› Sources (1)
- Hecht, S., Shlaer, S. & Pirenne, M. H. (1942). Energy, quanta, and vision. Journal of General Physiology 25(6): 819–840.
Energy
The limit on solar cells
A solar cell turns each photon above a threshold energy into one electron, and wastes the excess energy as heat. Shockley and Queisser used this to show that a single-junction cell cannot exceed about 33% efficiency, a limit the best silicon cells now approach.
› Sources (1)
- Shockley, W. & Queisser, H. J. (1961). Detailed balance limit of efficiency of p–n junction solar cells. Journal of Applied Physics 32(3): 510–519.

Open problems

Where the map runs out

Open

Why is the fine-structure constant about 1/137?

Open as of 2026. Its value is measured to better than one part in a billion but not explained.

Arnold Sommerfeld introduced the fine-structure constant $\alpha$ in 1916 to explain small splittings in hydrogen's spectral lines. It measures the strength of the electromagnetic force, is a pure number with no units, and equals $1/137.035999\ldots$ No theory predicts that value.

Why it is hard: In the Standard Model, $\alpha$ is an input, measured rather than derived, like the masses of the particles. Explaining it would need a deeper theory from which the constants follow, and no accepted theory does that. Numerological "derivations" have a long and unhappy history, including one by Arthur Eddington.
What resolving it unlocks: An explanation of why chemistry, and therefore life, is possible: a few per cent change in $\alpha$ would change how stars make carbon and how atoms bond.

› Sources (1)

Feynman, R. P. (1985). QED: The Strange Theory of Light and Matter, ch. 4. Princeton University Press.