Quantisation as an Eigenvalue Problem
An atom's electron is a standing wave — and the quantum numbers are just how many half-waves fit.
Trap a wave in a box and only certain shapes fit — that single fact is where the “quantum” in quantum physics comes from.
The big idea
By 1926 physicists knew atoms were strange: an electron could sit only at certain energies, and jumped between them, but nobody knew why. Schrödinger's answer was to stop picturing the electron as a tiny ball and start picturing it as a wave. And waves, when you pen them in, are picky — a guitar string fixed at both ends can vibrate only in a whole number of loops: one loop, two, three, never two-and-a-half.
His equation does the same thing for the electron. Confine it to an atom and only certain wave shapes “fit”. Each fitting shape has its own energy, so the allowed energies come out as a discrete set — automatically. The mysterious quantum numbers turn out to be nothing more exotic than how many loops the wave has. The whole numbers were never put in by hand; they fall out of fitting a wave into a space.
How it came about
The idea had a spark. In 1924 the young French physicist Louis de Broglie suggested that matter, like light, has a wavelength. Einstein took it seriously, and so did Schrödinger, who over the Christmas holiday of 1925–26 — at a Swiss alpine resort — worked out the wave equation that would carry his name. Within months he had solved the hydrogen atom and published a torrent of papers.
He was not alone. A year earlier Werner Heisenberg, with Max Born and Pascual Jordan, had built a rival “matrix mechanics” that got the same answers by manipulating arrays of numbers — abstract and, to many, unlovely. Schrödinger's waves felt visualisable and familiar, and physicists flocked to them; he then proved the two theories were secretly the same. But the comfort was misleading: Max Born showed the wave does not tell you where the electron is, only the probability of finding it there. Schrödinger disliked this so much that in 1935 he dreamed up his famous cat to mock it.
Why it mattered
This equation is, quite simply, how we calculate the world of atoms. Every chemical bond, the shape of every molecule, why neon glows red and sodium glows yellow, how a semiconductor carries current, how a laser works — all of it is, underneath, a solution of Schrödinger's equation. It turned chemistry from a catalogue of recipes into something you can compute, and a century later it remains the daily tool of physicists and chemists.
A way to picture it
Think of a guitar string. Pluck it and it can only sing certain notes — a fundamental and its overtones — because only a whole number of half-waves fits between the two fixed ends. You cannot have a fraction of a loop; the string won't allow it. Schrödinger's insight was that an electron trapped in an atom is just like that string. Its “notes” are the allowed energy levels, and the quantum numbers are the overtone numbers. Atoms have a discrete set of energies for the same reason a string has a discrete set of pitches.
Where it sits
Planck (1900) and Einstein (1905) had shown that light comes in lumps; Bohr (1913) guessed that atoms have fixed energy levels but could not say why. Schrödinger and Heisenberg supplied the why, founding quantum mechanics proper. The line runs on to Dirac, who added relativity and predicted antimatter, and to the transistor and the laser — and the same equation now underlies the qubits of quantum computers. In this Library it sits between Planck's quantum and the modern physics of Higgs and gravitational waves.
In this paper, I wish to consider, first, the simple case of the hydrogen atom (nonrelativistic and unperturbed), and show that the customary quantum conditions can be replaced by another postulate, in which the concept of ‘whole numbers’, merely as such, is not introduced. Rather, when integralness does appear, it arises in the same natural way as it does in the case of node-numbers of a vibrating string.