1900 – 1927 · Berlin · Bern · Copenhagen · Göttingen · Zürich
For a quarter of a century, every time physicists looked closely enough at light, heat, or atoms, the smooth continuum they expected shattered into discrete lines and lumps. This is the chain of experiments that forced it — and the single idea waiting underneath at the end.
Every step of the revolution was the same move. A classical theory predicted a smooth, continuous quantity; an experiment showed sharp steps or wave interference instead; and the cure was to quantize one more thing everyone had assumed was continuous. The punchline of the final step is why — underneath, everything is a wave, and “quantized” just means a wave that has to fit.
Read it as a relay. Each runner is handed a paradox by the last, quantizes something to fix it, and passes a sharper paradox forward. By the sixth, the thing being quantized stops looking like a patch and starts looking like the nature of matter itself.
Heat anything until it glows — an iron bar, a star, the filament of a lamp — and it emits a smooth rainbow of light whose shape depends only on temperature, not on what it's made of. Physics could explain almost all of that curve. The trouble was one end of it.
Measure how much light of each frequency \(\nu\) a hot body radiates. The curve rises, peaks, and falls — peaking bluer as it gets hotter. A clean, universal shape begging for a formula.
Treat the radiation as countless vibrating modes, each holding \(k_BT\) of energy (equipartition). But there are ever more modes at higher \(\nu\), so the predicted energy climbs without limit — infinite energy. The "ultraviolet catastrophe."
A mode of frequency \(\nu\) can only hold energy in whole steps of size \(h\nu\): \(E=nh\nu\). High-frequency modes can't afford even one step, so they stay dark. The new constant \(h\) was born.
Classical equipartition gives every electromagnetic mode an average energy \(\langle E\rangle = k_BT\). With the density of modes \(\propto \nu^2\), the Rayleigh–Jeans law follows: \(u(\nu)\,d\nu \propto \nu^2 k_BT\,d\nu\), which diverges as \(\int \nu^2\,d\nu\to\infty\).
Planck instead allowed each mode only the energies \(E_n=nh\nu\). Boltzmann-weighting them, \(p_n\propto e^{-nh\nu/k_BT}\), the average energy becomes a geometric sum: \[ \langle E\rangle = \frac{\sum_n nh\nu\,e^{-nh\nu/k_BT}}{\sum_n e^{-nh\nu/k_BT}} = \frac{h\nu}{e^{h\nu/k_BT}-1}. \] For \(h\nu\ll k_BT\) this returns the classical \(k_BT\); for \(h\nu\gg k_BT\) it collapses to \(\approx h\nu\,e^{-h\nu/k_BT}\to 0\). High-frequency modes freeze out, and \(u(\nu)\propto \nu^3/(e^{h\nu/k_BT}-1)\) — the Planck law — peaks and returns to zero. Planck called \(h\) "an act of desperation"; he did not yet believe energy was truly discrete.
Planck thought \(h\) was a bookkeeping trick about how matter trades energy. Einstein took it literally and aimed it at a puzzle nobody could crack: shine light on a metal and electrons fly off — but which light does it, and how hard, made no sense as a wave.
Light hits a metal; electrons are ejected. Vary the light's color (frequency \(\nu\)) and its brightness (intensity). Measure how many electrons come out and how fast.
Light is a wave; a brighter wave carries more energy, so any color should eventually shake electrons loose, and brighter light should make them faster. Wait long enough and even dim red light should work.
Light arrives as grains — photons — each carrying \(E=h\nu\). One photon kicks one electron. Below a threshold frequency a grain is simply too small, and no amount of them helps. The energy of escapees is \(h\nu - W\).
Einstein's relation: the maximum kinetic energy of an ejected electron is \[ K_{\max}=h\nu - W, \] where \(W\) is the work function (the energy binding the least-bound electron to the metal). Three consequences contradicted the wave picture and were all confirmed by Millikan by 1916: (i) a sharp threshold frequency \(\nu_0=W/h\) below which nothing is emitted regardless of intensity; (ii) \(K_{\max}\) grows linearly with \(\nu\), with slope exactly \(h\) — and independent of intensity; (iii) intensity sets only the number of electrons (the current), not their energy. This is the experiment Einstein's 1921 Nobel Prize cited — not relativity — and the first time \(h\) was claimed to describe light itself, not just matter's exchange of it.
Rutherford had just shown the atom is mostly empty: a tiny dense nucleus with electrons somewhere outside. Picture it as a little solar system — and it should die instantly. Yet heated hydrogen doesn't smear out light; it emits a handful of razor-sharp colored lines, the same ones every time.
Pass light from glowing hydrogen through a prism. Instead of a continuous rainbow you see a few bright lines at fixed wavelengths — 656, 486, 434, 410 nm — captured decades earlier by the empirical Rydberg formula.
An orbiting electron accelerates, so it must radiate, lose energy, and spiral into the nucleus in \(\sim\!10^{-11}\) s — emitting an ever-rising smear of all frequencies on the way down. Atoms shouldn't exist, and their light should be continuous.
Only certain orbits are allowed — those with quantized angular momentum \(L=n\hbar\). On them the electron simply doesn't radiate. Light is emitted only when it jumps between them: \(h\nu=E_i-E_f\). Discrete jumps, discrete lines.
Bohr's quantization condition \(L=m_e v r = n\hbar\) (with \(n=1,2,3,\dots\)) selects orbits with energies \[ E_n=-\frac{m_e e^4}{8\varepsilon_0^2 h^2}\,\frac{1}{n^2}=-\frac{13.6\ \text{eV}}{n^2}. \] A jump from level \(n_i\) to \(n_f\) emits a photon of energy \(E_{n_i}-E_{n_f}\), reproducing the Rydberg formula exactly: \[ \frac{1}{\lambda}=R_\infty\!\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right). \] The visible Balmer series is every jump landing on \(n_f=2\): \(3\!\to\!2\) gives 656 nm (red), \(4\!\to\!2\) 486 nm, \(5\!\to\!2\) 434 nm, \(6\!\to\!2\) 410 nm. Jumps to \(n_f=1\) (Lyman) are ultraviolet; to \(n_f=3\) (Paschen) infrared. The model nailed hydrogen but couldn't explain why only these orbits — that answer is Step 4.
Bohr's rule worked but was a naked guess: why is angular momentum quantized? De Broglie's answer was audacious and simple — if light, long thought a wave, can act like particles, then particles long thought solid might act like waves. Give the electron a wavelength, and Bohr's mystery dissolves.
Davisson and Germer fired electrons at a nickel crystal and got a diffraction pattern — bright and dark rings, the unmistakable signature of waves interfering. Electrons, supposedly tiny billiard balls, were diffracting like light.
Bohr's \(L=n\hbar\) had no derivation. Why integers? Why those radii and no others? The condition was pulled from thin air to match the spectrum.
Matter has a wavelength \(\lambda=h/p\). A stable orbit must hold a whole number of wavelengths — the wave has to meet itself in phase after one lap: \(2\pi r=n\lambda\). Substitute and out pops \(L=n\hbar\), for free.
The standing-wave condition on a circular orbit is \(2\pi r = n\lambda\) for integer \(n\): the only way the wave returns to its starting phase after one trip around. Insert de Broglie's \(\lambda = h/p = h/(m_e v)\): \[ 2\pi r = n\frac{h}{m_e v}\;\Longrightarrow\; m_e v r = n\frac{h}{2\pi}=n\hbar. \] That is exactly Bohr's quantization condition \(L=n\hbar\) — now derived, not postulated. The "allowed orbits" are simply the resonances of a confined wave, like the harmonics of a string fixed at both ends. For any non-integer number of wavelengths the wave interferes destructively with itself over many laps and cannot persist — which is why those orbits are forbidden. This is the hinge of the whole story: quantization is what waves do when you confine them.
De Broglie's standing wave explained one atom's circular orbits — but it was a picture, not a law. If matter really is a wave, something must govern that wave everywhere: in helium, in molecules, in any trap you build. Schrödinger wrote that equation, and something remarkable fell out of it: you no longer have to assume quantization anywhere. Demand only that a trapped wave die away outside its trap, and the equation has solutions at a few special energies — and at no others.
The "old quantum theory" was a grab-bag of patches: Bohr's rule nailed hydrogen but failed for helium, said nothing about how bright each line is, and de Broglie's "fit around the ring" only worked for circles. Physics had rules without a mechanics.
A wave needs an equation of motion — what Maxwell's equations are to light, or \(F=ma\) is to planets. Nothing said how the electron's wave \(\psi\) behaves in an arbitrary force field, or even what \(\psi\) physically is.
\(\hat H\psi = E\psi\): energy bookkeeping rewritten for waves. For a trapped wave — one that must vanish far away — solutions exist only at discrete energies \(E_1, E_2,\dots\) Quantization stops being an assumption and becomes a solvability condition. Born added the meaning: \(|\psi|^2\) is the probability of finding the particle.
Take de Broglie seriously: a free particle of momentum \(p\) is a wave \(\psi\sim e^{ikx}\) with \(p=\hbar k\) (that's \(\lambda=h/p\) restated). Note that differentiating extracts the momentum: \(-i\hbar\,\partial_x\psi = \hbar k\,\psi = p\,\psi\). So wherever classical mechanics uses \(p\), let the wave theory use the operator \(-i\hbar\,\partial_x\). Feeding that into the energy balance \(E = \frac{p^2}{2m}+V(x)\) turns a number equation into a wave equation:
\[ -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\,\psi \;=\; E\,\psi . \]
Each term is the old bookkeeping in new clothes: the derivative term is kinetic energy (a sharply curved wave = short wavelength = large \(p\)), \(V\psi\) is potential energy, and \(E\) is the total. For a bound particle we must also demand \(\psi\to 0\) far away (the electron is actually in the trap; the total probability must be finite). That boundary condition is everything: inside the well the solution oscillates, outside it is a mix of a decaying and an exploding exponential — and only at special energies \(E_n\) does the exploding piece cancel on both sides at once. The widget above is literally performing this "shooting method." Solve the same equation in three dimensions with the Coulomb potential \(V=-e^2/4\pi\varepsilon_0 r\) and the allowed energies come out \(E_n=-13.6\,\text{eV}/n^2\) — Bohr's ladder, now a theorem. Heisenberg, Born and Jordan had already reached equivalent rules in 1925 via matrix mechanics; Schrödinger proved the two formalisms identical. And in 1926 Born supplied the missing meaning of \(\psi\): it is not a smeared-out electron — \(|\psi(x)|^2\,dx\) is the probability of finding the whole electron near \(x\). That reading is tested in the strip at the bottom of the widget, and put on trial in Step 6.
Everything so far says matter is a wave that has to fit. But look at any detector: an electron always arrives as one sharp dot, never as a smear. The double slit is where the two truths collide — dots land one at a time, at unpredictable places, yet thousands of dots quietly draw the wave's interference stripes. This one image contains superposition, the Born rule, and what measurement does.
Fire electrons one at a time at a wall with two narrow slits and record where each lands. Every electron arrives as a single point — but the accumulated points form bright and dark fringes, with dark bands where no electron ever lands.
Bullets would pile up in one broad heap behind the slits — no dark bands. A classical wave would make fringes, but as a continuous smear arriving everywhere at once, never as lone dots. Nothing classical produces dots whose statistics are stripes.
Each electron's wave \(\psi\) passes through both slits and interferes with itself; \(|\psi|^2\) on the screen sets the odds for where that one dot lands. And if you watch which slit it used, you force the wave through one slit — the cross-term dies and the stripes vanish.
Call \(\psi_1\) and \(\psi_2\) the waves reaching a screen point \(y\) from slit 1 and slit 2. The path from the farther slit is longer by \(\delta \approx d\,y/L\) (slit separation \(d\), screen distance \(L\)), so \(\psi_2\) lags by a phase \(\varphi = 2\pi\delta/\lambda = 2\pi d y/\lambda L\). Quantum mechanics adds amplitudes, not probabilities:
\[ P(y) \;=\; |\psi_1+\psi_2|^2 \;=\; |\psi_1|^2+|\psi_2|^2+2\,\mathrm{Re}(\psi_1^*\psi_2) \;\propto\; \cos^2\!\frac{\pi d y}{\lambda L}, \]
maxima where the paths differ by a whole number of wavelengths, spaced \(\Delta y = \lambda L/d\) apart (the widget's geometry gives exactly 45 px). The last term — the interference cross-term — exists only while both paths are indistinguishable. Mark which slit the electron used (any which-path measurement, however gentle in intent) and the outcomes become distinguishable alternatives, so the probabilities add instead: \(P = |\psi_1|^2+|\psi_2|^2\), a smooth stripeless hump. Heisenberg's 1927 uncertainty relation \(\Delta x\,\Delta p \ge \hbar/2\) says the same thing mechanically: locating the electron to within \(d\) at the slits necessarily disturbs its transverse momentum by \(\gtrsim \hbar/d\) — precisely enough to smear the pattern by one full fringe. A historical honesty note: in 1927 the electron-wave evidence was Davisson–Germer's crystal diffraction (and G.P. Thomson's foils); the two-slit experiment itself was the founders' thought experiment, first done with electrons by Jönsson in 1961, and one-electron-at-a-time — dots visibly building fringes, exactly as in this widget — by Tonomura's team in 1989. It ran precisely as predicted in 1927.
Notice the quantity being chopped into steps marches inward — from the energy radiated by warm matter, to light, to the atom's insides, and finally to matter itself — until the last step reveals they were never separate fixes at all.
So the kernel holds all the way down. Six times, a continuum cracked into steps; six times the fix was to quantize. Only at the end does the reason surface: steps are what waves do when they're confined. A guitar string can't sound any pitch, only the ones that fit its length — and an electron bound to an atom is exactly that, a wave that has to close on itself. Quantum mechanics is the bookkeeping of confined waves.
A spectral timeline · built kernel-first · every number herein verified against the Planck, Rydberg, and Schrödinger relations
palette drawn from the true sRGB colors of the hydrogen Balmer lines — 656, 486, 434, 410 nm