The spectrum of a glowing body — the crack where energy first came in lumps. Classical physics predicted a hot cavity would blaze with infinite ultraviolet light (the catastrophe). Planck cured it with one quantum device: a single −1 in the denominator that makes the curve curl under and its total energy finite. Down the center, data flows: temperature goes in, the engine computes the blackbody curve, the proven spectrum comes out. The blue team builds and defends it; the red team tries to break it.
source M. Planck, Ueber das Gesetz der Energieverteilung im Normalspectrum (1901), Ann. Phys. 309, 553–563 — doi.org/10.1002/andp.19013090310. Rendered, not quoted.
Spectral energy density of a cavity at temperature T:
u(ν,T) = (8π h ν³ / c³) · 1/(e^{hν/kBT} − 1)
The Rayleigh–Jeans oscillator gets kBT of energy no matter the frequency — so uᴿᴶ = (8π ν²/c³) kBT climbs forever. Planck forced energy to arrive in packets E = nhν; the mean occupation of such an oscillator is the Bose factor 1/(e^{hν/kBT}−1). That single −1 is the whole cure.
Live, at the current T: Planck vs the classical curve at three frequencies —
| ν (Hz) | u Planck | u Rayleigh–Jeans |
|---|
Planck 1901 births the constant h: he quantized only the exchange of energy, calling it an act of desperation, and did not believe light itself was grainy.
Four years on, the photoelectric effect takes Planck's bookkeeping quantum and makes it a particle — the photon, energy hν, threshold φ/h. Planck's −1 that cured the UV catastrophe is the same h that then knocks electrons loose. Each sphere is the next one's premise.
The blue team's live re-check: recompute the dimensionless integral ∫x³/(eˣ−1)dx (must equal π⁴/15) and the fixed-cutoff doubling ratio (must stay ≈1, proving convergence). If red drops the −1, both fail and this badge turns red.
One input governs the whole spectrum: the absolute temperature 5772 K. Exact laws (SI):
h = 6.62607015e−34 J·s · kB = 1.380649e−23 J/K · c = 2.99792458e8 m/s
ℏ = h/2π · mₑ = 9.109e−31 kg · e = 1.602e−19 C
Only h, kB and c enter Planck's law. Feed T into the engine below.
Solid red = Planck. Dashed blue = classical Rayleigh–Jeans (shoots off the top: the UV catastrophe). Dashed grey = Wien tail. The dot marks the peak, and it slides right with T (Wien displacement).
Every number is integrated live from u(ν,T) — nothing is looked up.
What the machine proves, live: u(ν,T) is positive and finite for every ν>0; the low-frequency limit recovers Rayleigh–Jeans; the high-frequency tail falls exponentially (Wien) so the total is finite while the classical integral diverges; the total energy obeys Stefan–Boltzmann T⁴ — doubling T multiplies it by 16; and the peak frequency is linear in T.
The blue team's witness (left) confirms these live; the red team (right) tries to make the total diverge again.
It also treats the field as a continuum of independent modes at one temperature — real bodies have frequency-dependent emissivity (they are "grey", not "black"). The law is exact for the idealization; the idealization is the assumption a critic attacks.
"Planck believed light comes in particles." Cut. He quantized only energy exchange and resisted light-quanta for years; Einstein 1905 made light itself grainy.
"The peak wavelength and peak frequency are the same colour." Cut. They are not: ν-peak sits at x≈2.821, λ-peak at x≈4.965. Plotting per-frequency vs per-wavelength moves the maximum. The engine reports the ν-peak, labelled.
"Rayleigh–Jeans was just sloppy classical work." Kept, corrected. It is the rigorous classical prediction — equipartition applied honestly — which is exactly why its ∫ν²dν infinity was a crisis, not a bug.
The red team's move: drop the quantum −1. Replace the Bose factor with the classical equipartition value kBT per mode (Rayleigh–Jeans). The occupation stops falling at high ν — and ∫ν²dν diverges. The catastrophe returns; the witness (window 7) recomputes and turns red.
Drop the −1 and the total energy has no finite value — the witness disagrees with π⁴/15, the doubling-range ratio jumps to ~8, and it goes red. Nothing is faked; the attack is real and it is caught.