The quanta of energy

Planck's stroke of genius (1900)

To save the theory, Max Planck makes a hypothesis that he himself considers desperate, almost a mathematical sleight of hand. Yet it is going to change everything.

He postulates that the energy exchanges between matter (the walls of the cavity) and radiation do not happen continuously, but in discrete packets, quanta of energy. Each quantum has an energy proportional to the frequency:

   E = h × f

where h is a new fundamental constant, the Planck constant (h ≈ 6,63 × 10^-34 J·s). The exchanged energy can only be hf, 2hf, 3hf... never an intermediate value.

Why this resolves the catastrophe

Here is the decisive idea. The high-frequency modes require quanta of large energy (E = hf large). But at a given temperature, the available thermal energy is limited. Emitting an ultraviolet quantum costs so much energy that, statistically, it almost never happens.

   Classical : each mode receives the same energy
               -> the UV modes (innumerable) absorb infinity of it  -> CATASTROPHE

   Planck    : a UV mode requires an ENORMOUS quantum hf
               -> too costly, it is almost always "frozen"
               -> the UV emission collapses  -> the curve FALLS back to zero

The high frequencies are therefore "frozen": they cannot be excited for lack of enough energy to pay for a whole quantum. The divergence disappears, and the theoretical curve falls back to zero in the UV — exactly as in the experiment.

Planck's law

With this hypothesis, Planck obtains a formula — Planck's law — that describes the black-body curve perfectly, at any temperature and any wavelength:

   intensity
      ^
      |          PLANCK curve
      |         .-''''---..        (matches the experiment
      |      .-'          '..        over the WHOLE spectrum)
      |   .-'                '-..
      | -'                       '---....____
      +------------------------------------------> λ

At long wavelengths, it recovers the classical law (which worked). At short ones, it falls back to zero (where the classical one failed). It also contains Wien's law. Everything fits together.

A revolution in spite of its author

Planck, a conservative, saw his quantization as a mere computational trick, without deep physical reality. It would take Einstein (1905, photoelectric effect) to assert that these quanta of light — the photons — are real. But the breach is open: energy, at the microscopic scale, is quantized.

In summary

Planck resolves the catastrophe by postulating that energy is exchanged in quanta E = hf, and not continuously. The high frequencies, requiring quanta that are too costly in energy, are "frozen": the curve falls back to zero in the UV. His Planck's law describes the black body perfectly over the whole spectrum. This is the birth certificate of quantum physics — even if its author only half believed in it.