Results · §6 · the heuristic correspondence

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Radiation that behaves as if it consisted of energy quanta

Read with Boltzmann's principle, the entropy of monochromatic radiation gives the probability that all its energy is found in a part $v$ of the volume $v_0$ , and that probability has the form it has for independent points; so radiation of low density, where Wien's formula holds, behaves in thermal respects as if it consisted of mutually independent energy quanta of magnitude $R\beta\nu/N$ .

As printed

$W = \left( \frac{v}{v_0} \right)^{\frac{N}{R} \frac{E}{\beta \nu}}.$ Probability of the configurationVolumeVolumeAvogadro's numberGas constantRadiation energy in the bandSpectral constant BFrequency of the lightExplain this equation
Monochromatische Strahlung von geringer Dichte (innerhalb des Gültigkeitsbereiches der Wienschen Strahlungsformel) verhält sich in wärmetheoretischer Beziehung so, wie wenn sie aus voneinander unabhängigen Energiequanten von der Größe $R \beta \nu / N$ bestünde.

Display on page 143 of the German source, § 6, display 4 · Text on page 143 of the German source, § 6, paragraph 6

In modern notation

\Delta S_{\mathrm{rad}} = \frac{E}{B\,\nu}\,\ln\left(f\right)

\Delta S_{\mathrm{gas}} = n\,k_B\,\ln\left(f\right)

n_{\mathrm{eff}} = \frac{E}{k_B\,B\,\nu}

\frac{E}{n_{\mathrm{eff}}} = k_B\,B\,\nu = h\,\nu

Premise: Only for radiation of low density, within the range of validity of Wien's radiation formula; the sentence itself says so.
An inference beyond the argument: A heuristic conclusion from one thermal property, the dependence of the entropy on the volume. The paper next asks whether the laws of the generation and transformation of light are also such as they would be if light consisted of these quanta, and takes that question up in §§ 7 to 9.

Decoder

$W$: The probability that, at an arbitrarily chosen instant, the whole energy $E$ of radiation enclosed in the volume $v_0$ by reflecting walls is found in the part $v$ of it.
$\frac{N}{R}\frac{E}{\beta\nu}$: The exponent of $v/v_0$ : the energy $E$ divided by $R\beta\nu/N$ . For $n$ independent moving points of a gas the same probability is $(v/v_0)^n$ , which is the comparison the paper makes.
$R\beta\nu/N$: The magnitude of each energy quantum. $R/N$ is the gas constant per molecule and $\beta$ the constant of Wien's formula, so the product is what later texts write $h\nu$ ; the paper does not use the letter $h$ .
"wie wenn": As if: the sentence states how the radiation behaves in its thermal properties, not what light is made of.

Try it

LQ-06: Compare the radiation's entropy-volume law with that of a gas of independent points: how many points does the radiation behave like, and how much energy would each carry? Choose the preset “The Move (§6 Golden State)” there.

Where this stops: E/(hν) is a pure number that need not be a whole number, and it is never rounded to one. This does not derive Planck’s law or establish a general theory outside the Wien regime. Extending the picture to individual emission and absorption processes is a further hypothesis. Read the argument

Common wrong turns

Argument: A heuristic conclusion from a comparison
Model: Stated within the range of validity of Wien's formula
Evidence: An inference, not evidence
History: Introduced in this paper

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§6 · the heuristic correspondence

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