On a Heuristic Point of View Concerning the Generation and Transformation of Light

§ 6. Interpretation of the expression for the dependence of the entropy of monochromatic radiation on the volume according to Boltzmann’s principle.Show the German source

In § 4 we found for the dependence of the entropy of monochromatic radiation on the volume the expression:Show the German source $$S - S_0 = \frac{E}{\beta \nu} \lg \left( \frac{v}{v_0} \right).$$$S$Radiation entropy$S_0$Radiation entropy$E$Radiation energy in the band$\beta$Spectral constant B$\nu$Frequency of the light$v$Volume$v_0$VolumeExplain this equationShow the German source If one writes this formula in the form:Show the German source $$S - S_0 = \frac{R}{N} \lg \left[ \left( \frac{v}{v_0} \right)^{\frac{N}{R} \frac{E}{\beta \nu}} \right]$$$S$Radiation entropy$S_0$Radiation entropy$R$Gas constant$N$Avogadro's number$v$Volume$v_0$Volume$E$Radiation energy in the band$\beta$Spectral constant B$\nu$Frequency of the lightExplain this equationShow the German source and compares it with the general formula expressing Boltzmann’s principleShow the German source $$S - S_0 = \frac{R}{N} \lg W ,$$$S$Radiation entropy$S_0$Radiation entropy$R$Gas constant$N$Avogadro's number$W$Probability of the configurationExplain this equationShow the German source one arrives at the following conclusion:Show the German source

If monochromatic radiation of frequency $\nu$ and energy $E$ is enclosed in the volume $v_0$ (by reflecting walls), the probability that at an arbitrarily chosen instant of time the whole radiation energy is in the partial volume $v$ of the volume $v_0$ is:Show the German source $$W = \left( \frac{v}{v_0} \right)^{\frac{N}{R} \frac{E}{\beta \nu}}.$$$W$Probability of the configuration$v$Volume$v_0$Volume$N$Avogadro's number$R$Gas constant$E$Radiation energy in the band$\beta$Spectral constant B$\nu$Frequency of the lightExplain this equationShow the German source

From this we further conclude:Show the German source

Monochromatic radiation of low density (within the range of validity of Wien’s radiation formula) behaves in thermal respects as if it consisted of mutually independent energy quanta of magnitude $R \beta \nu / N$.Show the German source

We shall also compare the mean magnitude of the energy quanta of “black radiation” with the mean vis viva of the center-of-mass motion of a molecule at the same temperature.Show the German source The latter is $\frac{3}{2} (R/N) T$, while for the mean magnitude of the energy quantum one obtains, on the basis of Wien’s formula:Show the German source $$\frac{\int_0^\infty \alpha \nu^3 e^{-\frac{\beta \nu}{T}} d \nu}{\int_0^\infty \frac{N}{R \beta \nu} \alpha \nu^3 e^{-\frac{\beta \nu}{T}} d \nu} = 3 \frac{R}{N} T.$$$\alpha$Spectral constant A$\nu$Frequency of the light$\beta$Spectral constant B$T$Temperature of the radiation$N$Avogadro's number$R$Gas constantExplain this equationShow the German source

If, now, monochromatic radiation (of sufficiently small density) behaves, as regards the dependence of the entropy on the volume, like a discontinuous medium consisting of energy quanta of magnitude $R \beta \nu / N$, it is natural to investigate whether the laws of the generation and transformation of light are also such as they would be if light consisted of energy quanta of this kind.Show the German source We shall occupy ourselves with this question in what follows.Show the German source

Bern, 17 March 1905.Show the German source

(Received 18 March 1905.)Show the German source