Light Quanta and the Spatial Structure of Radiation.

Wolke. Phys. ZS 22, p. 375, 1921.

Starting from the conception of discrete independent light atoms, Einstein (1905) derived the spectral formula of W. Wien. Wolke (1913) showed that Planck’s formula can likewise be obtained from the hypothesis that radiant energy is distributed in the form of quanta \(h\nu\) among oscillations in space devoid of matter. From the polemic between Wolke and Krutkov (1914) it became clear that Wien’s formula is obtained if one assumes the spatial independence of light atoms, which holds in the case of low energy intensity (within the limits in which Wien’s formula is valid). If, however, one restricts oneself to acknowledging the independence of the existence of light atoms, while allowing their spatial combination, then one may arrive at Planck’s formula. (Similarly, Joffe (1911) indicated that, in order to obtain Planck’s formula from the hypothesis of radiation atoms, it is necessary to allow for the possibility of their association.) The present work of Wolke is devoted to the question of the spatial interrelation of light atoms at high energy densities.

As early as 1912, Ischiwara showed that Planck’s formula may be regarded as a sum of Wien formulas:

\[ u=\frac{8\pi h\nu^3}{c^3}\cdot\frac{1}{e^{\frac{h\nu}{kT}}-1} =\sum_{i=1}^{i=\infty} u_i,\quad \text{where } \quad u_i=\frac{8\pi h\nu^3}{c^3}\, e^{-\frac{i h\nu}{kT}} . \]

Calculating the sum of the entropies of the separate radiations \(u_i\), which are in thermodynamic equilibrium, Wolke shows that it is equal to the entropy of black radiation as calculated by Planck. Thus, radiation satisfying Planck’s formula may indeed be regarded as an aggregate of independent radiations with densities \(u_i\), where \(i=1,2,3,\ldots\). Einstein’s method, as applied to each such partial radiation, makes it possible to interpret the formulas obtained in the sense that the energy of the radiation corresponding to the index \(i\) consists of separate spatially independent elements whose magnitude is \(ih\nu\).

Thus, from Einstein’s point of view, monochromatic radiation in space must be regarded as composed of mutually spatially independent light molecules of the type \(h\nu, 2h\nu, 3h\nu,\ldots\). The relative number of more and less complex light molecules varies depending on the density of the radiation. With increasing density there occurs an association of light atoms into ever more complex light molecules, until, at very high density (the region of application of the Rayleigh-Jeans formula), the quanta merge into a continuum. Conversely, when the density decreases, “continuous” radiation dissociates into ever simpler light molecules, until it resolves into the simplest light atoms (the region of application of Wien’s formula).

G. S. Landsberg.