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

§ 1. On a difficulty concerning the theory of “black radiation”.Show the German source

We first take up the standpoint of Maxwell’s theory and the electron theory and consider the following case.Show the German source In a space enclosed by perfectly reflecting walls, let there be a number of gas molecules and electrons that are free to move and exert conservative forces on one another when they come very close to one another, i.e., that can collide with one another like gas molecules according to the kinetic theory of gases.^[1)]Show the German source Further, let a number of electrons be bound to points of the space far apart from one another, by forces directed toward these points and proportional to the elongations.Show the German source These electrons, too, are to enter into conservative interaction with the free molecules and electrons when the latter come very close to them.Show the German source We call the electrons bound to points in space “resonators”; they emit and absorb electromagnetic waves of definite period.Show the German source

According to the present view of the origin of light, the radiation in the space under consideration, which is found on the basis of Maxwell’s theory for the case of dynamic equilibrium, would have to be identical with the “black radiation” — at least if resonators of all frequencies that come into consideration are regarded as present.Show the German source

For the time being we disregard the radiation emitted and absorbed by the resonators and ask for the condition for dynamic equilibrium that corresponds to the interaction (the collisions) of molecules and electrons.Show the German source For the latter the kinetic theory of gases yields the condition that the mean vis viva of a resonator electron must be equal to the mean kinetic energy of the translational motion of a gas molecule.Show the German source If we resolve the motion of the resonator electron into three mutually perpendicular oscillatory motions, we find for the mean value $\overline{E}$ of the energy of one such rectilinear oscillatory motionShow the German source $$\overline{E} = \frac{R}{N} T,$$$\overline{E}$Mean resonator energy$R$Gas constant$N$Avogadro's number$T$Temperature of the radiationExplain this equationShow the German source where $R$ denotes the absolute gas constant, $N$ the number of “real molecules” in one gram-equivalent and $T$ the absolute temperature.Show the German source For, because the time averages of the kinetic and potential energy of the resonator are equal, the energy $\overline{E}$ is $2/3$ times as large as the vis viva of a free, monatomic gas molecule.Show the German source Now if some cause — in our case, radiation processes — were to bring it about that the energy of a resonator had a greater or smaller time average than $\overline{E}$, then the collisions with the free electrons and molecules would lead to a release of energy to the gas, or an uptake of energy from the gas, that differs from zero on average.Show the German source Thus, in the case we are considering, dynamic equilibrium is possible only if every resonator has the mean energy $\overline{E}$.Show the German source

We now make a similar consideration regarding the interaction of the resonators and the radiation present in the space.Show the German source Mr. Planck has derived the condition of dynamic equilibrium for this case^[1)] on the supposition that the radiation can be regarded as the most disordered process conceivable.^[2)]Show the German source He found:Show the German source $$\overline{E}_\nu = \frac{L^3}{8 \pi \nu^2} \varrho_\nu.$$${\overline{E}}_{\nu }$Mean resonator energy at a given frequency$L$Speed of light$\nu$Frequency of the light${\varrho }_{\nu }$Spectral energy densityExplain this equationShow the German source Here $\overline{E}_\nu$ is the mean energy of a resonator of natural frequency $\nu$ (per component of oscillation), $L$ the velocity of light, $\nu$ the frequency and $\varrho_\nu d \nu$ the energy per unit volume of that part of the radiation whose oscillation number lies between $\nu$ and $\nu + d \nu$.Show the German source

If the radiation energy of frequency $\nu$ is not to be continually either diminished or increased as a whole, then the following must hold:Show the German source $$\frac{R}{N} T = \overline{E} = \overline{E}_\nu = \frac{L^3}{8 \pi \nu^2} \varrho_\nu,$$$R$Gas constant$N$Avogadro's number$T$Temperature of the radiation$\overline{E}$Mean resonator energy${\overline{E}}_{\nu }$Mean resonator energy at a given frequency$L$Speed of light$\nu$Frequency of the light${\varrho }_{\nu }$Spectral energy densityExplain this equationShow the German source $$\varrho_\nu = \frac{R}{N} \frac{8 \pi \nu^2}{L^3} T.$$${\varrho }_{\nu }$Spectral energy density$R$Gas constant$N$Avogadro's number$\nu$Frequency of the light$L$Speed of light$T$Temperature of the radiationExplain this equationShow the German source This relation, found as the condition of dynamic equilibrium, not only lacks agreement with experience, but it also says that in our picture there can be no question of a definite distribution of energy between ether and matter.Show the German source For the wider the range of oscillation numbers of the resonators is chosen, the greater the radiation energy of the space becomes, and in the limit we obtainShow the German source $$\int_0^\infty \varrho_\nu d \nu = \frac{R}{N} \frac{8 \pi}{L^3} T \int_0^\infty \nu^2 d \nu = \infty.$$${\varrho }_{\nu }$Spectral energy density$\nu$Frequency of the light$R$Gas constant$N$Avogadro's number$L$Speed of light$T$Temperature of the radiationExplain this equationShow the German source

Bern, 17 March 1905.Show the German source

(Received 18 March 1905.)Show the German source