His derivation was of unparalleled boldness, but found brilliant confirmation. Not only the radiation formula proper and the calculated value of the elementary quantum in it was confirmed, but also the quantum-theoretically calculated value of Ē was confirmed by later investigations on specific heat. In this manner, equation (1), originally found by electromagnetic reasoning, was also confirmed. However, it remained unsatisfactory that the electromagnetic-mechanical analysis, which led to (1), is incompatible with quantum theory, and it is not surprising that Planck himself and all theoreticians who work on this topic incessantly tried to modify the theory such as to base it on noncontradictory foundations.

Since Bohr's theory of spectra has achieved its great successes, it seems no longer doubtful that the basic idea of quantum theory must be maintained. It so appears that the uniformity of the theory must be established such that the electromagneto-mechanical considerations, which led Planck to equation (1), are to be replaced by quantum-theoretical contemplations on the interaction between matter and radiation. In this endeavor I feel galvanized by the following consideration which is attractive both for its simplicity and generality.

The behavior of a monochromatic resonator in a field of radiation, according to the classical theory, can be easily understood if one recalls the manner of treatment that was first used in the theory of Brownian movement. Let E be the energy of the resonator at a given moment in time; we ask for the energy after time τ has elapsed. Hereby, τ is assumed to be large compared to the period of oscillation of the resonator, but still so small that the percentage change of E during τ can be treated as infinitely small. Two kinds of change can be distinguished. First the change

effected by emission; and second, the change Δ₂E caused by the work done by the electric field on the resonator. This second change increases with the radiation density and has a "chance"-dependent value and a "chance"-dependent sign. An electromagnetic, statistical consideration yields the mean-value relation

The constants A and B can be calculated in known manner. We call Δ₁E the energy change due to emitted radiation, Δ₂E the energy change due to incident radiation. Since the mean value of E, taken over many resonators, is supposed to be independent of time, there has to be

or

One obtains relation (1) if one calculates B and A for the monochromatic resonator in the known way with the help of electromagnetism and mechanics. We now want to undertake corresponding considerations, but on a quantum-theoretical basis and without specialized suppositions about the interaction between radiation and those structures which we want to call "molecules."

We consider a gas of identical molecules that are in static equilibrium with thermal radiation. Let each molecule be able to assume only a discrete sequence Z₁, Z₂, etc., of states with energy values ε₁, ε₂, respectively. Then it follows in known manner and in analogy to statistical mechanics, or directly from Boltzmann's principle, or finally from thermodynamic considerations, that the probability W_n of state Z_n (or the relative number of molecules which were in state Z_n) is given by

where k is the well-known Boltzmann constant. p_n is the statistical "weight" of state Z_n, i.e., a constant that is characteristic of the quantum state of the molecule but independent of the gas temperature T.

We shall now assume that a molecule can go from state Z_n to state Z_m by absorbing radiation of the distinct frequency ν = ν_nm ; and likewise from state Z_m to state Z_n by emitting such radiation. The radiation energy involved is ε_m - ε_n. In general, this is possible for any combination of two indices m and n. With respect to any of these elementary processes there must be a statistical equilibrium in thermal equilibrium. Therefore, we can confine ourselves to a single elementary process belonging to a distinct pair of indices (n,m).

At the thermal equilibrium, as many molecules per time unit will change from state Z_n to state Z_m under absorption of radiation, as molecules will go from state Z_m to state Z_n with emission of radiation. We shall state simple hypotheses about these transitions, where our guiding principle is the limiting case of classical theory, as it has been briefly outlined above.

We shall distinguish here also two types of transitions:

a) Emission of Radiation. This will be a transition from state Z_m to state Z_n with emission of the radiation energy ε_m - ε_n. This transition will take place without external influence. One can hardly imagine it to be other than similar to radioactive reactions. The number of transitions per time unit will have to be put at

where A_mⁿ is a constant that is characteristic of the combination of the states Z_m and Z_n, and N_m is the number of molecules in state Z_m.

b) Incidence of Radiation. Incidence is determined by the radiation within which the molecule resides; let it be proportional to the radiation density ρ of the effective frequency. In case of the resonator it may cause a loss in energy as well as an increase in energy; that is, in our case, it may cause a transition Z_n → Z_m as well as a transition Z_m → Z_n. The number of transitions
Z_n → Z_m per unit time is then

and the number of transitions Z_m → Z_n is to be expressed as

where B_n^m, B_mⁿ are constants related to the combination of states Z_n, Z_m.

As a condition for the statistical equilibrium between the reactions Z_n → Z_m and Z_m → Z_n one finds, therefore, the equation

Equation (2), on the other hand, yields

From (3) and (4) follows

While the three hypotheses concerning emission and incidence of radiation lead to Planck's radiation formula, I am of course very willing to admit that this does not elevate them to confirmed results. But the simplicity of the hypotheses, the generality with which the analysis can be carried out so effortlessly, and the natural connection to Planck's linear oscillator (as a limiting case of classical electrodynamics and mechanics) seem to make it highly probable that these are basic traits of a future theoretical representation. The postulated statistical law of emission is nothing but Rutherford's law of radioactive decay, and the law expressed by (8), in conjunction with (5a), is identical with the second basic hypothesis in Bohr's theory of spectra — this too speaks in favor of the theory presented here.

According to (2) and (5a), the state Z_m will be quite rare compared to state Z_n, and we shall assume that almost all gas molecules are in state Z_n. Aside from the previously considered process Z_m → Z_n, let the molecule in state Z_m also have the capability of another elementary "chemical" process, e.g., monomolecular dissociation. Let us furthermore assume that the reaction rate of this dissociation is large compared to the rate of occurrence of the reaction Z_m → Z_n.

What will happen now if we irradiate the gas with the effective frequency? Under absorption of the radiation energy ε_m - ε_n = hν, molecules will continually go from state Z_n to state Z_m. Only a very small fraction of these molecules will return to state Z_n by emission or absorption. Most, by far, will suffer chemical dissociation, corresponding to the postulated higher reaction rate of this process. This means that per dissociating molecule, we will practically find that the radiation energy hν has been absorbed, just as the law of equivalence demands.

The essence of this interpretation is that molecular dissociation is achieved by the absorption of light via the quantum state Z_m, but not directly without this intermediate state. In consequence, one need not distinguish between a chemically effective and a chemically ineffective absorption of radiation. The absorption of light and the chemical process appear as independent processes.

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