Albert Einstein - Emission and Absorption of Radiation in Quantum TheorySixteen years ago, when Planck created quantum theory by deriving his radiation formula, he took the following approach. He calculated the mean energy Ē of a resonator as a function of temperature according to his newly found quantum-theoretic basic principles, and determined from this the radiation density ρ as a function of frequency ν and temperature. He accomplished this by deriving — based upon electromagnetic considerations — a relation between radiation density and resonator energy Ē:
Ē = c3 ρ / 8 π ν2 (1)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.
§1. PLANCK's Resonator in a Field of RadiationThe 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
Δ1E = - A Eτeffected 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
< Δ2E > = - B ρ τThe 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
< E + Δ1E + Δ2E > = Ēor
Ē = - (B / A) ρOne 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."
§2. Quantum Theory and RadiationWe 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 Wn of state Zn (or the relative number of molecules which were in state Zn) is given by
Wn = pn e- εn / k T  (2)where k is the well-known Boltzmann constant. pn is the statistical "weight" of state Zn, 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 Zn to state Zm by absorbing radiation of the distinct frequency ν = νnm ; and likewise from state Zm to state Zn by emitting such radiation. The radiation energy involved is εm - εn. 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 Zn to state Zm under absorption of radiation, as molecules will go from state Zm to state Zn 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 Zm to state Zn with emission of the radiation energy εm - εn. 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
Amn Nm,where Amn is a constant that is characteristic of the combination of the states Zm and Zn, and Nm is the number of molecules in state Zm. 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 Zn → Zm as well as a transition Zm → Zn. The number of transitions
Zn → Zm per unit time is then
Bnm Nn ρ,and the number of transitions Zm → Zn is to be expressed as
Bmn Nm ρ,where Bnm, Bmn are constants related to the combination of states Zn, Zm. As a condition for the statistical equilibrium between the reactions Zn → Zm and Zm → Zn one finds, therefore, the equation
Amn Nm + Bmn Nm ρ = Bnm Nn ρ  (3)Equation (2), on the other hand, yields
Nn / Nm = (pn / pm) e ( εm - εn ) / k T (4)From (3) and (4) follows
Amn pm = ρ ( Bnm pn e ( εm - εn ) / k T - Bmn pm ) (5)ρ is the radiation density of that frequency which is emitted with the transition Zm → Zn and is absorbed with Zn → Zm. Our equation shows the relation between T and ρ at this frequency.
If we postulate that ρ must approach infinity with ever increasing T, then we necessarily have
Bnm pn = Bmn pm (6)Introducing the abbreviation
Amn / Bmn pm = αmn, (7)one finds
ρ = αmn / ( e ( εm - εn ) / k T - 1 ) (5a)This is Planck's relation between ρ and T with the constants left indeterminate. The constants Amn and Bmn could be calculated directly if we possessed a modified version of electrodynamics and mechanics that is in compliance with the quantum hypothesis. The fact that ρ must be a universal function of T and ν implies that αmn and εm - εn cannot depend upon the specific constitution of the molecule, but only upon the effective frequency ν. From Wien's law follows furthermore that αmn must be proportional to the third power, and εm - εn to the first power of ν . Consequently, one has
εm - εn = hν (8)where hν is a constant. 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.
§3. Remark on the Photochemical Law of EquivalenceThe photochemical law of equivalence falls in line with our train of thoughts in the following manner. Let there be a gas of such low temperature that the thermal radiation of frequency ν, which leads from state Zm to state Zn, does not practically occur. According to (2) and (5a), the state Zm will be quite rare compared to state Zn, and we shall assume that almost all gas molecules are in state Zn. Aside from the previously considered process Zm → Zn, let the molecule in state Zm 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 Zm → Zn. What will happen now if we irradiate the gas with the effective frequency? Under absorption of the radiation energy εm - εn = hν, molecules will continually go from state Zn to state Zm. Only a very small fraction of these molecules will return to state Zn 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 Zm, 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.