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Dieter Zeh Ernst Zermelo Wojciech Zurek Konrad Zuse Fritz Zwicky Presentations Biosemiotics Free Will Mental Causation James Symposium 
Albert Einstein  Emission and Absorption of Radiation in Quantum Theory
Sixteen 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 quantumtheoretic 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 = c^{3} ρ / 8 π ν^{2}
(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 quantumtheoretically 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 electromagneticmechanical 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 electromagnetomechanical considerations, which led Planck to equation (1), are to be replaced by quantumtheoretical 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 Radiation
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
Δ_{1}E =  A E_{τ}
effected by emission; and second, the change Δ_{2}E 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 meanvalue relation
< Δ_{2}E > =  B ρ τ
The constants A and B can be calculated in known manner. We call Δ_{1}E the energy change due to emitted radiation, Δ_{2}E 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 + Δ_{1}E + Δ_{2}E > = Ē
or
Ē =  (B / A) ρ
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 quantumtheoretical basis and without specialized suppositions about the interaction between radiation and those structures which we want to call "molecules."
§2. Quantum Theory and Radiation
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 Z_{1}, Z_{2}, etc., of states with energy values ε_{1}, ε_{2}, 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
W_{n} = p_{n} e^{ εn / k T} (2)
where k is the wellknown 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
A_{m}^{n} N_{m},
where A_{m}^{n} 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
B_{n}^{m} N_{n} ρ,
and the number of transitions Z_{m} → Z_{n} is to be expressed as
B_{m}^{n} N_{m} ρ,
where B_{n}^{m}, B_{m}^{n} 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
A_{m}^{n} N_{m} +
B_{m}^{n} N_{m} ρ =
B_{n}^{m} N_{n} ρ (3)
Equation (2), on the other hand, yields
N_{n} / N_{m} = (p_{n} / p_{m}) e ^{ ( εm  εn ) / k T} (4)
From (3) and (4) follows
A_{m}^{n} p_{m} = ρ ( B_{n}^{m} p_{n} e ^{ ( εm  εn ) / k T}  B_{m}^{n} p_{m} ) (5)
ρ is the radiation density of that frequency which is emitted with the transition
Z_{m} → Z_{n} and is absorbed with Z_{n} → Z_{m}. 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
B_{n}^{m} p_{n} = B_{m}^{n} p_{m}
(6)
Introducing the abbreviation
A_{m}^{n} / B_{m}^{n} p_{m} = α_{mn},
(7)
one finds
ρ = α_{mn} / ( e^{ ( εm  εn ) / k T}  1 )
(5a)
Einstein has derived Planck's blackbody law
This is Planck's relation between ρ and T with the constants left indeterminate.
The constants A_{m}^{n} and B_{m}^{n} could be calculated directly if we possessed a modified
version of electrodynamics and mechanics that is in compliance with the quantum
hypothesis.
Einstein derives Bohr's second postulate
The fact that ρ must be a universal function of T and ν implies that α_{mn} and
ε_{m}  ε_{n} cannot depend upon the specific constitution of the molecule, but only upon
the effective frequency ν. From Wien's law follows furthermore that α_{mn} must be
proportional to the third power, and ε_{m}  ε_{n} to the first power of ν . Consequently,
one has
ε_{m}  ε_{n} = 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 Equivalence
The 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 Z_{m} to state Z_{n}, does not practically
occur.
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. For Teachers
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