Annus MirabilisEinstein in 1905
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Read · Light quanta: from entropy to an energy scale · Results

§1 · classical energy allocation

Einstein argues that light, when it is emitted or absorbed, behaves as if its energy came in separate quanta, and calls the idea a heuristic point of view. Follow all nine sections, from the trouble with classical radiation to the photoelectric effect.

The paper's results

Page 136

The equilibrium the classical picture requires, and its infinite total

If each resonator has the mean energy RNT\frac{R}{N}T that the kinetic theory of gases requires and the mean energy Planck derived for equilibrium with the radiation, the radiation must have the density ϱν=RN8πν2L3T\varrho_\nu = \frac{R}{N}\frac{8\pi\nu^2}{L^3}T; the paper says this disagrees with experience, and that summed over all frequencies it gives an infinite energy, so the picture allows no definite distribution of energy between ether and matter.

As printed

Soll die Strahlungsenergie von der Frequenz ν\nu nicht beständig im Ganzen weder vermindert noch vermehrt werden, so muß gelten: RNT=E‾=E‾ν=L38πν2ϱν,\frac{R}{N} T = \overline{E} = \overline{E}_\nu = \frac{L^3}{8 \pi \nu^2} \varrho_\nu,Gas constantAvogadro's numberTemperature of the radiationMean resonator energyMean resonator energy at a given frequencySpeed of lightFrequency of the lightSpectral energy densityExplain this equation ϱν=RN8πν2L3T.\varrho_\nu = \frac{R}{N} \frac{8 \pi \nu^2}{L^3} T.Spectral energy densityGas constantAvogadro's numberFrequency of the lightSpeed of lightTemperature of the radiationExplain this equation Diese als Bedingung des dynamischen Gleichgewichtes gefundene Beziehung entbehrt nicht nur der Übereinstimmung mit der Erfahrung, sondern sie besagt auch, daß in unserem Bilde von einer bestimmten Energieverteilung zwischen Äther und Materie nicht die Rede sein kann. Je weiter nämlich der Schwingungszahlenbereich der Resonatoren gewählt wird, desto größer wird die Strahlungsenergie des Raumes, und wir erhalten in der Grenze ∫0∞ϱνdν=RN8πL3T∫0∞ν2dν=∞.\int_0^\infty \varrho_\nu d \nu = \frac{R}{N} \frac{8 \pi}{L^3} T \int_0^\infty \nu^2 d \nu = \infty.Spectral energy densityFrequency of the lightGas constantAvogadro's numberSpeed of lightTemperature of the radiationExplain this equation

Text on page 136 of the German source, § 1, paragraph 5

In modern notation

uν=8 π ν2c3 kB Tu_\nu = \frac{8\,\pi\,\nu^{2}}{c^{3}}\,k_B\,T
U(νc)=∫0νcuν dνU\left(\nu_{c}\right) = \int_{0}^{\nu_{c}} u_\nu\,\mathrm{d}\nu
U(νc)=8 π kB T3 c3 νc3U\left(\nu_{c}\right) = \frac{8\,\pi\,k_B\,T}{3\,c^{3}}\,\nu_{c}^{3}
  • Premise: Two conditions of equilibrium are combined: the resonators with the molecules and electrons of the gas, from the kinetic theory, and the resonators with the radiation, which Planck derived on the assumption that radiation is a process as disordered as can be.
  • A comparison, not a premise: § 2 shows that the same relation is the limit of Planck's formula for long wavelengths and large radiation densities, where it does agree with experience.

Decoder

E‾\overline{E}
The mean energy of one linear oscillation of a resonator electron, which the kinetic theory of gases sets equal to RNT\frac{R}{N}T in equilibrium with the gas.
E‾ν\overline{E}_\nu
The mean energy of a resonator of eigenfrequency ν\nu, per component of oscillation, in Planck's condition for equilibrium with the radiation.
ϱν\varrho_\nu
The energy density of the radiation per unit frequency: ϱν dν\varrho_\nu\,d\nu is the energy per unit volume of the part of the radiation between ν\nu and ν+dν\nu + d\nu.
LL
The speed of light, printed L in §§ 1 and 2 of this paper.
∞\infty
The total over all frequencies: the wider the range of the resonators' frequencies is taken, the greater the radiation energy of the space, without limit.

Try it

LQ-02: Give every resonator the same mean energy whatever its frequency: how much energy does the radiation hold as the range of frequencies widens?

Where this stops: The allocation is a classical model, not the spectrum measured across all frequencies. A finite cutoff is where the calculation stops, not a property of light. Read the argument

Where this stops: The passage explains what the footnotes state; it does not test whether real radiation in a cavity meets the disorder condition. Einstein cites Drude's theory of metals in support; the passage does not assess that theory. Read the argument

Common wrong turns

Argument
A consequence of two conditions of equilibrium, stated as a difficulty
Model
Within the picture of resonators, gas molecules and radiation that the section sets up
Evidence
A theoretical difficulty, which the paper says disagrees with experience
History
Its ingredients, the kinetic theory and Planck's resonator condition, were published before 1905; the paper states the difficulty

§1 · classical energy allocation

A finite window cannot cure an infinite total

At a positive temperature, classical allocation assigns more energy as the frequency cutoff rises. Its all-frequency total has no finite value; the plotting cutoff is not a physical explanation.

Read the argument

Two assumptions under the equilibrium of section 1

The equilibrium of §1 rests on two assumptions its footnotes spell out: gas molecules and electrons have the same mean kinetic energy, as Drude assumed for metals, and the radiation is as disordered as possible, meaning its Fourier amplitudes and phases are statistically independent of one another.

Read the argument