Annus MirabilisEinstein in 1905
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light quanta

The equation explained

ϱν=αν3eβνT−1,\varrho_\nu = \frac{\alpha \nu^3}{e^{\frac{\beta \nu}{T}} - 1},
In wordsThe radiation's energy per unit volume and unit frequency equals the constant α times the frequency cubed, divided by e raised to the constant β times the frequency over the absolute temperature, less one.OverviewPlanck's radiation formula, which the paper says satisfies all experience so far: the energy density at each frequency and temperature, with two constants α and β. Section 2 uses it to show that Planck's determination of the elementary quanta is to a degree independent of his theory of black radiation.Full explanationSection 2 sets out to show that Planck's determination of the elementary quanta is, to a certain degree, independent of his theory of black radiation. It starts from Planck's formula for ϱν\varrho_\nu, which the paper says satisfies all experience so far, citing Planck in the Annalen, vol. 4, p. 561, 1901. The numerator grows as ν3\nu^3, but the exponential eβνTe^{\frac{\beta \nu}{T}} in the denominator grows faster than any power once βν\beta \nu is large compared with TT, so at high frequency the density falls away to nothing and the total energy is finite, unlike section 1's result. At low frequency the exponential is close to 1+βνT1 + \frac{\beta \nu}{T} and the formula goes over into section 1's form, which is the next step of the argument. The values of α\alpha and β\beta follow on their own lines.Every stepeβνT−1e^{\frac{\beta \nu}{T}} - 1The denominator. At a fixed temperature it grows without limit as the frequency rises, faster than any power of the frequency.αν3eβνT−1\frac{\alpha \nu^3}{e^{\frac{\beta \nu}{T}} - 1}Divide αν3\alpha \nu^3 by it. At high frequency the exponential wins, so the density falls away to zero instead of growing without limit as section 1's did.ϱν=αν3eβνT−1\varrho_\nu = \frac{\alpha \nu^3}{e^{\frac{\beta \nu}{T}} - 1}Planck's formula as the paper quotes it. The paper takes it from Planck and from experience, and does not derive it.Also

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