Annus MirabilisEinstein in 1905
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light quanta

The equation explained

ϵ=Eneff=RβνN=hν\epsilon = \frac{E}{n_{\text{eff}}} = \frac{R\beta\nu}{N} = h\nu
In wordsThe size of one packet is the radiation's energy divided by the number of independent points it behaves as though it had, which is the gas constant times Wien's constant times the frequency, over Avogadro's number: the quantity later physics writes as Planck's constant times the frequency.OverviewWhat the exponent of the previous display means, read as a size: the radiation's energy divided into that many equal packets, each of them RβνN\frac{R\beta\nu}{N}.Full explanationThe previous display puts the entropy of the radiation into the form § 5 found for nn independent points, and the exponent standing where nn stood is NREβν\frac{N}{R}\frac{E}{\beta\nu}. Reading that exponent as a count is the whole of the step: if the radiation's entropy behaves as though it were made of that many independent things, then the energy of each of them is the total EE divided by the count, which is RβνN\frac{R\beta\nu}{N}. It depends on the frequency and on nothing else about the radiation, which is what makes it a property of the light rather than of how much of it there is. The last member is a rename: RβN\frac{R\beta}{N} is the constant Planck introduced in 1900, and modern notation writes it hh.Every stepW=(VV0)NREβνW = \left(\frac{V}{V_0}\right)^{\frac{N}{R}\frac{E}{\beta\nu}}The previous display. Compared with § 5's W=(vv0)nW = \left(\frac{v}{v_0}\right)^{n} for nn points moving independently in the volume, the exponent stands exactly where the count stands.neff=NREβνn_{\text{eff}} = \frac{N}{R}\frac{E}{\beta\nu}So this is the number of independent points the radiation's entropy behaves as though it had. It is a reading of the exponent, not a count of anything the paper has observed.ϵ=Eneff=ENREβν=RβνN\epsilon = \frac{E}{n_{\text{eff}}} = \frac{E}{\frac{N}{R}\frac{E}{\beta\nu}} = \frac{R\beta\nu}{N}Divide the energy by the count. The energy EE cancels, which is why the size of a packet does not depend on how much radiation there is.RβN=h⟹ϵ=hν\frac{R\beta}{N} = h \quad \Longrightarrow \quad \epsilon = h\nuThe three constants together are Planck's, so the packet's energy is his quantum of action times the frequency. The lab computes the product from Einstein's own values two displays below.Historian's marginThe chain is this edition's, not Einstein's: § 6 prints the entropy and the exponent and then states the conclusion in words, that monochromatic radiation of low density behaves, in respect of the theory of heat, as though it consisted of mutually independent energy quanta of magnitude RβνN\frac{R\beta\nu}{N}. Writing hh at the end is a rename and imports no later physics; the paper of 1905 has no hh in it. What the step does not establish is that light IS made of such quanta, and the paper does not claim it here: the entropy behaves as though it were, which is the heuristic point of view of the title.Also

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