Light quanta: from entropy to an energy scale: §6 · the heuristic correspondence

In the Wien regime, the radiation entropy coefficient has the form associated with independent elements. Reading an energy scale from that match is a heuristic inference, not a proof of every property of light.

The gas entropy change has coefficient n times Boltzmann's constant; the radiation change has coefficient E divided by B times frequency.

If the radiation is interpreted through the independent-element analogy, matching the coefficients suggests $n_\text{eff} = E/(k_B B\nu)$. Its corresponding energy per element is $k_B B\nu$. The algebra identifies a scale conditional on the analogy; it does not prove the analogy.

The effective coefficient is E divided by Boltzmann's constant times B times frequency. The implied energy scale is Boltzmann's constant times B times frequency, written h times frequency in modern notation.

Keep a non-integer effective coefficient as it stands. The entropy comparison concerns a macroscopic relation, not a directly observed list of individual packets. In the entropy-comparison laboratory, compare the coefficients before revealing their interpretation.

Return to where matching the two formulas is kept apart from what the match means. (included in this file)

Two results sit side by side. From §4: dilute radiation of energy E in a narrow band at ν changes its entropy by $\frac{E}{\beta\nu}\lg\frac{v}{v_0}$ when its volume goes from $v_0$ to v. From §5: n independent points change theirs by $\frac{R}{N}n\lg\frac{v}{v_0}$. In this explanation f is $v/v_0$, B is β, and $k_B$ is R/N.

The gas entropy change has coefficient n times Boltzmann's constant; the radiation change has coefficient E divided by B times frequency.

1. Both are a coefficient times ln f, the same function of the volume ratio. That shared form is what the comparison uses; letters that happen to look alike are not.
2. The paper rewrites the radiation result in the form of Boltzmann's principle, $S-S_0=\frac{R}{N}\lg W$, by moving the coefficient into the exponent, since $a\ln f=\ln f^{a}$. With $a=\frac{N}{R}\frac{E}{\beta\nu}$ this gives $S-S_0=\frac{R}{N}\lg\left[\left(\frac{v}{v_0}\right)^{\frac{N}{R}\frac{E}{\beta\nu}}\right]$.
3. Read the bracket as a probability, as Boltzmann's principle invites: the probability that all the radiation energy is found, at a moment chosen at random, in the part v of the volume $v_0$ is $W=(v/v_0)^{\frac{N}{R}\frac{E}{\beta\nu}}$.
4. Compare it with §5's $W=(v/v_0)^n$. The exponent $\frac{N}{R}\frac{E}{\beta\nu}$ stands where the number of independent points stood.
5. Set the coefficients equal, only as the proposed correspondence: $n_{\mathrm{eff}}k_B=E/(B\nu)$. Divide both sides by $k_B$.

The effective coefficient is E divided by Boltzmann's constant times B times frequency. The implied energy scale is Boltzmann's constant times B times frequency, written h times frequency in modern notation.

1. Divide E by that number to find the energy of one independent element: $k_BB\nu$, which the paper writes Rβν/N. The letter h for $k_BB$, in the formula's last step, is a later name and is not the paper's.
2. Worked size, from the paper's inputs: $R\beta/N$ is $8.31\times10^{7}\times4.866\times10^{-11}/(6.17\times10^{23})=6.55\times10^{-27}$ erg-seconds, with R the standard value of 1905. At a frequency of $10^{15}$ per second, in the ultraviolet, one element would carry $6.55\times10^{-12}$ erg.
3. A second comparison in §6: using Wien's formula, the paper finds the mean size of an energy element in black radiation to be $3\frac{R}{N}T$, twice the mean kinetic energy $\frac{3}{2}\frac{R}{N}T$ of a molecule's motion at the same temperature.
4. Keep $n_{\mathrm{eff}}$ as it comes out. It is a pure number that need not be whole, and nothing here rounds it.
5. Mark the status. The paper concludes that monochromatic radiation of low density, within the range of Wien's formula, behaves in thermodynamic respects as if it consisted of independent energy quanta of size Rβν/N. 'As if' is the claim: a heuristic reading of a matching form, not a proof of what light is.
6. What follows in the paper is a further question: whether the laws by which light is produced and transformed are also such as if light consisted of these quanta. §§7 to 9 test it on fluorescence, on electrons released by light, and on the ionization of gases.

Return to where matching the two formulas is kept apart from what the match means. (included in this file)

Modern qualifications

Einstein calls his proposal a heuristic point of view, and the paper’s title says so. A matching mathematical form can suggest a model without forcing it: Planck’s energy elements of 1900 belong to the oscillators in the walls, Einstein’s quanta to the radiation itself, and the two agree on some numbers while saying different things about light.