One thing Einstein didn’t do in 1905

As we look back on 2005, the 100th anniversary of Einstein’s most creative year, it is worthwhile to remember what he didn’t discover—or at least what he didn’t tell us he discovered—in 1905. Among the items is the photon.

A “photon”—the name was coined by the physical chemist Gilbert Lewis in 1926—is an elementary particle with energy hv, momentum hv/c, and mass zero. In 1905, Einstein discussed an energy quantum only. He did not discuss the quantum’s momentum until 1916. On the question of the mass, his 1905 paper contains an odd calculation. He noted that the average kinetic energy for a Maxwell–Boltzmann particle in a distribution at a temperature T is given by 3kT/2 (in our present notation, not his). He assumed that the Wien spectrum, which he used for high frequencies to exhibit the entropy of the radiation, is valid for all frequencies. He then calculated, in his notation

$$\begin{array}{c}\underset{0}{\overset{\infty }{\int }}\alpha v^3{e}^{-\beta v/T}dv/\underset{0}{\overset{\infty }{\int }}\frac{N}{R\beta v}\alpha v^3{e}^{-\beta v/T}dv\\ =3\frac{R}{N}T=3\kappa T\end{array}$$

Wilhelm Wien found his distribution using the analogy to the Maxwell–Boltzmann distribution, although it did not occur to him to ask why that particle analogy might be valid for radiation. The above result is what you would get if you replaced the classical kinetic energy by pc, which would be appropriate for a massless particle. Einstein never explained why he did this calculation or what its significance was.